%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SWW473^3 : TPTP v9.2.1. Released v5.3.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n005.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 : Wed Feb 25 09:48:10 AM UTC 2026
% Result : Theorem 4.90s 2.08s
% Output : Refutation 4.90s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 5
% Syntax : Number of formulae : 33 ( 11 unt; 0 typ; 0 def)
% Number of atoms : 216 ( 43 equ; 0 cnn)
% Maximal formula atoms : 6 ( 6 avg)
% Number of connectives : 252 ( 27 ~; 15 |; 7 &; 197 @)
% ( 3 <=>; 3 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 22 ( 22 >; 0 *; 0 +; 0 <<)
% Number of symbols : 394 ( 391 usr; 10 con; 0-4 aty)
% Number of variables : 33 ( 0 ^; 33 !; 0 ?; 33 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
pname: $tType ).
thf(type_def_6,type,
option_com: $tType ).
thf(type_def_8,type,
int: $tType ).
thf(type_def_9,type,
x_a: $tType ).
thf(type_def_10,type,
nat: $tType ).
thf(type_def_11,type,
com: $tType ).
thf(func_def_0,type,
x_a: $tType ).
thf(func_def_1,type,
com: $tType ).
thf(func_def_2,type,
pname: $tType ).
thf(func_def_3,type,
int: $tType ).
thf(func_def_4,type,
nat: $tType ).
thf(func_def_5,type,
option_com: $tType ).
thf(func_def_6,type,
body: pname > option_com ).
thf(func_def_7,type,
ex: ( int > $o ) > $o ).
thf(func_def_8,type,
finite_card_a_o: ( ( x_a > $o ) > $o ) > nat ).
thf(func_def_9,type,
finite_card_pname_o: ( ( pname > $o ) > $o ) > nat ).
thf(func_def_10,type,
finite_card_int_o: ( ( int > $o ) > $o ) > nat ).
thf(func_def_11,type,
finite_card_nat_o: ( ( nat > $o ) > $o ) > nat ).
thf(func_def_12,type,
finite_card_a: ( x_a > $o ) > nat ).
thf(func_def_13,type,
finite_card_pname: ( pname > $o ) > nat ).
thf(func_def_14,type,
finite_card_int: ( int > $o ) > nat ).
thf(func_def_15,type,
finite_card_nat: ( nat > $o ) > nat ).
thf(func_def_16,type,
finite_finite_a_o_o: ( ( ( x_a > $o ) > $o ) > $o ) > $o ).
thf(func_def_17,type,
finite1066544169me_o_o: ( ( ( pname > $o ) > $o ) > $o ) > $o ).
thf(func_def_18,type,
finite229719499nt_o_o: ( ( ( int > $o ) > $o ) > $o ) > $o ).
thf(func_def_19,type,
finite1676163439at_o_o: ( ( ( nat > $o ) > $o ) > $o ) > $o ).
thf(func_def_20,type,
finite_finite_a_o: ( ( x_a > $o ) > $o ) > $o ).
thf(func_def_21,type,
finite297249702name_o: ( ( pname > $o ) > $o ) > $o ).
thf(func_def_22,type,
finite_finite_int_o: ( ( int > $o ) > $o ) > $o ).
thf(func_def_23,type,
finite_finite_nat_o: ( ( nat > $o ) > $o ) > $o ).
thf(func_def_24,type,
finite_finite_a: ( x_a > $o ) > $o ).
thf(func_def_25,type,
finite_finite_pname: ( pname > $o ) > $o ).
thf(func_def_26,type,
finite_finite_int: ( int > $o ) > $o ).
thf(func_def_27,type,
finite_finite_nat: ( nat > $o ) > $o ).
thf(func_def_28,type,
finite_folding_one_a: ( x_a > x_a > x_a ) > ( ( x_a > $o ) > x_a ) > $o ).
thf(func_def_29,type,
finite1282449217_pname: ( pname > pname > pname ) > ( ( pname > $o ) > pname ) > $o ).
thf(func_def_30,type,
finite1626084323ne_int: ( int > int > int ) > ( ( int > $o ) > int ) > $o ).
thf(func_def_31,type,
finite988810631ne_nat: ( nat > nat > nat ) > ( ( nat > $o ) > nat ) > $o ).
thf(func_def_32,type,
finite1819937229idem_a: ( x_a > x_a > x_a ) > ( ( x_a > $o ) > x_a ) > $o ).
thf(func_def_33,type,
finite89670078_pname: ( pname > pname > pname ) > ( ( pname > $o ) > pname ) > $o ).
thf(func_def_34,type,
finite1432773856em_int: ( int > int > int ) > ( ( int > $o ) > int ) > $o ).
thf(func_def_35,type,
finite795500164em_nat: ( nat > nat > nat ) > ( ( nat > $o ) > nat ) > $o ).
thf(func_def_36,type,
abs_abs_int: int > int ).
thf(func_def_37,type,
minus_minus_a_o: ( x_a > $o ) > ( x_a > $o ) > x_a > $o ).
thf(func_def_38,type,
minus_minus_pname_o: ( pname > $o ) > ( pname > $o ) > pname > $o ).
thf(func_def_39,type,
minus_minus_int_o: ( int > $o ) > ( int > $o ) > int > $o ).
thf(func_def_40,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(func_def_41,type,
minus_minus_int: int > int > int ).
thf(func_def_42,type,
minus_minus_nat: nat > nat > nat ).
thf(func_def_43,type,
one_one_int: int ).
thf(func_def_44,type,
one_one_nat: nat ).
thf(func_def_45,type,
plus_plus_int: int > int > int ).
thf(func_def_46,type,
plus_plus_nat: nat > nat > nat ).
thf(func_def_47,type,
times_times_int: int > int > int ).
thf(func_def_48,type,
times_times_nat: nat > nat > nat ).
thf(func_def_49,type,
zero_zero_int: int ).
thf(func_def_50,type,
zero_zero_nat: nat ).
thf(func_def_51,type,
the_a: ( x_a > $o ) > x_a ).
thf(func_def_52,type,
the_int: ( int > $o ) > int ).
thf(func_def_53,type,
the_nat: ( nat > $o ) > nat ).
thf(func_def_54,type,
if_nat: $o > nat > nat > nat ).
thf(func_def_55,type,
bit1: int > int ).
thf(func_def_56,type,
pls: int ).
thf(func_def_57,type,
number_number_of_int: int > int ).
thf(func_def_58,type,
number_number_of_nat: int > nat ).
thf(func_def_59,type,
succ: int > int ).
thf(func_def_60,type,
suc: nat > nat ).
thf(func_def_61,type,
nat_case_o: $o > ( nat > $o ) > nat > $o ).
thf(func_def_62,type,
nat_case_nat: nat > ( nat > nat ) > nat > nat ).
thf(func_def_63,type,
semiri1621563631at_int: nat > int ).
thf(func_def_64,type,
nat_neg: int > $o ).
thf(func_def_65,type,
nat_tsub: int > int > int ).
thf(func_def_66,type,
the_com: option_com > com ).
thf(func_def_67,type,
bot_bot_a_o: x_a > $o ).
thf(func_def_68,type,
bot_bot_pname_o: pname > $o ).
thf(func_def_69,type,
bot_bot_int_o: int > $o ).
thf(func_def_70,type,
bot_bot_nat_o: nat > $o ).
thf(func_def_72,type,
bot_bot_nat: nat ).
thf(func_def_73,type,
ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_74,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_75,type,
ord_less_int: int > int > $o ).
thf(func_def_76,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_77,type,
ord_less_eq_a_o_o: ( ( x_a > $o ) > $o ) > ( ( x_a > $o ) > $o ) > $o ).
thf(func_def_78,type,
ord_le1205211808me_o_o: ( ( pname > $o ) > $o ) > ( ( pname > $o ) > $o ) > $o ).
thf(func_def_79,type,
ord_less_eq_int_o_o: ( ( int > $o ) > $o ) > ( ( int > $o ) > $o ) > $o ).
thf(func_def_80,type,
ord_less_eq_nat_o_o: ( ( nat > $o ) > $o ) > ( ( nat > $o ) > $o ) > $o ).
thf(func_def_81,type,
ord_less_eq_a_o: ( x_a > $o ) > ( x_a > $o ) > $o ).
thf(func_def_82,type,
ord_less_eq_pname_o: ( pname > $o ) > ( pname > $o ) > $o ).
thf(func_def_83,type,
ord_less_eq_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_84,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_85,type,
ord_less_eq_o: $o > $o > $o ).
thf(func_def_86,type,
ord_less_eq_int: int > int > $o ).
thf(func_def_87,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_88,type,
collect_a_o_o: ( ( ( x_a > $o ) > $o ) > $o ) > ( ( x_a > $o ) > $o ) > $o ).
thf(func_def_89,type,
collect_pname_o_o: ( ( ( pname > $o ) > $o ) > $o ) > ( ( pname > $o ) > $o ) > $o ).
thf(func_def_90,type,
collect_int_o_o: ( ( ( int > $o ) > $o ) > $o ) > ( ( int > $o ) > $o ) > $o ).
thf(func_def_91,type,
collect_nat_o_o: ( ( ( nat > $o ) > $o ) > $o ) > ( ( nat > $o ) > $o ) > $o ).
thf(func_def_92,type,
collect_a_o: ( ( x_a > $o ) > $o ) > ( x_a > $o ) > $o ).
thf(func_def_93,type,
collect_pname_o: ( ( pname > $o ) > $o ) > ( pname > $o ) > $o ).
thf(func_def_94,type,
collect_int_o: ( ( int > $o ) > $o ) > ( int > $o ) > $o ).
thf(func_def_95,type,
collect_nat_o: ( ( nat > $o ) > $o ) > ( nat > $o ) > $o ).
thf(func_def_96,type,
collect_a: ( x_a > $o ) > x_a > $o ).
thf(func_def_97,type,
collect_pname: ( pname > $o ) > pname > $o ).
thf(func_def_98,type,
collect_int: ( int > $o ) > int > $o ).
thf(func_def_99,type,
collect_nat: ( nat > $o ) > nat > $o ).
thf(func_def_100,type,
image_a_o_a: ( ( x_a > $o ) > x_a ) > ( ( x_a > $o ) > $o ) > x_a > $o ).
thf(func_def_101,type,
image_a_o_pname: ( ( x_a > $o ) > pname ) > ( ( x_a > $o ) > $o ) > pname > $o ).
thf(func_def_102,type,
image_a_o_int: ( ( x_a > $o ) > int ) > ( ( x_a > $o ) > $o ) > int > $o ).
thf(func_def_103,type,
image_a_o_nat: ( ( x_a > $o ) > nat ) > ( ( x_a > $o ) > $o ) > nat > $o ).
thf(func_def_104,type,
image_pname_o_a: ( ( pname > $o ) > x_a ) > ( ( pname > $o ) > $o ) > x_a > $o ).
thf(func_def_105,type,
image_pname_o_pname: ( ( pname > $o ) > pname ) > ( ( pname > $o ) > $o ) > pname > $o ).
thf(func_def_106,type,
image_pname_o_int: ( ( pname > $o ) > int ) > ( ( pname > $o ) > $o ) > int > $o ).
thf(func_def_107,type,
image_pname_o_nat: ( ( pname > $o ) > nat ) > ( ( pname > $o ) > $o ) > nat > $o ).
thf(func_def_108,type,
image_int_o_a: ( ( int > $o ) > x_a ) > ( ( int > $o ) > $o ) > x_a > $o ).
thf(func_def_109,type,
image_int_o_pname: ( ( int > $o ) > pname ) > ( ( int > $o ) > $o ) > pname > $o ).
thf(func_def_110,type,
image_int_o_int: ( ( int > $o ) > int ) > ( ( int > $o ) > $o ) > int > $o ).
thf(func_def_111,type,
image_int_o_nat: ( ( int > $o ) > nat ) > ( ( int > $o ) > $o ) > nat > $o ).
thf(func_def_112,type,
image_nat_o_a: ( ( nat > $o ) > x_a ) > ( ( nat > $o ) > $o ) > x_a > $o ).
thf(func_def_113,type,
image_nat_o_pname: ( ( nat > $o ) > pname ) > ( ( nat > $o ) > $o ) > pname > $o ).
thf(func_def_114,type,
image_nat_o_int: ( ( nat > $o ) > int ) > ( ( nat > $o ) > $o ) > int > $o ).
thf(func_def_115,type,
image_nat_o_nat: ( ( nat > $o ) > nat ) > ( ( nat > $o ) > $o ) > nat > $o ).
thf(func_def_116,type,
image_a_a_o: ( x_a > x_a > $o ) > ( x_a > $o ) > ( x_a > $o ) > $o ).
thf(func_def_117,type,
image_a_pname_o: ( x_a > pname > $o ) > ( x_a > $o ) > ( pname > $o ) > $o ).
thf(func_def_118,type,
image_a_int_o: ( x_a > int > $o ) > ( x_a > $o ) > ( int > $o ) > $o ).
thf(func_def_119,type,
image_a_nat_o: ( x_a > nat > $o ) > ( x_a > $o ) > ( nat > $o ) > $o ).
thf(func_def_120,type,
image_a_a: ( x_a > x_a ) > ( x_a > $o ) > x_a > $o ).
thf(func_def_121,type,
image_a_pname: ( x_a > pname ) > ( x_a > $o ) > pname > $o ).
thf(func_def_122,type,
image_a_int: ( x_a > int ) > ( x_a > $o ) > int > $o ).
thf(func_def_123,type,
image_a_nat: ( x_a > nat ) > ( x_a > $o ) > nat > $o ).
thf(func_def_124,type,
image_pname_a_o: ( pname > x_a > $o ) > ( pname > $o ) > ( x_a > $o ) > $o ).
thf(func_def_125,type,
image_pname_pname_o: ( pname > pname > $o ) > ( pname > $o ) > ( pname > $o ) > $o ).
thf(func_def_126,type,
image_pname_int_o: ( pname > int > $o ) > ( pname > $o ) > ( int > $o ) > $o ).
thf(func_def_127,type,
image_pname_nat_o: ( pname > nat > $o ) > ( pname > $o ) > ( nat > $o ) > $o ).
thf(func_def_128,type,
image_pname_a: ( pname > x_a ) > ( pname > $o ) > x_a > $o ).
thf(func_def_129,type,
image_pname_pname: ( pname > pname ) > ( pname > $o ) > pname > $o ).
thf(func_def_130,type,
image_pname_int: ( pname > int ) > ( pname > $o ) > int > $o ).
thf(func_def_131,type,
image_pname_nat: ( pname > nat ) > ( pname > $o ) > nat > $o ).
thf(func_def_132,type,
image_int_a_o: ( int > x_a > $o ) > ( int > $o ) > ( x_a > $o ) > $o ).
thf(func_def_133,type,
image_int_pname_o: ( int > pname > $o ) > ( int > $o ) > ( pname > $o ) > $o ).
thf(func_def_134,type,
image_int_int_o: ( int > int > $o ) > ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_135,type,
image_int_nat_o: ( int > nat > $o ) > ( int > $o ) > ( nat > $o ) > $o ).
thf(func_def_136,type,
image_int_a: ( int > x_a ) > ( int > $o ) > x_a > $o ).
thf(func_def_137,type,
image_int_pname: ( int > pname ) > ( int > $o ) > pname > $o ).
thf(func_def_138,type,
image_nat_a_o: ( nat > x_a > $o ) > ( nat > $o ) > ( x_a > $o ) > $o ).
thf(func_def_139,type,
image_nat_pname_o: ( nat > pname > $o ) > ( nat > $o ) > ( pname > $o ) > $o ).
thf(func_def_140,type,
image_nat_int_o: ( nat > int > $o ) > ( nat > $o ) > ( int > $o ) > $o ).
thf(func_def_141,type,
image_nat_nat_o: ( nat > nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_142,type,
image_nat_a: ( nat > x_a ) > ( nat > $o ) > x_a > $o ).
thf(func_def_143,type,
image_nat_pname: ( nat > pname ) > ( nat > $o ) > pname > $o ).
thf(func_def_144,type,
image_nat_int: ( nat > int ) > ( nat > $o ) > int > $o ).
thf(func_def_145,type,
insert_a_o: ( x_a > $o ) > ( ( x_a > $o ) > $o ) > ( x_a > $o ) > $o ).
thf(func_def_146,type,
insert_pname_o: ( pname > $o ) > ( ( pname > $o ) > $o ) > ( pname > $o ) > $o ).
thf(func_def_147,type,
insert_int_o: ( int > $o ) > ( ( int > $o ) > $o ) > ( int > $o ) > $o ).
thf(func_def_148,type,
insert_nat_o: ( nat > $o ) > ( ( nat > $o ) > $o ) > ( nat > $o ) > $o ).
thf(func_def_149,type,
insert_a: x_a > ( x_a > $o ) > x_a > $o ).
thf(func_def_150,type,
insert_pname: pname > ( pname > $o ) > pname > $o ).
thf(func_def_151,type,
insert_int: int > ( int > $o ) > int > $o ).
thf(func_def_152,type,
insert_nat: nat > ( nat > $o ) > nat > $o ).
thf(func_def_153,type,
the_elem_a: ( x_a > $o ) > x_a ).
thf(func_def_154,type,
the_elem_int: ( int > $o ) > int ).
thf(func_def_155,type,
the_elem_nat: ( nat > $o ) > nat ).
thf(func_def_156,type,
fequal_a: x_a > x_a > $o ).
thf(func_def_157,type,
fequal_int: int > int > $o ).
thf(func_def_158,type,
fequal_nat: nat > nat > $o ).
thf(func_def_159,type,
member_a_o: ( x_a > $o ) > ( ( x_a > $o ) > $o ) > $o ).
thf(func_def_160,type,
member_pname_o: ( pname > $o ) > ( ( pname > $o ) > $o ) > $o ).
thf(func_def_161,type,
member_int_o: ( int > $o ) > ( ( int > $o ) > $o ) > $o ).
thf(func_def_162,type,
member_nat_o: ( nat > $o ) > ( ( nat > $o ) > $o ) > $o ).
thf(func_def_163,type,
member_a: x_a > ( x_a > $o ) > $o ).
thf(func_def_164,type,
member_pname: pname > ( pname > $o ) > $o ).
thf(func_def_165,type,
member_int: int > ( int > $o ) > $o ).
thf(func_def_166,type,
member_nat: nat > ( nat > $o ) > $o ).
thf(func_def_167,type,
g: x_a > $o ).
thf(func_def_168,type,
p: ( x_a > $o ) > ( x_a > $o ) > $o ).
thf(func_def_169,type,
u: pname > $o ).
thf(func_def_170,type,
mgt: com > x_a ).
thf(func_def_171,type,
mgt_call: pname > x_a ).
thf(func_def_172,type,
na: nat ).
thf(func_def_173,type,
pn: pname ).
thf(func_def_174,type,
wt: com > $o ).
thf(func_def_185,type,
sP0: int > int > $o ).
thf(func_def_186,type,
sP1: int > ( int > $o ) > $o ).
thf(func_def_187,type,
sP2: nat > nat > ( nat > int ) > int > $o ).
thf(func_def_188,type,
sK3: ( nat > $o ) > ( nat > pname ) > ( pname > $o ) > nat > $o ).
thf(func_def_189,type,
sK4: ( int > $o ) > ( ( x_a > $o ) > $o ) > ( ( x_a > $o ) > int ) > ( x_a > $o ) > $o ).
thf(func_def_190,type,
sK5: ( int > $o ) > ( int > int > $o ) > int ).
thf(func_def_191,type,
sK6: ( nat > $o ) > nat ).
thf(func_def_192,type,
sK7: ( int > $o ) > ( ( int > $o ) > $o ) > ( ( int > $o ) > int ) > ( int > $o ) > $o ).
thf(func_def_193,type,
sK8: ( pname > pname > pname ) > pname ).
thf(func_def_194,type,
sK9: ( pname > pname > pname ) > pname ).
thf(func_def_195,type,
sK10: ( nat > $o ) > ( nat > nat > $o ) > nat ).
thf(func_def_196,type,
sK11: ( ( nat > $o ) > x_a ) > ( x_a > $o ) > ( ( nat > $o ) > $o ) > ( nat > $o ) > $o ).
thf(func_def_197,type,
sK12: ( x_a > $o ) > ( x_a > x_a ) > ( x_a > $o ) > x_a > $o ).
thf(func_def_198,type,
sK13: ( pname > x_a ) > ( x_a > $o ) > ( pname > $o ) > pname > $o ).
thf(func_def_199,type,
sK14: ( ( int > $o ) > $o ) > int ).
thf(func_def_200,type,
sK15: ( ( int > $o ) > $o ) > int > $o ).
thf(func_def_201,type,
sK16: ( x_a > $o ) > x_a > $o ).
thf(func_def_202,type,
sK17: ( x_a > $o ) > x_a ).
thf(func_def_203,type,
sK18: nat > nat > ( nat > $o ) > nat ).
thf(func_def_204,type,
sK19: ( ( x_a > $o ) > $o ) > x_a > $o ).
thf(func_def_205,type,
sK20: ( ( x_a > $o ) > $o ) > x_a ).
thf(func_def_206,type,
sK21: ( nat > $o ) > nat ).
thf(func_def_207,type,
sK22: ( int > $o ) > ( int > pname > $o ) > int ).
thf(func_def_208,type,
sK23: ( nat > $o ) > nat ).
thf(func_def_209,type,
sK24: ( nat > $o ) > nat > $o ).
thf(func_def_210,type,
sK25: ( int > $o ) > ( x_a > int ) > ( x_a > $o ) > x_a > $o ).
thf(func_def_211,type,
sK26: ( nat > $o ) > nat ).
thf(func_def_212,type,
sK27: int > ( nat > $o ) > ( nat > int ) > nat ).
thf(func_def_213,type,
sK28: ( pname > $o ) > ( ( pname > $o ) > pname ) > ( ( pname > $o ) > $o ) > ( pname > $o ) > $o ).
thf(func_def_214,type,
sK29: int > ( int > $o ) > int ).
thf(func_def_215,type,
sK30: ( int > $o ) > int ).
thf(func_def_216,type,
sK31: ( pname > $o ) > ( ( pname > $o ) > $o ) > ( pname > pname > $o ) > pname > $o ).
thf(func_def_217,type,
sK32: ( ( int > $o ) > $o ) > ( nat > $o ) > ( nat > int > $o ) > nat > $o ).
thf(func_def_218,type,
sK33: nat > nat > nat ).
thf(func_def_219,type,
sK34: ( int > $o ) > int ).
thf(func_def_220,type,
sK35: ( ( x_a > $o ) > $o ) > x_a > $o ).
thf(func_def_221,type,
sK36: ( ( x_a > $o ) > $o ) > x_a ).
thf(func_def_222,type,
sK37: ( pname > x_a ) > ( pname > $o ) > ( x_a > $o ) > pname ).
thf(func_def_223,type,
sK38: ( pname > $o ) > ( ( x_a > $o ) > $o ) > ( ( x_a > $o ) > pname ) > ( x_a > $o ) > $o ).
thf(func_def_224,type,
sK39: ( int > $o ) > int ).
thf(func_def_225,type,
sK40: ( ( int > $o ) > $o ) > int > $o ).
thf(func_def_226,type,
sK41: ( ( int > $o ) > $o ) > int ).
thf(func_def_227,type,
sK42: ( ( int > $o ) > $o ) > int ).
thf(func_def_228,type,
sK43: int > ( int > $o ) > int ).
thf(func_def_229,type,
sK44: ( nat > int ) > nat > nat ).
thf(func_def_230,type,
sK45: int > ( nat > int ) > nat > nat ).
thf(func_def_231,type,
sK46: ( pname > $o ) > ( ( pname > $o ) > $o ) > pname ).
thf(func_def_232,type,
sK47: ( pname > $o ) > ( ( pname > $o ) > $o ) > pname > $o ).
thf(func_def_233,type,
sK48: int > ( nat > int ) > ( nat > $o ) > nat ).
thf(func_def_234,type,
sK49: ( int > $o ) > ( int > $o ) > int ).
thf(func_def_235,type,
sK50: ( nat > x_a > $o ) > ( nat > $o ) > nat ).
thf(func_def_236,type,
sK51: ( int > $o ) > ( int > $o ) > int ).
thf(func_def_237,type,
sK52: ( ( pname > $o ) > $o ) > pname > $o ).
thf(func_def_238,type,
sK53: ( ( pname > $o ) > $o ) > pname ).
thf(func_def_239,type,
sK54: x_a > ( pname > x_a ) > ( pname > $o ) > pname ).
thf(func_def_240,type,
sK55: ( ( pname > $o ) > nat ) > ( ( pname > $o ) > $o ) > ( nat > $o ) > ( pname > $o ) > $o ).
thf(func_def_241,type,
sK56: nat > nat ).
thf(func_def_242,type,
sK57: nat > nat ).
thf(func_def_243,type,
sK58: ( nat > x_a > $o ) > nat ).
thf(func_def_244,type,
sK59: ( nat > nat ) > nat ).
thf(func_def_245,type,
sK60: nat > nat ).
thf(func_def_246,type,
sK61: ( nat > int ) > nat > nat ).
thf(func_def_247,type,
sK62: ( nat > int ) > int > nat > nat ).
thf(func_def_248,type,
sK63: ( int > $o ) > ( int > $o ) > int ).
thf(func_def_249,type,
sK64: ( nat > $o ) > nat ).
thf(func_def_250,type,
sK65: nat > nat > nat ).
thf(func_def_251,type,
sK66: ( int > nat > $o ) > ( int > $o ) > int ).
thf(func_def_252,type,
sK67: ( ( nat > $o ) > int ) > ( int > $o ) > ( ( nat > $o ) > $o ) > ( nat > $o ) > $o ).
thf(func_def_253,type,
sK68: ( nat > pname > $o ) > ( nat > $o ) > nat ).
thf(func_def_254,type,
sK69: ( nat > $o ) > ( nat > int ) > nat ).
thf(func_def_255,type,
sK70: ( pname > $o ) > ( pname > pname ) > ( pname > $o ) > pname > $o ).
thf(func_def_256,type,
sK71: int > ( int > $o ) > int ).
thf(func_def_257,type,
sK72: int > ( int > $o ) > int ).
thf(func_def_258,type,
sK73: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_259,type,
sK74: ( x_a > $o ) > x_a ).
thf(func_def_260,type,
sK75: ( x_a > $o ) > x_a ).
thf(func_def_261,type,
sK76: ( x_a > $o ) > ( x_a > pname > $o ) > ( ( pname > $o ) > $o ) > x_a > $o ).
thf(func_def_262,type,
sK77: ( ( nat > $o ) > $o ) > ( pname > $o ) > ( pname > nat > $o ) > pname > $o ).
thf(func_def_263,type,
sK78: ( int > $o ) > ( int > pname > $o ) > ( ( pname > $o ) > $o ) > int > $o ).
thf(func_def_264,type,
sK79: ( x_a > x_a > x_a ) > x_a ).
thf(func_def_265,type,
sK80: ( x_a > x_a > x_a ) > x_a ).
thf(func_def_266,type,
sK81: nat > ( nat > $o ) > nat ).
thf(func_def_267,type,
sK82: ( nat > nat > nat ) > nat ).
thf(func_def_268,type,
sK83: ( nat > nat > nat ) > nat ).
thf(func_def_269,type,
sK84: ( nat > nat ) > nat ).
thf(func_def_270,type,
sK85: ( int > $o ) > ( int > x_a > $o ) > int ).
thf(func_def_271,type,
sK86: ( pname > $o ) > pname ).
thf(func_def_272,type,
sK87: ( pname > $o ) > pname > $o ).
thf(func_def_273,type,
sK88: ( int > $o ) > ( int > $o ) > int ).
thf(func_def_274,type,
sK89: ( int > $o ) > ( pname > int ) > ( pname > $o ) > pname > $o ).
thf(func_def_275,type,
sK90: ( x_a > $o ) > ( ( nat > $o ) > $o ) > ( x_a > nat > $o ) > x_a > $o ).
thf(func_def_276,type,
sK91: int > nat ).
thf(func_def_277,type,
sK92: ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_278,type,
sK93: ( x_a > $o ) > ( nat > $o ) > ( nat > x_a ) > nat > $o ).
thf(func_def_279,type,
sK94: ( ( int > $o ) > $o ) > ( x_a > $o ) > ( x_a > int > $o ) > x_a > $o ).
thf(func_def_280,type,
sK95: ( nat > int ) > ( nat > $o ) > ( int > $o ) > nat ).
thf(func_def_281,type,
sK96: int > nat ).
thf(func_def_282,type,
sK97: ( ( pname > $o ) > $o ) > pname ).
thf(func_def_283,type,
sK98: ( ( pname > $o ) > $o ) > pname > $o ).
thf(func_def_284,type,
sK99: ( ( pname > $o ) > $o ) > pname ).
thf(func_def_285,type,
sK100: ( pname > x_a ) > ( pname > $o ) > x_a > pname ).
thf(func_def_286,type,
sK101: int > int > nat ).
thf(func_def_287,type,
sK102: ( int > int > int ) > int ).
thf(func_def_288,type,
sK103: ( int > int > int ) > int ).
thf(func_def_289,type,
sK104: ( x_a > $o ) > x_a ).
thf(func_def_290,type,
sK105: nat > nat ).
thf(func_def_291,type,
sK106: ( int > $o ) > int > int ).
thf(func_def_292,type,
sK107: nat > nat ).
thf(func_def_293,type,
sK108: ( pname > $o ) > pname ).
thf(func_def_294,type,
sK109: ( pname > $o ) > ( pname > $o ) > pname ).
thf(func_def_295,type,
sK110: int > ( int > $o ) > int ).
thf(func_def_296,type,
sK111: ( int > $o ) > int > int ).
thf(func_def_297,type,
sK112: int > nat ).
thf(func_def_298,type,
sK113: ( pname > $o ) > ( pname > x_a > $o ) > ( ( x_a > $o ) > $o ) > pname > $o ).
thf(func_def_299,type,
sK114: ( nat > pname > $o ) > nat ).
thf(func_def_300,type,
sK115: nat > nat > nat ).
thf(func_def_301,type,
sK116: ( ( nat > $o ) > $o ) > ( ( nat > $o ) > pname ) > ( pname > $o ) > ( nat > $o ) > $o ).
thf(func_def_302,type,
sK117: ( pname > $o ) > pname ).
thf(func_def_303,type,
sK118: ( pname > $o ) > pname > $o ).
thf(func_def_304,type,
sK119: ( x_a > $o ) > x_a ).
thf(func_def_305,type,
sK120: ( x_a > $o ) > x_a > $o ).
thf(func_def_306,type,
sK121: ( int > $o ) > ( nat > int ) > ( nat > $o ) > nat > $o ).
thf(func_def_307,type,
sK122: ( nat > $o ) > nat ).
thf(func_def_308,type,
sK123: ( ( pname > $o ) > $o ) > pname > $o ).
thf(func_def_309,type,
sK124: ( ( pname > $o ) > $o ) > pname ).
thf(func_def_310,type,
sK125: nat > nat > ( nat > int ) > int > nat ).
thf(func_def_311,type,
sK126: ( nat > int ) > nat > nat > nat ).
thf(func_def_312,type,
sK127: nat > nat > nat ).
thf(func_def_313,type,
sK128: nat > ( nat > $o ) > nat ).
thf(func_def_314,type,
sK129: ( int > $o ) > ( nat > $o ) > ( nat > int ) > nat > $o ).
thf(func_def_315,type,
sK130: ( pname > $o ) > pname ).
thf(func_def_316,type,
sK131: ( nat > nat ) > nat ).
thf(func_def_317,type,
sK132: ( nat > nat ) > nat ).
thf(func_def_318,type,
sK133: ( int > $o ) > int > int ).
thf(func_def_319,type,
sK134: ( pname > x_a ) > ( pname > $o ) > ( x_a > $o ) > pname > $o ).
thf(func_def_320,type,
sK135: ( ( pname > $o ) > $o ) > ( x_a > $o ) > ( ( pname > $o ) > x_a ) > ( pname > $o ) > $o ).
thf(func_def_321,type,
sK136: ( nat > int > $o ) > nat ).
thf(func_def_322,type,
sK137: ( nat > pname > $o ) > ( ( pname > $o ) > $o ) > ( nat > $o ) > nat > $o ).
thf(func_def_323,type,
sK138: ( ( x_a > $o ) > $o ) > ( x_a > $o ) > ( ( x_a > $o ) > x_a ) > ( x_a > $o ) > $o ).
thf(func_def_324,type,
sK139: ( int > $o ) > int ).
thf(func_def_325,type,
sK140: ( int > $o ) > int > $o ).
thf(func_def_326,type,
sK141: ( nat > $o ) > nat > nat ).
thf(func_def_327,type,
sK142: ( nat > $o ) > nat ).
thf(func_def_328,type,
sK143: nat > ( nat > $o ) > nat ).
thf(func_def_329,type,
sK144: ( nat > $o ) > nat ).
thf(func_def_330,type,
sK145: ( ( nat > $o ) > $o ) > nat > $o ).
thf(func_def_331,type,
sK146: ( ( nat > $o ) > $o ) > nat ).
thf(func_def_332,type,
sK147: ( nat > $o ) > ( ( nat > $o ) > $o ) > nat ).
thf(func_def_333,type,
sK148: ( nat > $o ) > ( ( nat > $o ) > $o ) > nat > $o ).
thf(func_def_334,type,
sK149: nat > ( nat > $o ) > nat ).
thf(func_def_335,type,
sK150: ( nat > $o ) > nat ).
thf(func_def_336,type,
sK151: ( ( int > $o ) > $o ) > int > $o ).
thf(func_def_337,type,
sK152: ( ( int > $o ) > $o ) > int ).
thf(func_def_338,type,
sK153: ( int > $o ) > int ).
thf(func_def_339,type,
sK154: ( int > $o ) > nat ).
thf(func_def_340,type,
sK155: ( int > $o ) > int ).
thf(func_def_341,type,
sK156: ( pname > x_a ) > ( pname > $o ) > pname ).
thf(func_def_342,type,
sK157: ( int > $o ) > ( ( int > $o ) > $o ) > int ).
thf(func_def_343,type,
sK158: ( int > $o ) > ( ( int > $o ) > $o ) > int > $o ).
thf(func_def_344,type,
sK159: ( x_a > $o ) > pname ).
thf(func_def_345,type,
sK160: int > ( int > $o ) > ( int > $o ) > int ).
thf(func_def_346,type,
sK161: int > ( int > $o ) > int ).
thf(func_def_347,type,
sK162: int > ( int > $o ) > int ).
thf(func_def_348,type,
sK163: ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_349,type,
sK164: ( x_a > $o ) > ( int > x_a ) > ( int > $o ) > int > $o ).
thf(func_def_350,type,
sK165: ( x_a > nat ) > ( nat > $o ) > ( x_a > $o ) > x_a > $o ).
thf(func_def_351,type,
sK166: ( nat > nat > $o ) > nat ).
thf(func_def_352,type,
sK167: ( ( int > $o ) > $o ) > ( int > int > $o ) > ( int > $o ) > int > $o ).
thf(func_def_353,type,
sK168: ( nat > $o ) > nat > nat > nat ).
thf(func_def_354,type,
sK169: ( nat > x_a > $o ) > ( nat > $o ) > ( ( x_a > $o ) > $o ) > nat > $o ).
thf(func_def_355,type,
sK170: ( x_a > $o ) > x_a ).
thf(func_def_356,type,
sK171: ( x_a > pname ) > ( x_a > $o ) > ( pname > $o ) > x_a > $o ).
thf(func_def_357,type,
sK172: ( ( int > $o ) > nat ) > ( nat > $o ) > ( ( int > $o ) > $o ) > ( int > $o ) > $o ).
thf(func_def_358,type,
sK173: ( x_a > x_a > $o ) > ( x_a > $o ) > ( ( x_a > $o ) > $o ) > x_a > $o ).
thf(func_def_359,type,
sK174: ( nat > $o ) > ( nat > nat > $o ) > ( ( nat > $o ) > $o ) > nat > $o ).
thf(func_def_360,type,
sK175: ( ( pname > $o ) > $o ) > ( int > $o ) > ( ( pname > $o ) > int ) > ( pname > $o ) > $o ).
thf(func_def_361,type,
sK176: ( nat > $o ) > nat ).
thf(func_def_362,type,
sK177: ( nat > $o ) > nat ).
thf(func_def_363,type,
sK178: ( int > $o ) > int > int ).
thf(func_def_364,type,
sK179: ( nat > int ) > nat ).
thf(func_def_365,type,
sK180: ( pname > $o ) > ( pname > nat ) > ( nat > $o ) > pname > $o ).
thf(func_def_366,type,
sK181: ( ( nat > $o ) > $o ) > ( nat > $o ) > ( ( nat > $o ) > nat ) > ( nat > $o ) > $o ).
thf(func_def_367,type,
sK182: ( int > $o ) > int > int ).
thf(func_def_368,type,
sK183: ( x_a > $o ) > ( x_a > $o ) > x_a ).
thf(func_def_369,type,
sK184: ( ( nat > $o ) > $o ) > nat ).
thf(func_def_370,type,
sK185: ( ( nat > $o ) > $o ) > nat > $o ).
thf(func_def_371,type,
sK186: ( ( x_a > $o ) > $o ) > x_a > $o ).
thf(func_def_372,type,
sK187: ( ( x_a > $o ) > $o ) > x_a ).
thf(func_def_373,type,
sK188: ( ( x_a > $o ) > $o ) > x_a ).
thf(func_def_374,type,
sK189: ( nat > $o ) > nat > nat ).
thf(func_def_375,type,
sK190: nat > nat > nat ).
thf(func_def_376,type,
sK191: int > nat ).
thf(func_def_377,type,
sK192: ( int > nat > $o ) > ( int > $o ) > ( ( nat > $o ) > $o ) > int > $o ).
thf(func_def_378,type,
sK193: ( ( int > $o ) > $o ) > ( pname > $o ) > ( ( int > $o ) > pname ) > ( int > $o ) > $o ).
thf(func_def_379,type,
sK194: ( nat > $o ) > nat ).
thf(func_def_380,type,
sK195: ( nat > $o ) > nat > $o ).
thf(func_def_381,type,
sK196: int > nat ).
thf(func_def_382,type,
sK197: int > nat ).
thf(func_def_383,type,
sK198: nat > nat > nat ).
thf(func_def_384,type,
sK199: ( x_a > $o ) > ( x_a > $o ) > x_a ).
thf(func_def_385,type,
sK200: ( int > $o ) > nat ).
thf(func_def_386,type,
sK201: ( int > $o ) > int ).
thf(func_def_387,type,
sK202: ( int > $o ) > ( int > x_a ) > int ).
thf(func_def_388,type,
sK203: ( int > pname ) > ( pname > $o ) > ( int > $o ) > int > $o ).
thf(func_def_389,type,
sK204: ( nat > nat ) > nat ).
thf(func_def_390,type,
sK205: ( nat > nat ) > nat ).
thf(func_def_391,type,
sK206: nat > nat > nat ).
thf(func_def_392,type,
sK207: int > int > nat ).
thf(func_def_393,type,
sK208: ( int > $o ) > int > $o ).
thf(func_def_394,type,
sK209: ( int > $o ) > int ).
thf(func_def_395,type,
sK210: ( ( nat > $o ) > $o ) > nat > $o ).
thf(func_def_396,type,
sK211: ( ( nat > $o ) > $o ) > nat ).
thf(func_def_397,type,
sK212: ( ( nat > $o ) > $o ) > nat ).
thf(func_def_398,type,
sK213: ( ( x_a > $o ) > $o ) > ( int > x_a > $o ) > ( int > $o ) > int > $o ).
thf(func_def_399,type,
sK214: ( ( x_a > $o ) > $o ) > ( x_a > $o ) > x_a ).
thf(func_def_400,type,
sK215: ( ( x_a > $o ) > $o ) > ( x_a > $o ) > x_a > $o ).
thf(func_def_401,type,
sK216: ( ( int > $o ) > x_a ) > ( ( int > $o ) > $o ) > ( x_a > $o ) > ( int > $o ) > $o ).
thf(func_def_402,type,
sK217: ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_403,type,
sK218: ( nat > $o ) > ( ( x_a > $o ) > $o ) > ( ( x_a > $o ) > nat ) > ( x_a > $o ) > $o ).
thf(func_def_404,type,
sK219: ( pname > $o ) > ( pname > int > $o ) > ( ( int > $o ) > $o ) > pname > $o ).
thf(func_def_405,type,
sK220: ( nat > $o ) > ( nat > int > $o ) > nat ).
thf(func_def_406,type,
sK221: ( x_a > $o ) > ( x_a > $o ) > x_a ).
thf(func_def_408,type,
ph223:
!>[X0: $tType] : X0 ).
thf(f8243,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f8241]) ).
thf(f8241,plain,
$true != $true,
inference(superposition,[status(thm)],[f8138,f5785]) ).
thf(f5785,plain,
( ( member_pname @ pn @ u )
= $true ),
inference(cnf_transformation,[status(thm)],[f2417]) ).
thf(f2417,plain,
( ( member_pname @ pn @ u )
= $true ),
inference(fool_elimination,[status(thm)],[f2416]) ).
thf(f2416,plain,
member_pname @ pn @ u,
inference(rectify,[status(thm)],[f1213]) ).
thf(f1213,axiom,
member_pname @ pn @ u,
file('/export/starexec/sandbox/tmp/tmp.KmoHMpFwsY/DTF2THF_2194.p',conj_4) ).
thf(f8138,plain,
( ( member_pname @ pn @ u )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f8136]) ).
thf(f8136,plain,
( ( ( member_pname @ pn @ u )
!= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f7814,f6903]) ).
thf(f6903,plain,
! [X2: pname,X0: pname > x_a,X1: pname > $o] :
( ( ( member_a @ ( X0 @ X2 ) @ ( image_pname_a @ X0 @ X1 ) )
= $true )
| ( ( member_pname @ X2 @ X1 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f3997]) ).
thf(f3997,plain,
! [X0: pname > x_a,X1: pname > $o,X2: pname] :
( ( ( member_a @ ( X0 @ X2 ) @ ( image_pname_a @ X0 @ X1 ) )
= $true )
| ( ( member_pname @ X2 @ X1 )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f1289]) ).
thf(f1289,plain,
! [X0: pname > x_a,X1: pname > $o,X2: pname] :
( ( ( member_pname @ X2 @ X1 )
= $true )
=> ( ( member_a @ ( X0 @ X2 ) @ ( image_pname_a @ X0 @ X1 ) )
= $true ) ),
inference(fool_elimination,[status(thm)],[f1288]) ).
thf(f1288,plain,
! [X0: pname > x_a,X1: pname > $o,X2: pname] :
( ( member_pname @ X2 @ X1 )
=> ( member_a @ ( X0 @ X2 ) @ ( image_pname_a @ X0 @ X1 ) ) ),
inference(rectify,[status(thm)],[f435]) ).
thf(f435,axiom,
! [X160: pname > x_a,X162: pname > $o,X161: pname] :
( ( member_pname @ X161 @ X162 )
=> ( member_a @ ( X160 @ X161 ) @ ( image_pname_a @ X160 @ X162 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.KmoHMpFwsY/DTF2THF_2194.p',fact_434_imageI) ).
thf(f7814,plain,
( $true
!= ( member_a @ ( mgt_call @ pn ) @ ( image_pname_a @ mgt_call @ u ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f7813]) ).
thf(f7813,plain,
( ( $true
!= ( member_a @ ( mgt_call @ pn ) @ ( image_pname_a @ mgt_call @ u ) ) )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f7801,f6937]) ).
thf(f6937,plain,
( ( ord_less_eq_a_o @ g @ ( image_pname_a @ mgt_call @ u ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f2737]) ).
thf(f2737,plain,
( ( ord_less_eq_a_o @ g @ ( image_pname_a @ mgt_call @ u ) )
= $true ),
inference(fool_elimination,[status(thm)],[f2736]) ).
thf(f2736,plain,
ord_less_eq_a_o @ g @ ( image_pname_a @ mgt_call @ u ),
inference(rectify,[status(thm)],[f1210]) ).
thf(f1210,axiom,
ord_less_eq_a_o @ g @ ( image_pname_a @ mgt_call @ u ),
file('/export/starexec/sandbox/tmp/tmp.KmoHMpFwsY/DTF2THF_2194.p',conj_1) ).
thf(f7801,plain,
( ( ( ord_less_eq_a_o @ g @ ( image_pname_a @ mgt_call @ u ) )
!= $true )
| ( $true
!= ( member_a @ ( mgt_call @ pn ) @ ( image_pname_a @ mgt_call @ u ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f7798]) ).
thf(f7798,plain,
( ( $true
!= ( member_a @ ( mgt_call @ pn ) @ ( image_pname_a @ mgt_call @ u ) ) )
| ( ( ord_less_eq_a_o @ g @ ( image_pname_a @ mgt_call @ u ) )
!= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f5434,f6060]) ).
thf(f6060,plain,
! [X2: x_a,X0: x_a > $o,X1: x_a > $o] :
( ( ( ord_less_eq_a_o @ ( insert_a @ X2 @ X1 ) @ X0 )
= $true )
| ( ( ord_less_eq_a_o @ X1 @ X0 )
!= $true )
| ( ( member_a @ X2 @ X0 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f4958]) ).
thf(f4958,plain,
! [X0: x_a > $o,X1: x_a > $o,X2: x_a] :
( ( ( ( ( member_a @ X2 @ X0 )
= $true )
& ( ( ord_less_eq_a_o @ X1 @ X0 )
= $true ) )
| ( ( ord_less_eq_a_o @ ( insert_a @ X2 @ X1 ) @ X0 )
!= $true ) )
& ( ( ( ord_less_eq_a_o @ ( insert_a @ X2 @ X1 ) @ X0 )
= $true )
| ( ( member_a @ X2 @ X0 )
!= $true )
| ( ( ord_less_eq_a_o @ X1 @ X0 )
!= $true ) ) ),
inference(flattening,[status(thm)],[f4957]) ).
thf(f4957,plain,
! [X0: x_a > $o,X1: x_a > $o,X2: x_a] :
( ( ( ( ( member_a @ X2 @ X0 )
= $true )
& ( ( ord_less_eq_a_o @ X1 @ X0 )
= $true ) )
| ( ( ord_less_eq_a_o @ ( insert_a @ X2 @ X1 ) @ X0 )
!= $true ) )
& ( ( ( ord_less_eq_a_o @ ( insert_a @ X2 @ X1 ) @ X0 )
= $true )
| ( ( member_a @ X2 @ X0 )
!= $true )
| ( ( ord_less_eq_a_o @ X1 @ X0 )
!= $true ) ) ),
inference(nnf_transformation,[status(thm)],[f1857]) ).
thf(f1857,plain,
! [X0: x_a > $o,X1: x_a > $o,X2: x_a] :
( ( ( ( member_a @ X2 @ X0 )
= $true )
& ( ( ord_less_eq_a_o @ X1 @ X0 )
= $true ) )
<=> ( ( ord_less_eq_a_o @ ( insert_a @ X2 @ X1 ) @ X0 )
= $true ) ),
inference(fool_elimination,[status(thm)],[f1856]) ).
thf(f1856,plain,
! [X0: x_a > $o,X1: x_a > $o,X2: x_a] :
( ( ord_less_eq_a_o @ ( insert_a @ X2 @ X1 ) @ X0 )
<=> ( ( member_a @ X2 @ X0 )
& ( ord_less_eq_a_o @ X1 @ X0 ) ) ),
inference(rectify,[status(thm)],[f454]) ).
thf(f454,axiom,
! [X181: x_a > $o,X180: x_a > $o,X179: x_a] :
( ( ord_less_eq_a_o @ ( insert_a @ X179 @ X180 ) @ X181 )
<=> ( ( member_a @ X179 @ X181 )
& ( ord_less_eq_a_o @ X180 @ X181 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.KmoHMpFwsY/DTF2THF_2194.p',fact_453_insert__subset) ).
thf(f5434,plain,
( ( ord_less_eq_a_o @ ( insert_a @ ( mgt_call @ pn ) @ g ) @ ( image_pname_a @ mgt_call @ u ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f3346]) ).
thf(f3346,plain,
( ( ord_less_eq_a_o @ ( insert_a @ ( mgt_call @ pn ) @ g ) @ ( image_pname_a @ mgt_call @ u ) )
!= $true ),
inference(flattening,[status(thm)],[f1807]) ).
thf(f1807,plain,
( ( ord_less_eq_a_o @ ( insert_a @ ( mgt_call @ pn ) @ g ) @ ( image_pname_a @ mgt_call @ u ) )
!= $true ),
inference(fool_elimination,[status(thm)],[f1806]) ).
thf(f1806,plain,
~ ( ord_less_eq_a_o @ ( insert_a @ ( mgt_call @ pn ) @ g ) @ ( image_pname_a @ mgt_call @ u ) ),
inference(rectify,[status(thm)],[f1216]) ).
thf(f1216,negated_conjecture,
~ ( ord_less_eq_a_o @ ( insert_a @ ( mgt_call @ pn ) @ g ) @ ( image_pname_a @ mgt_call @ u ) ),
inference(negated_conjecture,[status(cth)],[f1215]) ).
thf(f1215,conjecture,
ord_less_eq_a_o @ ( insert_a @ ( mgt_call @ pn ) @ g ) @ ( image_pname_a @ mgt_call @ u ),
file('/export/starexec/sandbox/tmp/tmp.KmoHMpFwsY/DTF2THF_2194.p',conj_6) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SWW473^3 : TPTP v9.2.1. Released v5.3.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.33 % Computer : n005.cluster.edu
% 0.17/0.33 % Model : x86_64 x86_64
% 0.17/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.33 % Memory : 8042.1875MB
% 0.17/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.33 % CPULimit : 300
% 0.17/0.33 % WCLimit : 300
% 0.17/0.33 % DateTime : Tue Feb 24 13:40:29 EST 2026
% 0.17/0.33 % CPUTime :
% 0.17/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.29/0.45 ---- Original DTF file ---
% 0.29/0.45 thf(spec,logic,$$dhol).
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 % File : SWW473^3 : TPTP v9.2.1. Released v5.3.0.
% 0.29/0.45 % Domain : Software Verification
% 0.29/0.45 % Problem : Hoare's Logic with Procedures line 383, 1000 axioms selected
% 0.29/0.45 % Version : Especial.
% 0.29/0.45 % English :
% 0.29/0.45
% 0.29/0.45 % Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% 0.29/0.45 % : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% 0.29/0.45 % Source : [Bla11]
% 0.29/0.45 % Names : hoare_1000_thf_l383 [Bla11]
% 0.29/0.45
% 0.29/0.45 % Status : ContradictoryAxioms
% 0.29/0.45 % Rating : 0.44 v9.1.0, 0.50 v8.2.0, 0.46 v8.1.0, 0.45 v7.5.0, 0.43 v7.4.0, 0.33 v7.3.0, 0.56 v7.2.0, 0.38 v7.1.0, 0.62 v7.0.0, 0.71 v6.4.0, 0.83 v6.3.0, 0.80 v6.2.0, 0.57 v6.1.0, 0.71 v5.5.0, 0.67 v5.4.0, 1.00 v5.3.0
% 0.29/0.45 % Syntax : Number of formulae : 1390 ( 234 unt; 175 typ; 0 def)
% 0.29/0.45 % Number of atoms : 4436 ( 862 equ; 70 cnn)
% 0.29/0.45 % Maximal formula atoms : 9 ( 3 avg)
% 0.29/0.45 % Number of connectives : 10367 ( 301 ~; 68 |; 270 &;8078 @)
% 0.29/0.45 % ( 253 <=>;1397 =>; 0 <=; 0 <~>)
% 0.29/0.45 % Maximal formula depth : 17 ( 7 avg)
% 0.29/0.45 % Number of types : 7 ( 6 usr)
% 0.29/0.45 % Number of type conns : 2454 (2454 >; 0 *; 0 +; 0 <<)
% 0.29/0.45 % Number of symbols : 176 ( 169 usr; 15 con; 0-3 aty)
% 0.29/0.45 % Number of variables : 2984 ( 93 ^;2786 !; 105 ?;2984 :)
% 0.29/0.45 % SPC : TH0_CAX_EQU_NAR
% 0.29/0.45
% 0.29/0.45 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.29/0.45 % 2011-08-09 19:59:45
% 0.29/0.45 %------------------------------------------------------------------------------
% 0.29/0.45 %----Should-be-implicit typings (6)
% 0.29/0.45 thf(ty_ty_t__a,type,
% 0.29/0.45 x_a: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_ty_tc__Com__Ocom,type,
% 0.29/0.45 com: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_ty_tc__Com__Opname,type,
% 0.29/0.45 pname: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_ty_tc__Int__Oint,type,
% 0.29/0.45 int: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_ty_tc__Nat__Onat,type,
% 0.29/0.45 nat: $tType ).
% 0.29/0.45
% 0.29/0.45 thf(ty_ty_tc__Option__Ooption_Itc__Com__Ocom_J,type,
% 0.29/0.45 option_com: $tType ).
% 0.29/0.45
% 0.29/0.45 %----Explicit typings (169)
% 0.29/0.45 thf(sy_c_Com_Obody,type,
% 0.29/0.45 body: pname > option_com ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Ex,type,
% 0.29/0.45 ex: ( int > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ocard_000_062_It__a_M_Eo_J,type,
% 0.29/0.45 finite_card_a_o: ( ( x_a > $o ) > $o ) > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ocard_000_062_Itc__Com__Opname_M_Eo_J,type,
% 0.29/0.45 finite_card_pname_o: ( ( pname > $o ) > $o ) > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ocard_000_062_Itc__Int__Oint_M_Eo_J,type,
% 0.29/0.45 finite_card_int_o: ( ( int > $o ) > $o ) > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ocard_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 0.29/0.45 finite_card_nat_o: ( ( nat > $o ) > $o ) > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ocard_000t__a,type,
% 0.29/0.45 finite_card_a: ( x_a > $o ) > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ocard_000tc__Com__Opname,type,
% 0.29/0.45 finite_card_pname: ( pname > $o ) > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ocard_000tc__Int__Oint,type,
% 0.29/0.45 finite_card_int: ( int > $o ) > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ocard_000tc__Nat__Onat,type,
% 0.29/0.45 finite_card_nat: ( nat > $o ) > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ofinite_000_062_I_062_It__a_M_Eo_J_M_Eo_J,type,
% 0.29/0.45 finite_finite_a_o_o: ( ( ( x_a > $o ) > $o ) > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ofinite_000_062_I_062_Itc__Com__Opname_M_Eo_J_M_Eo_J,type,
% 0.29/0.45 finite1066544169me_o_o: ( ( ( pname > $o ) > $o ) > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ofinite_000_062_I_062_Itc__Int__Oint_M_Eo_J_M_Eo_J,type,
% 0.29/0.45 finite229719499nt_o_o: ( ( ( int > $o ) > $o ) > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ofinite_000_062_I_062_Itc__Nat__Onat_M_Eo_J_M_Eo_J,type,
% 0.29/0.45 finite1676163439at_o_o: ( ( ( nat > $o ) > $o ) > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ofinite_000_062_It__a_M_Eo_J,type,
% 0.29/0.45 finite_finite_a_o: ( ( x_a > $o ) > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ofinite_000_062_Itc__Com__Opname_M_Eo_J,type,
% 0.29/0.45 finite297249702name_o: ( ( pname > $o ) > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ofinite_000_062_Itc__Int__Oint_M_Eo_J,type,
% 0.29/0.45 finite_finite_int_o: ( ( int > $o ) > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ofinite_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 0.29/0.45 finite_finite_nat_o: ( ( nat > $o ) > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ofinite_000t__a,type,
% 0.29/0.45 finite_finite_a: ( x_a > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ofinite_000tc__Com__Opname,type,
% 0.29/0.45 finite_finite_pname: ( pname > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ofinite_000tc__Int__Oint,type,
% 0.29/0.45 finite_finite_int: ( int > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ofinite_000tc__Nat__Onat,type,
% 0.29/0.45 finite_finite_nat: ( nat > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ofolding__one_000t__a,type,
% 0.29/0.45 finite_folding_one_a: ( x_a > x_a > x_a ) > ( ( x_a > $o ) > x_a ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ofolding__one_000tc__Com__Opname,type,
% 0.29/0.45 finite1282449217_pname: ( pname > pname > pname ) > ( ( pname > $o ) > pname ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ofolding__one_000tc__Int__Oint,type,
% 0.29/0.45 finite1626084323ne_int: ( int > int > int ) > ( ( int > $o ) > int ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ofolding__one_000tc__Nat__Onat,type,
% 0.29/0.45 finite988810631ne_nat: ( nat > nat > nat ) > ( ( nat > $o ) > nat ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ofolding__one__idem_000t__a,type,
% 0.29/0.45 finite1819937229idem_a: ( x_a > x_a > x_a ) > ( ( x_a > $o ) > x_a ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Com__Opname,type,
% 0.29/0.45 finite89670078_pname: ( pname > pname > pname ) > ( ( pname > $o ) > pname ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Int__Oint,type,
% 0.29/0.45 finite1432773856em_int: ( int > int > int ) > ( ( int > $o ) > int ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Nat__Onat,type,
% 0.29/0.45 finite795500164em_nat: ( nat > nat > nat ) > ( ( nat > $o ) > nat ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Oabs__class_Oabs_000tc__Int__Oint,type,
% 0.29/0.45 abs_abs_int: int > int ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Ominus__class_Ominus_000_062_It__a_M_Eo_J,type,
% 0.29/0.45 minus_minus_a_o: ( x_a > $o ) > ( x_a > $o ) > x_a > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Com__Opname_M_Eo_J,type,
% 0.29/0.45 minus_minus_pname_o: ( pname > $o ) > ( pname > $o ) > pname > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Int__Oint_M_Eo_J,type,
% 0.29/0.45 minus_minus_int_o: ( int > $o ) > ( int > $o ) > int > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 0.29/0.45 minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Ominus__class_Ominus_000tc__Int__Oint,type,
% 0.29/0.45 minus_minus_int: int > int > int ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
% 0.29/0.45 minus_minus_nat: nat > nat > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Oone__class_Oone_000tc__Int__Oint,type,
% 0.29/0.45 one_one_int: int ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
% 0.29/0.45 one_one_nat: nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Oplus__class_Oplus_000tc__Int__Oint,type,
% 0.29/0.45 plus_plus_int: int > int > int ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
% 0.29/0.45 plus_plus_nat: nat > nat > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Otimes__class_Otimes_000tc__Int__Oint,type,
% 0.29/0.45 times_times_int: int > int > int ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
% 0.29/0.45 times_times_nat: nat > nat > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Ozero__class_Ozero_000tc__Int__Oint,type,
% 0.29/0.45 zero_zero_int: int ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
% 0.29/0.45 zero_zero_nat: nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_HOL_OThe_000t__a,type,
% 0.29/0.45 the_a: ( x_a > $o ) > x_a ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_HOL_OThe_000tc__Int__Oint,type,
% 0.29/0.45 the_int: ( int > $o ) > int ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_HOL_OThe_000tc__Nat__Onat,type,
% 0.29/0.45 the_nat: ( nat > $o ) > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_If_000tc__Nat__Onat,type,
% 0.29/0.45 if_nat: $o > nat > nat > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Int_OBit1,type,
% 0.29/0.45 bit1: int > int ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Int_OPls,type,
% 0.29/0.45 pls: int ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Int_Onumber__class_Onumber__of_000tc__Int__Oint,type,
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% 0.29/0.45 thf(sy_c_Int_Onumber__class_Onumber__of_000tc__Nat__Onat,type,
% 0.29/0.45 number_number_of_nat: int > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Int_Osucc,type,
% 0.29/0.45 succ: int > int ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Nat_OSuc,type,
% 0.29/0.45 suc: nat > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Nat_Onat_Onat__case_000_Eo,type,
% 0.29/0.45 nat_case_o: $o > ( nat > $o ) > nat > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Nat_Onat_Onat__case_000tc__Nat__Onat,type,
% 0.29/0.45 nat_case_nat: nat > ( nat > nat ) > nat > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Int__Oint,type,
% 0.29/0.45 semiri1621563631at_int: nat > int ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Nat__Numeral_Oneg,type,
% 0.29/0.45 nat_neg: int > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Nat__Transfer_Otsub,type,
% 0.29/0.45 nat_tsub: int > int > int ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Option_Othe_000tc__Com__Ocom,type,
% 0.29/0.45 the_com: option_com > com ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Obot__class_Obot_000_062_It__a_M_Eo_J,type,
% 0.29/0.45 bot_bot_a_o: x_a > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Com__Opname_M_Eo_J,type,
% 0.29/0.45 bot_bot_pname_o: pname > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Int__Oint_M_Eo_J,type,
% 0.29/0.45 bot_bot_int_o: int > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 0.29/0.45 bot_bot_nat_o: nat > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Obot__class_Obot_000_Eo,type,
% 0.29/0.45 bot_bot_o: $o ).
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% 0.29/0.45 thf(sy_c_Orderings_Obot__class_Obot_000tc__Nat__Onat,type,
% 0.29/0.45 bot_bot_nat: nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless_000_062_Itc__Int__Oint_M_Eo_J,type,
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% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless_000_062_Itc__Nat__Onat_M_Eo_J,type,
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% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless_000tc__Int__Oint,type,
% 0.29/0.45 ord_less_int: int > int > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,
% 0.29/0.45 ord_less_nat: nat > nat > $o ).
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% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_I_062_It__a_M_Eo_J_M_Eo_J,type,
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% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_I_062_Itc__Com__Opname_M_Eo_J_M_Eo_,type,
% 0.29/0.45 ord_le1205211808me_o_o: ( ( pname > $o ) > $o ) > ( ( pname > $o ) > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_I_062_Itc__Int__Oint_M_Eo_J_M_Eo_J,type,
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% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_I_062_Itc__Nat__Onat_M_Eo_J_M_Eo_J,type,
% 0.29/0.45 ord_less_eq_nat_o_o: ( ( nat > $o ) > $o ) > ( ( nat > $o ) > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_It__a_M_Eo_J,type,
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% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Com__Opname_M_Eo_J,type,
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% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Int__Oint_M_Eo_J,type,
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% 0.29/0.45
% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_000_Eo,type,
% 0.29/0.45 ord_less_eq_o: $o > $o > $o ).
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% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Int__Oint,type,
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% 0.29/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
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% 0.29/0.45
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% 0.29/0.45
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% 0.29/0.45 thf(sy_c_Set_Oimage_000_062_Itc__Int__Oint_M_Eo_J_000tc__Int__Oint,type,
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% 0.29/0.45 thf(sy_c_Set_Oimage_000_062_Itc__Int__Oint_M_Eo_J_000tc__Nat__Onat,type,
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% 0.29/0.45
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% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oimage_000_062_Itc__Nat__Onat_M_Eo_J_000tc__Com__Opname,type,
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% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oimage_000_062_Itc__Nat__Onat_M_Eo_J_000tc__Int__Oint,type,
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% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oimage_000_062_Itc__Nat__Onat_M_Eo_J_000tc__Nat__Onat,type,
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% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oimage_000t__a_000_062_It__a_M_Eo_J,type,
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% 0.29/0.45 thf(sy_c_Set_Oimage_000t__a_000_062_Itc__Com__Opname_M_Eo_J,type,
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% 0.29/0.45 thf(sy_c_Set_Oimage_000t__a_000_062_Itc__Int__Oint_M_Eo_J,type,
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% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oimage_000t__a_000_062_Itc__Nat__Onat_M_Eo_J,type,
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% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oimage_000t__a_000t__a,type,
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% 0.29/0.45 thf(sy_c_Set_Oimage_000t__a_000tc__Com__Opname,type,
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% 0.29/0.45 thf(sy_c_Set_Oimage_000t__a_000tc__Int__Oint,type,
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% 0.29/0.45 thf(sy_c_Set_Oimage_000t__a_000tc__Nat__Onat,type,
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% 0.29/0.45
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% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oimage_000tc__Nat__Onat_000t__a,type,
% 0.29/0.45 image_nat_a: ( nat > x_a ) > ( nat > $o ) > x_a > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Com__Opname,type,
% 0.29/0.45 image_nat_pname: ( nat > pname ) > ( nat > $o ) > pname > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Int__Oint,type,
% 0.29/0.45 image_nat_int: ( nat > int ) > ( nat > $o ) > int > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oinsert_000_062_It__a_M_Eo_J,type,
% 0.29/0.45 insert_a_o: ( x_a > $o ) > ( ( x_a > $o ) > $o ) > ( x_a > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oinsert_000_062_Itc__Com__Opname_M_Eo_J,type,
% 0.29/0.45 insert_pname_o: ( pname > $o ) > ( ( pname > $o ) > $o ) > ( pname > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oinsert_000_062_Itc__Int__Oint_M_Eo_J,type,
% 0.29/0.45 insert_int_o: ( int > $o ) > ( ( int > $o ) > $o ) > ( int > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oinsert_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 0.29/0.45 insert_nat_o: ( nat > $o ) > ( ( nat > $o ) > $o ) > ( nat > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oinsert_000t__a,type,
% 0.29/0.45 insert_a: x_a > ( x_a > $o ) > x_a > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oinsert_000tc__Com__Opname,type,
% 0.29/0.45 insert_pname: pname > ( pname > $o ) > pname > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oinsert_000tc__Int__Oint,type,
% 0.29/0.45 insert_int: int > ( int > $o ) > int > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Oinsert_000tc__Nat__Onat,type,
% 0.29/0.45 insert_nat: nat > ( nat > $o ) > nat > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Othe__elem_000t__a,type,
% 0.29/0.45 the_elem_a: ( x_a > $o ) > x_a ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Othe__elem_000tc__Int__Oint,type,
% 0.29/0.45 the_elem_int: ( int > $o ) > int ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_Set_Othe__elem_000tc__Nat__Onat,type,
% 0.29/0.45 the_elem_nat: ( nat > $o ) > nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_fequal_000t__a,type,
% 0.29/0.45 fequal_a: x_a > x_a > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_fequal_000tc__Int__Oint,type,
% 0.29/0.45 fequal_int: int > int > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_fequal_000tc__Nat__Onat,type,
% 0.29/0.45 fequal_nat: nat > nat > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_member_000_062_It__a_M_Eo_J,type,
% 0.29/0.45 member_a_o: ( x_a > $o ) > ( ( x_a > $o ) > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_member_000_062_Itc__Com__Opname_M_Eo_J,type,
% 0.29/0.45 member_pname_o: ( pname > $o ) > ( ( pname > $o ) > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_member_000_062_Itc__Int__Oint_M_Eo_J,type,
% 0.29/0.45 member_int_o: ( int > $o ) > ( ( int > $o ) > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_member_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 0.29/0.45 member_nat_o: ( nat > $o ) > ( ( nat > $o ) > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_member_000t__a,type,
% 0.29/0.45 member_a: x_a > ( x_a > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_member_000tc__Com__Opname,type,
% 0.29/0.45 member_pname: pname > ( pname > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_member_000tc__Int__Oint,type,
% 0.29/0.45 member_int: int > ( int > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_c_member_000tc__Nat__Onat,type,
% 0.29/0.45 member_nat: nat > ( nat > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_G,type,
% 0.29/0.45 g: x_a > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_P,type,
% 0.29/0.45 p: ( x_a > $o ) > ( x_a > $o ) > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_U,type,
% 0.29/0.45 u: pname > $o ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_mgt,type,
% 0.29/0.45 mgt: com > x_a ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_mgt__call,type,
% 0.29/0.45 mgt_call: pname > x_a ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_na,type,
% 0.29/0.45 na: nat ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_pn,type,
% 0.29/0.45 pn: pname ).
% 0.29/0.45
% 0.29/0.45 thf(sy_v_wt,type,
% 0.29/0.45 wt: com > $o ).
% 0.29/0.45
% 0.29/0.45 %----Relevant facts (1199)
% 0.29/0.45 thf(fact_0_assms_I1_J,axiom,
% 0.29/0.45 ! [Ts: x_a > $o,G: x_a > $o] :
% 0.29/0.45 ( ( ord_less_eq_a_o @ Ts @ G )
% 0.29/0.45 => ( p @ G @ Ts ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_1_finite__Collect__subsets,axiom,
% 0.29/0.45 ! [A_161: ( int > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_int_o @ A_161 )
% 0.29/0.45 => ( finite229719499nt_o_o
% 0.29/0.45 @ ( collect_int_o_o
% 0.29/0.45 @ ^ [B_26: ( int > $o ) > $o] : ( ord_less_eq_int_o_o @ B_26 @ A_161 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_2_finite__Collect__subsets,axiom,
% 0.29/0.45 ! [A_161: ( nat > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_nat_o @ A_161 )
% 0.29/0.45 => ( finite1676163439at_o_o
% 0.29/0.45 @ ( collect_nat_o_o
% 0.29/0.45 @ ^ [B_26: ( nat > $o ) > $o] : ( ord_less_eq_nat_o_o @ B_26 @ A_161 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_3_finite__Collect__subsets,axiom,
% 0.29/0.45 ! [A_161: ( pname > $o ) > $o] :
% 0.29/0.45 ( ( finite297249702name_o @ A_161 )
% 0.29/0.45 => ( finite1066544169me_o_o
% 0.29/0.45 @ ( collect_pname_o_o
% 0.29/0.45 @ ^ [B_26: ( pname > $o ) > $o] : ( ord_le1205211808me_o_o @ B_26 @ A_161 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_4_finite__Collect__subsets,axiom,
% 0.29/0.45 ! [A_161: ( x_a > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_a_o @ A_161 )
% 0.29/0.45 => ( finite_finite_a_o_o
% 0.29/0.45 @ ( collect_a_o_o
% 0.29/0.45 @ ^ [B_26: ( x_a > $o ) > $o] : ( ord_less_eq_a_o_o @ B_26 @ A_161 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_5_finite__Collect__subsets,axiom,
% 0.29/0.45 ! [A_161: x_a > $o] :
% 0.29/0.45 ( ( finite_finite_a @ A_161 )
% 0.29/0.45 => ( finite_finite_a_o
% 0.29/0.45 @ ( collect_a_o
% 0.29/0.45 @ ^ [B_26: x_a > $o] : ( ord_less_eq_a_o @ B_26 @ A_161 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_6_finite__Collect__subsets,axiom,
% 0.29/0.45 ! [A_161: pname > $o] :
% 0.29/0.45 ( ( finite_finite_pname @ A_161 )
% 0.29/0.45 => ( finite297249702name_o
% 0.29/0.45 @ ( collect_pname_o
% 0.29/0.45 @ ^ [B_26: pname > $o] : ( ord_less_eq_pname_o @ B_26 @ A_161 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_7_finite__Collect__subsets,axiom,
% 0.29/0.45 ! [A_161: nat > $o] :
% 0.29/0.45 ( ( finite_finite_nat @ A_161 )
% 0.29/0.45 => ( finite_finite_nat_o
% 0.29/0.45 @ ( collect_nat_o
% 0.29/0.45 @ ^ [B_26: nat > $o] : ( ord_less_eq_nat_o @ B_26 @ A_161 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_8_finite__Collect__subsets,axiom,
% 0.29/0.45 ! [A_161: int > $o] :
% 0.29/0.45 ( ( finite_finite_int @ A_161 )
% 0.29/0.45 => ( finite_finite_int_o
% 0.29/0.45 @ ( collect_int_o
% 0.29/0.45 @ ^ [B_26: int > $o] : ( ord_less_eq_int_o @ B_26 @ A_161 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_9_finite__imageI,axiom,
% 0.29/0.45 ! [H: pname > int > $o,F_42: pname > $o] :
% 0.29/0.45 ( ( finite_finite_pname @ F_42 )
% 0.29/0.45 => ( finite_finite_int_o @ ( image_pname_int_o @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_10_finite__imageI,axiom,
% 0.29/0.45 ! [H: pname > nat > $o,F_42: pname > $o] :
% 0.29/0.45 ( ( finite_finite_pname @ F_42 )
% 0.29/0.45 => ( finite_finite_nat_o @ ( image_pname_nat_o @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_11_finite__imageI,axiom,
% 0.29/0.45 ! [H: pname > pname > $o,F_42: pname > $o] :
% 0.29/0.45 ( ( finite_finite_pname @ F_42 )
% 0.29/0.45 => ( finite297249702name_o @ ( image_pname_pname_o @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_12_finite__imageI,axiom,
% 0.29/0.45 ! [H: pname > x_a > $o,F_42: pname > $o] :
% 0.29/0.45 ( ( finite_finite_pname @ F_42 )
% 0.29/0.45 => ( finite_finite_a_o @ ( image_pname_a_o @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_13_finite__imageI,axiom,
% 0.29/0.45 ! [H: nat > x_a,F_42: nat > $o] :
% 0.29/0.45 ( ( finite_finite_nat @ F_42 )
% 0.29/0.45 => ( finite_finite_a @ ( image_nat_a @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_14_finite__imageI,axiom,
% 0.29/0.45 ! [H: nat > int > $o,F_42: nat > $o] :
% 0.29/0.45 ( ( finite_finite_nat @ F_42 )
% 0.29/0.45 => ( finite_finite_int_o @ ( image_nat_int_o @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_15_finite__imageI,axiom,
% 0.29/0.45 ! [H: nat > nat > $o,F_42: nat > $o] :
% 0.29/0.45 ( ( finite_finite_nat @ F_42 )
% 0.29/0.45 => ( finite_finite_nat_o @ ( image_nat_nat_o @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_16_finite__imageI,axiom,
% 0.29/0.45 ! [H: nat > pname > $o,F_42: nat > $o] :
% 0.29/0.45 ( ( finite_finite_nat @ F_42 )
% 0.29/0.45 => ( finite297249702name_o @ ( image_nat_pname_o @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_17_finite__imageI,axiom,
% 0.29/0.45 ! [H: nat > x_a > $o,F_42: nat > $o] :
% 0.29/0.45 ( ( finite_finite_nat @ F_42 )
% 0.29/0.45 => ( finite_finite_a_o @ ( image_nat_a_o @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_18_finite__imageI,axiom,
% 0.29/0.45 ! [H: int > x_a,F_42: int > $o] :
% 0.29/0.45 ( ( finite_finite_int @ F_42 )
% 0.29/0.45 => ( finite_finite_a @ ( image_int_a @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_19_finite__imageI,axiom,
% 0.29/0.45 ! [H: int > int > $o,F_42: int > $o] :
% 0.29/0.45 ( ( finite_finite_int @ F_42 )
% 0.29/0.45 => ( finite_finite_int_o @ ( image_int_int_o @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_20_finite__imageI,axiom,
% 0.29/0.45 ! [H: int > nat > $o,F_42: int > $o] :
% 0.29/0.45 ( ( finite_finite_int @ F_42 )
% 0.29/0.45 => ( finite_finite_nat_o @ ( image_int_nat_o @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_21_finite__imageI,axiom,
% 0.29/0.45 ! [H: int > pname > $o,F_42: int > $o] :
% 0.29/0.45 ( ( finite_finite_int @ F_42 )
% 0.29/0.45 => ( finite297249702name_o @ ( image_int_pname_o @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_22_finite__imageI,axiom,
% 0.29/0.45 ! [H: int > x_a > $o,F_42: int > $o] :
% 0.29/0.45 ( ( finite_finite_int @ F_42 )
% 0.29/0.45 => ( finite_finite_a_o @ ( image_int_a_o @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_23_finite__imageI,axiom,
% 0.29/0.45 ! [H: x_a > pname,F_42: x_a > $o] :
% 0.29/0.45 ( ( finite_finite_a @ F_42 )
% 0.29/0.45 => ( finite_finite_pname @ ( image_a_pname @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_24_finite__imageI,axiom,
% 0.29/0.45 ! [H: ( int > $o ) > pname,F_42: ( int > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_int_o @ F_42 )
% 0.29/0.45 => ( finite_finite_pname @ ( image_int_o_pname @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_25_finite__imageI,axiom,
% 0.29/0.45 ! [H: ( nat > $o ) > pname,F_42: ( nat > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_nat_o @ F_42 )
% 0.29/0.45 => ( finite_finite_pname @ ( image_nat_o_pname @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_26_finite__imageI,axiom,
% 0.29/0.45 ! [H: ( pname > $o ) > pname,F_42: ( pname > $o ) > $o] :
% 0.29/0.45 ( ( finite297249702name_o @ F_42 )
% 0.29/0.45 => ( finite_finite_pname @ ( image_pname_o_pname @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_27_finite__imageI,axiom,
% 0.29/0.45 ! [H: ( x_a > $o ) > pname,F_42: ( x_a > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_a_o @ F_42 )
% 0.29/0.45 => ( finite_finite_pname @ ( image_a_o_pname @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_28_finite__imageI,axiom,
% 0.29/0.45 ! [H: x_a > nat,F_42: x_a > $o] :
% 0.29/0.45 ( ( finite_finite_a @ F_42 )
% 0.29/0.45 => ( finite_finite_nat @ ( image_a_nat @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_29_finite__imageI,axiom,
% 0.29/0.45 ! [H: ( int > $o ) > nat,F_42: ( int > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_int_o @ F_42 )
% 0.29/0.45 => ( finite_finite_nat @ ( image_int_o_nat @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_30_finite__imageI,axiom,
% 0.29/0.45 ! [H: ( nat > $o ) > nat,F_42: ( nat > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_nat_o @ F_42 )
% 0.29/0.45 => ( finite_finite_nat @ ( image_nat_o_nat @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_31_finite__imageI,axiom,
% 0.29/0.45 ! [H: ( pname > $o ) > nat,F_42: ( pname > $o ) > $o] :
% 0.29/0.45 ( ( finite297249702name_o @ F_42 )
% 0.29/0.45 => ( finite_finite_nat @ ( image_pname_o_nat @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_32_finite__imageI,axiom,
% 0.29/0.45 ! [H: ( x_a > $o ) > nat,F_42: ( x_a > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_a_o @ F_42 )
% 0.29/0.45 => ( finite_finite_nat @ ( image_a_o_nat @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_33_finite__imageI,axiom,
% 0.29/0.45 ! [H: x_a > int,F_42: x_a > $o] :
% 0.29/0.45 ( ( finite_finite_a @ F_42 )
% 0.29/0.45 => ( finite_finite_int @ ( image_a_int @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_34_finite__imageI,axiom,
% 0.29/0.45 ! [H: ( int > $o ) > int,F_42: ( int > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_int_o @ F_42 )
% 0.29/0.45 => ( finite_finite_int @ ( image_int_o_int @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_35_finite__imageI,axiom,
% 0.29/0.45 ! [H: ( nat > $o ) > int,F_42: ( nat > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_nat_o @ F_42 )
% 0.29/0.45 => ( finite_finite_int @ ( image_nat_o_int @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_36_finite__imageI,axiom,
% 0.29/0.45 ! [H: ( pname > $o ) > int,F_42: ( pname > $o ) > $o] :
% 0.29/0.45 ( ( finite297249702name_o @ F_42 )
% 0.29/0.45 => ( finite_finite_int @ ( image_pname_o_int @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_37_finite__imageI,axiom,
% 0.29/0.45 ! [H: ( x_a > $o ) > int,F_42: ( x_a > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_a_o @ F_42 )
% 0.29/0.45 => ( finite_finite_int @ ( image_a_o_int @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_38_finite__imageI,axiom,
% 0.29/0.45 ! [H: pname > x_a,F_42: pname > $o] :
% 0.29/0.45 ( ( finite_finite_pname @ F_42 )
% 0.29/0.45 => ( finite_finite_a @ ( image_pname_a @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_39_finite__imageI,axiom,
% 0.29/0.45 ! [H: nat > int,F_42: nat > $o] :
% 0.29/0.45 ( ( finite_finite_nat @ F_42 )
% 0.29/0.45 => ( finite_finite_int @ ( image_nat_int @ H @ F_42 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_40_finite_OinsertI,axiom,
% 0.29/0.45 ! [A_160: int > $o,A_159: ( int > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_int_o @ A_159 )
% 0.29/0.45 => ( finite_finite_int_o @ ( insert_int_o @ A_160 @ A_159 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_41_finite_OinsertI,axiom,
% 0.29/0.45 ! [A_160: nat > $o,A_159: ( nat > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_nat_o @ A_159 )
% 0.29/0.45 => ( finite_finite_nat_o @ ( insert_nat_o @ A_160 @ A_159 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_42_finite_OinsertI,axiom,
% 0.29/0.45 ! [A_160: pname > $o,A_159: ( pname > $o ) > $o] :
% 0.29/0.45 ( ( finite297249702name_o @ A_159 )
% 0.29/0.45 => ( finite297249702name_o @ ( insert_pname_o @ A_160 @ A_159 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_43_finite_OinsertI,axiom,
% 0.29/0.45 ! [A_160: x_a > $o,A_159: ( x_a > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_a_o @ A_159 )
% 0.29/0.45 => ( finite_finite_a_o @ ( insert_a_o @ A_160 @ A_159 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_44_finite_OinsertI,axiom,
% 0.29/0.45 ! [A_160: pname,A_159: pname > $o] :
% 0.29/0.45 ( ( finite_finite_pname @ A_159 )
% 0.29/0.45 => ( finite_finite_pname @ ( insert_pname @ A_160 @ A_159 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_45_finite_OinsertI,axiom,
% 0.29/0.45 ! [A_160: nat,A_159: nat > $o] :
% 0.29/0.45 ( ( finite_finite_nat @ A_159 )
% 0.29/0.45 => ( finite_finite_nat @ ( insert_nat @ A_160 @ A_159 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_46_finite_OinsertI,axiom,
% 0.29/0.45 ! [A_160: int,A_159: int > $o] :
% 0.29/0.45 ( ( finite_finite_int @ A_159 )
% 0.29/0.45 => ( finite_finite_int @ ( insert_int @ A_160 @ A_159 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_47_finite_OinsertI,axiom,
% 0.29/0.45 ! [A_160: x_a,A_159: x_a > $o] :
% 0.29/0.45 ( ( finite_finite_a @ A_159 )
% 0.29/0.45 => ( finite_finite_a @ ( insert_a @ A_160 @ A_159 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_48_card__image__le,axiom,
% 0.29/0.45 ! [F_41: pname > pname,A_158: pname > $o] :
% 0.29/0.45 ( ( finite_finite_pname @ A_158 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_pname @ ( image_pname_pname @ F_41 @ A_158 ) ) @ ( finite_card_pname @ A_158 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_49_card__image__le,axiom,
% 0.29/0.45 ! [F_41: x_a > x_a,A_158: x_a > $o] :
% 0.29/0.45 ( ( finite_finite_a @ A_158 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_a @ ( image_a_a @ F_41 @ A_158 ) ) @ ( finite_card_a @ A_158 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_50_card__image__le,axiom,
% 0.29/0.45 ! [F_41: ( int > $o ) > x_a,A_158: ( int > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_int_o @ A_158 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_a @ ( image_int_o_a @ F_41 @ A_158 ) ) @ ( finite_card_int_o @ A_158 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_51_card__image__le,axiom,
% 0.29/0.45 ! [F_41: ( nat > $o ) > x_a,A_158: ( nat > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_nat_o @ A_158 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_a @ ( image_nat_o_a @ F_41 @ A_158 ) ) @ ( finite_card_nat_o @ A_158 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_52_card__image__le,axiom,
% 0.29/0.45 ! [F_41: ( pname > $o ) > x_a,A_158: ( pname > $o ) > $o] :
% 0.29/0.45 ( ( finite297249702name_o @ A_158 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_a @ ( image_pname_o_a @ F_41 @ A_158 ) ) @ ( finite_card_pname_o @ A_158 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_53_card__image__le,axiom,
% 0.29/0.45 ! [F_41: ( x_a > $o ) > x_a,A_158: ( x_a > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_a_o @ A_158 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_a @ ( image_a_o_a @ F_41 @ A_158 ) ) @ ( finite_card_a_o @ A_158 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_54_card__image__le,axiom,
% 0.29/0.45 ! [F_41: pname > nat,A_158: pname > $o] :
% 0.29/0.45 ( ( finite_finite_pname @ A_158 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_nat @ ( image_pname_nat @ F_41 @ A_158 ) ) @ ( finite_card_pname @ A_158 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_55_card__image__le,axiom,
% 0.29/0.45 ! [F_41: x_a > nat,A_158: x_a > $o] :
% 0.29/0.45 ( ( finite_finite_a @ A_158 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_nat @ ( image_a_nat @ F_41 @ A_158 ) ) @ ( finite_card_a @ A_158 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_56_card__image__le,axiom,
% 0.29/0.45 ! [F_41: ( int > $o ) > nat,A_158: ( int > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_int_o @ A_158 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_nat @ ( image_int_o_nat @ F_41 @ A_158 ) ) @ ( finite_card_int_o @ A_158 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_57_card__image__le,axiom,
% 0.29/0.45 ! [F_41: ( nat > $o ) > nat,A_158: ( nat > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_nat_o @ A_158 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_o_nat @ F_41 @ A_158 ) ) @ ( finite_card_nat_o @ A_158 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_58_card__image__le,axiom,
% 0.29/0.45 ! [F_41: ( pname > $o ) > nat,A_158: ( pname > $o ) > $o] :
% 0.29/0.45 ( ( finite297249702name_o @ A_158 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_nat @ ( image_pname_o_nat @ F_41 @ A_158 ) ) @ ( finite_card_pname_o @ A_158 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_59_card__image__le,axiom,
% 0.29/0.45 ! [F_41: ( x_a > $o ) > nat,A_158: ( x_a > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_a_o @ A_158 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_nat @ ( image_a_o_nat @ F_41 @ A_158 ) ) @ ( finite_card_a_o @ A_158 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_60_card__image__le,axiom,
% 0.29/0.45 ! [F_41: pname > int,A_158: pname > $o] :
% 0.29/0.45 ( ( finite_finite_pname @ A_158 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_int @ ( image_pname_int @ F_41 @ A_158 ) ) @ ( finite_card_pname @ A_158 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_61_card__image__le,axiom,
% 0.29/0.45 ! [F_41: x_a > int,A_158: x_a > $o] :
% 0.29/0.45 ( ( finite_finite_a @ A_158 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_int @ ( image_a_int @ F_41 @ A_158 ) ) @ ( finite_card_a @ A_158 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_62_card__image__le,axiom,
% 0.29/0.45 ! [F_41: ( int > $o ) > int,A_158: ( int > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_int_o @ A_158 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_int @ ( image_int_o_int @ F_41 @ A_158 ) ) @ ( finite_card_int_o @ A_158 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_63_card__image__le,axiom,
% 0.29/0.45 ! [F_41: ( nat > $o ) > int,A_158: ( nat > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_nat_o @ A_158 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_int @ ( image_nat_o_int @ F_41 @ A_158 ) ) @ ( finite_card_nat_o @ A_158 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_64_card__image__le,axiom,
% 0.29/0.45 ! [F_41: ( pname > $o ) > int,A_158: ( pname > $o ) > $o] :
% 0.29/0.45 ( ( finite297249702name_o @ A_158 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_int @ ( image_pname_o_int @ F_41 @ A_158 ) ) @ ( finite_card_pname_o @ A_158 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_65_card__image__le,axiom,
% 0.29/0.45 ! [F_41: ( x_a > $o ) > int,A_158: ( x_a > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_a_o @ A_158 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_int @ ( image_a_o_int @ F_41 @ A_158 ) ) @ ( finite_card_a_o @ A_158 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_66_card__image__le,axiom,
% 0.29/0.45 ! [F_41: x_a > pname,A_158: x_a > $o] :
% 0.29/0.45 ( ( finite_finite_a @ A_158 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_pname @ ( image_a_pname @ F_41 @ A_158 ) ) @ ( finite_card_a @ A_158 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_67_card__image__le,axiom,
% 0.29/0.45 ! [F_41: nat > pname,A_158: nat > $o] :
% 0.29/0.45 ( ( finite_finite_nat @ A_158 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_pname @ ( image_nat_pname @ F_41 @ A_158 ) ) @ ( finite_card_nat @ A_158 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_68_card__image__le,axiom,
% 0.29/0.45 ! [F_41: int > pname,A_158: int > $o] :
% 0.29/0.45 ( ( finite_finite_int @ A_158 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_pname @ ( image_int_pname @ F_41 @ A_158 ) ) @ ( finite_card_int @ A_158 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_69_card__image__le,axiom,
% 0.29/0.45 ! [F_41: pname > x_a,A_158: pname > $o] :
% 0.29/0.45 ( ( finite_finite_pname @ A_158 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_a @ ( image_pname_a @ F_41 @ A_158 ) ) @ ( finite_card_pname @ A_158 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_70_card__image__le,axiom,
% 0.29/0.45 ! [F_41: nat > int,A_158: nat > $o] :
% 0.29/0.45 ( ( finite_finite_nat @ A_158 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_int @ ( image_nat_int @ F_41 @ A_158 ) ) @ ( finite_card_nat @ A_158 ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_71_card__mono,axiom,
% 0.29/0.45 ! [A_157: ( int > $o ) > $o,B_89: ( int > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_int_o @ B_89 )
% 0.29/0.45 => ( ( ord_less_eq_int_o_o @ A_157 @ B_89 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_int_o @ A_157 ) @ ( finite_card_int_o @ B_89 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_72_card__mono,axiom,
% 0.29/0.45 ! [A_157: ( nat > $o ) > $o,B_89: ( nat > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_nat_o @ B_89 )
% 0.29/0.45 => ( ( ord_less_eq_nat_o_o @ A_157 @ B_89 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_nat_o @ A_157 ) @ ( finite_card_nat_o @ B_89 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_73_card__mono,axiom,
% 0.29/0.45 ! [A_157: ( pname > $o ) > $o,B_89: ( pname > $o ) > $o] :
% 0.29/0.45 ( ( finite297249702name_o @ B_89 )
% 0.29/0.45 => ( ( ord_le1205211808me_o_o @ A_157 @ B_89 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_pname_o @ A_157 ) @ ( finite_card_pname_o @ B_89 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_74_card__mono,axiom,
% 0.29/0.45 ! [A_157: ( x_a > $o ) > $o,B_89: ( x_a > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_a_o @ B_89 )
% 0.29/0.45 => ( ( ord_less_eq_a_o_o @ A_157 @ B_89 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_a_o @ A_157 ) @ ( finite_card_a_o @ B_89 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_75_card__mono,axiom,
% 0.29/0.45 ! [A_157: pname > $o,B_89: pname > $o] :
% 0.29/0.45 ( ( finite_finite_pname @ B_89 )
% 0.29/0.45 => ( ( ord_less_eq_pname_o @ A_157 @ B_89 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_pname @ A_157 ) @ ( finite_card_pname @ B_89 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_76_card__mono,axiom,
% 0.29/0.45 ! [A_157: x_a > $o,B_89: x_a > $o] :
% 0.29/0.45 ( ( finite_finite_a @ B_89 )
% 0.29/0.45 => ( ( ord_less_eq_a_o @ A_157 @ B_89 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_a @ A_157 ) @ ( finite_card_a @ B_89 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_77_card__mono,axiom,
% 0.29/0.45 ! [A_157: nat > $o,B_89: nat > $o] :
% 0.29/0.45 ( ( finite_finite_nat @ B_89 )
% 0.29/0.45 => ( ( ord_less_eq_nat_o @ A_157 @ B_89 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_nat @ A_157 ) @ ( finite_card_nat @ B_89 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_78_card__mono,axiom,
% 0.29/0.45 ! [A_157: int > $o,B_89: int > $o] :
% 0.29/0.45 ( ( finite_finite_int @ B_89 )
% 0.29/0.45 => ( ( ord_less_eq_int_o @ A_157 @ B_89 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_int @ A_157 ) @ ( finite_card_int @ B_89 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_79_card__seteq,axiom,
% 0.29/0.45 ! [A_156: ( int > $o ) > $o,B_88: ( int > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_int_o @ B_88 )
% 0.29/0.45 => ( ( ord_less_eq_int_o_o @ A_156 @ B_88 )
% 0.29/0.45 => ( ( ord_less_eq_nat @ ( finite_card_int_o @ B_88 ) @ ( finite_card_int_o @ A_156 ) )
% 0.29/0.45 => ( A_156 = B_88 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_80_card__seteq,axiom,
% 0.29/0.45 ! [A_156: ( nat > $o ) > $o,B_88: ( nat > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_nat_o @ B_88 )
% 0.29/0.45 => ( ( ord_less_eq_nat_o_o @ A_156 @ B_88 )
% 0.29/0.45 => ( ( ord_less_eq_nat @ ( finite_card_nat_o @ B_88 ) @ ( finite_card_nat_o @ A_156 ) )
% 0.29/0.45 => ( A_156 = B_88 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_81_card__seteq,axiom,
% 0.29/0.45 ! [A_156: ( pname > $o ) > $o,B_88: ( pname > $o ) > $o] :
% 0.29/0.45 ( ( finite297249702name_o @ B_88 )
% 0.29/0.45 => ( ( ord_le1205211808me_o_o @ A_156 @ B_88 )
% 0.29/0.45 => ( ( ord_less_eq_nat @ ( finite_card_pname_o @ B_88 ) @ ( finite_card_pname_o @ A_156 ) )
% 0.29/0.45 => ( A_156 = B_88 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_82_card__seteq,axiom,
% 0.29/0.45 ! [A_156: ( x_a > $o ) > $o,B_88: ( x_a > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_a_o @ B_88 )
% 0.29/0.45 => ( ( ord_less_eq_a_o_o @ A_156 @ B_88 )
% 0.29/0.45 => ( ( ord_less_eq_nat @ ( finite_card_a_o @ B_88 ) @ ( finite_card_a_o @ A_156 ) )
% 0.29/0.45 => ( A_156 = B_88 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_83_card__seteq,axiom,
% 0.29/0.45 ! [A_156: pname > $o,B_88: pname > $o] :
% 0.29/0.45 ( ( finite_finite_pname @ B_88 )
% 0.29/0.45 => ( ( ord_less_eq_pname_o @ A_156 @ B_88 )
% 0.29/0.45 => ( ( ord_less_eq_nat @ ( finite_card_pname @ B_88 ) @ ( finite_card_pname @ A_156 ) )
% 0.29/0.45 => ( A_156 = B_88 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_84_card__seteq,axiom,
% 0.29/0.45 ! [A_156: x_a > $o,B_88: x_a > $o] :
% 0.29/0.45 ( ( finite_finite_a @ B_88 )
% 0.29/0.45 => ( ( ord_less_eq_a_o @ A_156 @ B_88 )
% 0.29/0.45 => ( ( ord_less_eq_nat @ ( finite_card_a @ B_88 ) @ ( finite_card_a @ A_156 ) )
% 0.29/0.45 => ( A_156 = B_88 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_85_card__seteq,axiom,
% 0.29/0.45 ! [A_156: nat > $o,B_88: nat > $o] :
% 0.29/0.45 ( ( finite_finite_nat @ B_88 )
% 0.29/0.45 => ( ( ord_less_eq_nat_o @ A_156 @ B_88 )
% 0.29/0.45 => ( ( ord_less_eq_nat @ ( finite_card_nat @ B_88 ) @ ( finite_card_nat @ A_156 ) )
% 0.29/0.45 => ( A_156 = B_88 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_86_card__seteq,axiom,
% 0.29/0.45 ! [A_156: int > $o,B_88: int > $o] :
% 0.29/0.45 ( ( finite_finite_int @ B_88 )
% 0.29/0.45 => ( ( ord_less_eq_int_o @ A_156 @ B_88 )
% 0.29/0.45 => ( ( ord_less_eq_nat @ ( finite_card_int @ B_88 ) @ ( finite_card_int @ A_156 ) )
% 0.29/0.45 => ( A_156 = B_88 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_87_card__insert__le,axiom,
% 0.29/0.45 ! [X_53: int > $o,A_155: ( int > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_int_o @ A_155 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_int_o @ A_155 ) @ ( finite_card_int_o @ ( insert_int_o @ X_53 @ A_155 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_88_card__insert__le,axiom,
% 0.29/0.45 ! [X_53: nat > $o,A_155: ( nat > $o ) > $o] :
% 0.29/0.45 ( ( finite_finite_nat_o @ A_155 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_nat_o @ A_155 ) @ ( finite_card_nat_o @ ( insert_nat_o @ X_53 @ A_155 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_89_card__insert__le,axiom,
% 0.29/0.45 ! [X_53: pname > $o,A_155: ( pname > $o ) > $o] :
% 0.29/0.45 ( ( finite297249702name_o @ A_155 )
% 0.29/0.45 => ( ord_less_eq_nat @ ( finite_card_pname_o @ A_155 ) @ ( finite_card_pname_o @ ( insert_pname_o @ X_53 @ A_155 ) ) ) ) ).
% 0.29/0.45
% 0.29/0.45 thf(fact_90_card__insert__le,axiom,
% 0.29/0.46 ! [X_53: x_a > $o,A_155: ( x_a > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_a_o @ A_155 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( finite_card_a_o @ A_155 ) @ ( finite_card_a_o @ ( insert_a_o @ X_53 @ A_155 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_91_card__insert__le,axiom,
% 0.29/0.46 ! [X_53: pname,A_155: pname > $o] :
% 0.29/0.46 ( ( finite_finite_pname @ A_155 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( finite_card_pname @ A_155 ) @ ( finite_card_pname @ ( insert_pname @ X_53 @ A_155 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_92_card__insert__le,axiom,
% 0.29/0.46 ! [X_53: nat,A_155: nat > $o] :
% 0.29/0.46 ( ( finite_finite_nat @ A_155 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( finite_card_nat @ A_155 ) @ ( finite_card_nat @ ( insert_nat @ X_53 @ A_155 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_93_card__insert__le,axiom,
% 0.29/0.46 ! [X_53: int,A_155: int > $o] :
% 0.29/0.46 ( ( finite_finite_int @ A_155 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( finite_card_int @ A_155 ) @ ( finite_card_int @ ( insert_int @ X_53 @ A_155 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_94_card__insert__le,axiom,
% 0.29/0.46 ! [X_53: x_a,A_155: x_a > $o] :
% 0.29/0.46 ( ( finite_finite_a @ A_155 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( finite_card_a @ A_155 ) @ ( finite_card_a @ ( insert_a @ X_53 @ A_155 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_95_card__insert__if,axiom,
% 0.29/0.46 ! [X_52: int > $o,A_154: ( int > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_int_o @ A_154 )
% 0.29/0.46 => ( ( ( member_int_o @ X_52 @ A_154 )
% 0.29/0.46 => ( ( finite_card_int_o @ ( insert_int_o @ X_52 @ A_154 ) )
% 0.29/0.46 = ( finite_card_int_o @ A_154 ) ) )
% 0.29/0.46 & ( ~ ( member_int_o @ X_52 @ A_154 )
% 0.29/0.46 => ( ( finite_card_int_o @ ( insert_int_o @ X_52 @ A_154 ) )
% 0.29/0.46 = ( suc @ ( finite_card_int_o @ A_154 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_96_card__insert__if,axiom,
% 0.29/0.46 ! [X_52: nat > $o,A_154: ( nat > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_nat_o @ A_154 )
% 0.29/0.46 => ( ( ( member_nat_o @ X_52 @ A_154 )
% 0.29/0.46 => ( ( finite_card_nat_o @ ( insert_nat_o @ X_52 @ A_154 ) )
% 0.29/0.46 = ( finite_card_nat_o @ A_154 ) ) )
% 0.29/0.46 & ( ~ ( member_nat_o @ X_52 @ A_154 )
% 0.29/0.46 => ( ( finite_card_nat_o @ ( insert_nat_o @ X_52 @ A_154 ) )
% 0.29/0.46 = ( suc @ ( finite_card_nat_o @ A_154 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_97_card__insert__if,axiom,
% 0.29/0.46 ! [X_52: pname > $o,A_154: ( pname > $o ) > $o] :
% 0.29/0.46 ( ( finite297249702name_o @ A_154 )
% 0.29/0.46 => ( ( ( member_pname_o @ X_52 @ A_154 )
% 0.29/0.46 => ( ( finite_card_pname_o @ ( insert_pname_o @ X_52 @ A_154 ) )
% 0.29/0.46 = ( finite_card_pname_o @ A_154 ) ) )
% 0.29/0.46 & ( ~ ( member_pname_o @ X_52 @ A_154 )
% 0.29/0.46 => ( ( finite_card_pname_o @ ( insert_pname_o @ X_52 @ A_154 ) )
% 0.29/0.46 = ( suc @ ( finite_card_pname_o @ A_154 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_98_card__insert__if,axiom,
% 0.29/0.46 ! [X_52: x_a > $o,A_154: ( x_a > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_a_o @ A_154 )
% 0.29/0.46 => ( ( ( member_a_o @ X_52 @ A_154 )
% 0.29/0.46 => ( ( finite_card_a_o @ ( insert_a_o @ X_52 @ A_154 ) )
% 0.29/0.46 = ( finite_card_a_o @ A_154 ) ) )
% 0.29/0.46 & ( ~ ( member_a_o @ X_52 @ A_154 )
% 0.29/0.46 => ( ( finite_card_a_o @ ( insert_a_o @ X_52 @ A_154 ) )
% 0.29/0.46 = ( suc @ ( finite_card_a_o @ A_154 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_99_card__insert__if,axiom,
% 0.29/0.46 ! [X_52: pname,A_154: pname > $o] :
% 0.29/0.46 ( ( finite_finite_pname @ A_154 )
% 0.29/0.46 => ( ( ( member_pname @ X_52 @ A_154 )
% 0.29/0.46 => ( ( finite_card_pname @ ( insert_pname @ X_52 @ A_154 ) )
% 0.29/0.46 = ( finite_card_pname @ A_154 ) ) )
% 0.29/0.46 & ( ~ ( member_pname @ X_52 @ A_154 )
% 0.29/0.46 => ( ( finite_card_pname @ ( insert_pname @ X_52 @ A_154 ) )
% 0.29/0.46 = ( suc @ ( finite_card_pname @ A_154 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_100_card__insert__if,axiom,
% 0.29/0.46 ! [X_52: nat,A_154: nat > $o] :
% 0.29/0.46 ( ( finite_finite_nat @ A_154 )
% 0.29/0.46 => ( ( ( member_nat @ X_52 @ A_154 )
% 0.29/0.46 => ( ( finite_card_nat @ ( insert_nat @ X_52 @ A_154 ) )
% 0.29/0.46 = ( finite_card_nat @ A_154 ) ) )
% 0.29/0.46 & ( ~ ( member_nat @ X_52 @ A_154 )
% 0.29/0.46 => ( ( finite_card_nat @ ( insert_nat @ X_52 @ A_154 ) )
% 0.29/0.46 = ( suc @ ( finite_card_nat @ A_154 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_101_card__insert__if,axiom,
% 0.29/0.46 ! [X_52: int,A_154: int > $o] :
% 0.29/0.46 ( ( finite_finite_int @ A_154 )
% 0.29/0.46 => ( ( ( member_int @ X_52 @ A_154 )
% 0.29/0.46 => ( ( finite_card_int @ ( insert_int @ X_52 @ A_154 ) )
% 0.29/0.46 = ( finite_card_int @ A_154 ) ) )
% 0.29/0.46 & ( ~ ( member_int @ X_52 @ A_154 )
% 0.29/0.46 => ( ( finite_card_int @ ( insert_int @ X_52 @ A_154 ) )
% 0.29/0.46 = ( suc @ ( finite_card_int @ A_154 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_102_card__insert__if,axiom,
% 0.29/0.46 ! [X_52: x_a,A_154: x_a > $o] :
% 0.29/0.46 ( ( finite_finite_a @ A_154 )
% 0.29/0.46 => ( ( ( member_a @ X_52 @ A_154 )
% 0.29/0.46 => ( ( finite_card_a @ ( insert_a @ X_52 @ A_154 ) )
% 0.29/0.46 = ( finite_card_a @ A_154 ) ) )
% 0.29/0.46 & ( ~ ( member_a @ X_52 @ A_154 )
% 0.29/0.46 => ( ( finite_card_a @ ( insert_a @ X_52 @ A_154 ) )
% 0.29/0.46 = ( suc @ ( finite_card_a @ A_154 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_103_card__insert__disjoint,axiom,
% 0.29/0.46 ! [X_51: int > $o,A_153: ( int > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_int_o @ A_153 )
% 0.29/0.46 => ( ~ ( member_int_o @ X_51 @ A_153 )
% 0.29/0.46 => ( ( finite_card_int_o @ ( insert_int_o @ X_51 @ A_153 ) )
% 0.29/0.46 = ( suc @ ( finite_card_int_o @ A_153 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_104_card__insert__disjoint,axiom,
% 0.29/0.46 ! [X_51: nat > $o,A_153: ( nat > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_nat_o @ A_153 )
% 0.29/0.46 => ( ~ ( member_nat_o @ X_51 @ A_153 )
% 0.29/0.46 => ( ( finite_card_nat_o @ ( insert_nat_o @ X_51 @ A_153 ) )
% 0.29/0.46 = ( suc @ ( finite_card_nat_o @ A_153 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_105_card__insert__disjoint,axiom,
% 0.29/0.46 ! [X_51: pname > $o,A_153: ( pname > $o ) > $o] :
% 0.29/0.46 ( ( finite297249702name_o @ A_153 )
% 0.29/0.46 => ( ~ ( member_pname_o @ X_51 @ A_153 )
% 0.29/0.46 => ( ( finite_card_pname_o @ ( insert_pname_o @ X_51 @ A_153 ) )
% 0.29/0.46 = ( suc @ ( finite_card_pname_o @ A_153 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_106_card__insert__disjoint,axiom,
% 0.29/0.46 ! [X_51: x_a > $o,A_153: ( x_a > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_a_o @ A_153 )
% 0.29/0.46 => ( ~ ( member_a_o @ X_51 @ A_153 )
% 0.29/0.46 => ( ( finite_card_a_o @ ( insert_a_o @ X_51 @ A_153 ) )
% 0.29/0.46 = ( suc @ ( finite_card_a_o @ A_153 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_107_card__insert__disjoint,axiom,
% 0.29/0.46 ! [X_51: pname,A_153: pname > $o] :
% 0.29/0.46 ( ( finite_finite_pname @ A_153 )
% 0.29/0.46 => ( ~ ( member_pname @ X_51 @ A_153 )
% 0.29/0.46 => ( ( finite_card_pname @ ( insert_pname @ X_51 @ A_153 ) )
% 0.29/0.46 = ( suc @ ( finite_card_pname @ A_153 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_108_card__insert__disjoint,axiom,
% 0.29/0.46 ! [X_51: nat,A_153: nat > $o] :
% 0.29/0.46 ( ( finite_finite_nat @ A_153 )
% 0.29/0.46 => ( ~ ( member_nat @ X_51 @ A_153 )
% 0.29/0.46 => ( ( finite_card_nat @ ( insert_nat @ X_51 @ A_153 ) )
% 0.29/0.46 = ( suc @ ( finite_card_nat @ A_153 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_109_card__insert__disjoint,axiom,
% 0.29/0.46 ! [X_51: int,A_153: int > $o] :
% 0.29/0.46 ( ( finite_finite_int @ A_153 )
% 0.29/0.46 => ( ~ ( member_int @ X_51 @ A_153 )
% 0.29/0.46 => ( ( finite_card_int @ ( insert_int @ X_51 @ A_153 ) )
% 0.29/0.46 = ( suc @ ( finite_card_int @ A_153 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_110_card__insert__disjoint,axiom,
% 0.29/0.46 ! [X_51: x_a,A_153: x_a > $o] :
% 0.29/0.46 ( ( finite_finite_a @ A_153 )
% 0.29/0.46 => ( ~ ( member_a @ X_51 @ A_153 )
% 0.29/0.46 => ( ( finite_card_a @ ( insert_a @ X_51 @ A_153 ) )
% 0.29/0.46 = ( suc @ ( finite_card_a @ A_153 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_111_finite__Collect__conjI,axiom,
% 0.29/0.46 ! [Q_3: x_a > $o,P_13: x_a > $o] :
% 0.29/0.46 ( ( ( finite_finite_a @ ( collect_a @ P_13 ) )
% 0.29/0.46 | ( finite_finite_a @ ( collect_a @ Q_3 ) ) )
% 0.29/0.46 => ( finite_finite_a
% 0.29/0.46 @ ( collect_a
% 0.29/0.46 @ ^ [X_1: x_a] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_112_finite__Collect__conjI,axiom,
% 0.29/0.46 ! [Q_3: ( int > $o ) > $o,P_13: ( int > $o ) > $o] :
% 0.29/0.46 ( ( ( finite_finite_int_o @ ( collect_int_o @ P_13 ) )
% 0.29/0.46 | ( finite_finite_int_o @ ( collect_int_o @ Q_3 ) ) )
% 0.29/0.46 => ( finite_finite_int_o
% 0.29/0.46 @ ( collect_int_o
% 0.29/0.46 @ ^ [X_1: int > $o] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_113_finite__Collect__conjI,axiom,
% 0.29/0.46 ! [Q_3: ( nat > $o ) > $o,P_13: ( nat > $o ) > $o] :
% 0.29/0.46 ( ( ( finite_finite_nat_o @ ( collect_nat_o @ P_13 ) )
% 0.29/0.46 | ( finite_finite_nat_o @ ( collect_nat_o @ Q_3 ) ) )
% 0.29/0.46 => ( finite_finite_nat_o
% 0.29/0.46 @ ( collect_nat_o
% 0.29/0.46 @ ^ [X_1: nat > $o] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_114_finite__Collect__conjI,axiom,
% 0.29/0.46 ! [Q_3: ( pname > $o ) > $o,P_13: ( pname > $o ) > $o] :
% 0.29/0.46 ( ( ( finite297249702name_o @ ( collect_pname_o @ P_13 ) )
% 0.29/0.46 | ( finite297249702name_o @ ( collect_pname_o @ Q_3 ) ) )
% 0.29/0.46 => ( finite297249702name_o
% 0.29/0.46 @ ( collect_pname_o
% 0.29/0.46 @ ^ [X_1: pname > $o] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_115_finite__Collect__conjI,axiom,
% 0.29/0.46 ! [Q_3: ( x_a > $o ) > $o,P_13: ( x_a > $o ) > $o] :
% 0.29/0.46 ( ( ( finite_finite_a_o @ ( collect_a_o @ P_13 ) )
% 0.29/0.46 | ( finite_finite_a_o @ ( collect_a_o @ Q_3 ) ) )
% 0.29/0.46 => ( finite_finite_a_o
% 0.29/0.46 @ ( collect_a_o
% 0.29/0.46 @ ^ [X_1: x_a > $o] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_116_finite__Collect__conjI,axiom,
% 0.29/0.46 ! [Q_3: pname > $o,P_13: pname > $o] :
% 0.29/0.46 ( ( ( finite_finite_pname @ ( collect_pname @ P_13 ) )
% 0.29/0.46 | ( finite_finite_pname @ ( collect_pname @ Q_3 ) ) )
% 0.29/0.46 => ( finite_finite_pname
% 0.29/0.46 @ ( collect_pname
% 0.29/0.46 @ ^ [X_1: pname] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_117_finite__Collect__conjI,axiom,
% 0.29/0.46 ! [Q_3: nat > $o,P_13: nat > $o] :
% 0.29/0.46 ( ( ( finite_finite_nat @ ( collect_nat @ P_13 ) )
% 0.29/0.46 | ( finite_finite_nat @ ( collect_nat @ Q_3 ) ) )
% 0.29/0.46 => ( finite_finite_nat
% 0.29/0.46 @ ( collect_nat
% 0.29/0.46 @ ^ [X_1: nat] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_118_finite__Collect__conjI,axiom,
% 0.29/0.46 ! [Q_3: int > $o,P_13: int > $o] :
% 0.29/0.46 ( ( ( finite_finite_int @ ( collect_int @ P_13 ) )
% 0.29/0.46 | ( finite_finite_int @ ( collect_int @ Q_3 ) ) )
% 0.29/0.46 => ( finite_finite_int
% 0.29/0.46 @ ( collect_int
% 0.29/0.46 @ ^ [X_1: int] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_119_Suc__diff__le,axiom,
% 0.29/0.46 ! [N: nat,M: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.46 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 0.29/0.46 = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_120_finite__Collect__le__nat,axiom,
% 0.29/0.46 ! [K: nat] :
% 0.29/0.46 ( finite_finite_nat
% 0.29/0.46 @ ( collect_nat
% 0.29/0.46 @ ^ [N_1: nat] : ( ord_less_eq_nat @ N_1 @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_121_card__Collect__le__nat,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ( ( finite_card_nat
% 0.29/0.46 @ ( collect_nat
% 0.29/0.46 @ ^ [I: nat] : ( ord_less_eq_nat @ I @ N ) ) )
% 0.29/0.46 = ( suc @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_122_Suc__inject,axiom,
% 0.29/0.46 ! [X: nat,Y: nat] :
% 0.29/0.46 ( ( ( suc @ X )
% 0.29/0.46 = ( suc @ Y ) )
% 0.29/0.46 => ( X = Y ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_123_nat_Oinject,axiom,
% 0.29/0.46 ! [Nat_4: nat,Nat_1: nat] :
% 0.29/0.46 ( ( ( suc @ Nat_4 )
% 0.29/0.46 = ( suc @ Nat_1 ) )
% 0.29/0.46 <=> ( Nat_4 = Nat_1 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_124_Suc__n__not__n,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ( ( suc @ N )
% 0.29/0.46 != N ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_125_n__not__Suc__n,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ( N
% 0.29/0.46 != ( suc @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_126_le__antisym,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.46 => ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.46 => ( M = N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_127_le__trans,axiom,
% 0.29/0.46 ! [K: nat,I_1: nat,J: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ I_1 @ J )
% 0.29/0.46 => ( ( ord_less_eq_nat @ J @ K )
% 0.29/0.46 => ( ord_less_eq_nat @ I_1 @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_128_eq__imp__le,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( M = N )
% 0.29/0.46 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_129_nat__le__linear,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.46 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_130_le__refl,axiom,
% 0.29/0.46 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_131_diff__commute,axiom,
% 0.29/0.46 ! [I_1: nat,J: nat,K: nat] :
% 0.29/0.46 ( ( minus_minus_nat @ ( minus_minus_nat @ I_1 @ J ) @ K )
% 0.29/0.46 = ( minus_minus_nat @ ( minus_minus_nat @ I_1 @ K ) @ J ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_132_finite__Collect__disjI,axiom,
% 0.29/0.46 ! [P_12: x_a > $o,Q_2: x_a > $o] :
% 0.29/0.46 ( ( finite_finite_a
% 0.29/0.46 @ ( collect_a
% 0.29/0.46 @ ^ [X_1: x_a] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
% 0.29/0.46 <=> ( ( finite_finite_a @ ( collect_a @ P_12 ) )
% 0.29/0.46 & ( finite_finite_a @ ( collect_a @ Q_2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_133_finite__Collect__disjI,axiom,
% 0.29/0.46 ! [P_12: ( int > $o ) > $o,Q_2: ( int > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_int_o
% 0.29/0.46 @ ( collect_int_o
% 0.29/0.46 @ ^ [X_1: int > $o] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
% 0.29/0.46 <=> ( ( finite_finite_int_o @ ( collect_int_o @ P_12 ) )
% 0.29/0.46 & ( finite_finite_int_o @ ( collect_int_o @ Q_2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_134_finite__Collect__disjI,axiom,
% 0.29/0.46 ! [P_12: ( nat > $o ) > $o,Q_2: ( nat > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_nat_o
% 0.29/0.46 @ ( collect_nat_o
% 0.29/0.46 @ ^ [X_1: nat > $o] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
% 0.29/0.46 <=> ( ( finite_finite_nat_o @ ( collect_nat_o @ P_12 ) )
% 0.29/0.46 & ( finite_finite_nat_o @ ( collect_nat_o @ Q_2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_135_finite__Collect__disjI,axiom,
% 0.29/0.46 ! [P_12: ( pname > $o ) > $o,Q_2: ( pname > $o ) > $o] :
% 0.29/0.46 ( ( finite297249702name_o
% 0.29/0.46 @ ( collect_pname_o
% 0.29/0.46 @ ^ [X_1: pname > $o] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
% 0.29/0.46 <=> ( ( finite297249702name_o @ ( collect_pname_o @ P_12 ) )
% 0.29/0.46 & ( finite297249702name_o @ ( collect_pname_o @ Q_2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_136_finite__Collect__disjI,axiom,
% 0.29/0.46 ! [P_12: ( x_a > $o ) > $o,Q_2: ( x_a > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_a_o
% 0.29/0.46 @ ( collect_a_o
% 0.29/0.46 @ ^ [X_1: x_a > $o] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
% 0.29/0.46 <=> ( ( finite_finite_a_o @ ( collect_a_o @ P_12 ) )
% 0.29/0.46 & ( finite_finite_a_o @ ( collect_a_o @ Q_2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_137_finite__Collect__disjI,axiom,
% 0.29/0.46 ! [P_12: pname > $o,Q_2: pname > $o] :
% 0.29/0.46 ( ( finite_finite_pname
% 0.29/0.46 @ ( collect_pname
% 0.29/0.46 @ ^ [X_1: pname] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
% 0.29/0.46 <=> ( ( finite_finite_pname @ ( collect_pname @ P_12 ) )
% 0.29/0.46 & ( finite_finite_pname @ ( collect_pname @ Q_2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_138_finite__Collect__disjI,axiom,
% 0.29/0.46 ! [P_12: nat > $o,Q_2: nat > $o] :
% 0.29/0.46 ( ( finite_finite_nat
% 0.29/0.46 @ ( collect_nat
% 0.29/0.46 @ ^ [X_1: nat] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
% 0.29/0.46 <=> ( ( finite_finite_nat @ ( collect_nat @ P_12 ) )
% 0.29/0.46 & ( finite_finite_nat @ ( collect_nat @ Q_2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_139_finite__Collect__disjI,axiom,
% 0.29/0.46 ! [P_12: int > $o,Q_2: int > $o] :
% 0.29/0.46 ( ( finite_finite_int
% 0.29/0.46 @ ( collect_int
% 0.29/0.46 @ ^ [X_1: int] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
% 0.29/0.46 <=> ( ( finite_finite_int @ ( collect_int @ P_12 ) )
% 0.29/0.46 & ( finite_finite_int @ ( collect_int @ Q_2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_140_finite__insert,axiom,
% 0.29/0.46 ! [A_152: int > $o,A_151: ( int > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_int_o @ ( insert_int_o @ A_152 @ A_151 ) )
% 0.29/0.46 <=> ( finite_finite_int_o @ A_151 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_141_finite__insert,axiom,
% 0.29/0.46 ! [A_152: nat > $o,A_151: ( nat > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_nat_o @ ( insert_nat_o @ A_152 @ A_151 ) )
% 0.29/0.46 <=> ( finite_finite_nat_o @ A_151 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_142_finite__insert,axiom,
% 0.29/0.46 ! [A_152: pname > $o,A_151: ( pname > $o ) > $o] :
% 0.29/0.46 ( ( finite297249702name_o @ ( insert_pname_o @ A_152 @ A_151 ) )
% 0.29/0.46 <=> ( finite297249702name_o @ A_151 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_143_finite__insert,axiom,
% 0.29/0.46 ! [A_152: x_a > $o,A_151: ( x_a > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_a_o @ ( insert_a_o @ A_152 @ A_151 ) )
% 0.29/0.46 <=> ( finite_finite_a_o @ A_151 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_144_finite__insert,axiom,
% 0.29/0.46 ! [A_152: pname,A_151: pname > $o] :
% 0.29/0.46 ( ( finite_finite_pname @ ( insert_pname @ A_152 @ A_151 ) )
% 0.29/0.46 <=> ( finite_finite_pname @ A_151 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_145_finite__insert,axiom,
% 0.29/0.46 ! [A_152: nat,A_151: nat > $o] :
% 0.29/0.46 ( ( finite_finite_nat @ ( insert_nat @ A_152 @ A_151 ) )
% 0.29/0.46 <=> ( finite_finite_nat @ A_151 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_146_finite__insert,axiom,
% 0.29/0.46 ! [A_152: int,A_151: int > $o] :
% 0.29/0.46 ( ( finite_finite_int @ ( insert_int @ A_152 @ A_151 ) )
% 0.29/0.46 <=> ( finite_finite_int @ A_151 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_147_finite__insert,axiom,
% 0.29/0.46 ! [A_152: x_a,A_151: x_a > $o] :
% 0.29/0.46 ( ( finite_finite_a @ ( insert_a @ A_152 @ A_151 ) )
% 0.29/0.46 <=> ( finite_finite_a @ A_151 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_148_finite__subset,axiom,
% 0.29/0.46 ! [A_150: ( int > $o ) > $o,B_87: ( int > $o ) > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o_o @ A_150 @ B_87 )
% 0.29/0.46 => ( ( finite_finite_int_o @ B_87 )
% 0.29/0.46 => ( finite_finite_int_o @ A_150 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_149_finite__subset,axiom,
% 0.29/0.46 ! [A_150: ( nat > $o ) > $o,B_87: ( nat > $o ) > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o_o @ A_150 @ B_87 )
% 0.29/0.46 => ( ( finite_finite_nat_o @ B_87 )
% 0.29/0.46 => ( finite_finite_nat_o @ A_150 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_150_finite__subset,axiom,
% 0.29/0.46 ! [A_150: ( pname > $o ) > $o,B_87: ( pname > $o ) > $o] :
% 0.29/0.46 ( ( ord_le1205211808me_o_o @ A_150 @ B_87 )
% 0.29/0.46 => ( ( finite297249702name_o @ B_87 )
% 0.29/0.46 => ( finite297249702name_o @ A_150 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_151_finite__subset,axiom,
% 0.29/0.46 ! [A_150: ( x_a > $o ) > $o,B_87: ( x_a > $o ) > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o_o @ A_150 @ B_87 )
% 0.29/0.46 => ( ( finite_finite_a_o @ B_87 )
% 0.29/0.46 => ( finite_finite_a_o @ A_150 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_152_finite__subset,axiom,
% 0.29/0.46 ! [A_150: x_a > $o,B_87: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ A_150 @ B_87 )
% 0.29/0.46 => ( ( finite_finite_a @ B_87 )
% 0.29/0.46 => ( finite_finite_a @ A_150 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_153_finite__subset,axiom,
% 0.29/0.46 ! [A_150: pname > $o,B_87: pname > $o] :
% 0.29/0.46 ( ( ord_less_eq_pname_o @ A_150 @ B_87 )
% 0.29/0.46 => ( ( finite_finite_pname @ B_87 )
% 0.29/0.46 => ( finite_finite_pname @ A_150 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_154_finite__subset,axiom,
% 0.29/0.46 ! [A_150: nat > $o,B_87: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ A_150 @ B_87 )
% 0.29/0.46 => ( ( finite_finite_nat @ B_87 )
% 0.29/0.46 => ( finite_finite_nat @ A_150 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_155_finite__subset,axiom,
% 0.29/0.46 ! [A_150: int > $o,B_87: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ A_150 @ B_87 )
% 0.29/0.46 => ( ( finite_finite_int @ B_87 )
% 0.29/0.46 => ( finite_finite_int @ A_150 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_156_rev__finite__subset,axiom,
% 0.29/0.46 ! [A_149: ( int > $o ) > $o,B_86: ( int > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_int_o @ B_86 )
% 0.29/0.46 => ( ( ord_less_eq_int_o_o @ A_149 @ B_86 )
% 0.29/0.46 => ( finite_finite_int_o @ A_149 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_157_rev__finite__subset,axiom,
% 0.29/0.46 ! [A_149: ( nat > $o ) > $o,B_86: ( nat > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_nat_o @ B_86 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o_o @ A_149 @ B_86 )
% 0.29/0.46 => ( finite_finite_nat_o @ A_149 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_158_rev__finite__subset,axiom,
% 0.29/0.46 ! [A_149: ( pname > $o ) > $o,B_86: ( pname > $o ) > $o] :
% 0.29/0.46 ( ( finite297249702name_o @ B_86 )
% 0.29/0.46 => ( ( ord_le1205211808me_o_o @ A_149 @ B_86 )
% 0.29/0.46 => ( finite297249702name_o @ A_149 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_159_rev__finite__subset,axiom,
% 0.29/0.46 ! [A_149: ( x_a > $o ) > $o,B_86: ( x_a > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_a_o @ B_86 )
% 0.29/0.46 => ( ( ord_less_eq_a_o_o @ A_149 @ B_86 )
% 0.29/0.46 => ( finite_finite_a_o @ A_149 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_160_rev__finite__subset,axiom,
% 0.29/0.46 ! [A_149: x_a > $o,B_86: x_a > $o] :
% 0.29/0.46 ( ( finite_finite_a @ B_86 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ A_149 @ B_86 )
% 0.29/0.46 => ( finite_finite_a @ A_149 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_161_rev__finite__subset,axiom,
% 0.29/0.46 ! [A_149: pname > $o,B_86: pname > $o] :
% 0.29/0.46 ( ( finite_finite_pname @ B_86 )
% 0.29/0.46 => ( ( ord_less_eq_pname_o @ A_149 @ B_86 )
% 0.29/0.46 => ( finite_finite_pname @ A_149 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_162_rev__finite__subset,axiom,
% 0.29/0.46 ! [A_149: nat > $o,B_86: nat > $o] :
% 0.29/0.46 ( ( finite_finite_nat @ B_86 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o @ A_149 @ B_86 )
% 0.29/0.46 => ( finite_finite_nat @ A_149 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_163_rev__finite__subset,axiom,
% 0.29/0.46 ! [A_149: int > $o,B_86: int > $o] :
% 0.29/0.46 ( ( finite_finite_int @ B_86 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ A_149 @ B_86 )
% 0.29/0.46 => ( finite_finite_int @ A_149 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_164_Suc__leD,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.29/0.46 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_165_le__SucE,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.29/0.46 => ( ~ ( ord_less_eq_nat @ M @ N )
% 0.29/0.46 => ( M
% 0.29/0.46 = ( suc @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_166_le__SucI,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.46 => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_167_Suc__le__mono,axiom,
% 0.29/0.46 ! [N: nat,M: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.29/0.46 <=> ( ord_less_eq_nat @ N @ M ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_168_le__Suc__eq,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.29/0.46 <=> ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.46 | ( M
% 0.29/0.46 = ( suc @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_169_not__less__eq__eq,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ~ ( ord_less_eq_nat @ M @ N )
% 0.29/0.46 <=> ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_170_Suc__n__not__le__n,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_171_Suc__diff__diff,axiom,
% 0.29/0.46 ! [M: nat,N: nat,K: nat] :
% 0.29/0.46 ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 0.29/0.46 = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_172_diff__Suc__Suc,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.29/0.46 = ( minus_minus_nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_173_le__diff__iff,axiom,
% 0.29/0.46 ! [N: nat,K: nat,M: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ K @ M )
% 0.29/0.46 => ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.46 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.29/0.46 <=> ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_174_Nat_Odiff__diff__eq,axiom,
% 0.29/0.46 ! [N: nat,K: nat,M: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ K @ M )
% 0.29/0.46 => ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.46 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.29/0.46 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_175_eq__diff__iff,axiom,
% 0.29/0.46 ! [N: nat,K: nat,M: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ K @ M )
% 0.29/0.46 => ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.46 => ( ( ( minus_minus_nat @ M @ K )
% 0.29/0.46 = ( minus_minus_nat @ N @ K ) )
% 0.29/0.46 <=> ( M = N ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_176_diff__diff__cancel,axiom,
% 0.29/0.46 ! [I_1: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ I_1 @ N )
% 0.29/0.46 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I_1 ) )
% 0.29/0.46 = I_1 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_177_diff__le__mono,axiom,
% 0.29/0.46 ! [L: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.46 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_178_diff__le__mono2,axiom,
% 0.29/0.46 ! [L: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.46 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_179_diff__le__self,axiom,
% 0.29/0.46 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_180_finite__surj,axiom,
% 0.29/0.46 ! [B_85: x_a > $o,F_40: x_a > x_a,A_148: x_a > $o] :
% 0.29/0.46 ( ( finite_finite_a @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ B_85 @ ( image_a_a @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_a @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_181_finite__surj,axiom,
% 0.29/0.46 ! [B_85: x_a > $o,F_40: ( int > $o ) > x_a,A_148: ( int > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_int_o @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ B_85 @ ( image_int_o_a @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_a @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_182_finite__surj,axiom,
% 0.29/0.46 ! [B_85: x_a > $o,F_40: ( nat > $o ) > x_a,A_148: ( nat > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_nat_o @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ B_85 @ ( image_nat_o_a @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_a @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_183_finite__surj,axiom,
% 0.29/0.46 ! [B_85: x_a > $o,F_40: ( pname > $o ) > x_a,A_148: ( pname > $o ) > $o] :
% 0.29/0.46 ( ( finite297249702name_o @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ B_85 @ ( image_pname_o_a @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_a @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_184_finite__surj,axiom,
% 0.29/0.46 ! [B_85: x_a > $o,F_40: ( x_a > $o ) > x_a,A_148: ( x_a > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_a_o @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ B_85 @ ( image_a_o_a @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_a @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_185_finite__surj,axiom,
% 0.29/0.46 ! [B_85: ( int > $o ) > $o,F_40: pname > int > $o,A_148: pname > $o] :
% 0.29/0.46 ( ( finite_finite_pname @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_int_o_o @ B_85 @ ( image_pname_int_o @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_int_o @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_186_finite__surj,axiom,
% 0.29/0.46 ! [B_85: ( nat > $o ) > $o,F_40: pname > nat > $o,A_148: pname > $o] :
% 0.29/0.46 ( ( finite_finite_pname @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o_o @ B_85 @ ( image_pname_nat_o @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_nat_o @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_187_finite__surj,axiom,
% 0.29/0.46 ! [B_85: ( pname > $o ) > $o,F_40: pname > pname > $o,A_148: pname > $o] :
% 0.29/0.46 ( ( finite_finite_pname @ A_148 )
% 0.29/0.46 => ( ( ord_le1205211808me_o_o @ B_85 @ ( image_pname_pname_o @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite297249702name_o @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_188_finite__surj,axiom,
% 0.29/0.46 ! [B_85: ( x_a > $o ) > $o,F_40: pname > x_a > $o,A_148: pname > $o] :
% 0.29/0.46 ( ( finite_finite_pname @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_a_o_o @ B_85 @ ( image_pname_a_o @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_a_o @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_189_finite__surj,axiom,
% 0.29/0.46 ! [B_85: pname > $o,F_40: pname > pname,A_148: pname > $o] :
% 0.29/0.46 ( ( finite_finite_pname @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_pname_o @ B_85 @ ( image_pname_pname @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_pname @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_190_finite__surj,axiom,
% 0.29/0.46 ! [B_85: x_a > $o,F_40: nat > x_a,A_148: nat > $o] :
% 0.29/0.46 ( ( finite_finite_nat @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ B_85 @ ( image_nat_a @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_a @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_191_finite__surj,axiom,
% 0.29/0.46 ! [B_85: ( int > $o ) > $o,F_40: nat > int > $o,A_148: nat > $o] :
% 0.29/0.46 ( ( finite_finite_nat @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_int_o_o @ B_85 @ ( image_nat_int_o @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_int_o @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_192_finite__surj,axiom,
% 0.29/0.46 ! [B_85: ( nat > $o ) > $o,F_40: nat > nat > $o,A_148: nat > $o] :
% 0.29/0.46 ( ( finite_finite_nat @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o_o @ B_85 @ ( image_nat_nat_o @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_nat_o @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_193_finite__surj,axiom,
% 0.29/0.46 ! [B_85: ( pname > $o ) > $o,F_40: nat > pname > $o,A_148: nat > $o] :
% 0.29/0.46 ( ( finite_finite_nat @ A_148 )
% 0.29/0.46 => ( ( ord_le1205211808me_o_o @ B_85 @ ( image_nat_pname_o @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite297249702name_o @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_194_finite__surj,axiom,
% 0.29/0.46 ! [B_85: ( x_a > $o ) > $o,F_40: nat > x_a > $o,A_148: nat > $o] :
% 0.29/0.46 ( ( finite_finite_nat @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_a_o_o @ B_85 @ ( image_nat_a_o @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_a_o @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_195_finite__surj,axiom,
% 0.29/0.46 ! [B_85: pname > $o,F_40: nat > pname,A_148: nat > $o] :
% 0.29/0.46 ( ( finite_finite_nat @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_pname_o @ B_85 @ ( image_nat_pname @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_pname @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_196_finite__surj,axiom,
% 0.29/0.46 ! [B_85: x_a > $o,F_40: int > x_a,A_148: int > $o] :
% 0.29/0.46 ( ( finite_finite_int @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ B_85 @ ( image_int_a @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_a @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_197_finite__surj,axiom,
% 0.29/0.46 ! [B_85: ( int > $o ) > $o,F_40: int > int > $o,A_148: int > $o] :
% 0.29/0.46 ( ( finite_finite_int @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_int_o_o @ B_85 @ ( image_int_int_o @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_int_o @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_198_finite__surj,axiom,
% 0.29/0.46 ! [B_85: ( nat > $o ) > $o,F_40: int > nat > $o,A_148: int > $o] :
% 0.29/0.46 ( ( finite_finite_int @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o_o @ B_85 @ ( image_int_nat_o @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_nat_o @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_199_finite__surj,axiom,
% 0.29/0.46 ! [B_85: ( pname > $o ) > $o,F_40: int > pname > $o,A_148: int > $o] :
% 0.29/0.46 ( ( finite_finite_int @ A_148 )
% 0.29/0.46 => ( ( ord_le1205211808me_o_o @ B_85 @ ( image_int_pname_o @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite297249702name_o @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_200_finite__surj,axiom,
% 0.29/0.46 ! [B_85: ( x_a > $o ) > $o,F_40: int > x_a > $o,A_148: int > $o] :
% 0.29/0.46 ( ( finite_finite_int @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_a_o_o @ B_85 @ ( image_int_a_o @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_a_o @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_201_finite__surj,axiom,
% 0.29/0.46 ! [B_85: pname > $o,F_40: int > pname,A_148: int > $o] :
% 0.29/0.46 ( ( finite_finite_int @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_pname_o @ B_85 @ ( image_int_pname @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_pname @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_202_finite__surj,axiom,
% 0.29/0.46 ! [B_85: pname > $o,F_40: x_a > pname,A_148: x_a > $o] :
% 0.29/0.46 ( ( finite_finite_a @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_pname_o @ B_85 @ ( image_a_pname @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_pname @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_203_finite__surj,axiom,
% 0.29/0.46 ! [B_85: pname > $o,F_40: ( int > $o ) > pname,A_148: ( int > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_int_o @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_pname_o @ B_85 @ ( image_int_o_pname @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_pname @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_204_finite__surj,axiom,
% 0.29/0.46 ! [B_85: pname > $o,F_40: ( nat > $o ) > pname,A_148: ( nat > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_nat_o @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_pname_o @ B_85 @ ( image_nat_o_pname @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_pname @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_205_finite__surj,axiom,
% 0.29/0.46 ! [B_85: pname > $o,F_40: ( pname > $o ) > pname,A_148: ( pname > $o ) > $o] :
% 0.29/0.46 ( ( finite297249702name_o @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_pname_o @ B_85 @ ( image_pname_o_pname @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_pname @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_206_finite__surj,axiom,
% 0.29/0.46 ! [B_85: pname > $o,F_40: ( x_a > $o ) > pname,A_148: ( x_a > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_a_o @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_pname_o @ B_85 @ ( image_a_o_pname @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_pname @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_207_finite__surj,axiom,
% 0.29/0.46 ! [B_85: nat > $o,F_40: x_a > nat,A_148: x_a > $o] :
% 0.29/0.46 ( ( finite_finite_a @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o @ B_85 @ ( image_a_nat @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_nat @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_208_finite__surj,axiom,
% 0.29/0.46 ! [B_85: nat > $o,F_40: ( int > $o ) > nat,A_148: ( int > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_int_o @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o @ B_85 @ ( image_int_o_nat @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_nat @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_209_finite__surj,axiom,
% 0.29/0.46 ! [B_85: nat > $o,F_40: ( nat > $o ) > nat,A_148: ( nat > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_nat_o @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o @ B_85 @ ( image_nat_o_nat @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_nat @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_210_finite__surj,axiom,
% 0.29/0.46 ! [B_85: nat > $o,F_40: ( pname > $o ) > nat,A_148: ( pname > $o ) > $o] :
% 0.29/0.46 ( ( finite297249702name_o @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o @ B_85 @ ( image_pname_o_nat @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_nat @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_211_finite__surj,axiom,
% 0.29/0.46 ! [B_85: nat > $o,F_40: ( x_a > $o ) > nat,A_148: ( x_a > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_a_o @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o @ B_85 @ ( image_a_o_nat @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_nat @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_212_finite__surj,axiom,
% 0.29/0.46 ! [B_85: int > $o,F_40: x_a > int,A_148: x_a > $o] :
% 0.29/0.46 ( ( finite_finite_a @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ B_85 @ ( image_a_int @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_int @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_213_finite__surj,axiom,
% 0.29/0.46 ! [B_85: int > $o,F_40: ( int > $o ) > int,A_148: ( int > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_int_o @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ B_85 @ ( image_int_o_int @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_int @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_214_finite__surj,axiom,
% 0.29/0.46 ! [B_85: int > $o,F_40: ( nat > $o ) > int,A_148: ( nat > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_nat_o @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ B_85 @ ( image_nat_o_int @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_int @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_215_finite__surj,axiom,
% 0.29/0.46 ! [B_85: int > $o,F_40: ( pname > $o ) > int,A_148: ( pname > $o ) > $o] :
% 0.29/0.46 ( ( finite297249702name_o @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ B_85 @ ( image_pname_o_int @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_int @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_216_finite__surj,axiom,
% 0.29/0.46 ! [B_85: int > $o,F_40: ( x_a > $o ) > int,A_148: ( x_a > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_a_o @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ B_85 @ ( image_a_o_int @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_int @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_217_finite__surj,axiom,
% 0.29/0.46 ! [B_85: x_a > $o,F_40: pname > x_a,A_148: pname > $o] :
% 0.29/0.46 ( ( finite_finite_pname @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ B_85 @ ( image_pname_a @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_a @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_218_finite__surj,axiom,
% 0.29/0.46 ! [B_85: int > $o,F_40: nat > int,A_148: nat > $o] :
% 0.29/0.46 ( ( finite_finite_nat @ A_148 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ B_85 @ ( image_nat_int @ F_40 @ A_148 ) )
% 0.29/0.46 => ( finite_finite_int @ B_85 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_219_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: ( int > $o ) > x_a,A_147: ( int > $o ) > $o,B_84: x_a > $o] :
% 0.29/0.46 ( ( finite_finite_a @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ B_84 @ ( image_int_o_a @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: ( int > $o ) > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_int_o @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_int_o_a @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_220_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: ( nat > $o ) > x_a,A_147: ( nat > $o ) > $o,B_84: x_a > $o] :
% 0.29/0.46 ( ( finite_finite_a @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ B_84 @ ( image_nat_o_a @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: ( nat > $o ) > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_nat_o @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_nat_o_a @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_221_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: ( pname > $o ) > x_a,A_147: ( pname > $o ) > $o,B_84: x_a > $o] :
% 0.29/0.46 ( ( finite_finite_a @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ B_84 @ ( image_pname_o_a @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: ( pname > $o ) > $o] :
% 0.29/0.46 ( ( ord_le1205211808me_o_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite297249702name_o @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_pname_o_a @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_222_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: ( x_a > $o ) > x_a,A_147: ( x_a > $o ) > $o,B_84: x_a > $o] :
% 0.29/0.46 ( ( finite_finite_a @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ B_84 @ ( image_a_o_a @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: ( x_a > $o ) > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_a_o @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_a_o_a @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_223_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: x_a > x_a,A_147: x_a > $o,B_84: x_a > $o] :
% 0.29/0.46 ( ( finite_finite_a @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ B_84 @ ( image_a_a @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_a @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_a_a @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_224_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: x_a > int > $o,A_147: x_a > $o,B_84: ( int > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_int_o @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_int_o_o @ B_84 @ ( image_a_int_o @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_a @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_a_int_o @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_225_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: x_a > nat > $o,A_147: x_a > $o,B_84: ( nat > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_nat_o @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o_o @ B_84 @ ( image_a_nat_o @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_a @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_a_nat_o @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_226_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: x_a > pname > $o,A_147: x_a > $o,B_84: ( pname > $o ) > $o] :
% 0.29/0.46 ( ( finite297249702name_o @ B_84 )
% 0.29/0.46 => ( ( ord_le1205211808me_o_o @ B_84 @ ( image_a_pname_o @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_a @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_a_pname_o @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_227_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: x_a > x_a > $o,A_147: x_a > $o,B_84: ( x_a > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_a_o @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_a_o_o @ B_84 @ ( image_a_a_o @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_a @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_a_a_o @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_228_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: x_a > pname,A_147: x_a > $o,B_84: pname > $o] :
% 0.29/0.46 ( ( finite_finite_pname @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_pname_o @ B_84 @ ( image_a_pname @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_a @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_a_pname @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_229_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: ( int > $o ) > pname,A_147: ( int > $o ) > $o,B_84: pname > $o] :
% 0.29/0.46 ( ( finite_finite_pname @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_pname_o @ B_84 @ ( image_int_o_pname @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: ( int > $o ) > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_int_o @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_int_o_pname @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_230_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: ( nat > $o ) > pname,A_147: ( nat > $o ) > $o,B_84: pname > $o] :
% 0.29/0.46 ( ( finite_finite_pname @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_pname_o @ B_84 @ ( image_nat_o_pname @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: ( nat > $o ) > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_nat_o @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_nat_o_pname @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_231_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: ( pname > $o ) > pname,A_147: ( pname > $o ) > $o,B_84: pname > $o] :
% 0.29/0.46 ( ( finite_finite_pname @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_pname_o @ B_84 @ ( image_pname_o_pname @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: ( pname > $o ) > $o] :
% 0.29/0.46 ( ( ord_le1205211808me_o_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite297249702name_o @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_pname_o_pname @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_232_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: ( x_a > $o ) > pname,A_147: ( x_a > $o ) > $o,B_84: pname > $o] :
% 0.29/0.46 ( ( finite_finite_pname @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_pname_o @ B_84 @ ( image_a_o_pname @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: ( x_a > $o ) > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_a_o @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_a_o_pname @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_233_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: x_a > nat,A_147: x_a > $o,B_84: nat > $o] :
% 0.29/0.46 ( ( finite_finite_nat @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o @ B_84 @ ( image_a_nat @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_a @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_a_nat @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_234_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: ( int > $o ) > nat,A_147: ( int > $o ) > $o,B_84: nat > $o] :
% 0.29/0.46 ( ( finite_finite_nat @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o @ B_84 @ ( image_int_o_nat @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: ( int > $o ) > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_int_o @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_int_o_nat @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_235_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: ( nat > $o ) > nat,A_147: ( nat > $o ) > $o,B_84: nat > $o] :
% 0.29/0.46 ( ( finite_finite_nat @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o @ B_84 @ ( image_nat_o_nat @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: ( nat > $o ) > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_nat_o @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_nat_o_nat @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_236_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: ( pname > $o ) > nat,A_147: ( pname > $o ) > $o,B_84: nat > $o] :
% 0.29/0.46 ( ( finite_finite_nat @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o @ B_84 @ ( image_pname_o_nat @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: ( pname > $o ) > $o] :
% 0.29/0.46 ( ( ord_le1205211808me_o_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite297249702name_o @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_pname_o_nat @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_237_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: ( x_a > $o ) > nat,A_147: ( x_a > $o ) > $o,B_84: nat > $o] :
% 0.29/0.46 ( ( finite_finite_nat @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o @ B_84 @ ( image_a_o_nat @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: ( x_a > $o ) > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_a_o @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_a_o_nat @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_238_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: pname > nat,A_147: pname > $o,B_84: nat > $o] :
% 0.29/0.46 ( ( finite_finite_nat @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o @ B_84 @ ( image_pname_nat @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: pname > $o] :
% 0.29/0.46 ( ( ord_less_eq_pname_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_pname @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_pname_nat @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_239_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: x_a > int,A_147: x_a > $o,B_84: int > $o] :
% 0.29/0.46 ( ( finite_finite_int @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ B_84 @ ( image_a_int @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_a @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_a_int @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_240_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: ( int > $o ) > int,A_147: ( int > $o ) > $o,B_84: int > $o] :
% 0.29/0.46 ( ( finite_finite_int @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ B_84 @ ( image_int_o_int @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: ( int > $o ) > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_int_o @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_int_o_int @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_241_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: ( nat > $o ) > int,A_147: ( nat > $o ) > $o,B_84: int > $o] :
% 0.29/0.46 ( ( finite_finite_int @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ B_84 @ ( image_nat_o_int @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: ( nat > $o ) > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_nat_o @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_nat_o_int @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_242_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: ( pname > $o ) > int,A_147: ( pname > $o ) > $o,B_84: int > $o] :
% 0.29/0.46 ( ( finite_finite_int @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ B_84 @ ( image_pname_o_int @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: ( pname > $o ) > $o] :
% 0.29/0.46 ( ( ord_le1205211808me_o_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite297249702name_o @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_pname_o_int @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_243_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: ( x_a > $o ) > int,A_147: ( x_a > $o ) > $o,B_84: int > $o] :
% 0.29/0.46 ( ( finite_finite_int @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ B_84 @ ( image_a_o_int @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: ( x_a > $o ) > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_a_o @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_a_o_int @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_244_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: pname > int,A_147: pname > $o,B_84: int > $o] :
% 0.29/0.46 ( ( finite_finite_int @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ B_84 @ ( image_pname_int @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: pname > $o] :
% 0.29/0.46 ( ( ord_less_eq_pname_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_pname @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_pname_int @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_245_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: pname > int > $o,A_147: pname > $o,B_84: ( int > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_int_o @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_int_o_o @ B_84 @ ( image_pname_int_o @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: pname > $o] :
% 0.29/0.46 ( ( ord_less_eq_pname_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_pname @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_pname_int_o @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_246_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: pname > nat > $o,A_147: pname > $o,B_84: ( nat > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_nat_o @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o_o @ B_84 @ ( image_pname_nat_o @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: pname > $o] :
% 0.29/0.46 ( ( ord_less_eq_pname_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_pname @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_pname_nat_o @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_247_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: pname > pname > $o,A_147: pname > $o,B_84: ( pname > $o ) > $o] :
% 0.29/0.46 ( ( finite297249702name_o @ B_84 )
% 0.29/0.46 => ( ( ord_le1205211808me_o_o @ B_84 @ ( image_pname_pname_o @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: pname > $o] :
% 0.29/0.46 ( ( ord_less_eq_pname_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_pname @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_pname_pname_o @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_248_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: pname > x_a > $o,A_147: pname > $o,B_84: ( x_a > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_a_o @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_a_o_o @ B_84 @ ( image_pname_a_o @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: pname > $o] :
% 0.29/0.46 ( ( ord_less_eq_pname_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_pname @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_pname_a_o @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_249_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: pname > pname,A_147: pname > $o,B_84: pname > $o] :
% 0.29/0.46 ( ( finite_finite_pname @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_pname_o @ B_84 @ ( image_pname_pname @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: pname > $o] :
% 0.29/0.46 ( ( ord_less_eq_pname_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_pname @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_pname_pname @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_250_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: nat > x_a,A_147: nat > $o,B_84: x_a > $o] :
% 0.29/0.46 ( ( finite_finite_a @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ B_84 @ ( image_nat_a @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_nat @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_nat_a @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_251_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: nat > int > $o,A_147: nat > $o,B_84: ( int > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_int_o @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_int_o_o @ B_84 @ ( image_nat_int_o @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_nat @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_nat_int_o @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_252_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: nat > nat > $o,A_147: nat > $o,B_84: ( nat > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_nat_o @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o_o @ B_84 @ ( image_nat_nat_o @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_nat @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_nat_nat_o @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_253_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: nat > pname > $o,A_147: nat > $o,B_84: ( pname > $o ) > $o] :
% 0.29/0.46 ( ( finite297249702name_o @ B_84 )
% 0.29/0.46 => ( ( ord_le1205211808me_o_o @ B_84 @ ( image_nat_pname_o @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_nat @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_nat_pname_o @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_254_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: nat > x_a > $o,A_147: nat > $o,B_84: ( x_a > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_a_o @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_a_o_o @ B_84 @ ( image_nat_a_o @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_nat @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_nat_a_o @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_255_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: nat > pname,A_147: nat > $o,B_84: pname > $o] :
% 0.29/0.46 ( ( finite_finite_pname @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_pname_o @ B_84 @ ( image_nat_pname @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_nat @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_nat_pname @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_256_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: int > x_a,A_147: int > $o,B_84: x_a > $o] :
% 0.29/0.46 ( ( finite_finite_a @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ B_84 @ ( image_int_a @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_int @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_int_a @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_257_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: int > int > $o,A_147: int > $o,B_84: ( int > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_int_o @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_int_o_o @ B_84 @ ( image_int_int_o @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_int @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_int_int_o @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_258_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: int > nat > $o,A_147: int > $o,B_84: ( nat > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_nat_o @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o_o @ B_84 @ ( image_int_nat_o @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_int @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_int_nat_o @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_259_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: int > pname > $o,A_147: int > $o,B_84: ( pname > $o ) > $o] :
% 0.29/0.46 ( ( finite297249702name_o @ B_84 )
% 0.29/0.46 => ( ( ord_le1205211808me_o_o @ B_84 @ ( image_int_pname_o @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_int @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_int_pname_o @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_260_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: int > x_a > $o,A_147: int > $o,B_84: ( x_a > $o ) > $o] :
% 0.29/0.46 ( ( finite_finite_a_o @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_a_o_o @ B_84 @ ( image_int_a_o @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_int @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_int_a_o @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_261_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: int > pname,A_147: int > $o,B_84: pname > $o] :
% 0.29/0.46 ( ( finite_finite_pname @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_pname_o @ B_84 @ ( image_int_pname @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_int @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_int_pname @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_262_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: pname > x_a,A_147: pname > $o,B_84: x_a > $o] :
% 0.29/0.46 ( ( finite_finite_a @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ B_84 @ ( image_pname_a @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: pname > $o] :
% 0.29/0.46 ( ( ord_less_eq_pname_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_pname @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_pname_a @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_263_finite__subset__image,axiom,
% 0.29/0.46 ! [F_39: nat > int,A_147: nat > $o,B_84: int > $o] :
% 0.29/0.46 ( ( finite_finite_int @ B_84 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ B_84 @ ( image_nat_int @ F_39 @ A_147 ) )
% 0.29/0.46 => ? [C_34: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ C_34 @ A_147 )
% 0.29/0.46 & ( finite_finite_nat @ C_34 )
% 0.29/0.46 & ( B_84
% 0.29/0.46 = ( image_nat_int @ F_39 @ C_34 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_264_lift__Suc__mono__le,axiom,
% 0.29/0.46 ! [N_4: nat,N_3: nat,F_38: nat > pname > $o] :
% 0.29/0.46 ( ! [N_1: nat] : ( ord_less_eq_pname_o @ ( F_38 @ N_1 ) @ ( F_38 @ ( suc @ N_1 ) ) )
% 0.29/0.46 => ( ( ord_less_eq_nat @ N_4 @ N_3 )
% 0.29/0.46 => ( ord_less_eq_pname_o @ ( F_38 @ N_4 ) @ ( F_38 @ N_3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_265_lift__Suc__mono__le,axiom,
% 0.29/0.46 ! [N_4: nat,N_3: nat,F_38: nat > $o] :
% 0.29/0.46 ( ! [N_1: nat] : ( ord_less_eq_o @ ( F_38 @ N_1 ) @ ( F_38 @ ( suc @ N_1 ) ) )
% 0.29/0.46 => ( ( ord_less_eq_nat @ N_4 @ N_3 )
% 0.29/0.46 => ( ord_less_eq_o @ ( F_38 @ N_4 ) @ ( F_38 @ N_3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_266_lift__Suc__mono__le,axiom,
% 0.29/0.46 ! [N_4: nat,N_3: nat,F_38: nat > x_a > $o] :
% 0.29/0.46 ( ! [N_1: nat] : ( ord_less_eq_a_o @ ( F_38 @ N_1 ) @ ( F_38 @ ( suc @ N_1 ) ) )
% 0.29/0.46 => ( ( ord_less_eq_nat @ N_4 @ N_3 )
% 0.29/0.46 => ( ord_less_eq_a_o @ ( F_38 @ N_4 ) @ ( F_38 @ N_3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_267_lift__Suc__mono__le,axiom,
% 0.29/0.46 ! [N_4: nat,N_3: nat,F_38: nat > nat] :
% 0.29/0.46 ( ! [N_1: nat] : ( ord_less_eq_nat @ ( F_38 @ N_1 ) @ ( F_38 @ ( suc @ N_1 ) ) )
% 0.29/0.46 => ( ( ord_less_eq_nat @ N_4 @ N_3 )
% 0.29/0.46 => ( ord_less_eq_nat @ ( F_38 @ N_4 ) @ ( F_38 @ N_3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_268_lift__Suc__mono__le,axiom,
% 0.29/0.46 ! [N_4: nat,N_3: nat,F_38: nat > int] :
% 0.29/0.46 ( ! [N_1: nat] : ( ord_less_eq_int @ ( F_38 @ N_1 ) @ ( F_38 @ ( suc @ N_1 ) ) )
% 0.29/0.46 => ( ( ord_less_eq_nat @ N_4 @ N_3 )
% 0.29/0.46 => ( ord_less_eq_int @ ( F_38 @ N_4 ) @ ( F_38 @ N_3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_269_lift__Suc__mono__le,axiom,
% 0.29/0.46 ! [N_4: nat,N_3: nat,F_38: nat > nat > $o] :
% 0.29/0.46 ( ! [N_1: nat] : ( ord_less_eq_nat_o @ ( F_38 @ N_1 ) @ ( F_38 @ ( suc @ N_1 ) ) )
% 0.29/0.46 => ( ( ord_less_eq_nat @ N_4 @ N_3 )
% 0.29/0.46 => ( ord_less_eq_nat_o @ ( F_38 @ N_4 ) @ ( F_38 @ N_3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_270_lift__Suc__mono__le,axiom,
% 0.29/0.46 ! [N_4: nat,N_3: nat,F_38: nat > int > $o] :
% 0.29/0.46 ( ! [N_1: nat] : ( ord_less_eq_int_o @ ( F_38 @ N_1 ) @ ( F_38 @ ( suc @ N_1 ) ) )
% 0.29/0.46 => ( ( ord_less_eq_nat @ N_4 @ N_3 )
% 0.29/0.46 => ( ord_less_eq_int_o @ ( F_38 @ N_4 ) @ ( F_38 @ N_3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_271_pigeonhole__infinite,axiom,
% 0.29/0.46 ! [F_37: nat > int > $o,A_146: nat > $o] :
% 0.29/0.46 ( ~ ( finite_finite_nat @ A_146 )
% 0.29/0.46 => ( ( finite_finite_int_o @ ( image_nat_int_o @ F_37 @ A_146 ) )
% 0.29/0.46 => ? [X_1: nat] :
% 0.29/0.46 ( ( member_nat @ X_1 @ A_146 )
% 0.29/0.46 & ~ ( finite_finite_nat
% 0.29/0.46 @ ( collect_nat
% 0.29/0.46 @ ^ [A_37: nat] :
% 0.29/0.46 ( (&) @ ( member_nat @ A_37 @ A_146 )
% 0.29/0.46 @ ( ( F_37 @ A_37 )
% 0.29/0.46 = ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_272_pigeonhole__infinite,axiom,
% 0.29/0.46 ! [F_37: nat > nat > $o,A_146: nat > $o] :
% 0.29/0.46 ( ~ ( finite_finite_nat @ A_146 )
% 0.29/0.46 => ( ( finite_finite_nat_o @ ( image_nat_nat_o @ F_37 @ A_146 ) )
% 0.29/0.46 => ? [X_1: nat] :
% 0.29/0.46 ( ( member_nat @ X_1 @ A_146 )
% 0.29/0.46 & ~ ( finite_finite_nat
% 0.29/0.46 @ ( collect_nat
% 0.29/0.46 @ ^ [A_37: nat] :
% 0.29/0.46 ( (&) @ ( member_nat @ A_37 @ A_146 )
% 0.29/0.46 @ ( ( F_37 @ A_37 )
% 0.29/0.46 = ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_273_pigeonhole__infinite,axiom,
% 0.29/0.46 ! [F_37: nat > pname > $o,A_146: nat > $o] :
% 0.29/0.46 ( ~ ( finite_finite_nat @ A_146 )
% 0.29/0.46 => ( ( finite297249702name_o @ ( image_nat_pname_o @ F_37 @ A_146 ) )
% 0.29/0.46 => ? [X_1: nat] :
% 0.29/0.46 ( ( member_nat @ X_1 @ A_146 )
% 0.29/0.46 & ~ ( finite_finite_nat
% 0.29/0.46 @ ( collect_nat
% 0.29/0.46 @ ^ [A_37: nat] :
% 0.29/0.46 ( (&) @ ( member_nat @ A_37 @ A_146 )
% 0.29/0.46 @ ( ( F_37 @ A_37 )
% 0.29/0.46 = ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_274_pigeonhole__infinite,axiom,
% 0.29/0.46 ! [F_37: nat > x_a > $o,A_146: nat > $o] :
% 0.29/0.46 ( ~ ( finite_finite_nat @ A_146 )
% 0.29/0.46 => ( ( finite_finite_a_o @ ( image_nat_a_o @ F_37 @ A_146 ) )
% 0.29/0.46 => ? [X_1: nat] :
% 0.29/0.46 ( ( member_nat @ X_1 @ A_146 )
% 0.29/0.46 & ~ ( finite_finite_nat
% 0.29/0.46 @ ( collect_nat
% 0.29/0.46 @ ^ [A_37: nat] :
% 0.29/0.46 ( (&) @ ( member_nat @ A_37 @ A_146 )
% 0.29/0.46 @ ( ( F_37 @ A_37 )
% 0.29/0.46 = ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_275_pigeonhole__infinite,axiom,
% 0.29/0.46 ! [F_37: int > x_a,A_146: int > $o] :
% 0.29/0.46 ( ~ ( finite_finite_int @ A_146 )
% 0.29/0.46 => ( ( finite_finite_a @ ( image_int_a @ F_37 @ A_146 ) )
% 0.29/0.46 => ? [X_1: int] :
% 0.29/0.46 ( ( member_int @ X_1 @ A_146 )
% 0.29/0.46 & ~ ( finite_finite_int
% 0.29/0.46 @ ( collect_int
% 0.29/0.46 @ ^ [A_37: int] :
% 0.29/0.46 ( (&) @ ( member_int @ A_37 @ A_146 )
% 0.29/0.46 @ ( ( F_37 @ A_37 )
% 0.29/0.46 = ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_276_pigeonhole__infinite,axiom,
% 0.29/0.46 ! [F_37: int > int > $o,A_146: int > $o] :
% 0.29/0.46 ( ~ ( finite_finite_int @ A_146 )
% 0.29/0.46 => ( ( finite_finite_int_o @ ( image_int_int_o @ F_37 @ A_146 ) )
% 0.29/0.46 => ? [X_1: int] :
% 0.29/0.46 ( ( member_int @ X_1 @ A_146 )
% 0.29/0.46 & ~ ( finite_finite_int
% 0.29/0.46 @ ( collect_int
% 0.29/0.46 @ ^ [A_37: int] :
% 0.29/0.46 ( (&) @ ( member_int @ A_37 @ A_146 )
% 0.29/0.46 @ ( ( F_37 @ A_37 )
% 0.29/0.46 = ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_277_pigeonhole__infinite,axiom,
% 0.29/0.46 ! [F_37: int > nat > $o,A_146: int > $o] :
% 0.29/0.46 ( ~ ( finite_finite_int @ A_146 )
% 0.29/0.46 => ( ( finite_finite_nat_o @ ( image_int_nat_o @ F_37 @ A_146 ) )
% 0.29/0.46 => ? [X_1: int] :
% 0.29/0.46 ( ( member_int @ X_1 @ A_146 )
% 0.29/0.46 & ~ ( finite_finite_int
% 0.29/0.46 @ ( collect_int
% 0.29/0.46 @ ^ [A_37: int] :
% 0.29/0.46 ( (&) @ ( member_int @ A_37 @ A_146 )
% 0.29/0.46 @ ( ( F_37 @ A_37 )
% 0.29/0.46 = ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_278_pigeonhole__infinite,axiom,
% 0.29/0.46 ! [F_37: int > pname > $o,A_146: int > $o] :
% 0.29/0.46 ( ~ ( finite_finite_int @ A_146 )
% 0.29/0.46 => ( ( finite297249702name_o @ ( image_int_pname_o @ F_37 @ A_146 ) )
% 0.29/0.46 => ? [X_1: int] :
% 0.29/0.46 ( ( member_int @ X_1 @ A_146 )
% 0.29/0.46 & ~ ( finite_finite_int
% 0.29/0.46 @ ( collect_int
% 0.29/0.46 @ ^ [A_37: int] :
% 0.29/0.46 ( (&) @ ( member_int @ A_37 @ A_146 )
% 0.29/0.46 @ ( ( F_37 @ A_37 )
% 0.29/0.46 = ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_279_pigeonhole__infinite,axiom,
% 0.29/0.46 ! [F_37: int > x_a > $o,A_146: int > $o] :
% 0.29/0.46 ( ~ ( finite_finite_int @ A_146 )
% 0.29/0.46 => ( ( finite_finite_a_o @ ( image_int_a_o @ F_37 @ A_146 ) )
% 0.29/0.46 => ? [X_1: int] :
% 0.29/0.46 ( ( member_int @ X_1 @ A_146 )
% 0.29/0.46 & ~ ( finite_finite_int
% 0.29/0.46 @ ( collect_int
% 0.29/0.46 @ ^ [A_37: int] :
% 0.29/0.46 ( (&) @ ( member_int @ A_37 @ A_146 )
% 0.29/0.46 @ ( ( F_37 @ A_37 )
% 0.29/0.46 = ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_280_pigeonhole__infinite,axiom,
% 0.29/0.46 ! [F_37: pname > x_a,A_146: pname > $o] :
% 0.29/0.46 ( ~ ( finite_finite_pname @ A_146 )
% 0.29/0.46 => ( ( finite_finite_a @ ( image_pname_a @ F_37 @ A_146 ) )
% 0.29/0.46 => ? [X_1: pname] :
% 0.29/0.46 ( ( member_pname @ X_1 @ A_146 )
% 0.29/0.46 & ~ ( finite_finite_pname
% 0.29/0.46 @ ( collect_pname
% 0.29/0.46 @ ^ [A_37: pname] :
% 0.29/0.46 ( (&) @ ( member_pname @ A_37 @ A_146 )
% 0.29/0.46 @ ( ( F_37 @ A_37 )
% 0.29/0.46 = ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_281_pigeonhole__infinite,axiom,
% 0.29/0.46 ! [F_37: nat > int,A_146: nat > $o] :
% 0.29/0.46 ( ~ ( finite_finite_nat @ A_146 )
% 0.29/0.46 => ( ( finite_finite_int @ ( image_nat_int @ F_37 @ A_146 ) )
% 0.29/0.46 => ? [X_1: nat] :
% 0.29/0.46 ( ( member_nat @ X_1 @ A_146 )
% 0.29/0.46 & ~ ( finite_finite_nat
% 0.29/0.46 @ ( collect_nat
% 0.29/0.46 @ ^ [A_37: nat] :
% 0.29/0.46 ( (&) @ ( member_nat @ A_37 @ A_146 )
% 0.29/0.46 @ ( ( F_37 @ A_37 )
% 0.29/0.46 = ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_282_image__eqI,axiom,
% 0.29/0.46 ! [A_145: nat > $o,B_83: int,F_36: nat > int,X_50: nat] :
% 0.29/0.46 ( ( B_83
% 0.29/0.46 = ( F_36 @ X_50 ) )
% 0.29/0.46 => ( ( member_nat @ X_50 @ A_145 )
% 0.29/0.46 => ( member_int @ B_83 @ ( image_nat_int @ F_36 @ A_145 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_283_image__eqI,axiom,
% 0.29/0.46 ! [A_145: pname > $o,B_83: x_a,F_36: pname > x_a,X_50: pname] :
% 0.29/0.46 ( ( B_83
% 0.29/0.46 = ( F_36 @ X_50 ) )
% 0.29/0.46 => ( ( member_pname @ X_50 @ A_145 )
% 0.29/0.46 => ( member_a @ B_83 @ ( image_pname_a @ F_36 @ A_145 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_284_equalityI,axiom,
% 0.29/0.46 ! [A_144: int > $o,B_82: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ A_144 @ B_82 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ B_82 @ A_144 )
% 0.29/0.46 => ( A_144 = B_82 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_285_equalityI,axiom,
% 0.29/0.46 ! [A_144: nat > $o,B_82: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ A_144 @ B_82 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o @ B_82 @ A_144 )
% 0.29/0.46 => ( A_144 = B_82 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_286_equalityI,axiom,
% 0.29/0.46 ! [A_144: x_a > $o,B_82: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ A_144 @ B_82 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ B_82 @ A_144 )
% 0.29/0.46 => ( A_144 = B_82 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_287_subsetD,axiom,
% 0.29/0.46 ! [C_33: int,A_143: int > $o,B_81: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ A_143 @ B_81 )
% 0.29/0.46 => ( ( member_int @ C_33 @ A_143 )
% 0.29/0.46 => ( member_int @ C_33 @ B_81 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_288_subsetD,axiom,
% 0.29/0.46 ! [C_33: nat,A_143: nat > $o,B_81: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ A_143 @ B_81 )
% 0.29/0.46 => ( ( member_nat @ C_33 @ A_143 )
% 0.29/0.46 => ( member_nat @ C_33 @ B_81 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_289_subsetD,axiom,
% 0.29/0.46 ! [C_33: x_a,A_143: x_a > $o,B_81: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ A_143 @ B_81 )
% 0.29/0.46 => ( ( member_a @ C_33 @ A_143 )
% 0.29/0.46 => ( member_a @ C_33 @ B_81 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_290_subsetD,axiom,
% 0.29/0.46 ! [C_33: pname,A_143: pname > $o,B_81: pname > $o] :
% 0.29/0.46 ( ( ord_less_eq_pname_o @ A_143 @ B_81 )
% 0.29/0.46 => ( ( member_pname @ C_33 @ A_143 )
% 0.29/0.46 => ( member_pname @ C_33 @ B_81 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_291_insertCI,axiom,
% 0.29/0.46 ! [B_80: x_a,A_142: x_a,B_79: x_a > $o] :
% 0.29/0.46 ( ( ~ ( member_a @ A_142 @ B_79 )
% 0.29/0.46 => ( A_142 = B_80 ) )
% 0.29/0.46 => ( member_a @ A_142 @ ( insert_a @ B_80 @ B_79 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_292_insertCI,axiom,
% 0.29/0.46 ! [B_80: int,A_142: int,B_79: int > $o] :
% 0.29/0.46 ( ( ~ ( member_int @ A_142 @ B_79 )
% 0.29/0.46 => ( A_142 = B_80 ) )
% 0.29/0.46 => ( member_int @ A_142 @ ( insert_int @ B_80 @ B_79 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_293_insertCI,axiom,
% 0.29/0.46 ! [B_80: nat,A_142: nat,B_79: nat > $o] :
% 0.29/0.46 ( ( ~ ( member_nat @ A_142 @ B_79 )
% 0.29/0.46 => ( A_142 = B_80 ) )
% 0.29/0.46 => ( member_nat @ A_142 @ ( insert_nat @ B_80 @ B_79 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_294_insertCI,axiom,
% 0.29/0.46 ! [B_80: pname,A_142: pname,B_79: pname > $o] :
% 0.29/0.46 ( ( ~ ( member_pname @ A_142 @ B_79 )
% 0.29/0.46 => ( A_142 = B_80 ) )
% 0.29/0.46 => ( member_pname @ A_142 @ ( insert_pname @ B_80 @ B_79 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_295_insertE,axiom,
% 0.29/0.46 ! [A_141: x_a,B_78: x_a,A_140: x_a > $o] :
% 0.29/0.46 ( ( member_a @ A_141 @ ( insert_a @ B_78 @ A_140 ) )
% 0.29/0.46 => ( ( A_141 != B_78 )
% 0.29/0.46 => ( member_a @ A_141 @ A_140 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_296_insertE,axiom,
% 0.29/0.46 ! [A_141: int,B_78: int,A_140: int > $o] :
% 0.29/0.46 ( ( member_int @ A_141 @ ( insert_int @ B_78 @ A_140 ) )
% 0.29/0.46 => ( ( A_141 != B_78 )
% 0.29/0.46 => ( member_int @ A_141 @ A_140 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_297_insertE,axiom,
% 0.29/0.46 ! [A_141: nat,B_78: nat,A_140: nat > $o] :
% 0.29/0.46 ( ( member_nat @ A_141 @ ( insert_nat @ B_78 @ A_140 ) )
% 0.29/0.46 => ( ( A_141 != B_78 )
% 0.29/0.46 => ( member_nat @ A_141 @ A_140 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_298_insertE,axiom,
% 0.29/0.46 ! [A_141: pname,B_78: pname,A_140: pname > $o] :
% 0.29/0.46 ( ( member_pname @ A_141 @ ( insert_pname @ B_78 @ A_140 ) )
% 0.29/0.46 => ( ( A_141 != B_78 )
% 0.29/0.46 => ( member_pname @ A_141 @ A_140 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_299_zero__induct__lemma,axiom,
% 0.29/0.46 ! [I_1: nat,P: nat > $o,K: nat] :
% 0.29/0.46 ( ( P @ K )
% 0.29/0.46 => ( ! [N_1: nat] :
% 0.29/0.46 ( ( P @ ( suc @ N_1 ) )
% 0.29/0.46 => ( P @ N_1 ) )
% 0.29/0.46 => ( P @ ( minus_minus_nat @ K @ I_1 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_300_Suc__le__D,axiom,
% 0.29/0.46 ! [N: nat,M_3: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ ( suc @ N ) @ M_3 )
% 0.29/0.46 => ? [M_1: nat] :
% 0.29/0.46 ( M_3
% 0.29/0.46 = ( suc @ M_1 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_301_order__refl,axiom,
% 0.29/0.46 ! [X_49: int > $o] : ( ord_less_eq_int_o @ X_49 @ X_49 ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_302_order__refl,axiom,
% 0.29/0.46 ! [X_49: nat > $o] : ( ord_less_eq_nat_o @ X_49 @ X_49 ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_303_order__refl,axiom,
% 0.29/0.46 ! [X_49: int] : ( ord_less_eq_int @ X_49 @ X_49 ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_304_order__refl,axiom,
% 0.29/0.46 ! [X_49: nat] : ( ord_less_eq_nat @ X_49 @ X_49 ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_305_order__refl,axiom,
% 0.29/0.46 ! [X_49: x_a > $o] : ( ord_less_eq_a_o @ X_49 @ X_49 ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_306_linorder__linear,axiom,
% 0.29/0.46 ! [X_48: int,Y_12: int] :
% 0.29/0.46 ( ( ord_less_eq_int @ X_48 @ Y_12 )
% 0.29/0.46 | ( ord_less_eq_int @ Y_12 @ X_48 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_307_linorder__linear,axiom,
% 0.29/0.46 ! [X_48: nat,Y_12: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X_48 @ Y_12 )
% 0.29/0.46 | ( ord_less_eq_nat @ Y_12 @ X_48 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_308_order__eq__iff,axiom,
% 0.29/0.46 ! [X_47: int > $o,Y_11: int > $o] :
% 0.29/0.46 ( ( X_47 = Y_11 )
% 0.29/0.46 <=> ( ( ord_less_eq_int_o @ X_47 @ Y_11 )
% 0.29/0.46 & ( ord_less_eq_int_o @ Y_11 @ X_47 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_309_order__eq__iff,axiom,
% 0.29/0.46 ! [X_47: nat > $o,Y_11: nat > $o] :
% 0.29/0.46 ( ( X_47 = Y_11 )
% 0.29/0.46 <=> ( ( ord_less_eq_nat_o @ X_47 @ Y_11 )
% 0.29/0.46 & ( ord_less_eq_nat_o @ Y_11 @ X_47 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_310_order__eq__iff,axiom,
% 0.29/0.46 ! [X_47: int,Y_11: int] :
% 0.29/0.46 ( ( X_47 = Y_11 )
% 0.29/0.46 <=> ( ( ord_less_eq_int @ X_47 @ Y_11 )
% 0.29/0.46 & ( ord_less_eq_int @ Y_11 @ X_47 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_311_order__eq__iff,axiom,
% 0.29/0.46 ! [X_47: nat,Y_11: nat] :
% 0.29/0.46 ( ( X_47 = Y_11 )
% 0.29/0.46 <=> ( ( ord_less_eq_nat @ X_47 @ Y_11 )
% 0.29/0.46 & ( ord_less_eq_nat @ Y_11 @ X_47 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_312_order__eq__iff,axiom,
% 0.29/0.46 ! [X_47: x_a > $o,Y_11: x_a > $o] :
% 0.29/0.46 ( ( X_47 = Y_11 )
% 0.29/0.46 <=> ( ( ord_less_eq_a_o @ X_47 @ Y_11 )
% 0.29/0.46 & ( ord_less_eq_a_o @ Y_11 @ X_47 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_313_order__eq__refl,axiom,
% 0.29/0.46 ! [X_46: int > $o,Y_10: int > $o] :
% 0.29/0.46 ( ( X_46 = Y_10 )
% 0.29/0.46 => ( ord_less_eq_int_o @ X_46 @ Y_10 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_314_order__eq__refl,axiom,
% 0.29/0.46 ! [X_46: nat > $o,Y_10: nat > $o] :
% 0.29/0.46 ( ( X_46 = Y_10 )
% 0.29/0.46 => ( ord_less_eq_nat_o @ X_46 @ Y_10 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_315_order__eq__refl,axiom,
% 0.29/0.46 ! [X_46: int,Y_10: int] :
% 0.29/0.46 ( ( X_46 = Y_10 )
% 0.29/0.46 => ( ord_less_eq_int @ X_46 @ Y_10 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_316_order__eq__refl,axiom,
% 0.29/0.46 ! [X_46: nat,Y_10: nat] :
% 0.29/0.46 ( ( X_46 = Y_10 )
% 0.29/0.46 => ( ord_less_eq_nat @ X_46 @ Y_10 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_317_order__eq__refl,axiom,
% 0.29/0.46 ! [X_46: x_a > $o,Y_10: x_a > $o] :
% 0.29/0.46 ( ( X_46 = Y_10 )
% 0.29/0.46 => ( ord_less_eq_a_o @ X_46 @ Y_10 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_318_order__antisym__conv,axiom,
% 0.29/0.46 ! [Y_9: int > $o,X_45: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ Y_9 @ X_45 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ X_45 @ Y_9 )
% 0.29/0.46 <=> ( X_45 = Y_9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_319_order__antisym__conv,axiom,
% 0.29/0.46 ! [Y_9: nat > $o,X_45: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ Y_9 @ X_45 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o @ X_45 @ Y_9 )
% 0.29/0.46 <=> ( X_45 = Y_9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_320_order__antisym__conv,axiom,
% 0.29/0.46 ! [Y_9: int,X_45: int] :
% 0.29/0.46 ( ( ord_less_eq_int @ Y_9 @ X_45 )
% 0.29/0.46 => ( ( ord_less_eq_int @ X_45 @ Y_9 )
% 0.29/0.46 <=> ( X_45 = Y_9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_321_order__antisym__conv,axiom,
% 0.29/0.46 ! [Y_9: nat,X_45: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ Y_9 @ X_45 )
% 0.29/0.46 => ( ( ord_less_eq_nat @ X_45 @ Y_9 )
% 0.29/0.46 <=> ( X_45 = Y_9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_322_order__antisym__conv,axiom,
% 0.29/0.46 ! [Y_9: x_a > $o,X_45: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ Y_9 @ X_45 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ X_45 @ Y_9 )
% 0.29/0.46 <=> ( X_45 = Y_9 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_323_ord__eq__le__trans,axiom,
% 0.29/0.46 ! [C_32: int > $o,A_139: int > $o,B_77: int > $o] :
% 0.29/0.46 ( ( A_139 = B_77 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ B_77 @ C_32 )
% 0.29/0.46 => ( ord_less_eq_int_o @ A_139 @ C_32 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_324_ord__eq__le__trans,axiom,
% 0.29/0.46 ! [C_32: nat > $o,A_139: nat > $o,B_77: nat > $o] :
% 0.29/0.46 ( ( A_139 = B_77 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o @ B_77 @ C_32 )
% 0.29/0.46 => ( ord_less_eq_nat_o @ A_139 @ C_32 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_325_ord__eq__le__trans,axiom,
% 0.29/0.46 ! [C_32: int,A_139: int,B_77: int] :
% 0.29/0.46 ( ( A_139 = B_77 )
% 0.29/0.46 => ( ( ord_less_eq_int @ B_77 @ C_32 )
% 0.29/0.46 => ( ord_less_eq_int @ A_139 @ C_32 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_326_ord__eq__le__trans,axiom,
% 0.29/0.46 ! [C_32: nat,A_139: nat,B_77: nat] :
% 0.29/0.46 ( ( A_139 = B_77 )
% 0.29/0.46 => ( ( ord_less_eq_nat @ B_77 @ C_32 )
% 0.29/0.46 => ( ord_less_eq_nat @ A_139 @ C_32 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_327_ord__eq__le__trans,axiom,
% 0.29/0.46 ! [C_32: x_a > $o,A_139: x_a > $o,B_77: x_a > $o] :
% 0.29/0.46 ( ( A_139 = B_77 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ B_77 @ C_32 )
% 0.29/0.46 => ( ord_less_eq_a_o @ A_139 @ C_32 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_328_xt1_I3_J,axiom,
% 0.29/0.46 ! [C_31: int > $o,A_138: int > $o,B_76: int > $o] :
% 0.29/0.46 ( ( A_138 = B_76 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ C_31 @ B_76 )
% 0.29/0.46 => ( ord_less_eq_int_o @ C_31 @ A_138 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_329_xt1_I3_J,axiom,
% 0.29/0.46 ! [C_31: nat > $o,A_138: nat > $o,B_76: nat > $o] :
% 0.29/0.46 ( ( A_138 = B_76 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o @ C_31 @ B_76 )
% 0.29/0.46 => ( ord_less_eq_nat_o @ C_31 @ A_138 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_330_xt1_I3_J,axiom,
% 0.29/0.46 ! [C_31: int,A_138: int,B_76: int] :
% 0.29/0.46 ( ( A_138 = B_76 )
% 0.29/0.46 => ( ( ord_less_eq_int @ C_31 @ B_76 )
% 0.29/0.46 => ( ord_less_eq_int @ C_31 @ A_138 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_331_xt1_I3_J,axiom,
% 0.29/0.46 ! [C_31: nat,A_138: nat,B_76: nat] :
% 0.29/0.46 ( ( A_138 = B_76 )
% 0.29/0.46 => ( ( ord_less_eq_nat @ C_31 @ B_76 )
% 0.29/0.46 => ( ord_less_eq_nat @ C_31 @ A_138 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_332_xt1_I3_J,axiom,
% 0.29/0.46 ! [C_31: x_a > $o,A_138: x_a > $o,B_76: x_a > $o] :
% 0.29/0.46 ( ( A_138 = B_76 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ C_31 @ B_76 )
% 0.29/0.46 => ( ord_less_eq_a_o @ C_31 @ A_138 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_333_ord__le__eq__trans,axiom,
% 0.29/0.46 ! [C_30: int > $o,A_137: int > $o,B_75: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ A_137 @ B_75 )
% 0.29/0.46 => ( ( B_75 = C_30 )
% 0.29/0.46 => ( ord_less_eq_int_o @ A_137 @ C_30 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_334_ord__le__eq__trans,axiom,
% 0.29/0.46 ! [C_30: nat > $o,A_137: nat > $o,B_75: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ A_137 @ B_75 )
% 0.29/0.46 => ( ( B_75 = C_30 )
% 0.29/0.46 => ( ord_less_eq_nat_o @ A_137 @ C_30 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_335_ord__le__eq__trans,axiom,
% 0.29/0.46 ! [C_30: int,A_137: int,B_75: int] :
% 0.29/0.46 ( ( ord_less_eq_int @ A_137 @ B_75 )
% 0.29/0.46 => ( ( B_75 = C_30 )
% 0.29/0.46 => ( ord_less_eq_int @ A_137 @ C_30 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_336_ord__le__eq__trans,axiom,
% 0.29/0.46 ! [C_30: nat,A_137: nat,B_75: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ A_137 @ B_75 )
% 0.29/0.46 => ( ( B_75 = C_30 )
% 0.29/0.46 => ( ord_less_eq_nat @ A_137 @ C_30 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_337_ord__le__eq__trans,axiom,
% 0.29/0.46 ! [C_30: x_a > $o,A_137: x_a > $o,B_75: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ A_137 @ B_75 )
% 0.29/0.46 => ( ( B_75 = C_30 )
% 0.29/0.46 => ( ord_less_eq_a_o @ A_137 @ C_30 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_338_xt1_I4_J,axiom,
% 0.29/0.46 ! [C_29: int > $o,B_74: int > $o,A_136: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ B_74 @ A_136 )
% 0.29/0.46 => ( ( B_74 = C_29 )
% 0.29/0.46 => ( ord_less_eq_int_o @ C_29 @ A_136 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_339_xt1_I4_J,axiom,
% 0.29/0.46 ! [C_29: nat > $o,B_74: nat > $o,A_136: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ B_74 @ A_136 )
% 0.29/0.46 => ( ( B_74 = C_29 )
% 0.29/0.46 => ( ord_less_eq_nat_o @ C_29 @ A_136 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_340_xt1_I4_J,axiom,
% 0.29/0.46 ! [C_29: int,B_74: int,A_136: int] :
% 0.29/0.46 ( ( ord_less_eq_int @ B_74 @ A_136 )
% 0.29/0.46 => ( ( B_74 = C_29 )
% 0.29/0.46 => ( ord_less_eq_int @ C_29 @ A_136 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_341_xt1_I4_J,axiom,
% 0.29/0.46 ! [C_29: nat,B_74: nat,A_136: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ B_74 @ A_136 )
% 0.29/0.46 => ( ( B_74 = C_29 )
% 0.29/0.46 => ( ord_less_eq_nat @ C_29 @ A_136 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_342_xt1_I4_J,axiom,
% 0.29/0.46 ! [C_29: x_a > $o,B_74: x_a > $o,A_136: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ B_74 @ A_136 )
% 0.29/0.46 => ( ( B_74 = C_29 )
% 0.29/0.46 => ( ord_less_eq_a_o @ C_29 @ A_136 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_343_order__antisym,axiom,
% 0.29/0.46 ! [X_44: int > $o,Y_8: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ X_44 @ Y_8 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ Y_8 @ X_44 )
% 0.29/0.46 => ( X_44 = Y_8 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_344_order__antisym,axiom,
% 0.29/0.46 ! [X_44: nat > $o,Y_8: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ X_44 @ Y_8 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o @ Y_8 @ X_44 )
% 0.29/0.46 => ( X_44 = Y_8 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_345_order__antisym,axiom,
% 0.29/0.46 ! [X_44: int,Y_8: int] :
% 0.29/0.46 ( ( ord_less_eq_int @ X_44 @ Y_8 )
% 0.29/0.46 => ( ( ord_less_eq_int @ Y_8 @ X_44 )
% 0.29/0.46 => ( X_44 = Y_8 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_346_order__antisym,axiom,
% 0.29/0.46 ! [X_44: nat,Y_8: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X_44 @ Y_8 )
% 0.29/0.46 => ( ( ord_less_eq_nat @ Y_8 @ X_44 )
% 0.29/0.46 => ( X_44 = Y_8 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_347_order__antisym,axiom,
% 0.29/0.46 ! [X_44: x_a > $o,Y_8: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ X_44 @ Y_8 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ Y_8 @ X_44 )
% 0.29/0.46 => ( X_44 = Y_8 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_348_order__trans,axiom,
% 0.29/0.46 ! [Z_6: int > $o,X_43: int > $o,Y_7: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ X_43 @ Y_7 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ Y_7 @ Z_6 )
% 0.29/0.46 => ( ord_less_eq_int_o @ X_43 @ Z_6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_349_order__trans,axiom,
% 0.29/0.46 ! [Z_6: nat > $o,X_43: nat > $o,Y_7: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ X_43 @ Y_7 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o @ Y_7 @ Z_6 )
% 0.29/0.46 => ( ord_less_eq_nat_o @ X_43 @ Z_6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_350_order__trans,axiom,
% 0.29/0.46 ! [Z_6: int,X_43: int,Y_7: int] :
% 0.29/0.46 ( ( ord_less_eq_int @ X_43 @ Y_7 )
% 0.29/0.46 => ( ( ord_less_eq_int @ Y_7 @ Z_6 )
% 0.29/0.46 => ( ord_less_eq_int @ X_43 @ Z_6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_351_order__trans,axiom,
% 0.29/0.46 ! [Z_6: nat,X_43: nat,Y_7: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ X_43 @ Y_7 )
% 0.29/0.46 => ( ( ord_less_eq_nat @ Y_7 @ Z_6 )
% 0.29/0.46 => ( ord_less_eq_nat @ X_43 @ Z_6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_352_order__trans,axiom,
% 0.29/0.46 ! [Z_6: x_a > $o,X_43: x_a > $o,Y_7: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ X_43 @ Y_7 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ Y_7 @ Z_6 )
% 0.29/0.46 => ( ord_less_eq_a_o @ X_43 @ Z_6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_353_xt1_I5_J,axiom,
% 0.29/0.46 ! [Y_6: int > $o,X_42: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ Y_6 @ X_42 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ X_42 @ Y_6 )
% 0.29/0.46 => ( X_42 = Y_6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_354_xt1_I5_J,axiom,
% 0.29/0.46 ! [Y_6: nat > $o,X_42: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ Y_6 @ X_42 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o @ X_42 @ Y_6 )
% 0.29/0.46 => ( X_42 = Y_6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_355_xt1_I5_J,axiom,
% 0.29/0.46 ! [Y_6: int,X_42: int] :
% 0.29/0.46 ( ( ord_less_eq_int @ Y_6 @ X_42 )
% 0.29/0.46 => ( ( ord_less_eq_int @ X_42 @ Y_6 )
% 0.29/0.46 => ( X_42 = Y_6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_356_xt1_I5_J,axiom,
% 0.29/0.46 ! [Y_6: nat,X_42: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ Y_6 @ X_42 )
% 0.29/0.46 => ( ( ord_less_eq_nat @ X_42 @ Y_6 )
% 0.29/0.46 => ( X_42 = Y_6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_357_xt1_I5_J,axiom,
% 0.29/0.46 ! [Y_6: x_a > $o,X_42: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ Y_6 @ X_42 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ X_42 @ Y_6 )
% 0.29/0.46 => ( X_42 = Y_6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_358_xt1_I6_J,axiom,
% 0.29/0.46 ! [Z_5: int > $o,Y_5: int > $o,X_41: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ Y_5 @ X_41 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ Z_5 @ Y_5 )
% 0.29/0.46 => ( ord_less_eq_int_o @ Z_5 @ X_41 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_359_xt1_I6_J,axiom,
% 0.29/0.46 ! [Z_5: nat > $o,Y_5: nat > $o,X_41: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ Y_5 @ X_41 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o @ Z_5 @ Y_5 )
% 0.29/0.46 => ( ord_less_eq_nat_o @ Z_5 @ X_41 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_360_xt1_I6_J,axiom,
% 0.29/0.46 ! [Z_5: int,Y_5: int,X_41: int] :
% 0.29/0.46 ( ( ord_less_eq_int @ Y_5 @ X_41 )
% 0.29/0.46 => ( ( ord_less_eq_int @ Z_5 @ Y_5 )
% 0.29/0.46 => ( ord_less_eq_int @ Z_5 @ X_41 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_361_xt1_I6_J,axiom,
% 0.29/0.46 ! [Z_5: nat,Y_5: nat,X_41: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ Y_5 @ X_41 )
% 0.29/0.46 => ( ( ord_less_eq_nat @ Z_5 @ Y_5 )
% 0.29/0.46 => ( ord_less_eq_nat @ Z_5 @ X_41 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_362_xt1_I6_J,axiom,
% 0.29/0.46 ! [Z_5: x_a > $o,Y_5: x_a > $o,X_41: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ Y_5 @ X_41 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ Z_5 @ Y_5 )
% 0.29/0.46 => ( ord_less_eq_a_o @ Z_5 @ X_41 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_363_linorder__le__cases,axiom,
% 0.29/0.46 ! [X_40: int,Y_4: int] :
% 0.29/0.46 ( ~ ( ord_less_eq_int @ X_40 @ Y_4 )
% 0.29/0.46 => ( ord_less_eq_int @ Y_4 @ X_40 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_364_linorder__le__cases,axiom,
% 0.29/0.46 ! [X_40: nat,Y_4: nat] :
% 0.29/0.46 ( ~ ( ord_less_eq_nat @ X_40 @ Y_4 )
% 0.29/0.46 => ( ord_less_eq_nat @ Y_4 @ X_40 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_365_insertI1,axiom,
% 0.29/0.46 ! [A_135: x_a,B_73: x_a > $o] : ( member_a @ A_135 @ ( insert_a @ A_135 @ B_73 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_366_insertI1,axiom,
% 0.29/0.46 ! [A_135: int,B_73: int > $o] : ( member_int @ A_135 @ ( insert_int @ A_135 @ B_73 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_367_insertI1,axiom,
% 0.29/0.46 ! [A_135: nat,B_73: nat > $o] : ( member_nat @ A_135 @ ( insert_nat @ A_135 @ B_73 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_368_insertI1,axiom,
% 0.29/0.46 ! [A_135: pname,B_73: pname > $o] : ( member_pname @ A_135 @ ( insert_pname @ A_135 @ B_73 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_369_insert__compr,axiom,
% 0.29/0.46 ! [A_134: x_a,B_72: x_a > $o] :
% 0.29/0.46 ( ( insert_a @ A_134 @ B_72 )
% 0.29/0.46 = ( collect_a
% 0.29/0.46 @ ^ [X_1: x_a] : ( (|) @ ( X_1 = A_134 ) @ ( member_a @ X_1 @ B_72 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_370_insert__compr,axiom,
% 0.29/0.46 ! [A_134: int,B_72: int > $o] :
% 0.29/0.46 ( ( insert_int @ A_134 @ B_72 )
% 0.29/0.46 = ( collect_int
% 0.29/0.46 @ ^ [X_1: int] : ( (|) @ ( X_1 = A_134 ) @ ( member_int @ X_1 @ B_72 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_371_insert__compr,axiom,
% 0.29/0.46 ! [A_134: nat,B_72: nat > $o] :
% 0.29/0.46 ( ( insert_nat @ A_134 @ B_72 )
% 0.29/0.46 = ( collect_nat
% 0.29/0.46 @ ^ [X_1: nat] : ( (|) @ ( X_1 = A_134 ) @ ( member_nat @ X_1 @ B_72 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_372_insert__compr,axiom,
% 0.29/0.46 ! [A_134: pname,B_72: pname > $o] :
% 0.29/0.46 ( ( insert_pname @ A_134 @ B_72 )
% 0.29/0.46 = ( collect_pname
% 0.29/0.46 @ ^ [X_1: pname] : ( (|) @ ( X_1 = A_134 ) @ ( member_pname @ X_1 @ B_72 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_373_insert__Collect,axiom,
% 0.29/0.46 ! [A_133: x_a,P_11: x_a > $o] :
% 0.29/0.46 ( ( insert_a @ A_133 @ ( collect_a @ P_11 ) )
% 0.29/0.46 = ( collect_a
% 0.29/0.46 @ ^ [U_1: x_a] : ( (=>) @ ( (~) @ ( U_1 = A_133 ) ) @ ( P_11 @ U_1 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_374_insert__Collect,axiom,
% 0.29/0.46 ! [A_133: int,P_11: int > $o] :
% 0.29/0.46 ( ( insert_int @ A_133 @ ( collect_int @ P_11 ) )
% 0.29/0.46 = ( collect_int
% 0.29/0.46 @ ^ [U_1: int] : ( (=>) @ ( (~) @ ( U_1 = A_133 ) ) @ ( P_11 @ U_1 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_375_insert__Collect,axiom,
% 0.29/0.46 ! [A_133: nat,P_11: nat > $o] :
% 0.29/0.46 ( ( insert_nat @ A_133 @ ( collect_nat @ P_11 ) )
% 0.29/0.46 = ( collect_nat
% 0.29/0.46 @ ^ [U_1: nat] : ( (=>) @ ( (~) @ ( U_1 = A_133 ) ) @ ( P_11 @ U_1 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_376_mem__def,axiom,
% 0.29/0.46 ! [X_39: int,A_132: int > $o] :
% 0.29/0.46 ( ( member_int @ X_39 @ A_132 )
% 0.29/0.46 <=> ( A_132 @ X_39 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_377_mem__def,axiom,
% 0.29/0.46 ! [X_39: nat,A_132: nat > $o] :
% 0.29/0.46 ( ( member_nat @ X_39 @ A_132 )
% 0.29/0.46 <=> ( A_132 @ X_39 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_378_mem__def,axiom,
% 0.29/0.46 ! [X_39: x_a,A_132: x_a > $o] :
% 0.29/0.46 ( ( member_a @ X_39 @ A_132 )
% 0.29/0.46 <=> ( A_132 @ X_39 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_379_mem__def,axiom,
% 0.29/0.46 ! [X_39: pname,A_132: pname > $o] :
% 0.29/0.46 ( ( member_pname @ X_39 @ A_132 )
% 0.29/0.46 <=> ( A_132 @ X_39 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_380_Collect__def,axiom,
% 0.29/0.46 ! [P_10: int > $o] :
% 0.29/0.46 ( ( collect_int @ P_10 )
% 0.29/0.46 = P_10 ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_381_Collect__def,axiom,
% 0.29/0.46 ! [P_10: nat > $o] :
% 0.29/0.46 ( ( collect_nat @ P_10 )
% 0.29/0.46 = P_10 ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_382_insert__absorb2,axiom,
% 0.29/0.46 ! [X_38: x_a,A_131: x_a > $o] :
% 0.29/0.46 ( ( insert_a @ X_38 @ ( insert_a @ X_38 @ A_131 ) )
% 0.29/0.46 = ( insert_a @ X_38 @ A_131 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_383_insert__commute,axiom,
% 0.29/0.46 ! [X_37: x_a,Y_3: x_a,A_130: x_a > $o] :
% 0.29/0.46 ( ( insert_a @ X_37 @ ( insert_a @ Y_3 @ A_130 ) )
% 0.29/0.46 = ( insert_a @ Y_3 @ ( insert_a @ X_37 @ A_130 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_384_insert__iff,axiom,
% 0.29/0.46 ! [A_129: x_a,B_71: x_a,A_128: x_a > $o] :
% 0.29/0.46 ( ( member_a @ A_129 @ ( insert_a @ B_71 @ A_128 ) )
% 0.29/0.46 <=> ( ( A_129 = B_71 )
% 0.29/0.46 | ( member_a @ A_129 @ A_128 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_385_insert__iff,axiom,
% 0.29/0.46 ! [A_129: int,B_71: int,A_128: int > $o] :
% 0.29/0.46 ( ( member_int @ A_129 @ ( insert_int @ B_71 @ A_128 ) )
% 0.29/0.46 <=> ( ( A_129 = B_71 )
% 0.29/0.46 | ( member_int @ A_129 @ A_128 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_386_insert__iff,axiom,
% 0.29/0.46 ! [A_129: nat,B_71: nat,A_128: nat > $o] :
% 0.29/0.46 ( ( member_nat @ A_129 @ ( insert_nat @ B_71 @ A_128 ) )
% 0.29/0.46 <=> ( ( A_129 = B_71 )
% 0.29/0.46 | ( member_nat @ A_129 @ A_128 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_387_insert__iff,axiom,
% 0.29/0.46 ! [A_129: pname,B_71: pname,A_128: pname > $o] :
% 0.29/0.46 ( ( member_pname @ A_129 @ ( insert_pname @ B_71 @ A_128 ) )
% 0.29/0.46 <=> ( ( A_129 = B_71 )
% 0.29/0.46 | ( member_pname @ A_129 @ A_128 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_388_insert__code,axiom,
% 0.29/0.46 ! [Y_2: x_a,A_127: x_a > $o,X_36: x_a] :
% 0.29/0.46 ( ( insert_a @ Y_2 @ A_127 @ X_36 )
% 0.29/0.46 <=> ( ( Y_2 = X_36 )
% 0.29/0.46 | ( A_127 @ X_36 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_389_insert__ident,axiom,
% 0.29/0.46 ! [B_70: x_a > $o,X_35: x_a,A_126: x_a > $o] :
% 0.29/0.46 ( ~ ( member_a @ X_35 @ A_126 )
% 0.29/0.46 => ( ~ ( member_a @ X_35 @ B_70 )
% 0.29/0.46 => ( ( ( insert_a @ X_35 @ A_126 )
% 0.29/0.46 = ( insert_a @ X_35 @ B_70 ) )
% 0.29/0.46 <=> ( A_126 = B_70 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_390_insert__ident,axiom,
% 0.29/0.46 ! [B_70: int > $o,X_35: int,A_126: int > $o] :
% 0.29/0.46 ( ~ ( member_int @ X_35 @ A_126 )
% 0.29/0.46 => ( ~ ( member_int @ X_35 @ B_70 )
% 0.29/0.46 => ( ( ( insert_int @ X_35 @ A_126 )
% 0.29/0.46 = ( insert_int @ X_35 @ B_70 ) )
% 0.29/0.46 <=> ( A_126 = B_70 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_391_insert__ident,axiom,
% 0.29/0.46 ! [B_70: nat > $o,X_35: nat,A_126: nat > $o] :
% 0.29/0.46 ( ~ ( member_nat @ X_35 @ A_126 )
% 0.29/0.46 => ( ~ ( member_nat @ X_35 @ B_70 )
% 0.29/0.46 => ( ( ( insert_nat @ X_35 @ A_126 )
% 0.29/0.46 = ( insert_nat @ X_35 @ B_70 ) )
% 0.29/0.46 <=> ( A_126 = B_70 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_392_insert__ident,axiom,
% 0.29/0.46 ! [B_70: pname > $o,X_35: pname,A_126: pname > $o] :
% 0.29/0.46 ( ~ ( member_pname @ X_35 @ A_126 )
% 0.29/0.46 => ( ~ ( member_pname @ X_35 @ B_70 )
% 0.29/0.46 => ( ( ( insert_pname @ X_35 @ A_126 )
% 0.29/0.46 = ( insert_pname @ X_35 @ B_70 ) )
% 0.29/0.46 <=> ( A_126 = B_70 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_393_insertI2,axiom,
% 0.29/0.46 ! [B_69: x_a,A_125: x_a,B_68: x_a > $o] :
% 0.29/0.46 ( ( member_a @ A_125 @ B_68 )
% 0.29/0.46 => ( member_a @ A_125 @ ( insert_a @ B_69 @ B_68 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_394_insertI2,axiom,
% 0.29/0.46 ! [B_69: int,A_125: int,B_68: int > $o] :
% 0.29/0.46 ( ( member_int @ A_125 @ B_68 )
% 0.29/0.46 => ( member_int @ A_125 @ ( insert_int @ B_69 @ B_68 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_395_insertI2,axiom,
% 0.29/0.46 ! [B_69: nat,A_125: nat,B_68: nat > $o] :
% 0.29/0.46 ( ( member_nat @ A_125 @ B_68 )
% 0.29/0.46 => ( member_nat @ A_125 @ ( insert_nat @ B_69 @ B_68 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_396_insertI2,axiom,
% 0.29/0.46 ! [B_69: pname,A_125: pname,B_68: pname > $o] :
% 0.29/0.46 ( ( member_pname @ A_125 @ B_68 )
% 0.29/0.46 => ( member_pname @ A_125 @ ( insert_pname @ B_69 @ B_68 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_397_insert__absorb,axiom,
% 0.29/0.46 ! [A_124: x_a,A_123: x_a > $o] :
% 0.29/0.46 ( ( member_a @ A_124 @ A_123 )
% 0.29/0.46 => ( ( insert_a @ A_124 @ A_123 )
% 0.29/0.46 = A_123 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_398_insert__absorb,axiom,
% 0.29/0.46 ! [A_124: int,A_123: int > $o] :
% 0.29/0.46 ( ( member_int @ A_124 @ A_123 )
% 0.29/0.46 => ( ( insert_int @ A_124 @ A_123 )
% 0.29/0.46 = A_123 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_399_insert__absorb,axiom,
% 0.29/0.46 ! [A_124: nat,A_123: nat > $o] :
% 0.29/0.46 ( ( member_nat @ A_124 @ A_123 )
% 0.29/0.46 => ( ( insert_nat @ A_124 @ A_123 )
% 0.29/0.46 = A_123 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_400_insert__absorb,axiom,
% 0.29/0.46 ! [A_124: pname,A_123: pname > $o] :
% 0.29/0.46 ( ( member_pname @ A_124 @ A_123 )
% 0.29/0.46 => ( ( insert_pname @ A_124 @ A_123 )
% 0.29/0.46 = A_123 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_401_subset__refl,axiom,
% 0.29/0.46 ! [A_122: int > $o] : ( ord_less_eq_int_o @ A_122 @ A_122 ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_402_subset__refl,axiom,
% 0.29/0.46 ! [A_122: nat > $o] : ( ord_less_eq_nat_o @ A_122 @ A_122 ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_403_subset__refl,axiom,
% 0.29/0.46 ! [A_122: x_a > $o] : ( ord_less_eq_a_o @ A_122 @ A_122 ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_404_set__eq__subset,axiom,
% 0.29/0.46 ! [A_121: int > $o,B_67: int > $o] :
% 0.29/0.46 ( ( A_121 = B_67 )
% 0.29/0.46 <=> ( ( ord_less_eq_int_o @ A_121 @ B_67 )
% 0.29/0.46 & ( ord_less_eq_int_o @ B_67 @ A_121 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_405_set__eq__subset,axiom,
% 0.29/0.46 ! [A_121: nat > $o,B_67: nat > $o] :
% 0.29/0.46 ( ( A_121 = B_67 )
% 0.29/0.46 <=> ( ( ord_less_eq_nat_o @ A_121 @ B_67 )
% 0.29/0.46 & ( ord_less_eq_nat_o @ B_67 @ A_121 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_406_set__eq__subset,axiom,
% 0.29/0.46 ! [A_121: x_a > $o,B_67: x_a > $o] :
% 0.29/0.46 ( ( A_121 = B_67 )
% 0.29/0.46 <=> ( ( ord_less_eq_a_o @ A_121 @ B_67 )
% 0.29/0.46 & ( ord_less_eq_a_o @ B_67 @ A_121 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_407_equalityD1,axiom,
% 0.29/0.46 ! [A_120: int > $o,B_66: int > $o] :
% 0.29/0.46 ( ( A_120 = B_66 )
% 0.29/0.46 => ( ord_less_eq_int_o @ A_120 @ B_66 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_408_equalityD1,axiom,
% 0.29/0.46 ! [A_120: nat > $o,B_66: nat > $o] :
% 0.29/0.46 ( ( A_120 = B_66 )
% 0.29/0.46 => ( ord_less_eq_nat_o @ A_120 @ B_66 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_409_equalityD1,axiom,
% 0.29/0.46 ! [A_120: x_a > $o,B_66: x_a > $o] :
% 0.29/0.46 ( ( A_120 = B_66 )
% 0.29/0.46 => ( ord_less_eq_a_o @ A_120 @ B_66 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_410_equalityD2,axiom,
% 0.29/0.46 ! [A_119: int > $o,B_65: int > $o] :
% 0.29/0.46 ( ( A_119 = B_65 )
% 0.29/0.46 => ( ord_less_eq_int_o @ B_65 @ A_119 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_411_equalityD2,axiom,
% 0.29/0.46 ! [A_119: nat > $o,B_65: nat > $o] :
% 0.29/0.46 ( ( A_119 = B_65 )
% 0.29/0.46 => ( ord_less_eq_nat_o @ B_65 @ A_119 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_412_equalityD2,axiom,
% 0.29/0.46 ! [A_119: x_a > $o,B_65: x_a > $o] :
% 0.29/0.46 ( ( A_119 = B_65 )
% 0.29/0.46 => ( ord_less_eq_a_o @ B_65 @ A_119 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_413_in__mono,axiom,
% 0.29/0.46 ! [X_34: int,A_118: int > $o,B_64: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ A_118 @ B_64 )
% 0.29/0.46 => ( ( member_int @ X_34 @ A_118 )
% 0.29/0.46 => ( member_int @ X_34 @ B_64 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_414_in__mono,axiom,
% 0.29/0.46 ! [X_34: nat,A_118: nat > $o,B_64: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ A_118 @ B_64 )
% 0.29/0.46 => ( ( member_nat @ X_34 @ A_118 )
% 0.29/0.46 => ( member_nat @ X_34 @ B_64 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_415_in__mono,axiom,
% 0.29/0.46 ! [X_34: x_a,A_118: x_a > $o,B_64: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ A_118 @ B_64 )
% 0.29/0.46 => ( ( member_a @ X_34 @ A_118 )
% 0.29/0.46 => ( member_a @ X_34 @ B_64 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_416_in__mono,axiom,
% 0.29/0.46 ! [X_34: pname,A_118: pname > $o,B_64: pname > $o] :
% 0.29/0.46 ( ( ord_less_eq_pname_o @ A_118 @ B_64 )
% 0.29/0.46 => ( ( member_pname @ X_34 @ A_118 )
% 0.29/0.46 => ( member_pname @ X_34 @ B_64 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_417_set__rev__mp,axiom,
% 0.29/0.46 ! [B_63: int > $o,X_33: int,A_117: int > $o] :
% 0.29/0.46 ( ( member_int @ X_33 @ A_117 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ A_117 @ B_63 )
% 0.29/0.46 => ( member_int @ X_33 @ B_63 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_418_set__rev__mp,axiom,
% 0.29/0.46 ! [B_63: nat > $o,X_33: nat,A_117: nat > $o] :
% 0.29/0.46 ( ( member_nat @ X_33 @ A_117 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o @ A_117 @ B_63 )
% 0.29/0.46 => ( member_nat @ X_33 @ B_63 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_419_set__rev__mp,axiom,
% 0.29/0.46 ! [B_63: x_a > $o,X_33: x_a,A_117: x_a > $o] :
% 0.29/0.46 ( ( member_a @ X_33 @ A_117 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ A_117 @ B_63 )
% 0.29/0.46 => ( member_a @ X_33 @ B_63 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_420_set__rev__mp,axiom,
% 0.29/0.46 ! [B_63: pname > $o,X_33: pname,A_117: pname > $o] :
% 0.29/0.46 ( ( member_pname @ X_33 @ A_117 )
% 0.29/0.46 => ( ( ord_less_eq_pname_o @ A_117 @ B_63 )
% 0.29/0.46 => ( member_pname @ X_33 @ B_63 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_421_set__mp,axiom,
% 0.29/0.46 ! [X_32: int,A_116: int > $o,B_62: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ A_116 @ B_62 )
% 0.29/0.46 => ( ( member_int @ X_32 @ A_116 )
% 0.29/0.46 => ( member_int @ X_32 @ B_62 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_422_set__mp,axiom,
% 0.29/0.46 ! [X_32: nat,A_116: nat > $o,B_62: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ A_116 @ B_62 )
% 0.29/0.46 => ( ( member_nat @ X_32 @ A_116 )
% 0.29/0.46 => ( member_nat @ X_32 @ B_62 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_423_set__mp,axiom,
% 0.29/0.46 ! [X_32: x_a,A_116: x_a > $o,B_62: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ A_116 @ B_62 )
% 0.29/0.46 => ( ( member_a @ X_32 @ A_116 )
% 0.29/0.46 => ( member_a @ X_32 @ B_62 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_424_set__mp,axiom,
% 0.29/0.46 ! [X_32: pname,A_116: pname > $o,B_62: pname > $o] :
% 0.29/0.46 ( ( ord_less_eq_pname_o @ A_116 @ B_62 )
% 0.29/0.46 => ( ( member_pname @ X_32 @ A_116 )
% 0.29/0.46 => ( member_pname @ X_32 @ B_62 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_425_subset__trans,axiom,
% 0.29/0.46 ! [C_28: int > $o,A_115: int > $o,B_61: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ A_115 @ B_61 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ B_61 @ C_28 )
% 0.29/0.46 => ( ord_less_eq_int_o @ A_115 @ C_28 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_426_subset__trans,axiom,
% 0.29/0.46 ! [C_28: nat > $o,A_115: nat > $o,B_61: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ A_115 @ B_61 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o @ B_61 @ C_28 )
% 0.29/0.46 => ( ord_less_eq_nat_o @ A_115 @ C_28 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_427_subset__trans,axiom,
% 0.29/0.46 ! [C_28: x_a > $o,A_115: x_a > $o,B_61: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ A_115 @ B_61 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ B_61 @ C_28 )
% 0.29/0.46 => ( ord_less_eq_a_o @ A_115 @ C_28 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_428_equalityE,axiom,
% 0.29/0.46 ! [A_114: int > $o,B_60: int > $o] :
% 0.29/0.46 ( ( A_114 = B_60 )
% 0.29/0.46 => ~ ( ( ord_less_eq_int_o @ A_114 @ B_60 )
% 0.29/0.46 => ~ ( ord_less_eq_int_o @ B_60 @ A_114 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_429_equalityE,axiom,
% 0.29/0.46 ! [A_114: nat > $o,B_60: nat > $o] :
% 0.29/0.46 ( ( A_114 = B_60 )
% 0.29/0.46 => ~ ( ( ord_less_eq_nat_o @ A_114 @ B_60 )
% 0.29/0.46 => ~ ( ord_less_eq_nat_o @ B_60 @ A_114 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_430_equalityE,axiom,
% 0.29/0.46 ! [A_114: x_a > $o,B_60: x_a > $o] :
% 0.29/0.46 ( ( A_114 = B_60 )
% 0.29/0.46 => ~ ( ( ord_less_eq_a_o @ A_114 @ B_60 )
% 0.29/0.46 => ~ ( ord_less_eq_a_o @ B_60 @ A_114 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_431_image__iff,axiom,
% 0.29/0.46 ! [Z_4: int,F_35: nat > int,A_113: nat > $o] :
% 0.29/0.46 ( ( member_int @ Z_4 @ ( image_nat_int @ F_35 @ A_113 ) )
% 0.29/0.46 <=> ? [X_1: nat] :
% 0.29/0.46 ( ( member_nat @ X_1 @ A_113 )
% 0.29/0.46 & ( Z_4
% 0.29/0.46 = ( F_35 @ X_1 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_432_image__iff,axiom,
% 0.29/0.46 ! [Z_4: x_a,F_35: pname > x_a,A_113: pname > $o] :
% 0.29/0.46 ( ( member_a @ Z_4 @ ( image_pname_a @ F_35 @ A_113 ) )
% 0.29/0.46 <=> ? [X_1: pname] :
% 0.29/0.46 ( ( member_pname @ X_1 @ A_113 )
% 0.29/0.46 & ( Z_4
% 0.29/0.46 = ( F_35 @ X_1 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_433_imageI,axiom,
% 0.29/0.46 ! [F_34: nat > int,X_31: nat,A_112: nat > $o] :
% 0.29/0.46 ( ( member_nat @ X_31 @ A_112 )
% 0.29/0.46 => ( member_int @ ( F_34 @ X_31 ) @ ( image_nat_int @ F_34 @ A_112 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_434_imageI,axiom,
% 0.29/0.46 ! [F_34: pname > x_a,X_31: pname,A_112: pname > $o] :
% 0.29/0.46 ( ( member_pname @ X_31 @ A_112 )
% 0.29/0.46 => ( member_a @ ( F_34 @ X_31 ) @ ( image_pname_a @ F_34 @ A_112 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_435_rev__image__eqI,axiom,
% 0.29/0.46 ! [B_59: int,F_33: nat > int,X_30: nat,A_111: nat > $o] :
% 0.29/0.46 ( ( member_nat @ X_30 @ A_111 )
% 0.29/0.46 => ( ( B_59
% 0.29/0.46 = ( F_33 @ X_30 ) )
% 0.29/0.46 => ( member_int @ B_59 @ ( image_nat_int @ F_33 @ A_111 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_436_rev__image__eqI,axiom,
% 0.29/0.46 ! [B_59: x_a,F_33: pname > x_a,X_30: pname,A_111: pname > $o] :
% 0.29/0.46 ( ( member_pname @ X_30 @ A_111 )
% 0.29/0.46 => ( ( B_59
% 0.29/0.46 = ( F_33 @ X_30 ) )
% 0.29/0.46 => ( member_a @ B_59 @ ( image_pname_a @ F_33 @ A_111 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_437_insert__compr__raw,axiom,
% 0.29/0.46 ! [X_1: x_a,Xa: x_a > $o] :
% 0.29/0.46 ( ( insert_a @ X_1 @ Xa )
% 0.29/0.46 = ( collect_a
% 0.29/0.46 @ ^ [Y_1: x_a] : ( (|) @ ( Y_1 = X_1 ) @ ( member_a @ Y_1 @ Xa ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_438_insert__compr__raw,axiom,
% 0.29/0.46 ! [X_1: int,Xa: int > $o] :
% 0.29/0.46 ( ( insert_int @ X_1 @ Xa )
% 0.29/0.46 = ( collect_int
% 0.29/0.46 @ ^ [Y_1: int] : ( (|) @ ( Y_1 = X_1 ) @ ( member_int @ Y_1 @ Xa ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_439_insert__compr__raw,axiom,
% 0.29/0.46 ! [X_1: nat,Xa: nat > $o] :
% 0.29/0.46 ( ( insert_nat @ X_1 @ Xa )
% 0.29/0.46 = ( collect_nat
% 0.29/0.46 @ ^ [Y_1: nat] : ( (|) @ ( Y_1 = X_1 ) @ ( member_nat @ Y_1 @ Xa ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_440_insert__compr__raw,axiom,
% 0.29/0.46 ! [X_1: pname,Xa: pname > $o] :
% 0.29/0.46 ( ( insert_pname @ X_1 @ Xa )
% 0.29/0.46 = ( collect_pname
% 0.29/0.46 @ ^ [Y_1: pname] : ( (|) @ ( Y_1 = X_1 ) @ ( member_pname @ Y_1 @ Xa ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_441_le__fun__def,axiom,
% 0.29/0.46 ! [F_32: int > $o,G_4: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ F_32 @ G_4 )
% 0.29/0.46 <=> ! [X_1: int] : ( ord_less_eq_o @ ( F_32 @ X_1 ) @ ( G_4 @ X_1 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_442_le__fun__def,axiom,
% 0.29/0.46 ! [F_32: nat > $o,G_4: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ F_32 @ G_4 )
% 0.29/0.46 <=> ! [X_1: nat] : ( ord_less_eq_o @ ( F_32 @ X_1 ) @ ( G_4 @ X_1 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_443_le__fun__def,axiom,
% 0.29/0.46 ! [F_32: x_a > $o,G_4: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ F_32 @ G_4 )
% 0.29/0.46 <=> ! [X_1: x_a] : ( ord_less_eq_o @ ( F_32 @ X_1 ) @ ( G_4 @ X_1 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_444_le__funD,axiom,
% 0.29/0.46 ! [X_29: int,F_31: int > $o,G_3: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ F_31 @ G_3 )
% 0.29/0.46 => ( ord_less_eq_o @ ( F_31 @ X_29 ) @ ( G_3 @ X_29 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_445_le__funD,axiom,
% 0.29/0.46 ! [X_29: nat,F_31: nat > $o,G_3: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ F_31 @ G_3 )
% 0.29/0.46 => ( ord_less_eq_o @ ( F_31 @ X_29 ) @ ( G_3 @ X_29 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_446_le__funD,axiom,
% 0.29/0.46 ! [X_29: x_a,F_31: x_a > $o,G_3: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ F_31 @ G_3 )
% 0.29/0.46 => ( ord_less_eq_o @ ( F_31 @ X_29 ) @ ( G_3 @ X_29 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_447_le__funE,axiom,
% 0.29/0.46 ! [X_28: int,F_30: int > $o,G_2: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ F_30 @ G_2 )
% 0.29/0.46 => ( ord_less_eq_o @ ( F_30 @ X_28 ) @ ( G_2 @ X_28 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_448_le__funE,axiom,
% 0.29/0.46 ! [X_28: nat,F_30: nat > $o,G_2: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ F_30 @ G_2 )
% 0.29/0.46 => ( ord_less_eq_o @ ( F_30 @ X_28 ) @ ( G_2 @ X_28 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_449_le__funE,axiom,
% 0.29/0.46 ! [X_28: x_a,F_30: x_a > $o,G_2: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ F_30 @ G_2 )
% 0.29/0.46 => ( ord_less_eq_o @ ( F_30 @ X_28 ) @ ( G_2 @ X_28 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_450_subset__insertI,axiom,
% 0.29/0.46 ! [B_58: x_a > $o,A_110: x_a] : ( ord_less_eq_a_o @ B_58 @ ( insert_a @ A_110 @ B_58 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_451_subset__insertI,axiom,
% 0.29/0.46 ! [B_58: int > $o,A_110: int] : ( ord_less_eq_int_o @ B_58 @ ( insert_int @ A_110 @ B_58 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_452_subset__insertI,axiom,
% 0.29/0.46 ! [B_58: nat > $o,A_110: nat] : ( ord_less_eq_nat_o @ B_58 @ ( insert_nat @ A_110 @ B_58 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_453_insert__subset,axiom,
% 0.29/0.46 ! [X_27: x_a,A_109: x_a > $o,B_57: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ ( insert_a @ X_27 @ A_109 ) @ B_57 )
% 0.29/0.46 <=> ( ( member_a @ X_27 @ B_57 )
% 0.29/0.46 & ( ord_less_eq_a_o @ A_109 @ B_57 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_454_insert__subset,axiom,
% 0.29/0.46 ! [X_27: int,A_109: int > $o,B_57: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ ( insert_int @ X_27 @ A_109 ) @ B_57 )
% 0.29/0.46 <=> ( ( member_int @ X_27 @ B_57 )
% 0.29/0.46 & ( ord_less_eq_int_o @ A_109 @ B_57 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_455_insert__subset,axiom,
% 0.29/0.46 ! [X_27: nat,A_109: nat > $o,B_57: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ ( insert_nat @ X_27 @ A_109 ) @ B_57 )
% 0.29/0.46 <=> ( ( member_nat @ X_27 @ B_57 )
% 0.29/0.46 & ( ord_less_eq_nat_o @ A_109 @ B_57 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_456_insert__subset,axiom,
% 0.29/0.46 ! [X_27: pname,A_109: pname > $o,B_57: pname > $o] :
% 0.29/0.46 ( ( ord_less_eq_pname_o @ ( insert_pname @ X_27 @ A_109 ) @ B_57 )
% 0.29/0.46 <=> ( ( member_pname @ X_27 @ B_57 )
% 0.29/0.46 & ( ord_less_eq_pname_o @ A_109 @ B_57 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_457_subset__insert,axiom,
% 0.29/0.46 ! [B_56: x_a > $o,X_26: x_a,A_108: x_a > $o] :
% 0.29/0.46 ( ~ ( member_a @ X_26 @ A_108 )
% 0.29/0.46 => ( ( ord_less_eq_a_o @ A_108 @ ( insert_a @ X_26 @ B_56 ) )
% 0.29/0.46 <=> ( ord_less_eq_a_o @ A_108 @ B_56 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_458_subset__insert,axiom,
% 0.29/0.46 ! [B_56: int > $o,X_26: int,A_108: int > $o] :
% 0.29/0.46 ( ~ ( member_int @ X_26 @ A_108 )
% 0.29/0.46 => ( ( ord_less_eq_int_o @ A_108 @ ( insert_int @ X_26 @ B_56 ) )
% 0.29/0.46 <=> ( ord_less_eq_int_o @ A_108 @ B_56 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_459_subset__insert,axiom,
% 0.29/0.46 ! [B_56: nat > $o,X_26: nat,A_108: nat > $o] :
% 0.29/0.46 ( ~ ( member_nat @ X_26 @ A_108 )
% 0.29/0.46 => ( ( ord_less_eq_nat_o @ A_108 @ ( insert_nat @ X_26 @ B_56 ) )
% 0.29/0.46 <=> ( ord_less_eq_nat_o @ A_108 @ B_56 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_460_subset__insert,axiom,
% 0.29/0.46 ! [B_56: pname > $o,X_26: pname,A_108: pname > $o] :
% 0.29/0.46 ( ~ ( member_pname @ X_26 @ A_108 )
% 0.29/0.46 => ( ( ord_less_eq_pname_o @ A_108 @ ( insert_pname @ X_26 @ B_56 ) )
% 0.29/0.46 <=> ( ord_less_eq_pname_o @ A_108 @ B_56 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_461_subset__insertI2,axiom,
% 0.29/0.46 ! [B_55: x_a,A_107: x_a > $o,B_54: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ A_107 @ B_54 )
% 0.29/0.46 => ( ord_less_eq_a_o @ A_107 @ ( insert_a @ B_55 @ B_54 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_462_subset__insertI2,axiom,
% 0.29/0.46 ! [B_55: int,A_107: int > $o,B_54: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ A_107 @ B_54 )
% 0.29/0.46 => ( ord_less_eq_int_o @ A_107 @ ( insert_int @ B_55 @ B_54 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_463_subset__insertI2,axiom,
% 0.29/0.46 ! [B_55: nat,A_107: nat > $o,B_54: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ A_107 @ B_54 )
% 0.29/0.46 => ( ord_less_eq_nat_o @ A_107 @ ( insert_nat @ B_55 @ B_54 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_464_insert__mono,axiom,
% 0.29/0.46 ! [A_106: x_a,C_27: x_a > $o,D_7: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ C_27 @ D_7 )
% 0.29/0.46 => ( ord_less_eq_a_o @ ( insert_a @ A_106 @ C_27 ) @ ( insert_a @ A_106 @ D_7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_465_insert__mono,axiom,
% 0.29/0.46 ! [A_106: int,C_27: int > $o,D_7: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ C_27 @ D_7 )
% 0.29/0.46 => ( ord_less_eq_int_o @ ( insert_int @ A_106 @ C_27 ) @ ( insert_int @ A_106 @ D_7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_466_insert__mono,axiom,
% 0.29/0.46 ! [A_106: nat,C_27: nat > $o,D_7: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ C_27 @ D_7 )
% 0.29/0.46 => ( ord_less_eq_nat_o @ ( insert_nat @ A_106 @ C_27 ) @ ( insert_nat @ A_106 @ D_7 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_467_image__insert,axiom,
% 0.29/0.46 ! [F_29: nat > int,A_105: nat,B_53: nat > $o] :
% 0.29/0.46 ( ( image_nat_int @ F_29 @ ( insert_nat @ A_105 @ B_53 ) )
% 0.29/0.46 = ( insert_int @ ( F_29 @ A_105 ) @ ( image_nat_int @ F_29 @ B_53 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_468_image__insert,axiom,
% 0.29/0.46 ! [F_29: pname > x_a,A_105: pname,B_53: pname > $o] :
% 0.29/0.46 ( ( image_pname_a @ F_29 @ ( insert_pname @ A_105 @ B_53 ) )
% 0.29/0.46 = ( insert_a @ ( F_29 @ A_105 ) @ ( image_pname_a @ F_29 @ B_53 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_469_insert__image,axiom,
% 0.29/0.46 ! [F_28: nat > int,X_25: nat,A_104: nat > $o] :
% 0.29/0.46 ( ( member_nat @ X_25 @ A_104 )
% 0.29/0.46 => ( ( insert_int @ ( F_28 @ X_25 ) @ ( image_nat_int @ F_28 @ A_104 ) )
% 0.29/0.46 = ( image_nat_int @ F_28 @ A_104 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_470_insert__image,axiom,
% 0.29/0.46 ! [F_28: pname > x_a,X_25: pname,A_104: pname > $o] :
% 0.29/0.46 ( ( member_pname @ X_25 @ A_104 )
% 0.29/0.46 => ( ( insert_a @ ( F_28 @ X_25 ) @ ( image_pname_a @ F_28 @ A_104 ) )
% 0.29/0.46 = ( image_pname_a @ F_28 @ A_104 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_471_subset__image__iff,axiom,
% 0.29/0.46 ! [B_52: int > $o,F_27: nat > int,A_103: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ B_52 @ ( image_nat_int @ F_27 @ A_103 ) )
% 0.29/0.46 <=> ? [AA: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ AA @ A_103 )
% 0.29/0.46 & ( B_52
% 0.29/0.46 = ( image_nat_int @ F_27 @ AA ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_472_subset__image__iff,axiom,
% 0.29/0.46 ! [B_52: x_a > $o,F_27: pname > x_a,A_103: pname > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ B_52 @ ( image_pname_a @ F_27 @ A_103 ) )
% 0.29/0.46 <=> ? [AA: pname > $o] :
% 0.29/0.46 ( ( ord_less_eq_pname_o @ AA @ A_103 )
% 0.29/0.46 & ( B_52
% 0.29/0.46 = ( image_pname_a @ F_27 @ AA ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_473_image__mono,axiom,
% 0.29/0.46 ! [F_26: nat > int,A_102: nat > $o,B_51: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ A_102 @ B_51 )
% 0.29/0.46 => ( ord_less_eq_int_o @ ( image_nat_int @ F_26 @ A_102 ) @ ( image_nat_int @ F_26 @ B_51 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_474_image__mono,axiom,
% 0.29/0.46 ! [F_26: pname > x_a,A_102: pname > $o,B_51: pname > $o] :
% 0.29/0.46 ( ( ord_less_eq_pname_o @ A_102 @ B_51 )
% 0.29/0.46 => ( ord_less_eq_a_o @ ( image_pname_a @ F_26 @ A_102 ) @ ( image_pname_a @ F_26 @ B_51 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_475_imageE,axiom,
% 0.29/0.46 ! [B_50: int,F_25: nat > int,A_101: nat > $o] :
% 0.29/0.46 ( ( member_int @ B_50 @ ( image_nat_int @ F_25 @ A_101 ) )
% 0.29/0.46 => ~ ! [X_1: nat] :
% 0.29/0.46 ( ( B_50
% 0.29/0.46 = ( F_25 @ X_1 ) )
% 0.29/0.46 => ~ ( member_nat @ X_1 @ A_101 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_476_imageE,axiom,
% 0.29/0.46 ! [B_50: x_a,F_25: pname > x_a,A_101: pname > $o] :
% 0.29/0.46 ( ( member_a @ B_50 @ ( image_pname_a @ F_25 @ A_101 ) )
% 0.29/0.46 => ~ ! [X_1: pname] :
% 0.29/0.46 ( ( B_50
% 0.29/0.46 = ( F_25 @ X_1 ) )
% 0.29/0.46 => ~ ( member_pname @ X_1 @ A_101 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_477_subsetI,axiom,
% 0.29/0.46 ! [B_49: int > $o,A_100: int > $o] :
% 0.29/0.46 ( ! [X_1: int] :
% 0.29/0.46 ( ( member_int @ X_1 @ A_100 )
% 0.29/0.46 => ( member_int @ X_1 @ B_49 ) )
% 0.29/0.46 => ( ord_less_eq_int_o @ A_100 @ B_49 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_478_subsetI,axiom,
% 0.29/0.46 ! [B_49: nat > $o,A_100: nat > $o] :
% 0.29/0.46 ( ! [X_1: nat] :
% 0.29/0.46 ( ( member_nat @ X_1 @ A_100 )
% 0.29/0.46 => ( member_nat @ X_1 @ B_49 ) )
% 0.29/0.46 => ( ord_less_eq_nat_o @ A_100 @ B_49 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_479_subsetI,axiom,
% 0.29/0.46 ! [B_49: x_a > $o,A_100: x_a > $o] :
% 0.29/0.46 ( ! [X_1: x_a] :
% 0.29/0.46 ( ( member_a @ X_1 @ A_100 )
% 0.29/0.46 => ( member_a @ X_1 @ B_49 ) )
% 0.29/0.46 => ( ord_less_eq_a_o @ A_100 @ B_49 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_480_subsetI,axiom,
% 0.29/0.46 ! [B_49: pname > $o,A_100: pname > $o] :
% 0.29/0.46 ( ! [X_1: pname] :
% 0.29/0.46 ( ( member_pname @ X_1 @ A_100 )
% 0.29/0.46 => ( member_pname @ X_1 @ B_49 ) )
% 0.29/0.46 => ( ord_less_eq_pname_o @ A_100 @ B_49 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_481_image__subsetI,axiom,
% 0.29/0.46 ! [F_24: nat > int,B_48: int > $o,A_99: nat > $o] :
% 0.29/0.46 ( ! [X_1: nat] :
% 0.29/0.46 ( ( member_nat @ X_1 @ A_99 )
% 0.29/0.46 => ( member_int @ ( F_24 @ X_1 ) @ B_48 ) )
% 0.29/0.46 => ( ord_less_eq_int_o @ ( image_nat_int @ F_24 @ A_99 ) @ B_48 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_482_image__subsetI,axiom,
% 0.29/0.46 ! [F_24: pname > x_a,B_48: x_a > $o,A_99: pname > $o] :
% 0.29/0.46 ( ! [X_1: pname] :
% 0.29/0.46 ( ( member_pname @ X_1 @ A_99 )
% 0.29/0.46 => ( member_a @ ( F_24 @ X_1 ) @ B_48 ) )
% 0.29/0.46 => ( ord_less_eq_a_o @ ( image_pname_a @ F_24 @ A_99 ) @ B_48 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_483_le__funI,axiom,
% 0.29/0.46 ! [F_23: int > $o,G_1: int > $o] :
% 0.29/0.46 ( ! [X_1: int] : ( ord_less_eq_o @ ( F_23 @ X_1 ) @ ( G_1 @ X_1 ) )
% 0.29/0.46 => ( ord_less_eq_int_o @ F_23 @ G_1 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_484_le__funI,axiom,
% 0.29/0.46 ! [F_23: nat > $o,G_1: nat > $o] :
% 0.29/0.46 ( ! [X_1: nat] : ( ord_less_eq_o @ ( F_23 @ X_1 ) @ ( G_1 @ X_1 ) )
% 0.29/0.46 => ( ord_less_eq_nat_o @ F_23 @ G_1 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_485_le__funI,axiom,
% 0.29/0.46 ! [F_23: x_a > $o,G_1: x_a > $o] :
% 0.29/0.46 ( ! [X_1: x_a] : ( ord_less_eq_o @ ( F_23 @ X_1 ) @ ( G_1 @ X_1 ) )
% 0.29/0.46 => ( ord_less_eq_a_o @ F_23 @ G_1 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_486_finite__nat__set__iff__bounded__le,axiom,
% 0.29/0.46 ! [N_2: nat > $o] :
% 0.29/0.46 ( ( finite_finite_nat @ N_2 )
% 0.29/0.46 <=> ? [M_1: nat] :
% 0.29/0.46 ! [X_1: nat] :
% 0.29/0.46 ( ( member_nat @ X_1 @ N_2 )
% 0.29/0.46 => ( ord_less_eq_nat @ X_1 @ M_1 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_487_assms_I3_J,axiom,
% 0.29/0.46 ! [G: x_a > $o,C: com] :
% 0.29/0.46 ( ( wt @ C )
% 0.29/0.46 => ( ! [X_1: pname] :
% 0.29/0.46 ( ( member_pname @ X_1 @ u )
% 0.29/0.46 => ( p @ G @ ( insert_a @ ( mgt_call @ X_1 ) @ bot_bot_a_o ) ) )
% 0.29/0.46 => ( p @ G @ ( insert_a @ ( mgt @ C ) @ bot_bot_a_o ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_488_diff__eq__diff__less__eq,axiom,
% 0.29/0.46 ! [A_98: int,B_47: int,C_26: int,D_6: int] :
% 0.29/0.46 ( ( ( minus_minus_int @ A_98 @ B_47 )
% 0.29/0.46 = ( minus_minus_int @ C_26 @ D_6 ) )
% 0.29/0.46 => ( ( ord_less_eq_int @ A_98 @ B_47 )
% 0.29/0.46 <=> ( ord_less_eq_int @ C_26 @ D_6 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_489_less__eq__nat_Osimps_I2_J,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.29/0.46 <=> ( nat_case_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_490_emptyE,axiom,
% 0.29/0.46 ! [A_97: int] :
% 0.29/0.46 ~ ( member_int @ A_97 @ bot_bot_int_o ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_491_emptyE,axiom,
% 0.29/0.46 ! [A_97: nat] :
% 0.29/0.46 ~ ( member_nat @ A_97 @ bot_bot_nat_o ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_492_emptyE,axiom,
% 0.29/0.46 ! [A_97: x_a] :
% 0.29/0.46 ~ ( member_a @ A_97 @ bot_bot_a_o ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_493_emptyE,axiom,
% 0.29/0.46 ! [A_97: pname] :
% 0.29/0.46 ~ ( member_pname @ A_97 @ bot_bot_pname_o ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_494_finite__Diff,axiom,
% 0.29/0.46 ! [B_46: int > $o,A_96: int > $o] :
% 0.29/0.46 ( ( finite_finite_int @ A_96 )
% 0.29/0.46 => ( finite_finite_int @ ( minus_minus_int_o @ A_96 @ B_46 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_495_finite__Diff,axiom,
% 0.29/0.46 ! [B_46: nat > $o,A_96: nat > $o] :
% 0.29/0.46 ( ( finite_finite_nat @ A_96 )
% 0.29/0.46 => ( finite_finite_nat @ ( minus_minus_nat_o @ A_96 @ B_46 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_496_finite__Diff,axiom,
% 0.29/0.46 ! [B_46: pname > $o,A_96: pname > $o] :
% 0.29/0.46 ( ( finite_finite_pname @ A_96 )
% 0.29/0.46 => ( finite_finite_pname @ ( minus_minus_pname_o @ A_96 @ B_46 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_497_finite_OemptyI,axiom,
% 0.29/0.46 finite_finite_int @ bot_bot_int_o ).
% 0.29/0.46
% 0.29/0.46 thf(fact_498_finite_OemptyI,axiom,
% 0.29/0.46 finite_finite_nat @ bot_bot_nat_o ).
% 0.29/0.46
% 0.29/0.46 thf(fact_499_finite_OemptyI,axiom,
% 0.29/0.46 finite_finite_pname @ bot_bot_pname_o ).
% 0.29/0.46
% 0.29/0.46 thf(fact_500_finite_OemptyI,axiom,
% 0.29/0.46 finite_finite_a @ bot_bot_a_o ).
% 0.29/0.46
% 0.29/0.46 thf(fact_501_empty__subsetI,axiom,
% 0.29/0.46 ! [A_95: nat > $o] : ( ord_less_eq_nat_o @ bot_bot_nat_o @ A_95 ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_502_empty__subsetI,axiom,
% 0.29/0.46 ! [A_95: int > $o] : ( ord_less_eq_int_o @ bot_bot_int_o @ A_95 ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_503_empty__subsetI,axiom,
% 0.29/0.46 ! [A_95: x_a > $o] : ( ord_less_eq_a_o @ bot_bot_a_o @ A_95 ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_504_equals0D,axiom,
% 0.29/0.46 ! [A_94: int,A_93: int > $o] :
% 0.29/0.46 ( ( A_93 = bot_bot_int_o )
% 0.29/0.46 => ~ ( member_int @ A_94 @ A_93 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_505_equals0D,axiom,
% 0.29/0.46 ! [A_94: nat,A_93: nat > $o] :
% 0.29/0.46 ( ( A_93 = bot_bot_nat_o )
% 0.29/0.46 => ~ ( member_nat @ A_94 @ A_93 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_506_equals0D,axiom,
% 0.29/0.46 ! [A_94: x_a,A_93: x_a > $o] :
% 0.29/0.46 ( ( A_93 = bot_bot_a_o )
% 0.29/0.46 => ~ ( member_a @ A_94 @ A_93 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_507_equals0D,axiom,
% 0.29/0.46 ! [A_94: pname,A_93: pname > $o] :
% 0.29/0.46 ( ( A_93 = bot_bot_pname_o )
% 0.29/0.46 => ~ ( member_pname @ A_94 @ A_93 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_508_Collect__empty__eq,axiom,
% 0.29/0.46 ! [P_9: int > $o] :
% 0.29/0.46 ( ( ( collect_int @ P_9 )
% 0.29/0.46 = bot_bot_int_o )
% 0.29/0.46 <=> ! [X_1: int] :
% 0.29/0.46 ~ ( P_9 @ X_1 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_509_Collect__empty__eq,axiom,
% 0.29/0.46 ! [P_9: nat > $o] :
% 0.29/0.46 ( ( ( collect_nat @ P_9 )
% 0.29/0.46 = bot_bot_nat_o )
% 0.29/0.46 <=> ! [X_1: nat] :
% 0.29/0.46 ~ ( P_9 @ X_1 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_510_Collect__empty__eq,axiom,
% 0.29/0.46 ! [P_9: x_a > $o] :
% 0.29/0.46 ( ( ( collect_a @ P_9 )
% 0.29/0.46 = bot_bot_a_o )
% 0.29/0.46 <=> ! [X_1: x_a] :
% 0.29/0.46 ~ ( P_9 @ X_1 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_511_Diff__cancel,axiom,
% 0.29/0.46 ! [A_92: nat > $o] :
% 0.29/0.46 ( ( minus_minus_nat_o @ A_92 @ A_92 )
% 0.29/0.46 = bot_bot_nat_o ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_512_Diff__cancel,axiom,
% 0.29/0.46 ! [A_92: int > $o] :
% 0.29/0.46 ( ( minus_minus_int_o @ A_92 @ A_92 )
% 0.29/0.46 = bot_bot_int_o ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_513_Diff__cancel,axiom,
% 0.29/0.46 ! [A_92: x_a > $o] :
% 0.29/0.46 ( ( minus_minus_a_o @ A_92 @ A_92 )
% 0.29/0.46 = bot_bot_a_o ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_514_Diff__empty,axiom,
% 0.29/0.46 ! [A_91: nat > $o] :
% 0.29/0.46 ( ( minus_minus_nat_o @ A_91 @ bot_bot_nat_o )
% 0.29/0.46 = A_91 ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_515_Diff__empty,axiom,
% 0.29/0.46 ! [A_91: int > $o] :
% 0.29/0.46 ( ( minus_minus_int_o @ A_91 @ bot_bot_int_o )
% 0.29/0.46 = A_91 ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_516_Diff__empty,axiom,
% 0.29/0.46 ! [A_91: x_a > $o] :
% 0.29/0.46 ( ( minus_minus_a_o @ A_91 @ bot_bot_a_o )
% 0.29/0.46 = A_91 ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_517_empty__iff,axiom,
% 0.29/0.46 ! [C_25: int] :
% 0.29/0.46 ~ ( member_int @ C_25 @ bot_bot_int_o ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_518_empty__iff,axiom,
% 0.29/0.46 ! [C_25: nat] :
% 0.29/0.46 ~ ( member_nat @ C_25 @ bot_bot_nat_o ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_519_empty__iff,axiom,
% 0.29/0.46 ! [C_25: x_a] :
% 0.29/0.46 ~ ( member_a @ C_25 @ bot_bot_a_o ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_520_empty__iff,axiom,
% 0.29/0.46 ! [C_25: pname] :
% 0.29/0.46 ~ ( member_pname @ C_25 @ bot_bot_pname_o ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_521_empty__Collect__eq,axiom,
% 0.29/0.46 ! [P_8: int > $o] :
% 0.29/0.46 ( ( bot_bot_int_o
% 0.29/0.46 = ( collect_int @ P_8 ) )
% 0.29/0.46 <=> ! [X_1: int] :
% 0.29/0.46 ~ ( P_8 @ X_1 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_522_empty__Collect__eq,axiom,
% 0.29/0.46 ! [P_8: nat > $o] :
% 0.29/0.46 ( ( bot_bot_nat_o
% 0.29/0.46 = ( collect_nat @ P_8 ) )
% 0.29/0.46 <=> ! [X_1: nat] :
% 0.29/0.46 ~ ( P_8 @ X_1 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_523_empty__Collect__eq,axiom,
% 0.29/0.46 ! [P_8: x_a > $o] :
% 0.29/0.46 ( ( bot_bot_a_o
% 0.29/0.46 = ( collect_a @ P_8 ) )
% 0.29/0.46 <=> ! [X_1: x_a] :
% 0.29/0.46 ~ ( P_8 @ X_1 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_524_empty__Diff,axiom,
% 0.29/0.46 ! [A_90: nat > $o] :
% 0.29/0.46 ( ( minus_minus_nat_o @ bot_bot_nat_o @ A_90 )
% 0.29/0.46 = bot_bot_nat_o ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_525_empty__Diff,axiom,
% 0.29/0.46 ! [A_90: int > $o] :
% 0.29/0.46 ( ( minus_minus_int_o @ bot_bot_int_o @ A_90 )
% 0.29/0.46 = bot_bot_int_o ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_526_empty__Diff,axiom,
% 0.29/0.46 ! [A_90: x_a > $o] :
% 0.29/0.46 ( ( minus_minus_a_o @ bot_bot_a_o @ A_90 )
% 0.29/0.46 = bot_bot_a_o ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_527_ex__in__conv,axiom,
% 0.29/0.46 ! [A_89: int > $o] :
% 0.29/0.46 ( ? [X_1: int] : ( member_int @ X_1 @ A_89 )
% 0.29/0.46 <=> ( A_89 != bot_bot_int_o ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_528_ex__in__conv,axiom,
% 0.29/0.46 ! [A_89: nat > $o] :
% 0.29/0.46 ( ? [X_1: nat] : ( member_nat @ X_1 @ A_89 )
% 0.29/0.46 <=> ( A_89 != bot_bot_nat_o ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_529_ex__in__conv,axiom,
% 0.29/0.46 ! [A_89: x_a > $o] :
% 0.29/0.46 ( ? [X_1: x_a] : ( member_a @ X_1 @ A_89 )
% 0.29/0.46 <=> ( A_89 != bot_bot_a_o ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_530_ex__in__conv,axiom,
% 0.29/0.46 ! [A_89: pname > $o] :
% 0.29/0.46 ( ? [X_1: pname] : ( member_pname @ X_1 @ A_89 )
% 0.29/0.46 <=> ( A_89 != bot_bot_pname_o ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_531_all__not__in__conv,axiom,
% 0.29/0.46 ! [A_88: int > $o] :
% 0.29/0.46 ( ! [X_1: int] :
% 0.29/0.46 ~ ( member_int @ X_1 @ A_88 )
% 0.29/0.46 <=> ( A_88 = bot_bot_int_o ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_532_all__not__in__conv,axiom,
% 0.29/0.46 ! [A_88: nat > $o] :
% 0.29/0.46 ( ! [X_1: nat] :
% 0.29/0.46 ~ ( member_nat @ X_1 @ A_88 )
% 0.29/0.46 <=> ( A_88 = bot_bot_nat_o ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_533_all__not__in__conv,axiom,
% 0.29/0.46 ! [A_88: x_a > $o] :
% 0.29/0.46 ( ! [X_1: x_a] :
% 0.29/0.46 ~ ( member_a @ X_1 @ A_88 )
% 0.29/0.46 <=> ( A_88 = bot_bot_a_o ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_534_all__not__in__conv,axiom,
% 0.29/0.46 ! [A_88: pname > $o] :
% 0.29/0.46 ( ! [X_1: pname] :
% 0.29/0.46 ~ ( member_pname @ X_1 @ A_88 )
% 0.29/0.46 <=> ( A_88 = bot_bot_pname_o ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_535_bot__apply,axiom,
% 0.29/0.46 ! [X_24: nat] :
% 0.29/0.46 ( ( bot_bot_nat_o @ X_24 )
% 0.29/0.46 <=> bot_bot_o ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_536_bot__apply,axiom,
% 0.29/0.46 ! [X_24: int] :
% 0.29/0.46 ( ( bot_bot_int_o @ X_24 )
% 0.29/0.46 <=> bot_bot_o ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_537_bot__apply,axiom,
% 0.29/0.46 ! [X_24: x_a] :
% 0.29/0.46 ( ( bot_bot_a_o @ X_24 )
% 0.29/0.46 <=> bot_bot_o ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_538_empty__def,axiom,
% 0.29/0.46 ( bot_bot_int_o
% 0.29/0.46 = ( collect_int
% 0.29/0.46 @ ^ [X_1: int] : $false ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_539_empty__def,axiom,
% 0.29/0.46 ( bot_bot_nat_o
% 0.29/0.46 = ( collect_nat
% 0.29/0.46 @ ^ [X_1: nat] : $false ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_540_empty__def,axiom,
% 0.29/0.46 ( bot_bot_a_o
% 0.29/0.46 = ( collect_a
% 0.29/0.46 @ ^ [X_1: x_a] : $false ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_541_bot__fun__def,axiom,
% 0.29/0.46 ! [X_1: nat] :
% 0.29/0.46 ( ( bot_bot_nat_o @ X_1 )
% 0.29/0.46 <=> bot_bot_o ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_542_bot__fun__def,axiom,
% 0.29/0.46 ! [X_1: int] :
% 0.29/0.46 ( ( bot_bot_int_o @ X_1 )
% 0.29/0.46 <=> bot_bot_o ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_543_bot__fun__def,axiom,
% 0.29/0.46 ! [X_1: x_a] :
% 0.29/0.46 ( ( bot_bot_a_o @ X_1 )
% 0.29/0.46 <=> bot_bot_o ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_544_insert__Diff,axiom,
% 0.29/0.46 ! [A_87: x_a,A_86: x_a > $o] :
% 0.29/0.46 ( ( member_a @ A_87 @ A_86 )
% 0.29/0.46 => ( ( insert_a @ A_87 @ ( minus_minus_a_o @ A_86 @ ( insert_a @ A_87 @ bot_bot_a_o ) ) )
% 0.29/0.46 = A_86 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_545_insert__Diff,axiom,
% 0.29/0.46 ! [A_87: int,A_86: int > $o] :
% 0.29/0.46 ( ( member_int @ A_87 @ A_86 )
% 0.29/0.46 => ( ( insert_int @ A_87 @ ( minus_minus_int_o @ A_86 @ ( insert_int @ A_87 @ bot_bot_int_o ) ) )
% 0.29/0.46 = A_86 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_546_insert__Diff,axiom,
% 0.29/0.46 ! [A_87: nat,A_86: nat > $o] :
% 0.29/0.46 ( ( member_nat @ A_87 @ A_86 )
% 0.29/0.46 => ( ( insert_nat @ A_87 @ ( minus_minus_nat_o @ A_86 @ ( insert_nat @ A_87 @ bot_bot_nat_o ) ) )
% 0.29/0.46 = A_86 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_547_insert__Diff,axiom,
% 0.29/0.46 ! [A_87: pname,A_86: pname > $o] :
% 0.29/0.46 ( ( member_pname @ A_87 @ A_86 )
% 0.29/0.46 => ( ( insert_pname @ A_87 @ ( minus_minus_pname_o @ A_86 @ ( insert_pname @ A_87 @ bot_bot_pname_o ) ) )
% 0.29/0.46 = A_86 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_548_Diff__insert__absorb,axiom,
% 0.29/0.46 ! [X_23: x_a,A_85: x_a > $o] :
% 0.29/0.46 ( ~ ( member_a @ X_23 @ A_85 )
% 0.29/0.46 => ( ( minus_minus_a_o @ ( insert_a @ X_23 @ A_85 ) @ ( insert_a @ X_23 @ bot_bot_a_o ) )
% 0.29/0.46 = A_85 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_549_Diff__insert__absorb,axiom,
% 0.29/0.46 ! [X_23: int,A_85: int > $o] :
% 0.29/0.46 ( ~ ( member_int @ X_23 @ A_85 )
% 0.29/0.46 => ( ( minus_minus_int_o @ ( insert_int @ X_23 @ A_85 ) @ ( insert_int @ X_23 @ bot_bot_int_o ) )
% 0.29/0.46 = A_85 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_550_Diff__insert__absorb,axiom,
% 0.29/0.46 ! [X_23: nat,A_85: nat > $o] :
% 0.29/0.46 ( ~ ( member_nat @ X_23 @ A_85 )
% 0.29/0.46 => ( ( minus_minus_nat_o @ ( insert_nat @ X_23 @ A_85 ) @ ( insert_nat @ X_23 @ bot_bot_nat_o ) )
% 0.29/0.46 = A_85 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_551_Diff__insert__absorb,axiom,
% 0.29/0.46 ! [X_23: pname,A_85: pname > $o] :
% 0.29/0.46 ( ~ ( member_pname @ X_23 @ A_85 )
% 0.29/0.46 => ( ( minus_minus_pname_o @ ( insert_pname @ X_23 @ A_85 ) @ ( insert_pname @ X_23 @ bot_bot_pname_o ) )
% 0.29/0.46 = A_85 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_552_insert__Diff__single,axiom,
% 0.29/0.46 ! [A_84: x_a,A_83: x_a > $o] :
% 0.29/0.46 ( ( insert_a @ A_84 @ ( minus_minus_a_o @ A_83 @ ( insert_a @ A_84 @ bot_bot_a_o ) ) )
% 0.29/0.46 = ( insert_a @ A_84 @ A_83 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_553_insert__Diff__single,axiom,
% 0.29/0.46 ! [A_84: nat,A_83: nat > $o] :
% 0.29/0.46 ( ( insert_nat @ A_84 @ ( minus_minus_nat_o @ A_83 @ ( insert_nat @ A_84 @ bot_bot_nat_o ) ) )
% 0.29/0.46 = ( insert_nat @ A_84 @ A_83 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_554_insert__Diff__single,axiom,
% 0.29/0.46 ! [A_84: int,A_83: int > $o] :
% 0.29/0.46 ( ( insert_int @ A_84 @ ( minus_minus_int_o @ A_83 @ ( insert_int @ A_84 @ bot_bot_int_o ) ) )
% 0.29/0.46 = ( insert_int @ A_84 @ A_83 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_555_Diff__insert2,axiom,
% 0.29/0.46 ! [A_82: x_a > $o,A_81: x_a,B_45: x_a > $o] :
% 0.29/0.46 ( ( minus_minus_a_o @ A_82 @ ( insert_a @ A_81 @ B_45 ) )
% 0.29/0.46 = ( minus_minus_a_o @ ( minus_minus_a_o @ A_82 @ ( insert_a @ A_81 @ bot_bot_a_o ) ) @ B_45 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_556_Diff__insert2,axiom,
% 0.29/0.46 ! [A_82: nat > $o,A_81: nat,B_45: nat > $o] :
% 0.29/0.46 ( ( minus_minus_nat_o @ A_82 @ ( insert_nat @ A_81 @ B_45 ) )
% 0.29/0.46 = ( minus_minus_nat_o @ ( minus_minus_nat_o @ A_82 @ ( insert_nat @ A_81 @ bot_bot_nat_o ) ) @ B_45 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_557_Diff__insert2,axiom,
% 0.29/0.46 ! [A_82: int > $o,A_81: int,B_45: int > $o] :
% 0.29/0.46 ( ( minus_minus_int_o @ A_82 @ ( insert_int @ A_81 @ B_45 ) )
% 0.29/0.46 = ( minus_minus_int_o @ ( minus_minus_int_o @ A_82 @ ( insert_int @ A_81 @ bot_bot_int_o ) ) @ B_45 ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_558_Diff__insert,axiom,
% 0.29/0.46 ! [A_80: x_a > $o,A_79: x_a,B_44: x_a > $o] :
% 0.29/0.46 ( ( minus_minus_a_o @ A_80 @ ( insert_a @ A_79 @ B_44 ) )
% 0.29/0.46 = ( minus_minus_a_o @ ( minus_minus_a_o @ A_80 @ B_44 ) @ ( insert_a @ A_79 @ bot_bot_a_o ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_559_Diff__insert,axiom,
% 0.29/0.46 ! [A_80: nat > $o,A_79: nat,B_44: nat > $o] :
% 0.29/0.46 ( ( minus_minus_nat_o @ A_80 @ ( insert_nat @ A_79 @ B_44 ) )
% 0.29/0.46 = ( minus_minus_nat_o @ ( minus_minus_nat_o @ A_80 @ B_44 ) @ ( insert_nat @ A_79 @ bot_bot_nat_o ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_560_Diff__insert,axiom,
% 0.29/0.46 ! [A_80: int > $o,A_79: int,B_44: int > $o] :
% 0.29/0.46 ( ( minus_minus_int_o @ A_80 @ ( insert_int @ A_79 @ B_44 ) )
% 0.29/0.46 = ( minus_minus_int_o @ ( minus_minus_int_o @ A_80 @ B_44 ) @ ( insert_int @ A_79 @ bot_bot_int_o ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_561_diff__single__insert,axiom,
% 0.29/0.46 ! [A_78: x_a > $o,X_22: x_a,B_43: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ ( minus_minus_a_o @ A_78 @ ( insert_a @ X_22 @ bot_bot_a_o ) ) @ B_43 )
% 0.29/0.46 => ( ( member_a @ X_22 @ A_78 )
% 0.29/0.46 => ( ord_less_eq_a_o @ A_78 @ ( insert_a @ X_22 @ B_43 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_562_diff__single__insert,axiom,
% 0.29/0.46 ! [A_78: int > $o,X_22: int,B_43: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ ( minus_minus_int_o @ A_78 @ ( insert_int @ X_22 @ bot_bot_int_o ) ) @ B_43 )
% 0.29/0.46 => ( ( member_int @ X_22 @ A_78 )
% 0.29/0.46 => ( ord_less_eq_int_o @ A_78 @ ( insert_int @ X_22 @ B_43 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_563_diff__single__insert,axiom,
% 0.29/0.46 ! [A_78: nat > $o,X_22: nat,B_43: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_78 @ ( insert_nat @ X_22 @ bot_bot_nat_o ) ) @ B_43 )
% 0.29/0.46 => ( ( member_nat @ X_22 @ A_78 )
% 0.29/0.46 => ( ord_less_eq_nat_o @ A_78 @ ( insert_nat @ X_22 @ B_43 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_564_diff__single__insert,axiom,
% 0.29/0.46 ! [A_78: pname > $o,X_22: pname,B_43: pname > $o] :
% 0.29/0.46 ( ( ord_less_eq_pname_o @ ( minus_minus_pname_o @ A_78 @ ( insert_pname @ X_22 @ bot_bot_pname_o ) ) @ B_43 )
% 0.29/0.46 => ( ( member_pname @ X_22 @ A_78 )
% 0.29/0.46 => ( ord_less_eq_pname_o @ A_78 @ ( insert_pname @ X_22 @ B_43 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_565_subset__insert__iff,axiom,
% 0.29/0.46 ! [A_77: x_a > $o,X_21: x_a,B_42: x_a > $o] :
% 0.29/0.46 ( ( ord_less_eq_a_o @ A_77 @ ( insert_a @ X_21 @ B_42 ) )
% 0.29/0.46 <=> ( ( ( member_a @ X_21 @ A_77 )
% 0.29/0.46 => ( ord_less_eq_a_o @ ( minus_minus_a_o @ A_77 @ ( insert_a @ X_21 @ bot_bot_a_o ) ) @ B_42 ) )
% 0.29/0.46 & ( ~ ( member_a @ X_21 @ A_77 )
% 0.29/0.46 => ( ord_less_eq_a_o @ A_77 @ B_42 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_566_subset__insert__iff,axiom,
% 0.29/0.46 ! [A_77: int > $o,X_21: int,B_42: int > $o] :
% 0.29/0.46 ( ( ord_less_eq_int_o @ A_77 @ ( insert_int @ X_21 @ B_42 ) )
% 0.29/0.46 <=> ( ( ( member_int @ X_21 @ A_77 )
% 0.29/0.46 => ( ord_less_eq_int_o @ ( minus_minus_int_o @ A_77 @ ( insert_int @ X_21 @ bot_bot_int_o ) ) @ B_42 ) )
% 0.29/0.46 & ( ~ ( member_int @ X_21 @ A_77 )
% 0.29/0.46 => ( ord_less_eq_int_o @ A_77 @ B_42 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_567_subset__insert__iff,axiom,
% 0.29/0.46 ! [A_77: nat > $o,X_21: nat,B_42: nat > $o] :
% 0.29/0.46 ( ( ord_less_eq_nat_o @ A_77 @ ( insert_nat @ X_21 @ B_42 ) )
% 0.29/0.46 <=> ( ( ( member_nat @ X_21 @ A_77 )
% 0.29/0.46 => ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_77 @ ( insert_nat @ X_21 @ bot_bot_nat_o ) ) @ B_42 ) )
% 0.29/0.46 & ( ~ ( member_nat @ X_21 @ A_77 )
% 0.29/0.46 => ( ord_less_eq_nat_o @ A_77 @ B_42 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_568_subset__insert__iff,axiom,
% 0.29/0.46 ! [A_77: pname > $o,X_21: pname,B_42: pname > $o] :
% 0.29/0.46 ( ( ord_less_eq_pname_o @ A_77 @ ( insert_pname @ X_21 @ B_42 ) )
% 0.29/0.46 <=> ( ( ( member_pname @ X_21 @ A_77 )
% 0.29/0.46 => ( ord_less_eq_pname_o @ ( minus_minus_pname_o @ A_77 @ ( insert_pname @ X_21 @ bot_bot_pname_o ) ) @ B_42 ) )
% 0.29/0.46 & ( ~ ( member_pname @ X_21 @ A_77 )
% 0.29/0.46 => ( ord_less_eq_pname_o @ A_77 @ B_42 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_569_finite__Diff2,axiom,
% 0.29/0.46 ! [A_76: int > $o,B_41: int > $o] :
% 0.29/0.46 ( ( finite_finite_int @ B_41 )
% 0.29/0.46 => ( ( finite_finite_int @ ( minus_minus_int_o @ A_76 @ B_41 ) )
% 0.29/0.46 <=> ( finite_finite_int @ A_76 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_570_finite__Diff2,axiom,
% 0.29/0.46 ! [A_76: nat > $o,B_41: nat > $o] :
% 0.29/0.46 ( ( finite_finite_nat @ B_41 )
% 0.29/0.46 => ( ( finite_finite_nat @ ( minus_minus_nat_o @ A_76 @ B_41 ) )
% 0.29/0.46 <=> ( finite_finite_nat @ A_76 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_571_finite__Diff2,axiom,
% 0.29/0.46 ! [A_76: pname > $o,B_41: pname > $o] :
% 0.29/0.46 ( ( finite_finite_pname @ B_41 )
% 0.29/0.46 => ( ( finite_finite_pname @ ( minus_minus_pname_o @ A_76 @ B_41 ) )
% 0.29/0.46 <=> ( finite_finite_pname @ A_76 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_572_insert__Diff1,axiom,
% 0.29/0.46 ! [A_75: x_a > $o,X_20: x_a,B_40: x_a > $o] :
% 0.29/0.46 ( ( member_a @ X_20 @ B_40 )
% 0.29/0.46 => ( ( minus_minus_a_o @ ( insert_a @ X_20 @ A_75 ) @ B_40 )
% 0.29/0.46 = ( minus_minus_a_o @ A_75 @ B_40 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_573_insert__Diff1,axiom,
% 0.29/0.46 ! [A_75: int > $o,X_20: int,B_40: int > $o] :
% 0.29/0.46 ( ( member_int @ X_20 @ B_40 )
% 0.29/0.46 => ( ( minus_minus_int_o @ ( insert_int @ X_20 @ A_75 ) @ B_40 )
% 0.29/0.46 = ( minus_minus_int_o @ A_75 @ B_40 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_574_insert__Diff1,axiom,
% 0.29/0.46 ! [A_75: nat > $o,X_20: nat,B_40: nat > $o] :
% 0.29/0.46 ( ( member_nat @ X_20 @ B_40 )
% 0.29/0.46 => ( ( minus_minus_nat_o @ ( insert_nat @ X_20 @ A_75 ) @ B_40 )
% 0.29/0.46 = ( minus_minus_nat_o @ A_75 @ B_40 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_575_insert__Diff1,axiom,
% 0.29/0.46 ! [A_75: pname > $o,X_20: pname,B_40: pname > $o] :
% 0.29/0.46 ( ( member_pname @ X_20 @ B_40 )
% 0.29/0.46 => ( ( minus_minus_pname_o @ ( insert_pname @ X_20 @ A_75 ) @ B_40 )
% 0.29/0.46 = ( minus_minus_pname_o @ A_75 @ B_40 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_576_insert__Diff__if,axiom,
% 0.29/0.46 ! [A_74: x_a > $o,X_19: x_a,B_39: x_a > $o] :
% 0.29/0.46 ( ( ( member_a @ X_19 @ B_39 )
% 0.29/0.46 => ( ( minus_minus_a_o @ ( insert_a @ X_19 @ A_74 ) @ B_39 )
% 0.29/0.46 = ( minus_minus_a_o @ A_74 @ B_39 ) ) )
% 0.29/0.46 & ( ~ ( member_a @ X_19 @ B_39 )
% 0.29/0.46 => ( ( minus_minus_a_o @ ( insert_a @ X_19 @ A_74 ) @ B_39 )
% 0.29/0.46 = ( insert_a @ X_19 @ ( minus_minus_a_o @ A_74 @ B_39 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_577_insert__Diff__if,axiom,
% 0.29/0.46 ! [A_74: int > $o,X_19: int,B_39: int > $o] :
% 0.29/0.46 ( ( ( member_int @ X_19 @ B_39 )
% 0.29/0.46 => ( ( minus_minus_int_o @ ( insert_int @ X_19 @ A_74 ) @ B_39 )
% 0.29/0.46 = ( minus_minus_int_o @ A_74 @ B_39 ) ) )
% 0.29/0.46 & ( ~ ( member_int @ X_19 @ B_39 )
% 0.29/0.46 => ( ( minus_minus_int_o @ ( insert_int @ X_19 @ A_74 ) @ B_39 )
% 0.29/0.46 = ( insert_int @ X_19 @ ( minus_minus_int_o @ A_74 @ B_39 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_578_insert__Diff__if,axiom,
% 0.29/0.46 ! [A_74: nat > $o,X_19: nat,B_39: nat > $o] :
% 0.29/0.46 ( ( ( member_nat @ X_19 @ B_39 )
% 0.29/0.46 => ( ( minus_minus_nat_o @ ( insert_nat @ X_19 @ A_74 ) @ B_39 )
% 0.29/0.46 = ( minus_minus_nat_o @ A_74 @ B_39 ) ) )
% 0.29/0.46 & ( ~ ( member_nat @ X_19 @ B_39 )
% 0.29/0.46 => ( ( minus_minus_nat_o @ ( insert_nat @ X_19 @ A_74 ) @ B_39 )
% 0.29/0.46 = ( insert_nat @ X_19 @ ( minus_minus_nat_o @ A_74 @ B_39 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(fact_579_insert__Diff__if,axiom,
% 0.29/0.47 ! [A_74: pname > $o,X_19: pname,B_39: pname > $o] :
% 0.29/0.47 ( ( ( member_pname @ X_19 @ B_39 )
% 0.29/0.47 => ( ( minus_minus_pname_o @ ( insert_pname @ X_19 @ A_74 ) @ B_39 )
% 0.29/0.47 = ( minus_minus_pname_o @ A_74 @ B_39 ) ) )
% 0.29/0.47 & ( ~ ( member_pname @ X_19 @ B_39 )
% 0.29/0.47 => ( ( minus_minus_pname_o @ ( insert_pname @ X_19 @ A_74 ) @ B_39 )
% 0.29/0.47 = ( insert_pname @ X_19 @ ( minus_minus_pname_o @ A_74 @ B_39 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_580_double__diff,axiom,
% 0.29/0.47 ! [C_24: int > $o,A_73: int > $o,B_38: int > $o] :
% 0.29/0.47 ( ( ord_less_eq_int_o @ A_73 @ B_38 )
% 0.29/0.47 => ( ( ord_less_eq_int_o @ B_38 @ C_24 )
% 0.29/0.47 => ( ( minus_minus_int_o @ B_38 @ ( minus_minus_int_o @ C_24 @ A_73 ) )
% 0.29/0.47 = A_73 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_581_double__diff,axiom,
% 0.29/0.47 ! [C_24: nat > $o,A_73: nat > $o,B_38: nat > $o] :
% 0.29/0.47 ( ( ord_less_eq_nat_o @ A_73 @ B_38 )
% 0.29/0.47 => ( ( ord_less_eq_nat_o @ B_38 @ C_24 )
% 0.29/0.47 => ( ( minus_minus_nat_o @ B_38 @ ( minus_minus_nat_o @ C_24 @ A_73 ) )
% 0.29/0.47 = A_73 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_582_double__diff,axiom,
% 0.29/0.47 ! [C_24: x_a > $o,A_73: x_a > $o,B_38: x_a > $o] :
% 0.29/0.47 ( ( ord_less_eq_a_o @ A_73 @ B_38 )
% 0.29/0.47 => ( ( ord_less_eq_a_o @ B_38 @ C_24 )
% 0.29/0.47 => ( ( minus_minus_a_o @ B_38 @ ( minus_minus_a_o @ C_24 @ A_73 ) )
% 0.29/0.47 = A_73 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_583_Diff__mono,axiom,
% 0.29/0.47 ! [D_5: int > $o,B_37: int > $o,A_72: int > $o,C_23: int > $o] :
% 0.29/0.47 ( ( ord_less_eq_int_o @ A_72 @ C_23 )
% 0.29/0.47 => ( ( ord_less_eq_int_o @ D_5 @ B_37 )
% 0.29/0.47 => ( ord_less_eq_int_o @ ( minus_minus_int_o @ A_72 @ B_37 ) @ ( minus_minus_int_o @ C_23 @ D_5 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_584_Diff__mono,axiom,
% 0.29/0.47 ! [D_5: nat > $o,B_37: nat > $o,A_72: nat > $o,C_23: nat > $o] :
% 0.29/0.47 ( ( ord_less_eq_nat_o @ A_72 @ C_23 )
% 0.29/0.47 => ( ( ord_less_eq_nat_o @ D_5 @ B_37 )
% 0.29/0.47 => ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_72 @ B_37 ) @ ( minus_minus_nat_o @ C_23 @ D_5 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_585_Diff__mono,axiom,
% 0.29/0.47 ! [D_5: x_a > $o,B_37: x_a > $o,A_72: x_a > $o,C_23: x_a > $o] :
% 0.29/0.47 ( ( ord_less_eq_a_o @ A_72 @ C_23 )
% 0.29/0.47 => ( ( ord_less_eq_a_o @ D_5 @ B_37 )
% 0.29/0.47 => ( ord_less_eq_a_o @ ( minus_minus_a_o @ A_72 @ B_37 ) @ ( minus_minus_a_o @ C_23 @ D_5 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_586_Diff__subset,axiom,
% 0.29/0.47 ! [A_71: int > $o,B_36: int > $o] : ( ord_less_eq_int_o @ ( minus_minus_int_o @ A_71 @ B_36 ) @ A_71 ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_587_Diff__subset,axiom,
% 0.29/0.47 ! [A_71: nat > $o,B_36: nat > $o] : ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_71 @ B_36 ) @ A_71 ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_588_Diff__subset,axiom,
% 0.29/0.47 ! [A_71: x_a > $o,B_36: x_a > $o] : ( ord_less_eq_a_o @ ( minus_minus_a_o @ A_71 @ B_36 ) @ A_71 ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_589_singleton__inject,axiom,
% 0.29/0.47 ! [A_70: x_a,B_35: x_a] :
% 0.29/0.47 ( ( ( insert_a @ A_70 @ bot_bot_a_o )
% 0.29/0.47 = ( insert_a @ B_35 @ bot_bot_a_o ) )
% 0.29/0.47 => ( A_70 = B_35 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_590_singleton__inject,axiom,
% 0.29/0.47 ! [A_70: nat,B_35: nat] :
% 0.29/0.47 ( ( ( insert_nat @ A_70 @ bot_bot_nat_o )
% 0.29/0.47 = ( insert_nat @ B_35 @ bot_bot_nat_o ) )
% 0.29/0.47 => ( A_70 = B_35 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_591_singleton__inject,axiom,
% 0.29/0.47 ! [A_70: int,B_35: int] :
% 0.29/0.47 ( ( ( insert_int @ A_70 @ bot_bot_int_o )
% 0.29/0.47 = ( insert_int @ B_35 @ bot_bot_int_o ) )
% 0.29/0.47 => ( A_70 = B_35 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_592_singletonE,axiom,
% 0.29/0.47 ! [B_34: x_a,A_69: x_a] :
% 0.29/0.47 ( ( member_a @ B_34 @ ( insert_a @ A_69 @ bot_bot_a_o ) )
% 0.29/0.47 => ( B_34 = A_69 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_593_singletonE,axiom,
% 0.29/0.47 ! [B_34: int,A_69: int] :
% 0.29/0.47 ( ( member_int @ B_34 @ ( insert_int @ A_69 @ bot_bot_int_o ) )
% 0.29/0.47 => ( B_34 = A_69 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_594_singletonE,axiom,
% 0.29/0.47 ! [B_34: nat,A_69: nat] :
% 0.29/0.47 ( ( member_nat @ B_34 @ ( insert_nat @ A_69 @ bot_bot_nat_o ) )
% 0.29/0.47 => ( B_34 = A_69 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_595_singletonE,axiom,
% 0.29/0.47 ! [B_34: pname,A_69: pname] :
% 0.29/0.47 ( ( member_pname @ B_34 @ ( insert_pname @ A_69 @ bot_bot_pname_o ) )
% 0.29/0.47 => ( B_34 = A_69 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_596_doubleton__eq__iff,axiom,
% 0.29/0.47 ! [A_68: x_a,B_33: x_a,C_22: x_a,D_4: x_a] :
% 0.29/0.47 ( ( ( insert_a @ A_68 @ ( insert_a @ B_33 @ bot_bot_a_o ) )
% 0.29/0.47 = ( insert_a @ C_22 @ ( insert_a @ D_4 @ bot_bot_a_o ) ) )
% 0.29/0.47 <=> ( ( ( A_68 = C_22 )
% 0.29/0.47 & ( B_33 = D_4 ) )
% 0.29/0.47 | ( ( A_68 = D_4 )
% 0.29/0.47 & ( B_33 = C_22 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_597_doubleton__eq__iff,axiom,
% 0.29/0.47 ! [A_68: nat,B_33: nat,C_22: nat,D_4: nat] :
% 0.29/0.47 ( ( ( insert_nat @ A_68 @ ( insert_nat @ B_33 @ bot_bot_nat_o ) )
% 0.29/0.47 = ( insert_nat @ C_22 @ ( insert_nat @ D_4 @ bot_bot_nat_o ) ) )
% 0.29/0.47 <=> ( ( ( A_68 = C_22 )
% 0.29/0.47 & ( B_33 = D_4 ) )
% 0.29/0.47 | ( ( A_68 = D_4 )
% 0.29/0.47 & ( B_33 = C_22 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_598_doubleton__eq__iff,axiom,
% 0.29/0.47 ! [A_68: int,B_33: int,C_22: int,D_4: int] :
% 0.29/0.47 ( ( ( insert_int @ A_68 @ ( insert_int @ B_33 @ bot_bot_int_o ) )
% 0.29/0.47 = ( insert_int @ C_22 @ ( insert_int @ D_4 @ bot_bot_int_o ) ) )
% 0.29/0.47 <=> ( ( ( A_68 = C_22 )
% 0.29/0.47 & ( B_33 = D_4 ) )
% 0.29/0.47 | ( ( A_68 = D_4 )
% 0.29/0.47 & ( B_33 = C_22 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_599_singleton__iff,axiom,
% 0.29/0.47 ! [B_32: x_a,A_67: x_a] :
% 0.29/0.47 ( ( member_a @ B_32 @ ( insert_a @ A_67 @ bot_bot_a_o ) )
% 0.29/0.47 <=> ( B_32 = A_67 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_600_singleton__iff,axiom,
% 0.29/0.47 ! [B_32: int,A_67: int] :
% 0.29/0.47 ( ( member_int @ B_32 @ ( insert_int @ A_67 @ bot_bot_int_o ) )
% 0.29/0.47 <=> ( B_32 = A_67 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_601_singleton__iff,axiom,
% 0.29/0.47 ! [B_32: nat,A_67: nat] :
% 0.29/0.47 ( ( member_nat @ B_32 @ ( insert_nat @ A_67 @ bot_bot_nat_o ) )
% 0.29/0.47 <=> ( B_32 = A_67 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_602_singleton__iff,axiom,
% 0.29/0.47 ! [B_32: pname,A_67: pname] :
% 0.29/0.47 ( ( member_pname @ B_32 @ ( insert_pname @ A_67 @ bot_bot_pname_o ) )
% 0.29/0.47 <=> ( B_32 = A_67 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_603_insert__not__empty,axiom,
% 0.29/0.47 ! [A_66: x_a,A_65: x_a > $o] :
% 0.29/0.47 ( ( insert_a @ A_66 @ A_65 )
% 0.29/0.47 != bot_bot_a_o ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_604_insert__not__empty,axiom,
% 0.29/0.47 ! [A_66: nat,A_65: nat > $o] :
% 0.29/0.47 ( ( insert_nat @ A_66 @ A_65 )
% 0.29/0.47 != bot_bot_nat_o ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_605_insert__not__empty,axiom,
% 0.29/0.47 ! [A_66: int,A_65: int > $o] :
% 0.29/0.47 ( ( insert_int @ A_66 @ A_65 )
% 0.29/0.47 != bot_bot_int_o ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_606_empty__not__insert,axiom,
% 0.29/0.47 ! [A_64: x_a,A_63: x_a > $o] :
% 0.29/0.47 ( bot_bot_a_o
% 0.29/0.47 != ( insert_a @ A_64 @ A_63 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_607_empty__not__insert,axiom,
% 0.29/0.47 ! [A_64: nat,A_63: nat > $o] :
% 0.29/0.47 ( bot_bot_nat_o
% 0.29/0.47 != ( insert_nat @ A_64 @ A_63 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_608_empty__not__insert,axiom,
% 0.29/0.47 ! [A_64: int,A_63: int > $o] :
% 0.29/0.47 ( bot_bot_int_o
% 0.29/0.47 != ( insert_int @ A_64 @ A_63 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_609_subset__empty,axiom,
% 0.29/0.47 ! [A_62: nat > $o] :
% 0.29/0.47 ( ( ord_less_eq_nat_o @ A_62 @ bot_bot_nat_o )
% 0.29/0.47 <=> ( A_62 = bot_bot_nat_o ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_610_subset__empty,axiom,
% 0.29/0.47 ! [A_62: int > $o] :
% 0.29/0.47 ( ( ord_less_eq_int_o @ A_62 @ bot_bot_int_o )
% 0.29/0.47 <=> ( A_62 = bot_bot_int_o ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_611_subset__empty,axiom,
% 0.29/0.47 ! [A_62: x_a > $o] :
% 0.29/0.47 ( ( ord_less_eq_a_o @ A_62 @ bot_bot_a_o )
% 0.29/0.47 <=> ( A_62 = bot_bot_a_o ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_612_image__is__empty,axiom,
% 0.29/0.47 ! [F_22: nat > int,A_61: nat > $o] :
% 0.29/0.47 ( ( ( image_nat_int @ F_22 @ A_61 )
% 0.29/0.47 = bot_bot_int_o )
% 0.29/0.47 <=> ( A_61 = bot_bot_nat_o ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_613_image__is__empty,axiom,
% 0.29/0.47 ! [F_22: pname > x_a,A_61: pname > $o] :
% 0.29/0.47 ( ( ( image_pname_a @ F_22 @ A_61 )
% 0.29/0.47 = bot_bot_a_o )
% 0.29/0.47 <=> ( A_61 = bot_bot_pname_o ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_614_image__empty,axiom,
% 0.29/0.47 ! [F_21: nat > int] :
% 0.29/0.47 ( ( image_nat_int @ F_21 @ bot_bot_nat_o )
% 0.29/0.47 = bot_bot_int_o ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_615_image__empty,axiom,
% 0.29/0.47 ! [F_21: pname > x_a] :
% 0.29/0.47 ( ( image_pname_a @ F_21 @ bot_bot_pname_o )
% 0.29/0.47 = bot_bot_a_o ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_616_empty__is__image,axiom,
% 0.29/0.47 ! [F_20: nat > int,A_60: nat > $o] :
% 0.29/0.47 ( ( bot_bot_int_o
% 0.29/0.47 = ( image_nat_int @ F_20 @ A_60 ) )
% 0.29/0.47 <=> ( A_60 = bot_bot_nat_o ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_617_empty__is__image,axiom,
% 0.29/0.47 ! [F_20: pname > x_a,A_60: pname > $o] :
% 0.29/0.47 ( ( bot_bot_a_o
% 0.29/0.47 = ( image_pname_a @ F_20 @ A_60 ) )
% 0.29/0.47 <=> ( A_60 = bot_bot_pname_o ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_618_le__bot,axiom,
% 0.29/0.47 ! [A_59: nat > $o] :
% 0.29/0.47 ( ( ord_less_eq_nat_o @ A_59 @ bot_bot_nat_o )
% 0.29/0.47 => ( A_59 = bot_bot_nat_o ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_619_le__bot,axiom,
% 0.29/0.47 ! [A_59: int > $o] :
% 0.29/0.47 ( ( ord_less_eq_int_o @ A_59 @ bot_bot_int_o )
% 0.29/0.47 => ( A_59 = bot_bot_int_o ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_620_le__bot,axiom,
% 0.29/0.47 ! [A_59: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ A_59 @ bot_bot_nat )
% 0.29/0.47 => ( A_59 = bot_bot_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_621_le__bot,axiom,
% 0.29/0.47 ! [A_59: x_a > $o] :
% 0.29/0.47 ( ( ord_less_eq_a_o @ A_59 @ bot_bot_a_o )
% 0.29/0.47 => ( A_59 = bot_bot_a_o ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_622_bot__unique,axiom,
% 0.29/0.47 ! [A_58: nat > $o] :
% 0.29/0.47 ( ( ord_less_eq_nat_o @ A_58 @ bot_bot_nat_o )
% 0.29/0.47 <=> ( A_58 = bot_bot_nat_o ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_623_bot__unique,axiom,
% 0.29/0.47 ! [A_58: int > $o] :
% 0.29/0.47 ( ( ord_less_eq_int_o @ A_58 @ bot_bot_int_o )
% 0.29/0.47 <=> ( A_58 = bot_bot_int_o ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_624_bot__unique,axiom,
% 0.29/0.47 ! [A_58: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ A_58 @ bot_bot_nat )
% 0.29/0.47 <=> ( A_58 = bot_bot_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_625_bot__unique,axiom,
% 0.29/0.47 ! [A_58: x_a > $o] :
% 0.29/0.47 ( ( ord_less_eq_a_o @ A_58 @ bot_bot_a_o )
% 0.29/0.47 <=> ( A_58 = bot_bot_a_o ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_626_bot__least,axiom,
% 0.29/0.47 ! [A_57: nat > $o] : ( ord_less_eq_nat_o @ bot_bot_nat_o @ A_57 ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_627_bot__least,axiom,
% 0.29/0.47 ! [A_57: int > $o] : ( ord_less_eq_int_o @ bot_bot_int_o @ A_57 ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_628_bot__least,axiom,
% 0.29/0.47 ! [A_57: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A_57 ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_629_bot__least,axiom,
% 0.29/0.47 ! [A_57: x_a > $o] : ( ord_less_eq_a_o @ bot_bot_a_o @ A_57 ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_630_Collect__conv__if,axiom,
% 0.29/0.47 ! [P_7: x_a > $o,A_56: x_a] :
% 0.29/0.47 ( ( ( P_7 @ A_56 )
% 0.29/0.47 => ( ( collect_a
% 0.29/0.47 @ ^ [X_1: x_a] : ( (&) @ ( X_1 = A_56 ) @ ( P_7 @ X_1 ) ) )
% 0.29/0.47 = ( insert_a @ A_56 @ bot_bot_a_o ) ) )
% 0.29/0.47 & ( ~ ( P_7 @ A_56 )
% 0.29/0.47 => ( ( collect_a
% 0.29/0.47 @ ^ [X_1: x_a] : ( (&) @ ( X_1 = A_56 ) @ ( P_7 @ X_1 ) ) )
% 0.29/0.47 = bot_bot_a_o ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_631_Collect__conv__if,axiom,
% 0.29/0.47 ! [P_7: int > $o,A_56: int] :
% 0.29/0.47 ( ( ( P_7 @ A_56 )
% 0.29/0.47 => ( ( collect_int
% 0.29/0.47 @ ^ [X_1: int] : ( (&) @ ( X_1 = A_56 ) @ ( P_7 @ X_1 ) ) )
% 0.29/0.47 = ( insert_int @ A_56 @ bot_bot_int_o ) ) )
% 0.29/0.47 & ( ~ ( P_7 @ A_56 )
% 0.29/0.47 => ( ( collect_int
% 0.29/0.47 @ ^ [X_1: int] : ( (&) @ ( X_1 = A_56 ) @ ( P_7 @ X_1 ) ) )
% 0.29/0.47 = bot_bot_int_o ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_632_Collect__conv__if,axiom,
% 0.29/0.47 ! [P_7: nat > $o,A_56: nat] :
% 0.29/0.47 ( ( ( P_7 @ A_56 )
% 0.29/0.47 => ( ( collect_nat
% 0.29/0.47 @ ^ [X_1: nat] : ( (&) @ ( X_1 = A_56 ) @ ( P_7 @ X_1 ) ) )
% 0.29/0.47 = ( insert_nat @ A_56 @ bot_bot_nat_o ) ) )
% 0.29/0.47 & ( ~ ( P_7 @ A_56 )
% 0.29/0.47 => ( ( collect_nat
% 0.29/0.47 @ ^ [X_1: nat] : ( (&) @ ( X_1 = A_56 ) @ ( P_7 @ X_1 ) ) )
% 0.29/0.47 = bot_bot_nat_o ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_633_Collect__conv__if2,axiom,
% 0.29/0.47 ! [P_6: x_a > $o,A_55: x_a] :
% 0.29/0.47 ( ( ( P_6 @ A_55 )
% 0.29/0.47 => ( ( collect_a
% 0.29/0.47 @ ^ [X_1: x_a] : ( (&) @ ( A_55 = X_1 ) @ ( P_6 @ X_1 ) ) )
% 0.29/0.47 = ( insert_a @ A_55 @ bot_bot_a_o ) ) )
% 0.29/0.47 & ( ~ ( P_6 @ A_55 )
% 0.29/0.47 => ( ( collect_a
% 0.29/0.47 @ ^ [X_1: x_a] : ( (&) @ ( A_55 = X_1 ) @ ( P_6 @ X_1 ) ) )
% 0.29/0.47 = bot_bot_a_o ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_634_Collect__conv__if2,axiom,
% 0.29/0.47 ! [P_6: int > $o,A_55: int] :
% 0.29/0.47 ( ( ( P_6 @ A_55 )
% 0.29/0.47 => ( ( collect_int
% 0.29/0.47 @ ^ [X_1: int] : ( (&) @ ( A_55 = X_1 ) @ ( P_6 @ X_1 ) ) )
% 0.29/0.47 = ( insert_int @ A_55 @ bot_bot_int_o ) ) )
% 0.29/0.47 & ( ~ ( P_6 @ A_55 )
% 0.29/0.47 => ( ( collect_int
% 0.29/0.47 @ ^ [X_1: int] : ( (&) @ ( A_55 = X_1 ) @ ( P_6 @ X_1 ) ) )
% 0.29/0.47 = bot_bot_int_o ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_635_Collect__conv__if2,axiom,
% 0.29/0.47 ! [P_6: nat > $o,A_55: nat] :
% 0.29/0.47 ( ( ( P_6 @ A_55 )
% 0.29/0.47 => ( ( collect_nat
% 0.29/0.47 @ ^ [X_1: nat] : ( (&) @ ( A_55 = X_1 ) @ ( P_6 @ X_1 ) ) )
% 0.29/0.47 = ( insert_nat @ A_55 @ bot_bot_nat_o ) ) )
% 0.29/0.47 & ( ~ ( P_6 @ A_55 )
% 0.29/0.47 => ( ( collect_nat
% 0.29/0.47 @ ^ [X_1: nat] : ( (&) @ ( A_55 = X_1 ) @ ( P_6 @ X_1 ) ) )
% 0.29/0.47 = bot_bot_nat_o ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_636_singleton__conv,axiom,
% 0.29/0.47 ! [A_54: x_a] :
% 0.29/0.47 ( ( collect_a
% 0.29/0.47 @ ^ [X_1: x_a] : ( X_1 = A_54 ) )
% 0.29/0.47 = ( insert_a @ A_54 @ bot_bot_a_o ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_637_singleton__conv,axiom,
% 0.29/0.47 ! [A_54: int] :
% 0.29/0.47 ( ( collect_int
% 0.29/0.47 @ ^ [X_1: int] : ( X_1 = A_54 ) )
% 0.29/0.47 = ( insert_int @ A_54 @ bot_bot_int_o ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_638_singleton__conv,axiom,
% 0.29/0.47 ! [A_54: nat] :
% 0.29/0.47 ( ( collect_nat
% 0.29/0.47 @ ^ [X_1: nat] : ( X_1 = A_54 ) )
% 0.29/0.47 = ( insert_nat @ A_54 @ bot_bot_nat_o ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_639_singleton__conv2,axiom,
% 0.29/0.47 ! [A_53: x_a] :
% 0.29/0.47 ( ( collect_a @ ( fequal_a @ A_53 ) )
% 0.29/0.47 = ( insert_a @ A_53 @ bot_bot_a_o ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_640_singleton__conv2,axiom,
% 0.29/0.47 ! [A_53: int] :
% 0.29/0.47 ( ( collect_int @ ( fequal_int @ A_53 ) )
% 0.29/0.47 = ( insert_int @ A_53 @ bot_bot_int_o ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_641_singleton__conv2,axiom,
% 0.29/0.47 ! [A_53: nat] :
% 0.29/0.47 ( ( collect_nat @ ( fequal_nat @ A_53 ) )
% 0.29/0.47 = ( insert_nat @ A_53 @ bot_bot_nat_o ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_642_card__Suc__Diff1,axiom,
% 0.29/0.47 ! [X_18: x_a,A_52: x_a > $o] :
% 0.29/0.47 ( ( finite_finite_a @ A_52 )
% 0.29/0.47 => ( ( member_a @ X_18 @ A_52 )
% 0.29/0.47 => ( ( suc @ ( finite_card_a @ ( minus_minus_a_o @ A_52 @ ( insert_a @ X_18 @ bot_bot_a_o ) ) ) )
% 0.29/0.47 = ( finite_card_a @ A_52 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_643_card__Suc__Diff1,axiom,
% 0.29/0.47 ! [X_18: int,A_52: int > $o] :
% 0.29/0.47 ( ( finite_finite_int @ A_52 )
% 0.29/0.47 => ( ( member_int @ X_18 @ A_52 )
% 0.29/0.47 => ( ( suc @ ( finite_card_int @ ( minus_minus_int_o @ A_52 @ ( insert_int @ X_18 @ bot_bot_int_o ) ) ) )
% 0.29/0.47 = ( finite_card_int @ A_52 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_644_card__Suc__Diff1,axiom,
% 0.29/0.47 ! [X_18: nat,A_52: nat > $o] :
% 0.29/0.47 ( ( finite_finite_nat @ A_52 )
% 0.29/0.47 => ( ( member_nat @ X_18 @ A_52 )
% 0.29/0.47 => ( ( suc @ ( finite_card_nat @ ( minus_minus_nat_o @ A_52 @ ( insert_nat @ X_18 @ bot_bot_nat_o ) ) ) )
% 0.29/0.47 = ( finite_card_nat @ A_52 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_645_card__Suc__Diff1,axiom,
% 0.29/0.47 ! [X_18: pname,A_52: pname > $o] :
% 0.29/0.47 ( ( finite_finite_pname @ A_52 )
% 0.29/0.47 => ( ( member_pname @ X_18 @ A_52 )
% 0.29/0.47 => ( ( suc @ ( finite_card_pname @ ( minus_minus_pname_o @ A_52 @ ( insert_pname @ X_18 @ bot_bot_pname_o ) ) ) )
% 0.29/0.47 = ( finite_card_pname @ A_52 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_646_card__insert,axiom,
% 0.29/0.47 ! [X_17: x_a,A_51: x_a > $o] :
% 0.29/0.47 ( ( finite_finite_a @ A_51 )
% 0.29/0.47 => ( ( finite_card_a @ ( insert_a @ X_17 @ A_51 ) )
% 0.29/0.47 = ( suc @ ( finite_card_a @ ( minus_minus_a_o @ A_51 @ ( insert_a @ X_17 @ bot_bot_a_o ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_647_card__insert,axiom,
% 0.29/0.47 ! [X_17: int,A_51: int > $o] :
% 0.29/0.47 ( ( finite_finite_int @ A_51 )
% 0.29/0.47 => ( ( finite_card_int @ ( insert_int @ X_17 @ A_51 ) )
% 0.29/0.47 = ( suc @ ( finite_card_int @ ( minus_minus_int_o @ A_51 @ ( insert_int @ X_17 @ bot_bot_int_o ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_648_card__insert,axiom,
% 0.29/0.47 ! [X_17: nat,A_51: nat > $o] :
% 0.29/0.47 ( ( finite_finite_nat @ A_51 )
% 0.29/0.47 => ( ( finite_card_nat @ ( insert_nat @ X_17 @ A_51 ) )
% 0.29/0.47 = ( suc @ ( finite_card_nat @ ( minus_minus_nat_o @ A_51 @ ( insert_nat @ X_17 @ bot_bot_nat_o ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_649_card__insert,axiom,
% 0.29/0.47 ! [X_17: pname,A_51: pname > $o] :
% 0.29/0.47 ( ( finite_finite_pname @ A_51 )
% 0.29/0.47 => ( ( finite_card_pname @ ( insert_pname @ X_17 @ A_51 ) )
% 0.29/0.47 = ( suc @ ( finite_card_pname @ ( minus_minus_pname_o @ A_51 @ ( insert_pname @ X_17 @ bot_bot_pname_o ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_650_card__Diff1__le,axiom,
% 0.29/0.47 ! [X_16: x_a,A_50: x_a > $o] :
% 0.29/0.47 ( ( finite_finite_a @ A_50 )
% 0.29/0.47 => ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_a_o @ A_50 @ ( insert_a @ X_16 @ bot_bot_a_o ) ) ) @ ( finite_card_a @ A_50 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_651_card__Diff1__le,axiom,
% 0.29/0.47 ! [X_16: int,A_50: int > $o] :
% 0.29/0.47 ( ( finite_finite_int @ A_50 )
% 0.29/0.47 => ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_int_o @ A_50 @ ( insert_int @ X_16 @ bot_bot_int_o ) ) ) @ ( finite_card_int @ A_50 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_652_card__Diff1__le,axiom,
% 0.29/0.47 ! [X_16: nat,A_50: nat > $o] :
% 0.29/0.47 ( ( finite_finite_nat @ A_50 )
% 0.29/0.47 => ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_nat_o @ A_50 @ ( insert_nat @ X_16 @ bot_bot_nat_o ) ) ) @ ( finite_card_nat @ A_50 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_653_card__Diff1__le,axiom,
% 0.29/0.47 ! [X_16: pname,A_50: pname > $o] :
% 0.29/0.47 ( ( finite_finite_pname @ A_50 )
% 0.29/0.47 => ( ord_less_eq_nat @ ( finite_card_pname @ ( minus_minus_pname_o @ A_50 @ ( insert_pname @ X_16 @ bot_bot_pname_o ) ) ) @ ( finite_card_pname @ A_50 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_654_finite__Diff__insert,axiom,
% 0.29/0.47 ! [A_49: x_a > $o,A_48: x_a,B_31: x_a > $o] :
% 0.29/0.47 ( ( finite_finite_a @ ( minus_minus_a_o @ A_49 @ ( insert_a @ A_48 @ B_31 ) ) )
% 0.29/0.47 <=> ( finite_finite_a @ ( minus_minus_a_o @ A_49 @ B_31 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_655_finite__Diff__insert,axiom,
% 0.29/0.47 ! [A_49: int > $o,A_48: int,B_31: int > $o] :
% 0.29/0.47 ( ( finite_finite_int @ ( minus_minus_int_o @ A_49 @ ( insert_int @ A_48 @ B_31 ) ) )
% 0.29/0.47 <=> ( finite_finite_int @ ( minus_minus_int_o @ A_49 @ B_31 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_656_finite__Diff__insert,axiom,
% 0.29/0.47 ! [A_49: nat > $o,A_48: nat,B_31: nat > $o] :
% 0.29/0.47 ( ( finite_finite_nat @ ( minus_minus_nat_o @ A_49 @ ( insert_nat @ A_48 @ B_31 ) ) )
% 0.29/0.47 <=> ( finite_finite_nat @ ( minus_minus_nat_o @ A_49 @ B_31 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_657_finite__Diff__insert,axiom,
% 0.29/0.47 ! [A_49: pname > $o,A_48: pname,B_31: pname > $o] :
% 0.29/0.47 ( ( finite_finite_pname @ ( minus_minus_pname_o @ A_49 @ ( insert_pname @ A_48 @ B_31 ) ) )
% 0.29/0.47 <=> ( finite_finite_pname @ ( minus_minus_pname_o @ A_49 @ B_31 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_658_image__diff__subset,axiom,
% 0.29/0.47 ! [F_19: nat > int,A_47: nat > $o,B_30: nat > $o] : ( ord_less_eq_int_o @ ( minus_minus_int_o @ ( image_nat_int @ F_19 @ A_47 ) @ ( image_nat_int @ F_19 @ B_30 ) ) @ ( image_nat_int @ F_19 @ ( minus_minus_nat_o @ A_47 @ B_30 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_659_image__diff__subset,axiom,
% 0.29/0.47 ! [F_19: pname > x_a,A_47: pname > $o,B_30: pname > $o] : ( ord_less_eq_a_o @ ( minus_minus_a_o @ ( image_pname_a @ F_19 @ A_47 ) @ ( image_pname_a @ F_19 @ B_30 ) ) @ ( image_pname_a @ F_19 @ ( minus_minus_pname_o @ A_47 @ B_30 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_660_subset__singletonD,axiom,
% 0.29/0.47 ! [A_46: x_a > $o,X_15: x_a] :
% 0.29/0.47 ( ( ord_less_eq_a_o @ A_46 @ ( insert_a @ X_15 @ bot_bot_a_o ) )
% 0.29/0.47 => ( ( A_46 = bot_bot_a_o )
% 0.29/0.47 | ( A_46
% 0.29/0.47 = ( insert_a @ X_15 @ bot_bot_a_o ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_661_subset__singletonD,axiom,
% 0.29/0.47 ! [A_46: nat > $o,X_15: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat_o @ A_46 @ ( insert_nat @ X_15 @ bot_bot_nat_o ) )
% 0.29/0.47 => ( ( A_46 = bot_bot_nat_o )
% 0.29/0.47 | ( A_46
% 0.29/0.47 = ( insert_nat @ X_15 @ bot_bot_nat_o ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_662_subset__singletonD,axiom,
% 0.29/0.47 ! [A_46: int > $o,X_15: int] :
% 0.29/0.47 ( ( ord_less_eq_int_o @ A_46 @ ( insert_int @ X_15 @ bot_bot_int_o ) )
% 0.29/0.47 => ( ( A_46 = bot_bot_int_o )
% 0.29/0.47 | ( A_46
% 0.29/0.47 = ( insert_int @ X_15 @ bot_bot_int_o ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_663_nat__case__Suc,axiom,
% 0.29/0.47 ! [F1: nat,F2: nat > nat,Nat_3: nat] :
% 0.29/0.47 ( ( nat_case_nat @ F1 @ F2 @ ( suc @ Nat_3 ) )
% 0.29/0.47 = ( F2 @ Nat_3 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_664_nat__case__Suc,axiom,
% 0.29/0.47 ! [F1: $o,F2: nat > $o,Nat_3: nat] :
% 0.29/0.47 ( ( nat_case_o @ F1 @ F2 @ ( suc @ Nat_3 ) )
% 0.29/0.47 <=> ( F2 @ Nat_3 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_665_image__constant,axiom,
% 0.29/0.47 ! [C_21: int,X_14: nat,A_45: nat > $o] :
% 0.29/0.47 ( ( member_nat @ X_14 @ A_45 )
% 0.29/0.47 => ( ( image_nat_int
% 0.29/0.47 @ ^ [X_1: nat] : C_21
% 0.29/0.47 @ A_45 )
% 0.29/0.47 = ( insert_int @ C_21 @ bot_bot_int_o ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_666_image__constant,axiom,
% 0.29/0.47 ! [C_21: x_a,X_14: pname,A_45: pname > $o] :
% 0.29/0.47 ( ( member_pname @ X_14 @ A_45 )
% 0.29/0.47 => ( ( image_pname_a
% 0.29/0.47 @ ^ [X_1: pname] : C_21
% 0.29/0.47 @ A_45 )
% 0.29/0.47 = ( insert_a @ C_21 @ bot_bot_a_o ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_667_image__constant__conv,axiom,
% 0.29/0.47 ! [C_20: int,A_44: nat > $o] :
% 0.29/0.47 ( ( ( A_44 = bot_bot_nat_o )
% 0.29/0.47 => ( ( image_nat_int
% 0.29/0.47 @ ^ [X_1: nat] : C_20
% 0.29/0.47 @ A_44 )
% 0.29/0.47 = bot_bot_int_o ) )
% 0.29/0.47 & ( ( A_44 != bot_bot_nat_o )
% 0.29/0.47 => ( ( image_nat_int
% 0.29/0.47 @ ^ [X_1: nat] : C_20
% 0.29/0.47 @ A_44 )
% 0.29/0.47 = ( insert_int @ C_20 @ bot_bot_int_o ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_668_image__constant__conv,axiom,
% 0.29/0.47 ! [C_20: x_a,A_44: pname > $o] :
% 0.29/0.47 ( ( ( A_44 = bot_bot_pname_o )
% 0.29/0.47 => ( ( image_pname_a
% 0.29/0.47 @ ^ [X_1: pname] : C_20
% 0.29/0.47 @ A_44 )
% 0.29/0.47 = bot_bot_a_o ) )
% 0.29/0.47 & ( ( A_44 != bot_bot_pname_o )
% 0.29/0.47 => ( ( image_pname_a
% 0.29/0.47 @ ^ [X_1: pname] : C_20
% 0.29/0.47 @ A_44 )
% 0.29/0.47 = ( insert_a @ C_20 @ bot_bot_a_o ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_669_diff__eq__diff__eq,axiom,
% 0.29/0.47 ! [A_43: int,B_29: int,C_19: int,D_3: int] :
% 0.29/0.47 ( ( ( minus_minus_int @ A_43 @ B_29 )
% 0.29/0.47 = ( minus_minus_int @ C_19 @ D_3 ) )
% 0.29/0.47 => ( ( A_43 = B_29 )
% 0.29/0.47 <=> ( C_19 = D_3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_670_card__Diff__subset,axiom,
% 0.29/0.47 ! [A_42: int > $o,B_28: int > $o] :
% 0.29/0.47 ( ( finite_finite_int @ B_28 )
% 0.29/0.47 => ( ( ord_less_eq_int_o @ B_28 @ A_42 )
% 0.29/0.47 => ( ( finite_card_int @ ( minus_minus_int_o @ A_42 @ B_28 ) )
% 0.29/0.47 = ( minus_minus_nat @ ( finite_card_int @ A_42 ) @ ( finite_card_int @ B_28 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_671_card__Diff__subset,axiom,
% 0.29/0.47 ! [A_42: nat > $o,B_28: nat > $o] :
% 0.29/0.47 ( ( finite_finite_nat @ B_28 )
% 0.29/0.47 => ( ( ord_less_eq_nat_o @ B_28 @ A_42 )
% 0.29/0.47 => ( ( finite_card_nat @ ( minus_minus_nat_o @ A_42 @ B_28 ) )
% 0.29/0.47 = ( minus_minus_nat @ ( finite_card_nat @ A_42 ) @ ( finite_card_nat @ B_28 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_672_card__Diff__subset,axiom,
% 0.29/0.47 ! [A_42: x_a > $o,B_28: x_a > $o] :
% 0.29/0.47 ( ( finite_finite_a @ B_28 )
% 0.29/0.47 => ( ( ord_less_eq_a_o @ B_28 @ A_42 )
% 0.29/0.47 => ( ( finite_card_a @ ( minus_minus_a_o @ A_42 @ B_28 ) )
% 0.29/0.47 = ( minus_minus_nat @ ( finite_card_a @ A_42 ) @ ( finite_card_a @ B_28 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_673_card__Diff__subset,axiom,
% 0.29/0.47 ! [A_42: pname > $o,B_28: pname > $o] :
% 0.29/0.47 ( ( finite_finite_pname @ B_28 )
% 0.29/0.47 => ( ( ord_less_eq_pname_o @ B_28 @ A_42 )
% 0.29/0.47 => ( ( finite_card_pname @ ( minus_minus_pname_o @ A_42 @ B_28 ) )
% 0.29/0.47 = ( minus_minus_nat @ ( finite_card_pname @ A_42 ) @ ( finite_card_pname @ B_28 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_674_diff__card__le__card__Diff,axiom,
% 0.29/0.47 ! [A_41: int > $o,B_27: int > $o] :
% 0.29/0.47 ( ( finite_finite_int @ B_27 )
% 0.29/0.47 => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_int @ A_41 ) @ ( finite_card_int @ B_27 ) ) @ ( finite_card_int @ ( minus_minus_int_o @ A_41 @ B_27 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_675_diff__card__le__card__Diff,axiom,
% 0.29/0.47 ! [A_41: nat > $o,B_27: nat > $o] :
% 0.29/0.47 ( ( finite_finite_nat @ B_27 )
% 0.29/0.47 => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A_41 ) @ ( finite_card_nat @ B_27 ) ) @ ( finite_card_nat @ ( minus_minus_nat_o @ A_41 @ B_27 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_676_diff__card__le__card__Diff,axiom,
% 0.29/0.47 ! [A_41: x_a > $o,B_27: x_a > $o] :
% 0.29/0.47 ( ( finite_finite_a @ B_27 )
% 0.29/0.47 => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_a @ A_41 ) @ ( finite_card_a @ B_27 ) ) @ ( finite_card_a @ ( minus_minus_a_o @ A_41 @ B_27 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_677_diff__card__le__card__Diff,axiom,
% 0.29/0.47 ! [A_41: pname > $o,B_27: pname > $o] :
% 0.29/0.47 ( ( finite_finite_pname @ B_27 )
% 0.29/0.47 => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_pname @ A_41 ) @ ( finite_card_pname @ B_27 ) ) @ ( finite_card_pname @ ( minus_minus_pname_o @ A_41 @ B_27 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_678_finite__subset__induct,axiom,
% 0.29/0.47 ! [P_5: ( x_a > $o ) > $o,A_40: x_a > $o,F_18: x_a > $o] :
% 0.29/0.47 ( ( finite_finite_a @ F_18 )
% 0.29/0.47 => ( ( ord_less_eq_a_o @ F_18 @ A_40 )
% 0.29/0.47 => ( ( P_5 @ bot_bot_a_o )
% 0.29/0.47 => ( ! [A_37: x_a,F_2: x_a > $o] :
% 0.29/0.47 ( ( finite_finite_a @ F_2 )
% 0.29/0.47 => ( ( member_a @ A_37 @ A_40 )
% 0.29/0.47 => ( ~ ( member_a @ A_37 @ F_2 )
% 0.29/0.47 => ( ( P_5 @ F_2 )
% 0.29/0.47 => ( P_5 @ ( insert_a @ A_37 @ F_2 ) ) ) ) ) )
% 0.29/0.47 => ( P_5 @ F_18 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_679_finite__subset__induct,axiom,
% 0.29/0.47 ! [P_5: ( int > $o ) > $o,A_40: int > $o,F_18: int > $o] :
% 0.29/0.47 ( ( finite_finite_int @ F_18 )
% 0.29/0.47 => ( ( ord_less_eq_int_o @ F_18 @ A_40 )
% 0.29/0.47 => ( ( P_5 @ bot_bot_int_o )
% 0.29/0.47 => ( ! [A_37: int,F_2: int > $o] :
% 0.29/0.47 ( ( finite_finite_int @ F_2 )
% 0.29/0.47 => ( ( member_int @ A_37 @ A_40 )
% 0.29/0.47 => ( ~ ( member_int @ A_37 @ F_2 )
% 0.29/0.47 => ( ( P_5 @ F_2 )
% 0.29/0.47 => ( P_5 @ ( insert_int @ A_37 @ F_2 ) ) ) ) ) )
% 0.29/0.47 => ( P_5 @ F_18 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_680_finite__subset__induct,axiom,
% 0.29/0.47 ! [P_5: ( nat > $o ) > $o,A_40: nat > $o,F_18: nat > $o] :
% 0.29/0.47 ( ( finite_finite_nat @ F_18 )
% 0.29/0.47 => ( ( ord_less_eq_nat_o @ F_18 @ A_40 )
% 0.29/0.47 => ( ( P_5 @ bot_bot_nat_o )
% 0.29/0.47 => ( ! [A_37: nat,F_2: nat > $o] :
% 0.29/0.47 ( ( finite_finite_nat @ F_2 )
% 0.29/0.47 => ( ( member_nat @ A_37 @ A_40 )
% 0.29/0.47 => ( ~ ( member_nat @ A_37 @ F_2 )
% 0.29/0.47 => ( ( P_5 @ F_2 )
% 0.29/0.47 => ( P_5 @ ( insert_nat @ A_37 @ F_2 ) ) ) ) ) )
% 0.29/0.47 => ( P_5 @ F_18 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_681_finite__subset__induct,axiom,
% 0.29/0.47 ! [P_5: ( pname > $o ) > $o,A_40: pname > $o,F_18: pname > $o] :
% 0.29/0.47 ( ( finite_finite_pname @ F_18 )
% 0.29/0.47 => ( ( ord_less_eq_pname_o @ F_18 @ A_40 )
% 0.29/0.47 => ( ( P_5 @ bot_bot_pname_o )
% 0.29/0.47 => ( ! [A_37: pname,F_2: pname > $o] :
% 0.29/0.47 ( ( finite_finite_pname @ F_2 )
% 0.29/0.47 => ( ( member_pname @ A_37 @ A_40 )
% 0.29/0.47 => ( ~ ( member_pname @ A_37 @ F_2 )
% 0.29/0.47 => ( ( P_5 @ F_2 )
% 0.29/0.47 => ( P_5 @ ( insert_pname @ A_37 @ F_2 ) ) ) ) ) )
% 0.29/0.47 => ( P_5 @ F_18 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_682_assms_I2_J,axiom,
% 0.29/0.47 ! [Pn: pname,G: x_a > $o] :
% 0.29/0.47 ( ( p @ ( insert_a @ ( mgt_call @ Pn ) @ G ) @ ( insert_a @ ( mgt @ ( the_com @ ( body @ Pn ) ) ) @ bot_bot_a_o ) )
% 0.29/0.47 => ( p @ G @ ( insert_a @ ( mgt_call @ Pn ) @ bot_bot_a_o ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_683_finite__empty__induct,axiom,
% 0.29/0.47 ! [P_4: ( x_a > $o ) > $o,A_39: x_a > $o] :
% 0.29/0.47 ( ( finite_finite_a @ A_39 )
% 0.29/0.47 => ( ( P_4 @ A_39 )
% 0.29/0.47 => ( ! [A_37: x_a,A_38: x_a > $o] :
% 0.29/0.47 ( ( finite_finite_a @ A_38 )
% 0.29/0.47 => ( ( member_a @ A_37 @ A_38 )
% 0.29/0.47 => ( ( P_4 @ A_38 )
% 0.29/0.47 => ( P_4 @ ( minus_minus_a_o @ A_38 @ ( insert_a @ A_37 @ bot_bot_a_o ) ) ) ) ) )
% 0.29/0.47 => ( P_4 @ bot_bot_a_o ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_684_finite__empty__induct,axiom,
% 0.29/0.47 ! [P_4: ( int > $o ) > $o,A_39: int > $o] :
% 0.29/0.47 ( ( finite_finite_int @ A_39 )
% 0.29/0.47 => ( ( P_4 @ A_39 )
% 0.29/0.47 => ( ! [A_37: int,A_38: int > $o] :
% 0.29/0.47 ( ( finite_finite_int @ A_38 )
% 0.29/0.47 => ( ( member_int @ A_37 @ A_38 )
% 0.29/0.47 => ( ( P_4 @ A_38 )
% 0.29/0.47 => ( P_4 @ ( minus_minus_int_o @ A_38 @ ( insert_int @ A_37 @ bot_bot_int_o ) ) ) ) ) )
% 0.29/0.47 => ( P_4 @ bot_bot_int_o ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_685_finite__empty__induct,axiom,
% 0.29/0.47 ! [P_4: ( nat > $o ) > $o,A_39: nat > $o] :
% 0.29/0.47 ( ( finite_finite_nat @ A_39 )
% 0.29/0.47 => ( ( P_4 @ A_39 )
% 0.29/0.47 => ( ! [A_37: nat,A_38: nat > $o] :
% 0.29/0.47 ( ( finite_finite_nat @ A_38 )
% 0.29/0.47 => ( ( member_nat @ A_37 @ A_38 )
% 0.29/0.47 => ( ( P_4 @ A_38 )
% 0.29/0.47 => ( P_4 @ ( minus_minus_nat_o @ A_38 @ ( insert_nat @ A_37 @ bot_bot_nat_o ) ) ) ) ) )
% 0.29/0.47 => ( P_4 @ bot_bot_nat_o ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_686_finite__empty__induct,axiom,
% 0.29/0.47 ! [P_4: ( pname > $o ) > $o,A_39: pname > $o] :
% 0.29/0.47 ( ( finite_finite_pname @ A_39 )
% 0.29/0.47 => ( ( P_4 @ A_39 )
% 0.29/0.47 => ( ! [A_37: pname,A_38: pname > $o] :
% 0.29/0.47 ( ( finite_finite_pname @ A_38 )
% 0.29/0.47 => ( ( member_pname @ A_37 @ A_38 )
% 0.29/0.47 => ( ( P_4 @ A_38 )
% 0.29/0.47 => ( P_4 @ ( minus_minus_pname_o @ A_38 @ ( insert_pname @ A_37 @ bot_bot_pname_o ) ) ) ) ) )
% 0.29/0.47 => ( P_4 @ bot_bot_pname_o ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_687_finite__induct,axiom,
% 0.29/0.47 ! [P_3: ( x_a > $o ) > $o,F_17: x_a > $o] :
% 0.29/0.47 ( ( finite_finite_a @ F_17 )
% 0.29/0.47 => ( ( P_3 @ bot_bot_a_o )
% 0.29/0.47 => ( ! [X_1: x_a,F_2: x_a > $o] :
% 0.29/0.47 ( ( finite_finite_a @ F_2 )
% 0.29/0.47 => ( ~ ( member_a @ X_1 @ F_2 )
% 0.29/0.47 => ( ( P_3 @ F_2 )
% 0.29/0.47 => ( P_3 @ ( insert_a @ X_1 @ F_2 ) ) ) ) )
% 0.29/0.47 => ( P_3 @ F_17 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_688_finite__induct,axiom,
% 0.29/0.47 ! [P_3: ( int > $o ) > $o,F_17: int > $o] :
% 0.29/0.47 ( ( finite_finite_int @ F_17 )
% 0.29/0.47 => ( ( P_3 @ bot_bot_int_o )
% 0.29/0.47 => ( ! [X_1: int,F_2: int > $o] :
% 0.29/0.47 ( ( finite_finite_int @ F_2 )
% 0.29/0.47 => ( ~ ( member_int @ X_1 @ F_2 )
% 0.29/0.47 => ( ( P_3 @ F_2 )
% 0.29/0.47 => ( P_3 @ ( insert_int @ X_1 @ F_2 ) ) ) ) )
% 0.29/0.47 => ( P_3 @ F_17 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_689_finite__induct,axiom,
% 0.29/0.47 ! [P_3: ( nat > $o ) > $o,F_17: nat > $o] :
% 0.29/0.47 ( ( finite_finite_nat @ F_17 )
% 0.29/0.47 => ( ( P_3 @ bot_bot_nat_o )
% 0.29/0.47 => ( ! [X_1: nat,F_2: nat > $o] :
% 0.29/0.47 ( ( finite_finite_nat @ F_2 )
% 0.29/0.47 => ( ~ ( member_nat @ X_1 @ F_2 )
% 0.29/0.47 => ( ( P_3 @ F_2 )
% 0.29/0.47 => ( P_3 @ ( insert_nat @ X_1 @ F_2 ) ) ) ) )
% 0.29/0.47 => ( P_3 @ F_17 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_690_finite__induct,axiom,
% 0.29/0.47 ! [P_3: ( pname > $o ) > $o,F_17: pname > $o] :
% 0.29/0.47 ( ( finite_finite_pname @ F_17 )
% 0.29/0.47 => ( ( P_3 @ bot_bot_pname_o )
% 0.29/0.47 => ( ! [X_1: pname,F_2: pname > $o] :
% 0.29/0.47 ( ( finite_finite_pname @ F_2 )
% 0.29/0.47 => ( ~ ( member_pname @ X_1 @ F_2 )
% 0.29/0.47 => ( ( P_3 @ F_2 )
% 0.29/0.47 => ( P_3 @ ( insert_pname @ X_1 @ F_2 ) ) ) ) )
% 0.29/0.47 => ( P_3 @ F_17 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_691_finite_Osimps,axiom,
% 0.29/0.47 ! [A_36: x_a > $o] :
% 0.29/0.47 ( ( finite_finite_a @ A_36 )
% 0.29/0.47 <=> ( ( A_36 = bot_bot_a_o )
% 0.29/0.47 | ? [A_38: x_a > $o,A_37: x_a] :
% 0.29/0.47 ( ( A_36
% 0.29/0.47 = ( insert_a @ A_37 @ A_38 ) )
% 0.29/0.47 & ( finite_finite_a @ A_38 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_692_finite_Osimps,axiom,
% 0.29/0.47 ! [A_36: int > $o] :
% 0.29/0.47 ( ( finite_finite_int @ A_36 )
% 0.29/0.47 <=> ( ( A_36 = bot_bot_int_o )
% 0.29/0.47 | ? [A_38: int > $o,A_37: int] :
% 0.29/0.47 ( ( A_36
% 0.29/0.47 = ( insert_int @ A_37 @ A_38 ) )
% 0.29/0.47 & ( finite_finite_int @ A_38 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_693_finite_Osimps,axiom,
% 0.29/0.47 ! [A_36: nat > $o] :
% 0.29/0.47 ( ( finite_finite_nat @ A_36 )
% 0.29/0.47 <=> ( ( A_36 = bot_bot_nat_o )
% 0.29/0.47 | ? [A_38: nat > $o,A_37: nat] :
% 0.29/0.47 ( ( A_36
% 0.29/0.47 = ( insert_nat @ A_37 @ A_38 ) )
% 0.29/0.47 & ( finite_finite_nat @ A_38 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_694_finite_Osimps,axiom,
% 0.29/0.47 ! [A_36: pname > $o] :
% 0.29/0.47 ( ( finite_finite_pname @ A_36 )
% 0.29/0.47 <=> ( ( A_36 = bot_bot_pname_o )
% 0.29/0.47 | ? [A_38: pname > $o,A_37: pname] :
% 0.29/0.47 ( ( A_36
% 0.29/0.47 = ( insert_pname @ A_37 @ A_38 ) )
% 0.29/0.47 & ( finite_finite_pname @ A_38 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_695_finite__less__ub,axiom,
% 0.29/0.47 ! [U: nat,F: nat > nat] :
% 0.29/0.47 ( ! [N_1: nat] : ( ord_less_eq_nat @ N_1 @ ( F @ N_1 ) )
% 0.29/0.47 => ( finite_finite_nat
% 0.29/0.47 @ ( collect_nat
% 0.29/0.47 @ ^ [N_1: nat] : ( ord_less_eq_nat @ ( F @ N_1 ) @ U ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_696_the__elem__eq,axiom,
% 0.29/0.47 ! [X_13: x_a] :
% 0.29/0.47 ( ( the_elem_a @ ( insert_a @ X_13 @ bot_bot_a_o ) )
% 0.29/0.47 = X_13 ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_697_the__elem__eq,axiom,
% 0.29/0.47 ! [X_13: nat] :
% 0.29/0.47 ( ( the_elem_nat @ ( insert_nat @ X_13 @ bot_bot_nat_o ) )
% 0.29/0.47 = X_13 ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_698_the__elem__eq,axiom,
% 0.29/0.47 ! [X_13: int] :
% 0.29/0.47 ( ( the_elem_int @ ( insert_int @ X_13 @ bot_bot_int_o ) )
% 0.29/0.47 = X_13 ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_699_nonempty__iff,axiom,
% 0.29/0.47 ! [A_35: x_a > $o] :
% 0.29/0.47 ( ( A_35 != bot_bot_a_o )
% 0.29/0.47 <=> ? [X_1: x_a,B_26: x_a > $o] :
% 0.29/0.47 ( ( A_35
% 0.29/0.47 = ( insert_a @ X_1 @ B_26 ) )
% 0.29/0.47 & ~ ( member_a @ X_1 @ B_26 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_700_nonempty__iff,axiom,
% 0.29/0.47 ! [A_35: int > $o] :
% 0.29/0.47 ( ( A_35 != bot_bot_int_o )
% 0.29/0.47 <=> ? [X_1: int,B_26: int > $o] :
% 0.29/0.47 ( ( A_35
% 0.29/0.47 = ( insert_int @ X_1 @ B_26 ) )
% 0.29/0.47 & ~ ( member_int @ X_1 @ B_26 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_701_nonempty__iff,axiom,
% 0.29/0.47 ! [A_35: nat > $o] :
% 0.29/0.47 ( ( A_35 != bot_bot_nat_o )
% 0.29/0.47 <=> ? [X_1: nat,B_26: nat > $o] :
% 0.29/0.47 ( ( A_35
% 0.29/0.47 = ( insert_nat @ X_1 @ B_26 ) )
% 0.29/0.47 & ~ ( member_nat @ X_1 @ B_26 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_702_nonempty__iff,axiom,
% 0.29/0.47 ! [A_35: pname > $o] :
% 0.29/0.47 ( ( A_35 != bot_bot_pname_o )
% 0.29/0.47 <=> ? [X_1: pname,B_26: pname > $o] :
% 0.29/0.47 ( ( A_35
% 0.29/0.47 = ( insert_pname @ X_1 @ B_26 ) )
% 0.29/0.47 & ~ ( member_pname @ X_1 @ B_26 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_703_assms_I4_J,axiom,
% 0.29/0.47 ! [Pn: pname] :
% 0.29/0.47 ( ( member_pname @ Pn @ u )
% 0.29/0.47 => ( wt @ ( the_com @ ( body @ Pn ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_704_DiffE,axiom,
% 0.29/0.47 ! [C_18: int,A_34: int > $o,B_25: int > $o] :
% 0.29/0.47 ( ( member_int @ C_18 @ ( minus_minus_int_o @ A_34 @ B_25 ) )
% 0.29/0.47 => ~ ( ( member_int @ C_18 @ A_34 )
% 0.29/0.47 => ( member_int @ C_18 @ B_25 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_705_DiffE,axiom,
% 0.29/0.47 ! [C_18: nat,A_34: nat > $o,B_25: nat > $o] :
% 0.29/0.47 ( ( member_nat @ C_18 @ ( minus_minus_nat_o @ A_34 @ B_25 ) )
% 0.29/0.47 => ~ ( ( member_nat @ C_18 @ A_34 )
% 0.29/0.47 => ( member_nat @ C_18 @ B_25 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_706_DiffE,axiom,
% 0.29/0.47 ! [C_18: x_a,A_34: x_a > $o,B_25: x_a > $o] :
% 0.29/0.47 ( ( member_a @ C_18 @ ( minus_minus_a_o @ A_34 @ B_25 ) )
% 0.29/0.47 => ~ ( ( member_a @ C_18 @ A_34 )
% 0.29/0.47 => ( member_a @ C_18 @ B_25 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_707_DiffE,axiom,
% 0.29/0.47 ! [C_18: pname,A_34: pname > $o,B_25: pname > $o] :
% 0.29/0.47 ( ( member_pname @ C_18 @ ( minus_minus_pname_o @ A_34 @ B_25 ) )
% 0.29/0.47 => ~ ( ( member_pname @ C_18 @ A_34 )
% 0.29/0.47 => ( member_pname @ C_18 @ B_25 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_708_DiffI,axiom,
% 0.29/0.47 ! [B_24: int > $o,C_17: int,A_33: int > $o] :
% 0.29/0.47 ( ( member_int @ C_17 @ A_33 )
% 0.29/0.47 => ( ~ ( member_int @ C_17 @ B_24 )
% 0.29/0.47 => ( member_int @ C_17 @ ( minus_minus_int_o @ A_33 @ B_24 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_709_DiffI,axiom,
% 0.29/0.47 ! [B_24: nat > $o,C_17: nat,A_33: nat > $o] :
% 0.29/0.47 ( ( member_nat @ C_17 @ A_33 )
% 0.29/0.47 => ( ~ ( member_nat @ C_17 @ B_24 )
% 0.29/0.47 => ( member_nat @ C_17 @ ( minus_minus_nat_o @ A_33 @ B_24 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_710_DiffI,axiom,
% 0.29/0.47 ! [B_24: x_a > $o,C_17: x_a,A_33: x_a > $o] :
% 0.29/0.47 ( ( member_a @ C_17 @ A_33 )
% 0.29/0.47 => ( ~ ( member_a @ C_17 @ B_24 )
% 0.29/0.47 => ( member_a @ C_17 @ ( minus_minus_a_o @ A_33 @ B_24 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_711_DiffI,axiom,
% 0.29/0.47 ! [B_24: pname > $o,C_17: pname,A_33: pname > $o] :
% 0.29/0.47 ( ( member_pname @ C_17 @ A_33 )
% 0.29/0.47 => ( ~ ( member_pname @ C_17 @ B_24 )
% 0.29/0.47 => ( member_pname @ C_17 @ ( minus_minus_pname_o @ A_33 @ B_24 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_712_DiffD2,axiom,
% 0.29/0.47 ! [C_16: int,A_32: int > $o,B_23: int > $o] :
% 0.29/0.47 ( ( member_int @ C_16 @ ( minus_minus_int_o @ A_32 @ B_23 ) )
% 0.29/0.47 => ~ ( member_int @ C_16 @ B_23 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_713_DiffD2,axiom,
% 0.29/0.47 ! [C_16: nat,A_32: nat > $o,B_23: nat > $o] :
% 0.29/0.47 ( ( member_nat @ C_16 @ ( minus_minus_nat_o @ A_32 @ B_23 ) )
% 0.29/0.47 => ~ ( member_nat @ C_16 @ B_23 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_714_DiffD2,axiom,
% 0.29/0.47 ! [C_16: x_a,A_32: x_a > $o,B_23: x_a > $o] :
% 0.29/0.47 ( ( member_a @ C_16 @ ( minus_minus_a_o @ A_32 @ B_23 ) )
% 0.29/0.47 => ~ ( member_a @ C_16 @ B_23 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_715_DiffD2,axiom,
% 0.29/0.47 ! [C_16: pname,A_32: pname > $o,B_23: pname > $o] :
% 0.29/0.47 ( ( member_pname @ C_16 @ ( minus_minus_pname_o @ A_32 @ B_23 ) )
% 0.29/0.47 => ~ ( member_pname @ C_16 @ B_23 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_716_DiffD1,axiom,
% 0.29/0.47 ! [C_15: int,A_31: int > $o,B_22: int > $o] :
% 0.29/0.47 ( ( member_int @ C_15 @ ( minus_minus_int_o @ A_31 @ B_22 ) )
% 0.29/0.47 => ( member_int @ C_15 @ A_31 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_717_DiffD1,axiom,
% 0.29/0.47 ! [C_15: nat,A_31: nat > $o,B_22: nat > $o] :
% 0.29/0.47 ( ( member_nat @ C_15 @ ( minus_minus_nat_o @ A_31 @ B_22 ) )
% 0.29/0.47 => ( member_nat @ C_15 @ A_31 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_718_DiffD1,axiom,
% 0.29/0.47 ! [C_15: x_a,A_31: x_a > $o,B_22: x_a > $o] :
% 0.29/0.47 ( ( member_a @ C_15 @ ( minus_minus_a_o @ A_31 @ B_22 ) )
% 0.29/0.47 => ( member_a @ C_15 @ A_31 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_719_DiffD1,axiom,
% 0.29/0.47 ! [C_15: pname,A_31: pname > $o,B_22: pname > $o] :
% 0.29/0.47 ( ( member_pname @ C_15 @ ( minus_minus_pname_o @ A_31 @ B_22 ) )
% 0.29/0.47 => ( member_pname @ C_15 @ A_31 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_720_Diff__iff,axiom,
% 0.29/0.47 ! [C_14: int,A_30: int > $o,B_21: int > $o] :
% 0.29/0.47 ( ( member_int @ C_14 @ ( minus_minus_int_o @ A_30 @ B_21 ) )
% 0.29/0.47 <=> ( ( member_int @ C_14 @ A_30 )
% 0.29/0.47 & ~ ( member_int @ C_14 @ B_21 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_721_Diff__iff,axiom,
% 0.29/0.47 ! [C_14: nat,A_30: nat > $o,B_21: nat > $o] :
% 0.29/0.47 ( ( member_nat @ C_14 @ ( minus_minus_nat_o @ A_30 @ B_21 ) )
% 0.29/0.47 <=> ( ( member_nat @ C_14 @ A_30 )
% 0.29/0.47 & ~ ( member_nat @ C_14 @ B_21 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_722_Diff__iff,axiom,
% 0.29/0.47 ! [C_14: x_a,A_30: x_a > $o,B_21: x_a > $o] :
% 0.29/0.47 ( ( member_a @ C_14 @ ( minus_minus_a_o @ A_30 @ B_21 ) )
% 0.29/0.47 <=> ( ( member_a @ C_14 @ A_30 )
% 0.29/0.47 & ~ ( member_a @ C_14 @ B_21 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_723_Diff__iff,axiom,
% 0.29/0.47 ! [C_14: pname,A_30: pname > $o,B_21: pname > $o] :
% 0.29/0.47 ( ( member_pname @ C_14 @ ( minus_minus_pname_o @ A_30 @ B_21 ) )
% 0.29/0.47 <=> ( ( member_pname @ C_14 @ A_30 )
% 0.29/0.47 & ~ ( member_pname @ C_14 @ B_21 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_724_set__diff__eq,axiom,
% 0.29/0.47 ! [A_29: int > $o,B_20: int > $o] :
% 0.29/0.47 ( ( minus_minus_int_o @ A_29 @ B_20 )
% 0.29/0.47 = ( collect_int
% 0.29/0.47 @ ^ [X_1: int] : ( (&) @ ( member_int @ X_1 @ A_29 ) @ ( (~) @ ( member_int @ X_1 @ B_20 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_725_set__diff__eq,axiom,
% 0.29/0.47 ! [A_29: nat > $o,B_20: nat > $o] :
% 0.29/0.47 ( ( minus_minus_nat_o @ A_29 @ B_20 )
% 0.29/0.47 = ( collect_nat
% 0.29/0.47 @ ^ [X_1: nat] : ( (&) @ ( member_nat @ X_1 @ A_29 ) @ ( (~) @ ( member_nat @ X_1 @ B_20 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_726_set__diff__eq,axiom,
% 0.29/0.47 ! [A_29: x_a > $o,B_20: x_a > $o] :
% 0.29/0.47 ( ( minus_minus_a_o @ A_29 @ B_20 )
% 0.29/0.47 = ( collect_a
% 0.29/0.47 @ ^ [X_1: x_a] : ( (&) @ ( member_a @ X_1 @ A_29 ) @ ( (~) @ ( member_a @ X_1 @ B_20 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_727_set__diff__eq,axiom,
% 0.29/0.47 ! [A_29: pname > $o,B_20: pname > $o] :
% 0.29/0.47 ( ( minus_minus_pname_o @ A_29 @ B_20 )
% 0.29/0.47 = ( collect_pname
% 0.29/0.47 @ ^ [X_1: pname] : ( (&) @ ( member_pname @ X_1 @ A_29 ) @ ( (~) @ ( member_pname @ X_1 @ B_20 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_728_folding__one_Oinsert__remove,axiom,
% 0.29/0.47 ! [X_12: x_a,A_28: x_a > $o,F_16: x_a > x_a > x_a,F_15: ( x_a > $o ) > x_a] :
% 0.29/0.47 ( ( finite_folding_one_a @ F_16 @ F_15 )
% 0.29/0.47 => ( ( finite_finite_a @ A_28 )
% 0.29/0.47 => ( ( ( ( minus_minus_a_o @ A_28 @ ( insert_a @ X_12 @ bot_bot_a_o ) )
% 0.29/0.47 = bot_bot_a_o )
% 0.29/0.47 => ( ( F_15 @ ( insert_a @ X_12 @ A_28 ) )
% 0.29/0.47 = X_12 ) )
% 0.29/0.47 & ( ( ( minus_minus_a_o @ A_28 @ ( insert_a @ X_12 @ bot_bot_a_o ) )
% 0.29/0.47 != bot_bot_a_o )
% 0.29/0.47 => ( ( F_15 @ ( insert_a @ X_12 @ A_28 ) )
% 0.29/0.47 = ( F_16 @ X_12 @ ( F_15 @ ( minus_minus_a_o @ A_28 @ ( insert_a @ X_12 @ bot_bot_a_o ) ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_729_folding__one_Oinsert__remove,axiom,
% 0.29/0.47 ! [X_12: int,A_28: int > $o,F_16: int > int > int,F_15: ( int > $o ) > int] :
% 0.29/0.47 ( ( finite1626084323ne_int @ F_16 @ F_15 )
% 0.29/0.47 => ( ( finite_finite_int @ A_28 )
% 0.29/0.47 => ( ( ( ( minus_minus_int_o @ A_28 @ ( insert_int @ X_12 @ bot_bot_int_o ) )
% 0.29/0.47 = bot_bot_int_o )
% 0.29/0.47 => ( ( F_15 @ ( insert_int @ X_12 @ A_28 ) )
% 0.29/0.47 = X_12 ) )
% 0.29/0.47 & ( ( ( minus_minus_int_o @ A_28 @ ( insert_int @ X_12 @ bot_bot_int_o ) )
% 0.29/0.47 != bot_bot_int_o )
% 0.29/0.47 => ( ( F_15 @ ( insert_int @ X_12 @ A_28 ) )
% 0.29/0.47 = ( F_16 @ X_12 @ ( F_15 @ ( minus_minus_int_o @ A_28 @ ( insert_int @ X_12 @ bot_bot_int_o ) ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_730_folding__one_Oinsert__remove,axiom,
% 0.29/0.47 ! [X_12: nat,A_28: nat > $o,F_16: nat > nat > nat,F_15: ( nat > $o ) > nat] :
% 0.29/0.47 ( ( finite988810631ne_nat @ F_16 @ F_15 )
% 0.29/0.47 => ( ( finite_finite_nat @ A_28 )
% 0.29/0.47 => ( ( ( ( minus_minus_nat_o @ A_28 @ ( insert_nat @ X_12 @ bot_bot_nat_o ) )
% 0.29/0.47 = bot_bot_nat_o )
% 0.29/0.47 => ( ( F_15 @ ( insert_nat @ X_12 @ A_28 ) )
% 0.29/0.47 = X_12 ) )
% 0.29/0.47 & ( ( ( minus_minus_nat_o @ A_28 @ ( insert_nat @ X_12 @ bot_bot_nat_o ) )
% 0.29/0.47 != bot_bot_nat_o )
% 0.29/0.47 => ( ( F_15 @ ( insert_nat @ X_12 @ A_28 ) )
% 0.29/0.47 = ( F_16 @ X_12 @ ( F_15 @ ( minus_minus_nat_o @ A_28 @ ( insert_nat @ X_12 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_731_folding__one_Oinsert__remove,axiom,
% 0.29/0.47 ! [X_12: pname,A_28: pname > $o,F_16: pname > pname > pname,F_15: ( pname > $o ) > pname] :
% 0.29/0.47 ( ( finite1282449217_pname @ F_16 @ F_15 )
% 0.29/0.47 => ( ( finite_finite_pname @ A_28 )
% 0.29/0.47 => ( ( ( ( minus_minus_pname_o @ A_28 @ ( insert_pname @ X_12 @ bot_bot_pname_o ) )
% 0.29/0.47 = bot_bot_pname_o )
% 0.29/0.47 => ( ( F_15 @ ( insert_pname @ X_12 @ A_28 ) )
% 0.29/0.47 = X_12 ) )
% 0.29/0.47 & ( ( ( minus_minus_pname_o @ A_28 @ ( insert_pname @ X_12 @ bot_bot_pname_o ) )
% 0.29/0.47 != bot_bot_pname_o )
% 0.29/0.47 => ( ( F_15 @ ( insert_pname @ X_12 @ A_28 ) )
% 0.29/0.47 = ( F_16 @ X_12 @ ( F_15 @ ( minus_minus_pname_o @ A_28 @ ( insert_pname @ X_12 @ bot_bot_pname_o ) ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_732_folding__one_Oremove,axiom,
% 0.29/0.47 ! [X_11: x_a,A_27: x_a > $o,F_14: x_a > x_a > x_a,F_13: ( x_a > $o ) > x_a] :
% 0.29/0.47 ( ( finite_folding_one_a @ F_14 @ F_13 )
% 0.29/0.47 => ( ( finite_finite_a @ A_27 )
% 0.29/0.47 => ( ( member_a @ X_11 @ A_27 )
% 0.29/0.47 => ( ( ( ( minus_minus_a_o @ A_27 @ ( insert_a @ X_11 @ bot_bot_a_o ) )
% 0.29/0.47 = bot_bot_a_o )
% 0.29/0.47 => ( ( F_13 @ A_27 )
% 0.29/0.47 = X_11 ) )
% 0.29/0.47 & ( ( ( minus_minus_a_o @ A_27 @ ( insert_a @ X_11 @ bot_bot_a_o ) )
% 0.29/0.47 != bot_bot_a_o )
% 0.29/0.47 => ( ( F_13 @ A_27 )
% 0.29/0.47 = ( F_14 @ X_11 @ ( F_13 @ ( minus_minus_a_o @ A_27 @ ( insert_a @ X_11 @ bot_bot_a_o ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_733_folding__one_Oremove,axiom,
% 0.29/0.47 ! [X_11: int,A_27: int > $o,F_14: int > int > int,F_13: ( int > $o ) > int] :
% 0.29/0.47 ( ( finite1626084323ne_int @ F_14 @ F_13 )
% 0.29/0.47 => ( ( finite_finite_int @ A_27 )
% 0.29/0.47 => ( ( member_int @ X_11 @ A_27 )
% 0.29/0.47 => ( ( ( ( minus_minus_int_o @ A_27 @ ( insert_int @ X_11 @ bot_bot_int_o ) )
% 0.29/0.47 = bot_bot_int_o )
% 0.29/0.47 => ( ( F_13 @ A_27 )
% 0.29/0.47 = X_11 ) )
% 0.29/0.47 & ( ( ( minus_minus_int_o @ A_27 @ ( insert_int @ X_11 @ bot_bot_int_o ) )
% 0.29/0.47 != bot_bot_int_o )
% 0.29/0.47 => ( ( F_13 @ A_27 )
% 0.29/0.47 = ( F_14 @ X_11 @ ( F_13 @ ( minus_minus_int_o @ A_27 @ ( insert_int @ X_11 @ bot_bot_int_o ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_734_folding__one_Oremove,axiom,
% 0.29/0.47 ! [X_11: nat,A_27: nat > $o,F_14: nat > nat > nat,F_13: ( nat > $o ) > nat] :
% 0.29/0.47 ( ( finite988810631ne_nat @ F_14 @ F_13 )
% 0.29/0.47 => ( ( finite_finite_nat @ A_27 )
% 0.29/0.47 => ( ( member_nat @ X_11 @ A_27 )
% 0.29/0.47 => ( ( ( ( minus_minus_nat_o @ A_27 @ ( insert_nat @ X_11 @ bot_bot_nat_o ) )
% 0.29/0.47 = bot_bot_nat_o )
% 0.29/0.47 => ( ( F_13 @ A_27 )
% 0.29/0.47 = X_11 ) )
% 0.29/0.47 & ( ( ( minus_minus_nat_o @ A_27 @ ( insert_nat @ X_11 @ bot_bot_nat_o ) )
% 0.29/0.47 != bot_bot_nat_o )
% 0.29/0.47 => ( ( F_13 @ A_27 )
% 0.29/0.47 = ( F_14 @ X_11 @ ( F_13 @ ( minus_minus_nat_o @ A_27 @ ( insert_nat @ X_11 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_735_folding__one_Oremove,axiom,
% 0.29/0.47 ! [X_11: pname,A_27: pname > $o,F_14: pname > pname > pname,F_13: ( pname > $o ) > pname] :
% 0.29/0.47 ( ( finite1282449217_pname @ F_14 @ F_13 )
% 0.29/0.47 => ( ( finite_finite_pname @ A_27 )
% 0.29/0.47 => ( ( member_pname @ X_11 @ A_27 )
% 0.29/0.47 => ( ( ( ( minus_minus_pname_o @ A_27 @ ( insert_pname @ X_11 @ bot_bot_pname_o ) )
% 0.29/0.47 = bot_bot_pname_o )
% 0.29/0.47 => ( ( F_13 @ A_27 )
% 0.29/0.47 = X_11 ) )
% 0.29/0.47 & ( ( ( minus_minus_pname_o @ A_27 @ ( insert_pname @ X_11 @ bot_bot_pname_o ) )
% 0.29/0.47 != bot_bot_pname_o )
% 0.29/0.47 => ( ( F_13 @ A_27 )
% 0.29/0.47 = ( F_14 @ X_11 @ ( F_13 @ ( minus_minus_pname_o @ A_27 @ ( insert_pname @ X_11 @ bot_bot_pname_o ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_736_card__Diff__singleton__if,axiom,
% 0.29/0.47 ! [X_10: x_a,A_26: x_a > $o] :
% 0.29/0.47 ( ( finite_finite_a @ A_26 )
% 0.29/0.47 => ( ( ( member_a @ X_10 @ A_26 )
% 0.29/0.47 => ( ( finite_card_a @ ( minus_minus_a_o @ A_26 @ ( insert_a @ X_10 @ bot_bot_a_o ) ) )
% 0.29/0.47 = ( minus_minus_nat @ ( finite_card_a @ A_26 ) @ one_one_nat ) ) )
% 0.29/0.47 & ( ~ ( member_a @ X_10 @ A_26 )
% 0.29/0.47 => ( ( finite_card_a @ ( minus_minus_a_o @ A_26 @ ( insert_a @ X_10 @ bot_bot_a_o ) ) )
% 0.29/0.47 = ( finite_card_a @ A_26 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_737_card__Diff__singleton__if,axiom,
% 0.29/0.47 ! [X_10: int,A_26: int > $o] :
% 0.29/0.47 ( ( finite_finite_int @ A_26 )
% 0.29/0.47 => ( ( ( member_int @ X_10 @ A_26 )
% 0.29/0.47 => ( ( finite_card_int @ ( minus_minus_int_o @ A_26 @ ( insert_int @ X_10 @ bot_bot_int_o ) ) )
% 0.29/0.47 = ( minus_minus_nat @ ( finite_card_int @ A_26 ) @ one_one_nat ) ) )
% 0.29/0.47 & ( ~ ( member_int @ X_10 @ A_26 )
% 0.29/0.47 => ( ( finite_card_int @ ( minus_minus_int_o @ A_26 @ ( insert_int @ X_10 @ bot_bot_int_o ) ) )
% 0.29/0.47 = ( finite_card_int @ A_26 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_738_card__Diff__singleton__if,axiom,
% 0.29/0.47 ! [X_10: nat,A_26: nat > $o] :
% 0.29/0.47 ( ( finite_finite_nat @ A_26 )
% 0.29/0.47 => ( ( ( member_nat @ X_10 @ A_26 )
% 0.29/0.47 => ( ( finite_card_nat @ ( minus_minus_nat_o @ A_26 @ ( insert_nat @ X_10 @ bot_bot_nat_o ) ) )
% 0.29/0.47 = ( minus_minus_nat @ ( finite_card_nat @ A_26 ) @ one_one_nat ) ) )
% 0.29/0.47 & ( ~ ( member_nat @ X_10 @ A_26 )
% 0.29/0.47 => ( ( finite_card_nat @ ( minus_minus_nat_o @ A_26 @ ( insert_nat @ X_10 @ bot_bot_nat_o ) ) )
% 0.29/0.47 = ( finite_card_nat @ A_26 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_739_card__Diff__singleton__if,axiom,
% 0.29/0.47 ! [X_10: pname,A_26: pname > $o] :
% 0.29/0.47 ( ( finite_finite_pname @ A_26 )
% 0.29/0.47 => ( ( ( member_pname @ X_10 @ A_26 )
% 0.29/0.47 => ( ( finite_card_pname @ ( minus_minus_pname_o @ A_26 @ ( insert_pname @ X_10 @ bot_bot_pname_o ) ) )
% 0.29/0.47 = ( minus_minus_nat @ ( finite_card_pname @ A_26 ) @ one_one_nat ) ) )
% 0.29/0.47 & ( ~ ( member_pname @ X_10 @ A_26 )
% 0.29/0.47 => ( ( finite_card_pname @ ( minus_minus_pname_o @ A_26 @ ( insert_pname @ X_10 @ bot_bot_pname_o ) ) )
% 0.29/0.47 = ( finite_card_pname @ A_26 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_740_card__Diff__singleton,axiom,
% 0.29/0.47 ! [X_9: x_a,A_25: x_a > $o] :
% 0.29/0.47 ( ( finite_finite_a @ A_25 )
% 0.29/0.47 => ( ( member_a @ X_9 @ A_25 )
% 0.29/0.47 => ( ( finite_card_a @ ( minus_minus_a_o @ A_25 @ ( insert_a @ X_9 @ bot_bot_a_o ) ) )
% 0.29/0.47 = ( minus_minus_nat @ ( finite_card_a @ A_25 ) @ one_one_nat ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_741_card__Diff__singleton,axiom,
% 0.29/0.47 ! [X_9: int,A_25: int > $o] :
% 0.29/0.47 ( ( finite_finite_int @ A_25 )
% 0.29/0.47 => ( ( member_int @ X_9 @ A_25 )
% 0.29/0.47 => ( ( finite_card_int @ ( minus_minus_int_o @ A_25 @ ( insert_int @ X_9 @ bot_bot_int_o ) ) )
% 0.29/0.47 = ( minus_minus_nat @ ( finite_card_int @ A_25 ) @ one_one_nat ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_742_card__Diff__singleton,axiom,
% 0.29/0.47 ! [X_9: nat,A_25: nat > $o] :
% 0.29/0.47 ( ( finite_finite_nat @ A_25 )
% 0.29/0.47 => ( ( member_nat @ X_9 @ A_25 )
% 0.29/0.47 => ( ( finite_card_nat @ ( minus_minus_nat_o @ A_25 @ ( insert_nat @ X_9 @ bot_bot_nat_o ) ) )
% 0.29/0.47 = ( minus_minus_nat @ ( finite_card_nat @ A_25 ) @ one_one_nat ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_743_card__Diff__singleton,axiom,
% 0.29/0.47 ! [X_9: pname,A_25: pname > $o] :
% 0.29/0.47 ( ( finite_finite_pname @ A_25 )
% 0.29/0.47 => ( ( member_pname @ X_9 @ A_25 )
% 0.29/0.47 => ( ( finite_card_pname @ ( minus_minus_pname_o @ A_25 @ ( insert_pname @ X_9 @ bot_bot_pname_o ) ) )
% 0.29/0.47 = ( minus_minus_nat @ ( finite_card_pname @ A_25 ) @ one_one_nat ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_744_one__reorient,axiom,
% 0.29/0.47 ! [X_8: int] :
% 0.29/0.47 ( ( one_one_int = X_8 )
% 0.29/0.47 <=> ( X_8 = one_one_int ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_745_one__reorient,axiom,
% 0.29/0.47 ! [X_8: nat] :
% 0.29/0.47 ( ( one_one_nat = X_8 )
% 0.29/0.47 <=> ( X_8 = one_one_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_746_diff__Suc__1,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
% 0.29/0.47 = N ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_747_diff__Suc__eq__diff__pred,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 0.29/0.47 = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_748_folding__one_Osingleton,axiom,
% 0.29/0.47 ! [X_7: x_a,F_12: x_a > x_a > x_a,F_11: ( x_a > $o ) > x_a] :
% 0.29/0.47 ( ( finite_folding_one_a @ F_12 @ F_11 )
% 0.29/0.47 => ( ( F_11 @ ( insert_a @ X_7 @ bot_bot_a_o ) )
% 0.29/0.47 = X_7 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_749_folding__one_Osingleton,axiom,
% 0.29/0.47 ! [X_7: nat,F_12: nat > nat > nat,F_11: ( nat > $o ) > nat] :
% 0.29/0.47 ( ( finite988810631ne_nat @ F_12 @ F_11 )
% 0.29/0.47 => ( ( F_11 @ ( insert_nat @ X_7 @ bot_bot_nat_o ) )
% 0.29/0.47 = X_7 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_750_folding__one_Osingleton,axiom,
% 0.29/0.47 ! [X_7: int,F_12: int > int > int,F_11: ( int > $o ) > int] :
% 0.29/0.47 ( ( finite1626084323ne_int @ F_12 @ F_11 )
% 0.29/0.47 => ( ( F_11 @ ( insert_int @ X_7 @ bot_bot_int_o ) )
% 0.29/0.47 = X_7 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_751_card__Diff__insert,axiom,
% 0.29/0.47 ! [B_19: x_a > $o,A_24: x_a,A_23: x_a > $o] :
% 0.29/0.47 ( ( finite_finite_a @ A_23 )
% 0.29/0.47 => ( ( member_a @ A_24 @ A_23 )
% 0.29/0.47 => ( ~ ( member_a @ A_24 @ B_19 )
% 0.29/0.47 => ( ( finite_card_a @ ( minus_minus_a_o @ A_23 @ ( insert_a @ A_24 @ B_19 ) ) )
% 0.29/0.47 = ( minus_minus_nat @ ( finite_card_a @ ( minus_minus_a_o @ A_23 @ B_19 ) ) @ one_one_nat ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_752_card__Diff__insert,axiom,
% 0.29/0.47 ! [B_19: int > $o,A_24: int,A_23: int > $o] :
% 0.29/0.47 ( ( finite_finite_int @ A_23 )
% 0.29/0.47 => ( ( member_int @ A_24 @ A_23 )
% 0.29/0.47 => ( ~ ( member_int @ A_24 @ B_19 )
% 0.29/0.47 => ( ( finite_card_int @ ( minus_minus_int_o @ A_23 @ ( insert_int @ A_24 @ B_19 ) ) )
% 0.29/0.47 = ( minus_minus_nat @ ( finite_card_int @ ( minus_minus_int_o @ A_23 @ B_19 ) ) @ one_one_nat ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_753_card__Diff__insert,axiom,
% 0.29/0.47 ! [B_19: nat > $o,A_24: nat,A_23: nat > $o] :
% 0.29/0.47 ( ( finite_finite_nat @ A_23 )
% 0.29/0.47 => ( ( member_nat @ A_24 @ A_23 )
% 0.29/0.47 => ( ~ ( member_nat @ A_24 @ B_19 )
% 0.29/0.47 => ( ( finite_card_nat @ ( minus_minus_nat_o @ A_23 @ ( insert_nat @ A_24 @ B_19 ) ) )
% 0.29/0.47 = ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_nat_o @ A_23 @ B_19 ) ) @ one_one_nat ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_754_card__Diff__insert,axiom,
% 0.29/0.47 ! [B_19: pname > $o,A_24: pname,A_23: pname > $o] :
% 0.29/0.47 ( ( finite_finite_pname @ A_23 )
% 0.29/0.47 => ( ( member_pname @ A_24 @ A_23 )
% 0.29/0.47 => ( ~ ( member_pname @ A_24 @ B_19 )
% 0.29/0.47 => ( ( finite_card_pname @ ( minus_minus_pname_o @ A_23 @ ( insert_pname @ A_24 @ B_19 ) ) )
% 0.29/0.47 = ( minus_minus_nat @ ( finite_card_pname @ ( minus_minus_pname_o @ A_23 @ B_19 ) ) @ one_one_nat ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_755_folding__one_Oinsert,axiom,
% 0.29/0.47 ! [X_6: x_a,A_22: x_a > $o,F_10: x_a > x_a > x_a,F_9: ( x_a > $o ) > x_a] :
% 0.29/0.47 ( ( finite_folding_one_a @ F_10 @ F_9 )
% 0.29/0.47 => ( ( finite_finite_a @ A_22 )
% 0.29/0.47 => ( ~ ( member_a @ X_6 @ A_22 )
% 0.29/0.47 => ( ( A_22 != bot_bot_a_o )
% 0.29/0.47 => ( ( F_9 @ ( insert_a @ X_6 @ A_22 ) )
% 0.29/0.47 = ( F_10 @ X_6 @ ( F_9 @ A_22 ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_756_folding__one_Oinsert,axiom,
% 0.29/0.47 ! [X_6: int,A_22: int > $o,F_10: int > int > int,F_9: ( int > $o ) > int] :
% 0.29/0.47 ( ( finite1626084323ne_int @ F_10 @ F_9 )
% 0.29/0.47 => ( ( finite_finite_int @ A_22 )
% 0.29/0.47 => ( ~ ( member_int @ X_6 @ A_22 )
% 0.29/0.47 => ( ( A_22 != bot_bot_int_o )
% 0.29/0.47 => ( ( F_9 @ ( insert_int @ X_6 @ A_22 ) )
% 0.29/0.47 = ( F_10 @ X_6 @ ( F_9 @ A_22 ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_757_folding__one_Oinsert,axiom,
% 0.29/0.47 ! [X_6: nat,A_22: nat > $o,F_10: nat > nat > nat,F_9: ( nat > $o ) > nat] :
% 0.29/0.47 ( ( finite988810631ne_nat @ F_10 @ F_9 )
% 0.29/0.47 => ( ( finite_finite_nat @ A_22 )
% 0.29/0.47 => ( ~ ( member_nat @ X_6 @ A_22 )
% 0.29/0.47 => ( ( A_22 != bot_bot_nat_o )
% 0.29/0.47 => ( ( F_9 @ ( insert_nat @ X_6 @ A_22 ) )
% 0.29/0.47 = ( F_10 @ X_6 @ ( F_9 @ A_22 ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_758_folding__one_Oinsert,axiom,
% 0.29/0.47 ! [X_6: pname,A_22: pname > $o,F_10: pname > pname > pname,F_9: ( pname > $o ) > pname] :
% 0.29/0.47 ( ( finite1282449217_pname @ F_10 @ F_9 )
% 0.29/0.47 => ( ( finite_finite_pname @ A_22 )
% 0.29/0.47 => ( ~ ( member_pname @ X_6 @ A_22 )
% 0.29/0.47 => ( ( A_22 != bot_bot_pname_o )
% 0.29/0.47 => ( ( F_9 @ ( insert_pname @ X_6 @ A_22 ) )
% 0.29/0.47 = ( F_10 @ X_6 @ ( F_9 @ A_22 ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_759_folding__one_Oclosed,axiom,
% 0.29/0.47 ! [A_21: x_a > $o,F_8: x_a > x_a > x_a,F_7: ( x_a > $o ) > x_a] :
% 0.29/0.47 ( ( finite_folding_one_a @ F_8 @ F_7 )
% 0.29/0.47 => ( ( finite_finite_a @ A_21 )
% 0.29/0.47 => ( ( A_21 != bot_bot_a_o )
% 0.29/0.47 => ( ! [X_1: x_a,Y_1: x_a] : ( member_a @ ( F_8 @ X_1 @ Y_1 ) @ ( insert_a @ X_1 @ ( insert_a @ Y_1 @ bot_bot_a_o ) ) )
% 0.29/0.47 => ( member_a @ ( F_7 @ A_21 ) @ A_21 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_760_folding__one_Oclosed,axiom,
% 0.29/0.47 ! [A_21: int > $o,F_8: int > int > int,F_7: ( int > $o ) > int] :
% 0.29/0.47 ( ( finite1626084323ne_int @ F_8 @ F_7 )
% 0.29/0.47 => ( ( finite_finite_int @ A_21 )
% 0.29/0.47 => ( ( A_21 != bot_bot_int_o )
% 0.29/0.47 => ( ! [X_1: int,Y_1: int] : ( member_int @ ( F_8 @ X_1 @ Y_1 ) @ ( insert_int @ X_1 @ ( insert_int @ Y_1 @ bot_bot_int_o ) ) )
% 0.29/0.47 => ( member_int @ ( F_7 @ A_21 ) @ A_21 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_761_folding__one_Oclosed,axiom,
% 0.29/0.47 ! [A_21: nat > $o,F_8: nat > nat > nat,F_7: ( nat > $o ) > nat] :
% 0.29/0.47 ( ( finite988810631ne_nat @ F_8 @ F_7 )
% 0.29/0.47 => ( ( finite_finite_nat @ A_21 )
% 0.29/0.47 => ( ( A_21 != bot_bot_nat_o )
% 0.29/0.47 => ( ! [X_1: nat,Y_1: nat] : ( member_nat @ ( F_8 @ X_1 @ Y_1 ) @ ( insert_nat @ X_1 @ ( insert_nat @ Y_1 @ bot_bot_nat_o ) ) )
% 0.29/0.47 => ( member_nat @ ( F_7 @ A_21 ) @ A_21 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_762_folding__one_Oclosed,axiom,
% 0.29/0.47 ! [A_21: pname > $o,F_8: pname > pname > pname,F_7: ( pname > $o ) > pname] :
% 0.29/0.47 ( ( finite1282449217_pname @ F_8 @ F_7 )
% 0.29/0.47 => ( ( finite_finite_pname @ A_21 )
% 0.29/0.47 => ( ( A_21 != bot_bot_pname_o )
% 0.29/0.47 => ( ! [X_1: pname,Y_1: pname] : ( member_pname @ ( F_8 @ X_1 @ Y_1 ) @ ( insert_pname @ X_1 @ ( insert_pname @ Y_1 @ bot_bot_pname_o ) ) )
% 0.29/0.47 => ( member_pname @ ( F_7 @ A_21 ) @ A_21 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_763_card_Oremove,axiom,
% 0.29/0.47 ! [X_5: x_a,A_20: x_a > $o] :
% 0.29/0.47 ( ( finite_finite_a @ A_20 )
% 0.29/0.47 => ( ( member_a @ X_5 @ A_20 )
% 0.29/0.47 => ( ( finite_card_a @ A_20 )
% 0.29/0.47 = ( plus_plus_nat @ one_one_nat @ ( finite_card_a @ ( minus_minus_a_o @ A_20 @ ( insert_a @ X_5 @ bot_bot_a_o ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_764_card_Oremove,axiom,
% 0.29/0.47 ! [X_5: int,A_20: int > $o] :
% 0.29/0.47 ( ( finite_finite_int @ A_20 )
% 0.29/0.47 => ( ( member_int @ X_5 @ A_20 )
% 0.29/0.47 => ( ( finite_card_int @ A_20 )
% 0.29/0.47 = ( plus_plus_nat @ one_one_nat @ ( finite_card_int @ ( minus_minus_int_o @ A_20 @ ( insert_int @ X_5 @ bot_bot_int_o ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_765_card_Oremove,axiom,
% 0.29/0.47 ! [X_5: nat,A_20: nat > $o] :
% 0.29/0.47 ( ( finite_finite_nat @ A_20 )
% 0.29/0.47 => ( ( member_nat @ X_5 @ A_20 )
% 0.29/0.47 => ( ( finite_card_nat @ A_20 )
% 0.29/0.47 = ( plus_plus_nat @ one_one_nat @ ( finite_card_nat @ ( minus_minus_nat_o @ A_20 @ ( insert_nat @ X_5 @ bot_bot_nat_o ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_766_card_Oremove,axiom,
% 0.29/0.47 ! [X_5: pname,A_20: pname > $o] :
% 0.29/0.47 ( ( finite_finite_pname @ A_20 )
% 0.29/0.47 => ( ( member_pname @ X_5 @ A_20 )
% 0.29/0.47 => ( ( finite_card_pname @ A_20 )
% 0.29/0.47 = ( plus_plus_nat @ one_one_nat @ ( finite_card_pname @ ( minus_minus_pname_o @ A_20 @ ( insert_pname @ X_5 @ bot_bot_pname_o ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_767_card_Oinsert__remove,axiom,
% 0.29/0.47 ! [X_4: x_a,A_19: x_a > $o] :
% 0.29/0.47 ( ( finite_finite_a @ A_19 )
% 0.29/0.47 => ( ( finite_card_a @ ( insert_a @ X_4 @ A_19 ) )
% 0.29/0.47 = ( plus_plus_nat @ one_one_nat @ ( finite_card_a @ ( minus_minus_a_o @ A_19 @ ( insert_a @ X_4 @ bot_bot_a_o ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_768_card_Oinsert__remove,axiom,
% 0.29/0.47 ! [X_4: int,A_19: int > $o] :
% 0.29/0.47 ( ( finite_finite_int @ A_19 )
% 0.29/0.47 => ( ( finite_card_int @ ( insert_int @ X_4 @ A_19 ) )
% 0.29/0.47 = ( plus_plus_nat @ one_one_nat @ ( finite_card_int @ ( minus_minus_int_o @ A_19 @ ( insert_int @ X_4 @ bot_bot_int_o ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_769_card_Oinsert__remove,axiom,
% 0.29/0.47 ! [X_4: nat,A_19: nat > $o] :
% 0.29/0.47 ( ( finite_finite_nat @ A_19 )
% 0.29/0.47 => ( ( finite_card_nat @ ( insert_nat @ X_4 @ A_19 ) )
% 0.29/0.47 = ( plus_plus_nat @ one_one_nat @ ( finite_card_nat @ ( minus_minus_nat_o @ A_19 @ ( insert_nat @ X_4 @ bot_bot_nat_o ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_770_card_Oinsert__remove,axiom,
% 0.29/0.47 ! [X_4: pname,A_19: pname > $o] :
% 0.29/0.47 ( ( finite_finite_pname @ A_19 )
% 0.29/0.47 => ( ( finite_card_pname @ ( insert_pname @ X_4 @ A_19 ) )
% 0.29/0.47 = ( plus_plus_nat @ one_one_nat @ ( finite_card_pname @ ( minus_minus_pname_o @ A_19 @ ( insert_pname @ X_4 @ bot_bot_pname_o ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_771_folding__one__idem_Osubset__idem,axiom,
% 0.29/0.47 ! [B_18: int > $o,A_18: int > $o,F_6: int > int > int,F_5: ( int > $o ) > int] :
% 0.29/0.47 ( ( finite1432773856em_int @ F_6 @ F_5 )
% 0.29/0.47 => ( ( finite_finite_int @ A_18 )
% 0.29/0.47 => ( ( B_18 != bot_bot_int_o )
% 0.29/0.47 => ( ( ord_less_eq_int_o @ B_18 @ A_18 )
% 0.29/0.47 => ( ( F_6 @ ( F_5 @ B_18 ) @ ( F_5 @ A_18 ) )
% 0.29/0.47 = ( F_5 @ A_18 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_772_folding__one__idem_Osubset__idem,axiom,
% 0.29/0.47 ! [B_18: nat > $o,A_18: nat > $o,F_6: nat > nat > nat,F_5: ( nat > $o ) > nat] :
% 0.29/0.47 ( ( finite795500164em_nat @ F_6 @ F_5 )
% 0.29/0.47 => ( ( finite_finite_nat @ A_18 )
% 0.29/0.47 => ( ( B_18 != bot_bot_nat_o )
% 0.29/0.47 => ( ( ord_less_eq_nat_o @ B_18 @ A_18 )
% 0.29/0.47 => ( ( F_6 @ ( F_5 @ B_18 ) @ ( F_5 @ A_18 ) )
% 0.29/0.47 = ( F_5 @ A_18 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_773_folding__one__idem_Osubset__idem,axiom,
% 0.29/0.47 ! [B_18: pname > $o,A_18: pname > $o,F_6: pname > pname > pname,F_5: ( pname > $o ) > pname] :
% 0.29/0.47 ( ( finite89670078_pname @ F_6 @ F_5 )
% 0.29/0.47 => ( ( finite_finite_pname @ A_18 )
% 0.29/0.47 => ( ( B_18 != bot_bot_pname_o )
% 0.29/0.47 => ( ( ord_less_eq_pname_o @ B_18 @ A_18 )
% 0.29/0.47 => ( ( F_6 @ ( F_5 @ B_18 ) @ ( F_5 @ A_18 ) )
% 0.29/0.47 = ( F_5 @ A_18 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_774_folding__one__idem_Osubset__idem,axiom,
% 0.29/0.47 ! [B_18: x_a > $o,A_18: x_a > $o,F_6: x_a > x_a > x_a,F_5: ( x_a > $o ) > x_a] :
% 0.29/0.47 ( ( finite1819937229idem_a @ F_6 @ F_5 )
% 0.29/0.47 => ( ( finite_finite_a @ A_18 )
% 0.29/0.47 => ( ( B_18 != bot_bot_a_o )
% 0.29/0.47 => ( ( ord_less_eq_a_o @ B_18 @ A_18 )
% 0.29/0.47 => ( ( F_6 @ ( F_5 @ B_18 ) @ ( F_5 @ A_18 ) )
% 0.29/0.47 = ( F_5 @ A_18 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_775_folding__one__idem_Oinsert__idem,axiom,
% 0.29/0.47 ! [X_3: x_a,A_17: x_a > $o,F_4: x_a > x_a > x_a,F_3: ( x_a > $o ) > x_a] :
% 0.29/0.47 ( ( finite1819937229idem_a @ F_4 @ F_3 )
% 0.29/0.47 => ( ( finite_finite_a @ A_17 )
% 0.29/0.47 => ( ( A_17 != bot_bot_a_o )
% 0.29/0.47 => ( ( F_3 @ ( insert_a @ X_3 @ A_17 ) )
% 0.29/0.47 = ( F_4 @ X_3 @ ( F_3 @ A_17 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_776_folding__one__idem_Oinsert__idem,axiom,
% 0.29/0.47 ! [X_3: int,A_17: int > $o,F_4: int > int > int,F_3: ( int > $o ) > int] :
% 0.29/0.47 ( ( finite1432773856em_int @ F_4 @ F_3 )
% 0.29/0.47 => ( ( finite_finite_int @ A_17 )
% 0.29/0.47 => ( ( A_17 != bot_bot_int_o )
% 0.29/0.47 => ( ( F_3 @ ( insert_int @ X_3 @ A_17 ) )
% 0.29/0.47 = ( F_4 @ X_3 @ ( F_3 @ A_17 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_777_folding__one__idem_Oinsert__idem,axiom,
% 0.29/0.47 ! [X_3: nat,A_17: nat > $o,F_4: nat > nat > nat,F_3: ( nat > $o ) > nat] :
% 0.29/0.47 ( ( finite795500164em_nat @ F_4 @ F_3 )
% 0.29/0.47 => ( ( finite_finite_nat @ A_17 )
% 0.29/0.47 => ( ( A_17 != bot_bot_nat_o )
% 0.29/0.47 => ( ( F_3 @ ( insert_nat @ X_3 @ A_17 ) )
% 0.29/0.47 = ( F_4 @ X_3 @ ( F_3 @ A_17 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_778_folding__one__idem_Oinsert__idem,axiom,
% 0.29/0.47 ! [X_3: pname,A_17: pname > $o,F_4: pname > pname > pname,F_3: ( pname > $o ) > pname] :
% 0.29/0.47 ( ( finite89670078_pname @ F_4 @ F_3 )
% 0.29/0.47 => ( ( finite_finite_pname @ A_17 )
% 0.29/0.47 => ( ( A_17 != bot_bot_pname_o )
% 0.29/0.47 => ( ( F_3 @ ( insert_pname @ X_3 @ A_17 ) )
% 0.29/0.47 = ( F_4 @ X_3 @ ( F_3 @ A_17 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_779_finite__ne__induct,axiom,
% 0.29/0.47 ! [P_2: ( x_a > $o ) > $o,F_1: x_a > $o] :
% 0.29/0.47 ( ( finite_finite_a @ F_1 )
% 0.29/0.47 => ( ( F_1 != bot_bot_a_o )
% 0.29/0.47 => ( ! [X_1: x_a] : ( P_2 @ ( insert_a @ X_1 @ bot_bot_a_o ) )
% 0.29/0.47 => ( ! [X_1: x_a,F_2: x_a > $o] :
% 0.29/0.47 ( ( finite_finite_a @ F_2 )
% 0.29/0.47 => ( ( F_2 != bot_bot_a_o )
% 0.29/0.47 => ( ~ ( member_a @ X_1 @ F_2 )
% 0.29/0.47 => ( ( P_2 @ F_2 )
% 0.29/0.47 => ( P_2 @ ( insert_a @ X_1 @ F_2 ) ) ) ) ) )
% 0.29/0.47 => ( P_2 @ F_1 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_780_finite__ne__induct,axiom,
% 0.29/0.47 ! [P_2: ( int > $o ) > $o,F_1: int > $o] :
% 0.29/0.47 ( ( finite_finite_int @ F_1 )
% 0.29/0.47 => ( ( F_1 != bot_bot_int_o )
% 0.29/0.47 => ( ! [X_1: int] : ( P_2 @ ( insert_int @ X_1 @ bot_bot_int_o ) )
% 0.29/0.47 => ( ! [X_1: int,F_2: int > $o] :
% 0.29/0.47 ( ( finite_finite_int @ F_2 )
% 0.29/0.47 => ( ( F_2 != bot_bot_int_o )
% 0.29/0.47 => ( ~ ( member_int @ X_1 @ F_2 )
% 0.29/0.47 => ( ( P_2 @ F_2 )
% 0.29/0.47 => ( P_2 @ ( insert_int @ X_1 @ F_2 ) ) ) ) ) )
% 0.29/0.47 => ( P_2 @ F_1 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_781_finite__ne__induct,axiom,
% 0.29/0.47 ! [P_2: ( nat > $o ) > $o,F_1: nat > $o] :
% 0.29/0.47 ( ( finite_finite_nat @ F_1 )
% 0.29/0.47 => ( ( F_1 != bot_bot_nat_o )
% 0.29/0.47 => ( ! [X_1: nat] : ( P_2 @ ( insert_nat @ X_1 @ bot_bot_nat_o ) )
% 0.29/0.47 => ( ! [X_1: nat,F_2: nat > $o] :
% 0.29/0.47 ( ( finite_finite_nat @ F_2 )
% 0.29/0.47 => ( ( F_2 != bot_bot_nat_o )
% 0.29/0.47 => ( ~ ( member_nat @ X_1 @ F_2 )
% 0.29/0.47 => ( ( P_2 @ F_2 )
% 0.29/0.47 => ( P_2 @ ( insert_nat @ X_1 @ F_2 ) ) ) ) ) )
% 0.29/0.47 => ( P_2 @ F_1 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_782_finite__ne__induct,axiom,
% 0.29/0.47 ! [P_2: ( pname > $o ) > $o,F_1: pname > $o] :
% 0.29/0.47 ( ( finite_finite_pname @ F_1 )
% 0.29/0.47 => ( ( F_1 != bot_bot_pname_o )
% 0.29/0.47 => ( ! [X_1: pname] : ( P_2 @ ( insert_pname @ X_1 @ bot_bot_pname_o ) )
% 0.29/0.47 => ( ! [X_1: pname,F_2: pname > $o] :
% 0.29/0.47 ( ( finite_finite_pname @ F_2 )
% 0.29/0.47 => ( ( F_2 != bot_bot_pname_o )
% 0.29/0.47 => ( ~ ( member_pname @ X_1 @ F_2 )
% 0.29/0.47 => ( ( P_2 @ F_2 )
% 0.29/0.47 => ( P_2 @ ( insert_pname @ X_1 @ F_2 ) ) ) ) ) )
% 0.29/0.47 => ( P_2 @ F_1 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_783_the__elem__def,axiom,
% 0.29/0.47 ! [X_2: x_a > $o] :
% 0.29/0.47 ( ( the_elem_a @ X_2 )
% 0.29/0.47 = ( the_a
% 0.29/0.47 @ ^ [X_1: x_a] :
% 0.29/0.47 ( X_2
% 0.29/0.47 = ( insert_a @ X_1 @ bot_bot_a_o ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_784_the__elem__def,axiom,
% 0.29/0.47 ! [X_2: nat > $o] :
% 0.29/0.47 ( ( the_elem_nat @ X_2 )
% 0.29/0.47 = ( the_nat
% 0.29/0.47 @ ^ [X_1: nat] :
% 0.29/0.47 ( X_2
% 0.29/0.47 = ( insert_nat @ X_1 @ bot_bot_nat_o ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_785_the__elem__def,axiom,
% 0.29/0.47 ! [X_2: int > $o] :
% 0.29/0.47 ( ( the_elem_int @ X_2 )
% 0.29/0.47 = ( the_int
% 0.29/0.47 @ ^ [X_1: int] :
% 0.29/0.47 ( X_2
% 0.29/0.47 = ( insert_int @ X_1 @ bot_bot_int_o ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_786_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.29/0.47 ! [A_16: int,B_17: int,C_13: int] :
% 0.29/0.47 ( ( plus_plus_int @ ( plus_plus_int @ A_16 @ B_17 ) @ C_13 )
% 0.29/0.47 = ( plus_plus_int @ A_16 @ ( plus_plus_int @ B_17 @ C_13 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_787_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.29/0.47 ! [A_16: nat,B_17: nat,C_13: nat] :
% 0.29/0.47 ( ( plus_plus_nat @ ( plus_plus_nat @ A_16 @ B_17 ) @ C_13 )
% 0.29/0.47 = ( plus_plus_nat @ A_16 @ ( plus_plus_nat @ B_17 @ C_13 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_788_add__left__cancel,axiom,
% 0.29/0.47 ! [A_15: int,B_16: int,C_12: int] :
% 0.29/0.47 ( ( ( plus_plus_int @ A_15 @ B_16 )
% 0.29/0.47 = ( plus_plus_int @ A_15 @ C_12 ) )
% 0.29/0.47 <=> ( B_16 = C_12 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_789_add__left__cancel,axiom,
% 0.29/0.47 ! [A_15: nat,B_16: nat,C_12: nat] :
% 0.29/0.47 ( ( ( plus_plus_nat @ A_15 @ B_16 )
% 0.29/0.47 = ( plus_plus_nat @ A_15 @ C_12 ) )
% 0.29/0.47 <=> ( B_16 = C_12 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_790_add__right__cancel,axiom,
% 0.29/0.47 ! [B_15: int,A_14: int,C_11: int] :
% 0.29/0.47 ( ( ( plus_plus_int @ B_15 @ A_14 )
% 0.29/0.47 = ( plus_plus_int @ C_11 @ A_14 ) )
% 0.29/0.47 <=> ( B_15 = C_11 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_791_add__right__cancel,axiom,
% 0.29/0.47 ! [B_15: nat,A_14: nat,C_11: nat] :
% 0.29/0.47 ( ( ( plus_plus_nat @ B_15 @ A_14 )
% 0.29/0.47 = ( plus_plus_nat @ C_11 @ A_14 ) )
% 0.29/0.47 <=> ( B_15 = C_11 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_792_add__left__imp__eq,axiom,
% 0.29/0.47 ! [A_13: int,B_14: int,C_10: int] :
% 0.29/0.47 ( ( ( plus_plus_int @ A_13 @ B_14 )
% 0.29/0.47 = ( plus_plus_int @ A_13 @ C_10 ) )
% 0.29/0.47 => ( B_14 = C_10 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_793_add__left__imp__eq,axiom,
% 0.29/0.47 ! [A_13: nat,B_14: nat,C_10: nat] :
% 0.29/0.47 ( ( ( plus_plus_nat @ A_13 @ B_14 )
% 0.29/0.47 = ( plus_plus_nat @ A_13 @ C_10 ) )
% 0.29/0.47 => ( B_14 = C_10 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_794_add__imp__eq,axiom,
% 0.29/0.47 ! [A_12: int,B_13: int,C_9: int] :
% 0.29/0.47 ( ( ( plus_plus_int @ A_12 @ B_13 )
% 0.29/0.47 = ( plus_plus_int @ A_12 @ C_9 ) )
% 0.29/0.47 => ( B_13 = C_9 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_795_add__imp__eq,axiom,
% 0.29/0.47 ! [A_12: nat,B_13: nat,C_9: nat] :
% 0.29/0.47 ( ( ( plus_plus_nat @ A_12 @ B_13 )
% 0.29/0.47 = ( plus_plus_nat @ A_12 @ C_9 ) )
% 0.29/0.47 => ( B_13 = C_9 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_796_add__right__imp__eq,axiom,
% 0.29/0.47 ! [B_12: int,A_11: int,C_8: int] :
% 0.29/0.47 ( ( ( plus_plus_int @ B_12 @ A_11 )
% 0.29/0.47 = ( plus_plus_int @ C_8 @ A_11 ) )
% 0.29/0.47 => ( B_12 = C_8 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_797_add__right__imp__eq,axiom,
% 0.29/0.47 ! [B_12: nat,A_11: nat,C_8: nat] :
% 0.29/0.47 ( ( ( plus_plus_nat @ B_12 @ A_11 )
% 0.29/0.47 = ( plus_plus_nat @ C_8 @ A_11 ) )
% 0.29/0.47 => ( B_12 = C_8 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_798_nat__add__commute,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( plus_plus_nat @ M @ N )
% 0.29/0.47 = ( plus_plus_nat @ N @ M ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_799_nat__add__left__commute,axiom,
% 0.29/0.47 ! [X: nat,Y: nat,Z: nat] :
% 0.29/0.47 ( ( plus_plus_nat @ X @ ( plus_plus_nat @ Y @ Z ) )
% 0.29/0.47 = ( plus_plus_nat @ Y @ ( plus_plus_nat @ X @ Z ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_800_nat__add__assoc,axiom,
% 0.29/0.47 ! [M: nat,N: nat,K: nat] :
% 0.29/0.47 ( ( plus_plus_nat @ ( plus_plus_nat @ M @ N ) @ K )
% 0.29/0.47 = ( plus_plus_nat @ M @ ( plus_plus_nat @ N @ K ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_801_nat__add__left__cancel,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ( plus_plus_nat @ K @ M )
% 0.29/0.47 = ( plus_plus_nat @ K @ N ) )
% 0.29/0.47 <=> ( M = N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_802_nat__add__right__cancel,axiom,
% 0.29/0.47 ! [M: nat,K: nat,N: nat] :
% 0.29/0.47 ( ( ( plus_plus_nat @ M @ K )
% 0.29/0.47 = ( plus_plus_nat @ N @ K ) )
% 0.29/0.47 <=> ( M = N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_803_add__le__cancel__right,axiom,
% 0.29/0.47 ! [A_10: int,C_7: int,B_11: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ ( plus_plus_int @ A_10 @ C_7 ) @ ( plus_plus_int @ B_11 @ C_7 ) )
% 0.29/0.47 <=> ( ord_less_eq_int @ A_10 @ B_11 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_804_add__le__cancel__right,axiom,
% 0.29/0.47 ! [A_10: nat,C_7: nat,B_11: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A_10 @ C_7 ) @ ( plus_plus_nat @ B_11 @ C_7 ) )
% 0.29/0.47 <=> ( ord_less_eq_nat @ A_10 @ B_11 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_805_add__le__cancel__left,axiom,
% 0.29/0.47 ! [C_6: int,A_9: int,B_10: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ ( plus_plus_int @ C_6 @ A_9 ) @ ( plus_plus_int @ C_6 @ B_10 ) )
% 0.29/0.47 <=> ( ord_less_eq_int @ A_9 @ B_10 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_806_add__le__cancel__left,axiom,
% 0.29/0.47 ! [C_6: nat,A_9: nat,B_10: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C_6 @ A_9 ) @ ( plus_plus_nat @ C_6 @ B_10 ) )
% 0.29/0.47 <=> ( ord_less_eq_nat @ A_9 @ B_10 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_807_add__right__mono,axiom,
% 0.29/0.47 ! [C_5: int,A_8: int,B_9: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ A_8 @ B_9 )
% 0.29/0.47 => ( ord_less_eq_int @ ( plus_plus_int @ A_8 @ C_5 ) @ ( plus_plus_int @ B_9 @ C_5 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_808_add__right__mono,axiom,
% 0.29/0.47 ! [C_5: nat,A_8: nat,B_9: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ A_8 @ B_9 )
% 0.29/0.47 => ( ord_less_eq_nat @ ( plus_plus_nat @ A_8 @ C_5 ) @ ( plus_plus_nat @ B_9 @ C_5 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_809_add__left__mono,axiom,
% 0.29/0.47 ! [C_4: int,A_7: int,B_8: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ A_7 @ B_8 )
% 0.29/0.47 => ( ord_less_eq_int @ ( plus_plus_int @ C_4 @ A_7 ) @ ( plus_plus_int @ C_4 @ B_8 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_810_add__left__mono,axiom,
% 0.29/0.47 ! [C_4: nat,A_7: nat,B_8: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ A_7 @ B_8 )
% 0.29/0.47 => ( ord_less_eq_nat @ ( plus_plus_nat @ C_4 @ A_7 ) @ ( plus_plus_nat @ C_4 @ B_8 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_811_add__mono,axiom,
% 0.29/0.47 ! [C_3: int,D_2: int,A_6: int,B_7: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ A_6 @ B_7 )
% 0.29/0.47 => ( ( ord_less_eq_int @ C_3 @ D_2 )
% 0.29/0.47 => ( ord_less_eq_int @ ( plus_plus_int @ A_6 @ C_3 ) @ ( plus_plus_int @ B_7 @ D_2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_812_add__mono,axiom,
% 0.29/0.47 ! [C_3: nat,D_2: nat,A_6: nat,B_7: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ A_6 @ B_7 )
% 0.29/0.47 => ( ( ord_less_eq_nat @ C_3 @ D_2 )
% 0.29/0.47 => ( ord_less_eq_nat @ ( plus_plus_nat @ A_6 @ C_3 ) @ ( plus_plus_nat @ B_7 @ D_2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_813_add__le__imp__le__right,axiom,
% 0.29/0.47 ! [A_5: int,C_2: int,B_6: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ ( plus_plus_int @ A_5 @ C_2 ) @ ( plus_plus_int @ B_6 @ C_2 ) )
% 0.29/0.47 => ( ord_less_eq_int @ A_5 @ B_6 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_814_add__le__imp__le__right,axiom,
% 0.29/0.47 ! [A_5: nat,C_2: nat,B_6: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A_5 @ C_2 ) @ ( plus_plus_nat @ B_6 @ C_2 ) )
% 0.29/0.47 => ( ord_less_eq_nat @ A_5 @ B_6 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_815_add__le__imp__le__left,axiom,
% 0.29/0.47 ! [C_1: int,A_4: int,B_5: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ ( plus_plus_int @ C_1 @ A_4 ) @ ( plus_plus_int @ C_1 @ B_5 ) )
% 0.29/0.47 => ( ord_less_eq_int @ A_4 @ B_5 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_816_add__le__imp__le__left,axiom,
% 0.29/0.47 ! [C_1: nat,A_4: nat,B_5: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C_1 @ A_4 ) @ ( plus_plus_nat @ C_1 @ B_5 ) )
% 0.29/0.47 => ( ord_less_eq_nat @ A_4 @ B_5 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_817_diff__add__cancel,axiom,
% 0.29/0.47 ! [A_3: int,B_4: int] :
% 0.29/0.47 ( ( plus_plus_int @ ( minus_minus_int @ A_3 @ B_4 ) @ B_4 )
% 0.29/0.47 = A_3 ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_818_add__diff__cancel,axiom,
% 0.29/0.47 ! [A_2: int,B_3: int] :
% 0.29/0.47 ( ( minus_minus_int @ ( plus_plus_int @ A_2 @ B_3 ) @ B_3 )
% 0.29/0.47 = A_2 ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_819_add__Suc__right,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( plus_plus_nat @ M @ ( suc @ N ) )
% 0.29/0.47 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_820_add__Suc,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.29/0.47 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_821_add__Suc__shift,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.29/0.47 = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_822_le__add2,axiom,
% 0.29/0.47 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_823_le__add1,axiom,
% 0.29/0.47 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_824_le__iff__add,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.47 <=> ? [K_1: nat] :
% 0.29/0.47 ( N
% 0.29/0.47 = ( plus_plus_nat @ M @ K_1 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_825_nat__add__left__cancel__le,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.29/0.47 <=> ( ord_less_eq_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_826_trans__le__add1,axiom,
% 0.29/0.47 ! [M: nat,I_1: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ I_1 @ J )
% 0.29/0.47 => ( ord_less_eq_nat @ I_1 @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_827_trans__le__add2,axiom,
% 0.29/0.47 ! [M: nat,I_1: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ I_1 @ J )
% 0.29/0.47 => ( ord_less_eq_nat @ I_1 @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_828_add__le__mono1,axiom,
% 0.29/0.47 ! [K: nat,I_1: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ I_1 @ J )
% 0.29/0.47 => ( ord_less_eq_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_829_add__le__mono,axiom,
% 0.29/0.47 ! [K: nat,L: nat,I_1: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ I_1 @ J )
% 0.29/0.47 => ( ( ord_less_eq_nat @ K @ L )
% 0.29/0.47 => ( ord_less_eq_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_830_add__leD2,axiom,
% 0.29/0.47 ! [M: nat,K: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.29/0.47 => ( ord_less_eq_nat @ K @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_831_add__leD1,axiom,
% 0.29/0.47 ! [M: nat,K: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.29/0.47 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_832_add__leE,axiom,
% 0.29/0.47 ! [M: nat,K: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.29/0.47 => ~ ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.47 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_833_diff__add__inverse2,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 0.29/0.47 = M ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_834_diff__add__inverse,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 0.29/0.47 = M ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_835_diff__diff__left,axiom,
% 0.29/0.47 ! [I_1: nat,J: nat,K: nat] :
% 0.29/0.47 ( ( minus_minus_nat @ ( minus_minus_nat @ I_1 @ J ) @ K )
% 0.29/0.47 = ( minus_minus_nat @ I_1 @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_836_diff__cancel,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.29/0.47 = ( minus_minus_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_837_diff__cancel2,axiom,
% 0.29/0.47 ! [M: nat,K: nat,N: nat] :
% 0.29/0.47 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 0.29/0.47 = ( minus_minus_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_838_diff__diff__right,axiom,
% 0.29/0.47 ! [I_1: nat,K: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ K @ J )
% 0.29/0.47 => ( ( minus_minus_nat @ I_1 @ ( minus_minus_nat @ J @ K ) )
% 0.29/0.47 = ( minus_minus_nat @ ( plus_plus_nat @ I_1 @ K ) @ J ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_839_le__diff__conv,axiom,
% 0.29/0.47 ! [J: nat,K: nat,I_1: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I_1 )
% 0.29/0.47 <=> ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I_1 @ K ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_840_le__add__diff,axiom,
% 0.29/0.47 ! [M: nat,K: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.47 => ( ord_less_eq_nat @ M @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_841_le__add__diff__inverse,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.47 => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 0.29/0.47 = M ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_842_add__diff__assoc,axiom,
% 0.29/0.47 ! [I_1: nat,K: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ K @ J )
% 0.29/0.47 => ( ( plus_plus_nat @ I_1 @ ( minus_minus_nat @ J @ K ) )
% 0.29/0.47 = ( minus_minus_nat @ ( plus_plus_nat @ I_1 @ J ) @ K ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_843_le__diff__conv2,axiom,
% 0.29/0.47 ! [I_1: nat,K: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ K @ J )
% 0.29/0.47 => ( ( ord_less_eq_nat @ I_1 @ ( minus_minus_nat @ J @ K ) )
% 0.29/0.47 <=> ( ord_less_eq_nat @ ( plus_plus_nat @ I_1 @ K ) @ J ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_844_le__add__diff__inverse2,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.47 => ( ( plus_plus_nat @ ( minus_minus_nat @ M @ N ) @ N )
% 0.29/0.47 = M ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_845_le__imp__diff__is__add,axiom,
% 0.29/0.47 ! [K: nat,I_1: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ I_1 @ J )
% 0.29/0.47 => ( ( ( minus_minus_nat @ J @ I_1 )
% 0.29/0.47 = K )
% 0.29/0.47 <=> ( J
% 0.29/0.47 = ( plus_plus_nat @ K @ I_1 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_846_diff__add__assoc,axiom,
% 0.29/0.47 ! [I_1: nat,K: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ K @ J )
% 0.29/0.47 => ( ( minus_minus_nat @ ( plus_plus_nat @ I_1 @ J ) @ K )
% 0.29/0.47 = ( plus_plus_nat @ I_1 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_847_add__diff__assoc2,axiom,
% 0.29/0.47 ! [I_1: nat,K: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ K @ J )
% 0.29/0.47 => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I_1 )
% 0.29/0.47 = ( minus_minus_nat @ ( plus_plus_nat @ J @ I_1 ) @ K ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_848_diff__add__assoc2,axiom,
% 0.29/0.47 ! [I_1: nat,K: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ K @ J )
% 0.29/0.47 => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I_1 ) @ K )
% 0.29/0.47 = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I_1 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_849_Suc__eq__plus1__left,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( suc @ N )
% 0.29/0.47 = ( plus_plus_nat @ one_one_nat @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_850_Suc__eq__plus1,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( suc @ N )
% 0.29/0.47 = ( plus_plus_nat @ N @ one_one_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_851_diff__Suc__diff__eq1,axiom,
% 0.29/0.47 ! [M: nat,K: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ K @ J )
% 0.29/0.47 => ( ( minus_minus_nat @ M @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
% 0.29/0.47 = ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( suc @ J ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_852_diff__Suc__diff__eq2,axiom,
% 0.29/0.47 ! [M: nat,K: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ K @ J )
% 0.29/0.47 => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ M )
% 0.29/0.47 = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ M ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_853_termination__basic__simps_I4_J,axiom,
% 0.29/0.47 ! [Y: nat,X: nat,Z: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ X @ Z )
% 0.29/0.47 => ( ord_less_eq_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_854_termination__basic__simps_I3_J,axiom,
% 0.29/0.47 ! [Z: nat,X: nat,Y: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ X @ Y )
% 0.29/0.47 => ( ord_less_eq_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_855_lessI,axiom,
% 0.29/0.47 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_856_Suc__mono,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ N )
% 0.29/0.47 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_857_finite__Collect__less__nat,axiom,
% 0.29/0.47 ! [K: nat] :
% 0.29/0.47 ( finite_finite_nat
% 0.29/0.47 @ ( collect_nat
% 0.29/0.47 @ ^ [N_1: nat] : ( ord_less_nat @ N_1 @ K ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_858_less__not__refl,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ~ ( ord_less_nat @ N @ N ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_859_nat__neq__iff,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( M != N )
% 0.29/0.47 <=> ( ( ord_less_nat @ M @ N )
% 0.29/0.47 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_860_linorder__neqE__nat,axiom,
% 0.29/0.47 ! [X: nat,Y: nat] :
% 0.29/0.47 ( ( X != Y )
% 0.29/0.47 => ( ~ ( ord_less_nat @ X @ Y )
% 0.29/0.47 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_861_less__irrefl__nat,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ~ ( ord_less_nat @ N @ N ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_862_less__not__refl2,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ( ord_less_nat @ N @ M )
% 0.29/0.47 => ( M != N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_863_less__not__refl3,axiom,
% 0.29/0.47 ! [S: nat,T: nat] :
% 0.29/0.47 ( ( ord_less_nat @ S @ T )
% 0.29/0.47 => ( S != T ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_864_nat__less__cases,axiom,
% 0.29/0.47 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.29/0.47 ( ( ( ord_less_nat @ M @ N )
% 0.29/0.47 => ( P @ N @ M ) )
% 0.29/0.47 => ( ( ( M = N )
% 0.29/0.47 => ( P @ N @ M ) )
% 0.29/0.47 => ( ( ( ord_less_nat @ N @ M )
% 0.29/0.47 => ( P @ N @ M ) )
% 0.29/0.47 => ( P @ N @ M ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_865_not__less__eq,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ~ ( ord_less_nat @ M @ N )
% 0.29/0.47 <=> ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_866_less__Suc__eq,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.29/0.47 <=> ( ( ord_less_nat @ M @ N )
% 0.29/0.47 | ( M = N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_867_Suc__less__eq,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.29/0.47 <=> ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_868_not__less__less__Suc__eq,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ~ ( ord_less_nat @ N @ M )
% 0.29/0.47 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.29/0.47 <=> ( N = M ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_869_less__antisym,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ~ ( ord_less_nat @ N @ M )
% 0.29/0.47 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.29/0.47 => ( M = N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_870_less__SucI,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ N )
% 0.29/0.47 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_871_Suc__lessI,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ N )
% 0.29/0.47 => ( ( ( suc @ M )
% 0.29/0.47 != N )
% 0.29/0.47 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_872_less__trans__Suc,axiom,
% 0.29/0.47 ! [K: nat,I_1: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I_1 @ J )
% 0.29/0.47 => ( ( ord_less_nat @ J @ K )
% 0.29/0.47 => ( ord_less_nat @ ( suc @ I_1 ) @ K ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_873_less__SucE,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.29/0.47 => ( ~ ( ord_less_nat @ M @ N )
% 0.29/0.47 => ( M = N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_874_Suc__lessD,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 0.29/0.47 => ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_875_Suc__less__SucD,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.29/0.47 => ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_876_not__add__less1,axiom,
% 0.29/0.47 ! [I_1: nat,J: nat] :
% 0.29/0.47 ~ ( ord_less_nat @ ( plus_plus_nat @ I_1 @ J ) @ I_1 ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_877_not__add__less2,axiom,
% 0.29/0.47 ! [J: nat,I_1: nat] :
% 0.29/0.47 ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I_1 ) @ I_1 ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_878_nat__add__left__cancel__less,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.29/0.47 <=> ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_879_trans__less__add1,axiom,
% 0.29/0.47 ! [M: nat,I_1: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I_1 @ J )
% 0.29/0.47 => ( ord_less_nat @ I_1 @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_880_trans__less__add2,axiom,
% 0.29/0.47 ! [M: nat,I_1: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I_1 @ J )
% 0.29/0.47 => ( ord_less_nat @ I_1 @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_881_add__less__mono1,axiom,
% 0.29/0.47 ! [K: nat,I_1: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I_1 @ J )
% 0.29/0.47 => ( ord_less_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_882_add__less__mono,axiom,
% 0.29/0.47 ! [K: nat,L: nat,I_1: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I_1 @ J )
% 0.29/0.47 => ( ( ord_less_nat @ K @ L )
% 0.29/0.47 => ( ord_less_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_883_less__add__eq__less,axiom,
% 0.29/0.47 ! [M: nat,N: nat,K: nat,L: nat] :
% 0.29/0.47 ( ( ord_less_nat @ K @ L )
% 0.29/0.47 => ( ( ( plus_plus_nat @ M @ L )
% 0.29/0.47 = ( plus_plus_nat @ K @ N ) )
% 0.29/0.47 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_884_add__lessD1,axiom,
% 0.29/0.47 ! [I_1: nat,J: nat,K: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( plus_plus_nat @ I_1 @ J ) @ K )
% 0.29/0.47 => ( ord_less_nat @ I_1 @ K ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_885_termination__basic__simps_I1_J,axiom,
% 0.29/0.47 ! [Z: nat,X: nat,Y: nat] :
% 0.29/0.47 ( ( ord_less_nat @ X @ Y )
% 0.29/0.47 => ( ord_less_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_886_termination__basic__simps_I2_J,axiom,
% 0.29/0.47 ! [Y: nat,X: nat,Z: nat] :
% 0.29/0.47 ( ( ord_less_nat @ X @ Z )
% 0.29/0.47 => ( ord_less_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_887_nat__less__le,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ N )
% 0.29/0.47 <=> ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.47 & ( M != N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_888_le__eq__less__or__eq,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.47 <=> ( ( ord_less_nat @ M @ N )
% 0.29/0.47 | ( M = N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_889_less__imp__le__nat,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ N )
% 0.29/0.47 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_890_le__neq__implies__less,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.47 => ( ( M != N )
% 0.29/0.47 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_891_less__or__eq__imp__le,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ( ord_less_nat @ M @ N )
% 0.29/0.47 | ( M = N ) )
% 0.29/0.47 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_892_termination__basic__simps_I5_J,axiom,
% 0.29/0.47 ! [X: nat,Y: nat] :
% 0.29/0.47 ( ( ord_less_nat @ X @ Y )
% 0.29/0.47 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_893_less__imp__diff__less,axiom,
% 0.29/0.47 ! [N: nat,J: nat,K: nat] :
% 0.29/0.47 ( ( ord_less_nat @ J @ K )
% 0.29/0.47 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_894_diff__less__mono2,axiom,
% 0.29/0.47 ! [L: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ N )
% 0.29/0.47 => ( ( ord_less_nat @ M @ L )
% 0.29/0.47 => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_895_finite__nat__set__iff__bounded,axiom,
% 0.29/0.47 ! [N_2: nat > $o] :
% 0.29/0.47 ( ( finite_finite_nat @ N_2 )
% 0.29/0.47 <=> ? [M_1: nat] :
% 0.29/0.47 ! [X_1: nat] :
% 0.29/0.47 ( ( member_nat @ X_1 @ N_2 )
% 0.29/0.47 => ( ord_less_nat @ X_1 @ M_1 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_896_card__Collect__less__nat,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( finite_card_nat
% 0.29/0.47 @ ( collect_nat
% 0.29/0.47 @ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) )
% 0.29/0.47 = N ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_897_finite__M__bounded__by__nat,axiom,
% 0.29/0.47 ! [P: nat > $o,I_1: nat] :
% 0.29/0.47 ( finite_finite_nat
% 0.29/0.47 @ ( collect_nat
% 0.29/0.47 @ ^ [K_1: nat] : ( (&) @ ( P @ K_1 ) @ ( ord_less_nat @ K_1 @ I_1 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_898_less__add__Suc1,axiom,
% 0.29/0.47 ! [I_1: nat,M: nat] : ( ord_less_nat @ I_1 @ ( suc @ ( plus_plus_nat @ I_1 @ M ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_899_less__add__Suc2,axiom,
% 0.29/0.47 ! [I_1: nat,M: nat] : ( ord_less_nat @ I_1 @ ( suc @ ( plus_plus_nat @ M @ I_1 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_900_less__iff__Suc__add,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ N )
% 0.29/0.47 <=> ? [K_1: nat] :
% 0.29/0.47 ( N
% 0.29/0.47 = ( suc @ ( plus_plus_nat @ M @ K_1 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_901_less__eq__Suc__le,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ( ord_less_nat @ N @ M )
% 0.29/0.47 <=> ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_902_less__Suc__eq__le,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.29/0.47 <=> ( ord_less_eq_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_903_Suc__le__eq,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.29/0.47 <=> ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_904_le__imp__less__Suc,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.47 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_905_Suc__leI,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ N )
% 0.29/0.47 => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_906_le__less__Suc__eq,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.47 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.29/0.47 <=> ( N = M ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_907_Suc__le__lessD,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.29/0.47 => ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_908_diff__less__Suc,axiom,
% 0.29/0.47 ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_909_less__diff__conv,axiom,
% 0.29/0.47 ! [I_1: nat,J: nat,K: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I_1 @ ( minus_minus_nat @ J @ K ) )
% 0.29/0.47 <=> ( ord_less_nat @ ( plus_plus_nat @ I_1 @ K ) @ J ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_910_add__diff__inverse,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ~ ( ord_less_nat @ M @ N )
% 0.29/0.47 => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 0.29/0.47 = M ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_911_diff__less__mono,axiom,
% 0.29/0.47 ! [C: nat,A_1: nat,B_2: nat] :
% 0.29/0.47 ( ( ord_less_nat @ A_1 @ B_2 )
% 0.29/0.47 => ( ( ord_less_eq_nat @ C @ A_1 )
% 0.29/0.47 => ( ord_less_nat @ ( minus_minus_nat @ A_1 @ C ) @ ( minus_minus_nat @ B_2 @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_912_less__diff__iff,axiom,
% 0.29/0.47 ! [N: nat,K: nat,M: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ K @ M )
% 0.29/0.47 => ( ( ord_less_eq_nat @ K @ N )
% 0.29/0.47 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.29/0.47 <=> ( ord_less_nat @ M @ N ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_913_less__eq__Suc__le__raw,axiom,
% 0.29/0.47 ! [X_1: nat] :
% 0.29/0.47 ( ( ord_less_nat @ X_1 )
% 0.29/0.47 = ( ord_less_eq_nat @ ( suc @ X_1 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_914_mono__nat__linear__lb,axiom,
% 0.29/0.47 ! [M: nat,K: nat,F: nat > nat] :
% 0.29/0.47 ( ! [M_1: nat,N_1: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M_1 @ N_1 )
% 0.29/0.47 => ( ord_less_nat @ ( F @ M_1 ) @ ( F @ N_1 ) ) )
% 0.29/0.47 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_915_inc__induct,axiom,
% 0.29/0.47 ! [P: nat > $o,I_1: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ I_1 @ J )
% 0.29/0.47 => ( ( P @ J )
% 0.29/0.47 => ( ! [I: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I @ J )
% 0.29/0.47 => ( ( P @ ( suc @ I ) )
% 0.29/0.47 => ( P @ I ) ) )
% 0.29/0.47 => ( P @ I_1 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_916_less__imp__Suc__add,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ N )
% 0.29/0.47 => ? [K_1: nat] :
% 0.29/0.47 ( N
% 0.29/0.47 = ( suc @ ( plus_plus_nat @ M @ K_1 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_917_bounded__nat__set__is__finite,axiom,
% 0.29/0.47 ! [N: nat,N_2: nat > $o] :
% 0.29/0.47 ( ! [X_1: nat] :
% 0.29/0.47 ( ( member_nat @ X_1 @ N_2 )
% 0.29/0.47 => ( ord_less_nat @ X_1 @ N ) )
% 0.29/0.47 => ( finite_finite_nat @ N_2 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_918_less__mono__imp__le__mono,axiom,
% 0.29/0.47 ! [I_1: nat,J: nat,F: nat > nat] :
% 0.29/0.47 ( ! [I: nat,J_1: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I @ J_1 )
% 0.29/0.47 => ( ord_less_nat @ ( F @ I ) @ ( F @ J_1 ) ) )
% 0.29/0.47 => ( ( ord_less_eq_nat @ I_1 @ J )
% 0.29/0.47 => ( ord_less_eq_nat @ ( F @ I_1 ) @ ( F @ J ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_919_lessE,axiom,
% 0.29/0.47 ! [I_1: nat,K: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I_1 @ K )
% 0.29/0.47 => ( ( K
% 0.29/0.47 != ( suc @ I_1 ) )
% 0.29/0.47 => ~ ! [J_1: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I_1 @ J_1 )
% 0.29/0.47 => ( K
% 0.29/0.47 != ( suc @ J_1 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_920_Suc__lessE,axiom,
% 0.29/0.47 ! [I_1: nat,K: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( suc @ I_1 ) @ K )
% 0.29/0.47 => ~ ! [J_1: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I_1 @ J_1 )
% 0.29/0.47 => ( K
% 0.29/0.47 != ( suc @ J_1 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_921_less__zeroE,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_922_le0,axiom,
% 0.29/0.47 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_923_zero__less__Suc,axiom,
% 0.29/0.47 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_924_le__0__eq,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.29/0.47 <=> ( N = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_925_less__eq__nat_Osimps_I1_J,axiom,
% 0.29/0.47 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_926_diffs0__imp__equal,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ( minus_minus_nat @ M @ N )
% 0.29/0.47 = zero_zero_nat )
% 0.29/0.47 => ( ( ( minus_minus_nat @ N @ M )
% 0.29/0.47 = zero_zero_nat )
% 0.29/0.47 => ( M = N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_927_diff__self__eq__0,axiom,
% 0.29/0.47 ! [M: nat] :
% 0.29/0.47 ( ( minus_minus_nat @ M @ M )
% 0.29/0.47 = zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_928_minus__nat_Odiff__0,axiom,
% 0.29/0.47 ! [M: nat] :
% 0.29/0.47 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 0.29/0.47 = M ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_929_diff__0__eq__0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 0.29/0.47 = zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_930_bot__nat__def,axiom,
% 0.29/0.47 bot_bot_nat = zero_zero_nat ).
% 0.29/0.47
% 0.29/0.47 thf(fact_931_add__eq__self__zero,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ( plus_plus_nat @ M @ N )
% 0.29/0.47 = M )
% 0.29/0.47 => ( N = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_932_add__is__0,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ( plus_plus_nat @ M @ N )
% 0.29/0.47 = zero_zero_nat )
% 0.29/0.47 <=> ( ( M = zero_zero_nat )
% 0.29/0.47 & ( N = zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_933_Nat_Oadd__0__right,axiom,
% 0.29/0.47 ! [M: nat] :
% 0.29/0.47 ( ( plus_plus_nat @ M @ zero_zero_nat )
% 0.29/0.47 = M ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_934_plus__nat_Oadd__0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( plus_plus_nat @ zero_zero_nat @ N )
% 0.29/0.47 = N ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_935_gr0I,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( N != zero_zero_nat )
% 0.29/0.47 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_936_gr__implies__not0,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ N )
% 0.29/0.47 => ( N != zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_937_less__nat__zero__code,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_938_neq0__conv,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( N != zero_zero_nat )
% 0.29/0.47 <=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_939_not__less0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_940_Suc__neq__Zero,axiom,
% 0.29/0.47 ! [M: nat] :
% 0.29/0.47 ( ( suc @ M )
% 0.29/0.47 != zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_941_Zero__neq__Suc,axiom,
% 0.29/0.47 ! [M: nat] :
% 0.29/0.47 ( zero_zero_nat
% 0.29/0.47 != ( suc @ M ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_942_nat_Osimps_I3_J,axiom,
% 0.29/0.47 ! [Nat_2: nat] :
% 0.29/0.47 ( ( suc @ Nat_2 )
% 0.29/0.47 != zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_943_Suc__not__Zero,axiom,
% 0.29/0.47 ! [M: nat] :
% 0.29/0.47 ( ( suc @ M )
% 0.29/0.47 != zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_944_nat_Osimps_I2_J,axiom,
% 0.29/0.47 ! [Nat_1: nat] :
% 0.29/0.47 ( zero_zero_nat
% 0.29/0.47 != ( suc @ Nat_1 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_945_Zero__not__Suc,axiom,
% 0.29/0.47 ! [M: nat] :
% 0.29/0.47 ( zero_zero_nat
% 0.29/0.47 != ( suc @ M ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_946_gr0__conv__Suc,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.47 <=> ? [M_1: nat] :
% 0.29/0.47 ( N
% 0.29/0.47 = ( suc @ M_1 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_947_less__Suc0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.29/0.47 <=> ( N = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_948_less__Suc__eq__0__disj,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.29/0.47 <=> ( ( M = zero_zero_nat )
% 0.29/0.47 | ? [J_1: nat] :
% 0.29/0.47 ( ( M
% 0.29/0.47 = ( suc @ J_1 ) )
% 0.29/0.47 & ( ord_less_nat @ J_1 @ N ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_949_one__is__add,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ( suc @ zero_zero_nat )
% 0.29/0.47 = ( plus_plus_nat @ M @ N ) )
% 0.29/0.47 <=> ( ( ( M
% 0.29/0.47 = ( suc @ zero_zero_nat ) )
% 0.29/0.47 & ( N = zero_zero_nat ) )
% 0.29/0.47 | ( ( M = zero_zero_nat )
% 0.29/0.47 & ( N
% 0.29/0.47 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_950_add__is__1,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ( plus_plus_nat @ M @ N )
% 0.29/0.47 = ( suc @ zero_zero_nat ) )
% 0.29/0.47 <=> ( ( ( M
% 0.29/0.47 = ( suc @ zero_zero_nat ) )
% 0.29/0.47 & ( N = zero_zero_nat ) )
% 0.29/0.47 | ( ( M = zero_zero_nat )
% 0.29/0.47 & ( N
% 0.29/0.47 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_951_add__gr__0,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 0.29/0.47 <=> ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.47 | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_952_zero__less__diff,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 0.29/0.47 <=> ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_953_diff__less,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.47 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.47 => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_954_diff__add__0,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 0.29/0.47 = zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_955_diff__is__0__eq,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ( minus_minus_nat @ M @ N )
% 0.29/0.47 = zero_zero_nat )
% 0.29/0.47 <=> ( ord_less_eq_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_956_diff__is__0__eq_H,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ M @ N )
% 0.29/0.47 => ( ( minus_minus_nat @ M @ N )
% 0.29/0.47 = zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_957_One__nat__def,axiom,
% 0.29/0.47 ( one_one_nat
% 0.29/0.47 = ( suc @ zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_958_diff__Suc,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 0.29/0.47 = ( nat_case_nat @ zero_zero_nat
% 0.29/0.47 @ ^ [K_1: nat] : K_1
% 0.29/0.47 @ ( minus_minus_nat @ M @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_959_Suc__pred,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.47 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 0.29/0.47 = N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_960_diff__Suc__less,axiom,
% 0.29/0.47 ! [I_1: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.47 => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I_1 ) ) @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_961_nat__diff__split,axiom,
% 0.29/0.47 ! [P: nat > $o,A_1: nat,B_2: nat] :
% 0.29/0.47 ( ( P @ ( minus_minus_nat @ A_1 @ B_2 ) )
% 0.29/0.47 <=> ( ( ( ord_less_nat @ A_1 @ B_2 )
% 0.29/0.47 => ( P @ zero_zero_nat ) )
% 0.29/0.47 & ! [D_1: nat] :
% 0.29/0.47 ( ( A_1
% 0.29/0.47 = ( plus_plus_nat @ B_2 @ D_1 ) )
% 0.29/0.47 => ( P @ D_1 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_962_nat__diff__split__asm,axiom,
% 0.29/0.47 ! [P: nat > $o,A_1: nat,B_2: nat] :
% 0.29/0.47 ( ( P @ ( minus_minus_nat @ A_1 @ B_2 ) )
% 0.29/0.47 <=> ~ ( ( ( ord_less_nat @ A_1 @ B_2 )
% 0.29/0.47 & ~ ( P @ zero_zero_nat ) )
% 0.29/0.47 | ? [D_1: nat] :
% 0.29/0.47 ( ( A_1
% 0.29/0.47 = ( plus_plus_nat @ B_2 @ D_1 ) )
% 0.29/0.47 & ~ ( P @ D_1 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_963_card__less__Suc,axiom,
% 0.29/0.47 ! [I_1: nat,M_2: nat > $o] :
% 0.29/0.47 ( ( member_nat @ zero_zero_nat @ M_2 )
% 0.29/0.47 => ( ( suc
% 0.29/0.47 @ ( finite_card_nat
% 0.29/0.47 @ ( collect_nat
% 0.29/0.47 @ ^ [K_1: nat] : ( (&) @ ( member_nat @ ( suc @ K_1 ) @ M_2 ) @ ( ord_less_nat @ K_1 @ I_1 ) ) ) ) )
% 0.29/0.47 = ( finite_card_nat
% 0.29/0.47 @ ( collect_nat
% 0.29/0.47 @ ^ [K_1: nat] : ( (&) @ ( member_nat @ K_1 @ M_2 ) @ ( ord_less_nat @ K_1 @ ( suc @ I_1 ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_964_card__less,axiom,
% 0.29/0.47 ! [I_1: nat,M_2: nat > $o] :
% 0.29/0.47 ( ( member_nat @ zero_zero_nat @ M_2 )
% 0.29/0.47 => ( ( finite_card_nat
% 0.29/0.47 @ ( collect_nat
% 0.29/0.47 @ ^ [K_1: nat] : ( (&) @ ( member_nat @ K_1 @ M_2 ) @ ( ord_less_nat @ K_1 @ ( suc @ I_1 ) ) ) ) )
% 0.29/0.47 != zero_zero_nat ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_965_card__less__Suc2,axiom,
% 0.29/0.47 ! [I_1: nat,M_2: nat > $o] :
% 0.29/0.47 ( ~ ( member_nat @ zero_zero_nat @ M_2 )
% 0.29/0.47 => ( ( finite_card_nat
% 0.29/0.47 @ ( collect_nat
% 0.29/0.47 @ ^ [K_1: nat] : ( (&) @ ( member_nat @ ( suc @ K_1 ) @ M_2 ) @ ( ord_less_nat @ K_1 @ I_1 ) ) ) )
% 0.29/0.47 = ( finite_card_nat
% 0.29/0.47 @ ( collect_nat
% 0.29/0.47 @ ^ [K_1: nat] : ( (&) @ ( member_nat @ K_1 @ M_2 ) @ ( ord_less_nat @ K_1 @ ( suc @ I_1 ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_966_Suc__diff__1,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.47 => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
% 0.29/0.47 = N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_967_Suc__pred_H,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.47 => ( N
% 0.29/0.47 = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_968_add__eq__if,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ( ( M = zero_zero_nat )
% 0.29/0.47 => ( ( plus_plus_nat @ M @ N )
% 0.29/0.47 = N ) )
% 0.29/0.47 & ( ( M != zero_zero_nat )
% 0.29/0.47 => ( ( plus_plus_nat @ M @ N )
% 0.29/0.47 = ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_969_ex__least__nat__less,axiom,
% 0.29/0.47 ! [N: nat,P: nat > $o] :
% 0.29/0.47 ( ~ ( P @ zero_zero_nat )
% 0.29/0.47 => ( ( P @ N )
% 0.29/0.47 => ? [K_1: nat] :
% 0.29/0.47 ( ( ord_less_nat @ K_1 @ N )
% 0.29/0.47 & ! [I: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ I @ K_1 )
% 0.29/0.47 => ~ ( P @ I ) )
% 0.29/0.47 & ( P @ ( plus_plus_nat @ K_1 @ one_one_nat ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_970_ex__least__nat__le,axiom,
% 0.29/0.47 ! [N: nat,P: nat > $o] :
% 0.29/0.47 ( ~ ( P @ zero_zero_nat )
% 0.29/0.47 => ( ( P @ N )
% 0.29/0.47 => ? [K_1: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ K_1 @ N )
% 0.29/0.47 & ! [I: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I @ K_1 )
% 0.29/0.47 => ~ ( P @ I ) )
% 0.29/0.47 & ( P @ K_1 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_971_less__imp__add__positive,axiom,
% 0.29/0.47 ! [I_1: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I_1 @ J )
% 0.29/0.47 => ? [K_1: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ K_1 )
% 0.29/0.47 & ( ( plus_plus_nat @ I_1 @ K_1 )
% 0.29/0.47 = J ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_972_gr0__implies__Suc,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.29/0.47 => ? [M_1: nat] :
% 0.29/0.47 ( N
% 0.29/0.47 = ( suc @ M_1 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_973_nat_Oexhaust,axiom,
% 0.29/0.47 ! [Y: nat] :
% 0.29/0.47 ( ( Y != zero_zero_nat )
% 0.29/0.47 => ~ ! [Nat: nat] :
% 0.29/0.47 ( Y
% 0.29/0.47 != ( suc @ Nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_974_zero__induct,axiom,
% 0.29/0.47 ! [P: nat > $o,K: nat] :
% 0.29/0.47 ( ( P @ K )
% 0.29/0.47 => ( ! [N_1: nat] :
% 0.29/0.47 ( ( P @ ( suc @ N_1 ) )
% 0.29/0.47 => ( P @ N_1 ) )
% 0.29/0.47 => ( P @ zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_975_not0__implies__Suc,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( N != zero_zero_nat )
% 0.29/0.47 => ? [M_1: nat] :
% 0.29/0.47 ( N
% 0.29/0.47 = ( suc @ M_1 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_976_nat__induct,axiom,
% 0.29/0.47 ! [N: nat,P: nat > $o] :
% 0.29/0.47 ( ( P @ zero_zero_nat )
% 0.29/0.47 => ( ! [N_1: nat] :
% 0.29/0.47 ( ( P @ N_1 )
% 0.29/0.47 => ( P @ ( suc @ N_1 ) ) )
% 0.29/0.47 => ( P @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_977_expand__Suc,axiom,
% 0.29/0.47 ! [V: int] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ ( number_number_of_nat @ V ) )
% 0.29/0.47 => ( ( number_number_of_nat @ V )
% 0.29/0.47 = ( suc @ ( minus_minus_nat @ ( number_number_of_nat @ V ) @ one_one_nat ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_978_mult__0,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( times_times_nat @ zero_zero_nat @ N )
% 0.29/0.47 = zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_979_mult__0__right,axiom,
% 0.29/0.47 ! [M: nat] :
% 0.29/0.47 ( ( times_times_nat @ M @ zero_zero_nat )
% 0.29/0.47 = zero_zero_nat ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_980_mult__is__0,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ( times_times_nat @ M @ N )
% 0.29/0.47 = zero_zero_nat )
% 0.29/0.47 <=> ( ( M = zero_zero_nat )
% 0.29/0.47 | ( N = zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_981_mult__cancel1,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ( times_times_nat @ K @ M )
% 0.29/0.47 = ( times_times_nat @ K @ N ) )
% 0.29/0.47 <=> ( ( M = N )
% 0.29/0.47 | ( K = zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_982_mult__cancel2,axiom,
% 0.29/0.47 ! [M: nat,K: nat,N: nat] :
% 0.29/0.47 ( ( ( times_times_nat @ M @ K )
% 0.29/0.47 = ( times_times_nat @ N @ K ) )
% 0.29/0.47 <=> ( ( M = N )
% 0.29/0.47 | ( K = zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_983_Suc__mult__cancel1,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 0.29/0.47 = ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.29/0.47 <=> ( M = N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_984_add__mult__distrib,axiom,
% 0.29/0.47 ! [M: nat,N: nat,K: nat] :
% 0.29/0.47 ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
% 0.29/0.47 = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_985_add__mult__distrib2,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
% 0.29/0.47 = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_986_mult__le__mono,axiom,
% 0.29/0.47 ! [K: nat,L: nat,I_1: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ I_1 @ J )
% 0.29/0.47 => ( ( ord_less_eq_nat @ K @ L )
% 0.29/0.47 => ( ord_less_eq_nat @ ( times_times_nat @ I_1 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_987_mult__le__mono2,axiom,
% 0.29/0.47 ! [K: nat,I_1: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ I_1 @ J )
% 0.29/0.47 => ( ord_less_eq_nat @ ( times_times_nat @ K @ I_1 ) @ ( times_times_nat @ K @ J ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_988_mult__le__mono1,axiom,
% 0.29/0.47 ! [K: nat,I_1: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ I_1 @ J )
% 0.29/0.47 => ( ord_less_eq_nat @ ( times_times_nat @ I_1 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_989_le__cube,axiom,
% 0.29/0.47 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_990_le__square,axiom,
% 0.29/0.47 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_991_diff__mult__distrib,axiom,
% 0.29/0.47 ! [M: nat,N: nat,K: nat] :
% 0.29/0.47 ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
% 0.29/0.47 = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_992_diff__mult__distrib2,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.29/0.47 = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_993_nat__mult__eq__1__iff,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ( times_times_nat @ M @ N )
% 0.29/0.47 = one_one_nat )
% 0.29/0.47 <=> ( ( M = one_one_nat )
% 0.29/0.47 & ( N = one_one_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_994_nat__mult__1__right,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( times_times_nat @ N @ one_one_nat )
% 0.29/0.47 = N ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_995_nat__1__eq__mult__iff,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( one_one_nat
% 0.29/0.47 = ( times_times_nat @ M @ N ) )
% 0.29/0.47 <=> ( ( M = one_one_nat )
% 0.29/0.47 & ( N = one_one_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_996_nat__mult__1,axiom,
% 0.29/0.47 ! [N: nat] :
% 0.29/0.47 ( ( times_times_nat @ one_one_nat @ N )
% 0.29/0.47 = N ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_997_mult__eq__1__iff,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ( times_times_nat @ M @ N )
% 0.29/0.47 = ( suc @ zero_zero_nat ) )
% 0.29/0.47 <=> ( ( M
% 0.29/0.47 = ( suc @ zero_zero_nat ) )
% 0.29/0.47 & ( N
% 0.29/0.47 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_998_mult__less__mono2,axiom,
% 0.29/0.47 ! [K: nat,I_1: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I_1 @ J )
% 0.29/0.47 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.47 => ( ord_less_nat @ ( times_times_nat @ K @ I_1 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_999_mult__less__mono1,axiom,
% 0.29/0.47 ! [K: nat,I_1: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_nat @ I_1 @ J )
% 0.29/0.47 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.47 => ( ord_less_nat @ ( times_times_nat @ I_1 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1000_mult__less__cancel2,axiom,
% 0.29/0.47 ! [M: nat,K: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.29/0.47 <=> ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.47 & ( ord_less_nat @ M @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1001_mult__less__cancel1,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.29/0.47 <=> ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.47 & ( ord_less_nat @ M @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1002_nat__0__less__mult__iff,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 0.29/0.47 <=> ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.47 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1003_Suc__mult__less__cancel1,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.29/0.47 <=> ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1004_mult__Suc__right,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( times_times_nat @ M @ ( suc @ N ) )
% 0.29/0.47 = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1005_mult__Suc,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( times_times_nat @ ( suc @ M ) @ N )
% 0.29/0.47 = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1006_Suc__mult__le__cancel1,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.29/0.47 <=> ( ord_less_eq_nat @ M @ N ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1007_mult__eq__self__implies__10,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( M
% 0.29/0.47 = ( times_times_nat @ M @ N ) )
% 0.29/0.47 => ( ( N = one_one_nat )
% 0.29/0.47 | ( M = zero_zero_nat ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1008_n__less__m__mult__n,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.29/0.47 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.29/0.47 => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1009_n__less__n__mult__m,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.29/0.47 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.29/0.47 => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1010_one__less__mult,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.29/0.47 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.29/0.47 => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1011_one__le__mult__iff,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
% 0.29/0.47 <=> ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.29/0.47 & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1012_mult__le__cancel1,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.29/0.47 <=> ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.47 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1013_mult__le__cancel2,axiom,
% 0.29/0.47 ! [M: nat,K: nat,N: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.29/0.47 <=> ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.47 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1014_mult__eq__if,axiom,
% 0.29/0.47 ! [N: nat,M: nat] :
% 0.29/0.47 ( ( ( M = zero_zero_nat )
% 0.29/0.47 => ( ( times_times_nat @ M @ N )
% 0.29/0.47 = zero_zero_nat ) )
% 0.29/0.47 & ( ( M != zero_zero_nat )
% 0.29/0.47 => ( ( times_times_nat @ M @ N )
% 0.29/0.47 = ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1015_nat__less__add__iff2,axiom,
% 0.29/0.47 ! [U: nat,M: nat,N: nat,I_1: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ I_1 @ J )
% 0.29/0.47 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.29/0.47 <=> ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I_1 ) @ U ) @ N ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1016_nat__mult__commute,axiom,
% 0.29/0.47 ! [M: nat,N: nat] :
% 0.29/0.47 ( ( times_times_nat @ M @ N )
% 0.29/0.47 = ( times_times_nat @ N @ M ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1017_nat__mult__assoc,axiom,
% 0.29/0.47 ! [M: nat,N: nat,K: nat] :
% 0.29/0.47 ( ( times_times_nat @ ( times_times_nat @ M @ N ) @ K )
% 0.29/0.47 = ( times_times_nat @ M @ ( times_times_nat @ N @ K ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1018_less__number__of__int__code,axiom,
% 0.29/0.47 ! [K: int,L: int] :
% 0.29/0.47 ( ( ord_less_int @ ( number_number_of_int @ K ) @ ( number_number_of_int @ L ) )
% 0.29/0.47 <=> ( ord_less_int @ K @ L ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1019_zmult__1,axiom,
% 0.29/0.47 ! [Z: int] :
% 0.29/0.47 ( ( times_times_int @ one_one_int @ Z )
% 0.29/0.47 = Z ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1020_zmult__1__right,axiom,
% 0.29/0.47 ! [Z: int] :
% 0.29/0.47 ( ( times_times_int @ Z @ one_one_int )
% 0.29/0.47 = Z ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1021_zmult__commute,axiom,
% 0.29/0.47 ! [Z: int,W: int] :
% 0.29/0.47 ( ( times_times_int @ Z @ W )
% 0.29/0.47 = ( times_times_int @ W @ Z ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1022_times__numeral__code_I5_J,axiom,
% 0.29/0.47 ! [V: int,W: int] :
% 0.29/0.47 ( ( times_times_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ W ) )
% 0.29/0.47 = ( number_number_of_int @ ( times_times_int @ V @ W ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1023_zmult__assoc,axiom,
% 0.29/0.47 ! [Z1: int,Z2: int,Z3: int] :
% 0.29/0.47 ( ( times_times_int @ ( times_times_int @ Z1 @ Z2 ) @ Z3 )
% 0.29/0.47 = ( times_times_int @ Z1 @ ( times_times_int @ Z2 @ Z3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1024_less__eq__number__of__int__code,axiom,
% 0.29/0.47 ! [K: int,L: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ ( number_number_of_int @ K ) @ ( number_number_of_int @ L ) )
% 0.29/0.47 <=> ( ord_less_eq_int @ K @ L ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1025_zdiff__zmult__distrib,axiom,
% 0.29/0.47 ! [Z1: int,Z2: int,W: int] :
% 0.29/0.47 ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z2 ) @ W )
% 0.29/0.47 = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1026_zdiff__zmult__distrib2,axiom,
% 0.29/0.47 ! [W: int,Z1: int,Z2: int] :
% 0.29/0.47 ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z2 ) )
% 0.29/0.47 = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1027_zadd__zmult__distrib2,axiom,
% 0.29/0.47 ! [W: int,Z1: int,Z2: int] :
% 0.29/0.47 ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z2 ) )
% 0.29/0.47 = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1028_plus__numeral__code_I9_J,axiom,
% 0.29/0.47 ! [V: int,W: int] :
% 0.29/0.47 ( ( plus_plus_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ W ) )
% 0.29/0.47 = ( number_number_of_int @ ( plus_plus_int @ V @ W ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1029_zadd__zmult__distrib,axiom,
% 0.29/0.47 ! [Z1: int,Z2: int,W: int] :
% 0.29/0.47 ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z2 ) @ W )
% 0.29/0.47 = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1030_zmult__zless__mono2,axiom,
% 0.29/0.47 ! [K: int,I_1: int,J: int] :
% 0.29/0.47 ( ( ord_less_int @ I_1 @ J )
% 0.29/0.47 => ( ( ord_less_int @ zero_zero_int @ K )
% 0.29/0.47 => ( ord_less_int @ ( times_times_int @ K @ I_1 ) @ ( times_times_int @ K @ J ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1031_pos__zmult__eq__1__iff,axiom,
% 0.29/0.47 ! [N: int,M: int] :
% 0.29/0.47 ( ( ord_less_int @ zero_zero_int @ M )
% 0.29/0.47 => ( ( ( times_times_int @ M @ N )
% 0.29/0.47 = one_one_int )
% 0.29/0.47 <=> ( ( M = one_one_int )
% 0.29/0.47 & ( N = one_one_int ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1032_odd__nonzero,axiom,
% 0.29/0.47 ! [Z: int] :
% 0.29/0.47 ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
% 0.29/0.47 != zero_zero_int ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1033_zadd__0__right,axiom,
% 0.29/0.47 ! [Z: int] :
% 0.29/0.47 ( ( plus_plus_int @ Z @ zero_zero_int )
% 0.29/0.47 = Z ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1034_zadd__0,axiom,
% 0.29/0.47 ! [Z: int] :
% 0.29/0.47 ( ( plus_plus_int @ zero_zero_int @ Z )
% 0.29/0.47 = Z ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1035_int__0__less__1,axiom,
% 0.29/0.47 ord_less_int @ zero_zero_int @ one_one_int ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1036_int__one__le__iff__zero__less,axiom,
% 0.29/0.47 ! [Z: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ one_one_int @ Z )
% 0.29/0.47 <=> ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1037_less__bin__lemma,axiom,
% 0.29/0.47 ! [K: int,L: int] :
% 0.29/0.47 ( ( ord_less_int @ K @ L )
% 0.29/0.47 <=> ( ord_less_int @ ( minus_minus_int @ K @ L ) @ zero_zero_int ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1038_le__imp__0__less,axiom,
% 0.29/0.47 ! [Z: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.29/0.47 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1039_odd__less__0,axiom,
% 0.29/0.47 ! [Z: int] :
% 0.29/0.47 ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
% 0.29/0.47 <=> ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1040_zadd__left__mono,axiom,
% 0.29/0.47 ! [K: int,I_1: int,J: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ I_1 @ J )
% 0.29/0.47 => ( ord_less_eq_int @ ( plus_plus_int @ K @ I_1 ) @ ( plus_plus_int @ K @ J ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1041_zadd__assoc,axiom,
% 0.29/0.47 ! [Z1: int,Z2: int,Z3: int] :
% 0.29/0.47 ( ( plus_plus_int @ ( plus_plus_int @ Z1 @ Z2 ) @ Z3 )
% 0.29/0.47 = ( plus_plus_int @ Z1 @ ( plus_plus_int @ Z2 @ Z3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1042_zadd__left__commute,axiom,
% 0.29/0.47 ! [X: int,Y: int,Z: int] :
% 0.29/0.47 ( ( plus_plus_int @ X @ ( plus_plus_int @ Y @ Z ) )
% 0.29/0.47 = ( plus_plus_int @ Y @ ( plus_plus_int @ X @ Z ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1043_zadd__commute,axiom,
% 0.29/0.47 ! [Z: int,W: int] :
% 0.29/0.47 ( ( plus_plus_int @ Z @ W )
% 0.29/0.47 = ( plus_plus_int @ W @ Z ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1044_zle__refl,axiom,
% 0.29/0.47 ! [W: int] : ( ord_less_eq_int @ W @ W ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1045_zle__linear,axiom,
% 0.29/0.47 ! [Z: int,W: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ Z @ W )
% 0.29/0.47 | ( ord_less_eq_int @ W @ Z ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1046_zle__trans,axiom,
% 0.29/0.47 ! [K: int,I_1: int,J: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ I_1 @ J )
% 0.29/0.47 => ( ( ord_less_eq_int @ J @ K )
% 0.29/0.47 => ( ord_less_eq_int @ I_1 @ K ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1047_zle__antisym,axiom,
% 0.29/0.47 ! [Z: int,W: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ Z @ W )
% 0.29/0.47 => ( ( ord_less_eq_int @ W @ Z )
% 0.29/0.47 => ( Z = W ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1048_zle__diff1__eq,axiom,
% 0.29/0.47 ! [W: int,Z: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
% 0.29/0.47 <=> ( ord_less_int @ W @ Z ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1049_zless__linear,axiom,
% 0.29/0.47 ! [X: int,Y: int] :
% 0.29/0.47 ( ( ord_less_int @ X @ Y )
% 0.29/0.47 | ( X = Y )
% 0.29/0.47 | ( ord_less_int @ Y @ X ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1050_zless__le,axiom,
% 0.29/0.47 ! [Z: int,W: int] :
% 0.29/0.47 ( ( ord_less_int @ Z @ W )
% 0.29/0.47 <=> ( ( ord_less_eq_int @ Z @ W )
% 0.29/0.47 & ( Z != W ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1051_zadd__strict__right__mono,axiom,
% 0.29/0.47 ! [K: int,I_1: int,J: int] :
% 0.29/0.47 ( ( ord_less_int @ I_1 @ J )
% 0.29/0.47 => ( ord_less_int @ ( plus_plus_int @ I_1 @ K ) @ ( plus_plus_int @ J @ K ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1052_zadd__zless__mono,axiom,
% 0.29/0.47 ! [Z_3: int,Z: int,W_1: int,W: int] :
% 0.29/0.47 ( ( ord_less_int @ W_1 @ W )
% 0.29/0.47 => ( ( ord_less_eq_int @ Z_3 @ Z )
% 0.29/0.47 => ( ord_less_int @ ( plus_plus_int @ W_1 @ Z_3 ) @ ( plus_plus_int @ W @ Z ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1053_zle__add1__eq__le,axiom,
% 0.29/0.47 ! [W: int,Z: int] :
% 0.29/0.47 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 0.29/0.47 <=> ( ord_less_eq_int @ W @ Z ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1054_add1__zle__eq,axiom,
% 0.29/0.47 ! [W: int,Z: int] :
% 0.29/0.47 ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
% 0.29/0.47 <=> ( ord_less_int @ W @ Z ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1055_zless__imp__add1__zle,axiom,
% 0.29/0.47 ! [W: int,Z: int] :
% 0.29/0.47 ( ( ord_less_int @ W @ Z )
% 0.29/0.47 => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1056_zless__add1__eq,axiom,
% 0.29/0.47 ! [W: int,Z: int] :
% 0.29/0.47 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 0.29/0.47 <=> ( ( ord_less_int @ W @ Z )
% 0.29/0.47 | ( W = Z ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1057_nat__mult__eq__cancel__disj,axiom,
% 0.29/0.47 ! [K: nat,M: nat,N: nat] :
% 0.29/0.47 ( ( ( times_times_nat @ K @ M )
% 0.29/0.47 = ( times_times_nat @ K @ N ) )
% 0.29/0.47 <=> ( ( K = zero_zero_nat )
% 0.29/0.47 | ( M = N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1058_left__add__mult__distrib,axiom,
% 0.29/0.47 ! [I_1: nat,U: nat,J: nat,K: nat] :
% 0.29/0.47 ( ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
% 0.29/0.47 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I_1 @ J ) @ U ) @ K ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1059_nat__mult__eq__cancel1,axiom,
% 0.29/0.47 ! [M: nat,N: nat,K: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.47 => ( ( ( times_times_nat @ K @ M )
% 0.29/0.47 = ( times_times_nat @ K @ N ) )
% 0.29/0.47 <=> ( M = N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1060_nat__mult__less__cancel1,axiom,
% 0.29/0.47 ! [M: nat,N: nat,K: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.47 => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.29/0.47 <=> ( ord_less_nat @ M @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1061_nat__mult__le__cancel1,axiom,
% 0.29/0.47 ! [M: nat,N: nat,K: nat] :
% 0.29/0.47 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.47 => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.29/0.47 <=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1062_nat__le__add__iff1,axiom,
% 0.29/0.47 ! [U: nat,M: nat,N: nat,J: nat,I_1: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ J @ I_1 )
% 0.29/0.47 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.29/0.47 <=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1063_nat__diff__add__eq1,axiom,
% 0.29/0.47 ! [U: nat,M: nat,N: nat,J: nat,I_1: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ J @ I_1 )
% 0.29/0.47 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.29/0.47 = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1064_nat__eq__add__iff1,axiom,
% 0.29/0.47 ! [U: nat,M: nat,N: nat,J: nat,I_1: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ J @ I_1 )
% 0.29/0.47 => ( ( ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M )
% 0.29/0.47 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.29/0.47 <=> ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J ) @ U ) @ M )
% 0.29/0.47 = N ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1065_nat__le__add__iff2,axiom,
% 0.29/0.47 ! [U: nat,M: nat,N: nat,I_1: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ I_1 @ J )
% 0.29/0.47 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.29/0.47 <=> ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I_1 ) @ U ) @ N ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1066_nat__diff__add__eq2,axiom,
% 0.29/0.47 ! [U: nat,M: nat,N: nat,I_1: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ I_1 @ J )
% 0.29/0.47 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.29/0.47 = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I_1 ) @ U ) @ N ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1067_nat__eq__add__iff2,axiom,
% 0.29/0.47 ! [U: nat,M: nat,N: nat,I_1: nat,J: nat] :
% 0.29/0.47 ( ( ord_less_eq_nat @ I_1 @ J )
% 0.29/0.47 => ( ( ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M )
% 0.29/0.47 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.29/0.47 <=> ( M
% 0.29/0.47 = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I_1 ) @ U ) @ N ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(fact_1068_nat__less__add__iff1,axiom,
% 0.29/0.47 ! [U: nat,M: nat,N: nat,J: nat,I_1: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ J @ I_1 )
% 0.29/0.48 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.29/0.48 <=> ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1069_zdiv__mono2__neg__lemma,axiom,
% 0.29/0.48 ! [B_2: int,Q_1: int,R_1: int,B_1: int,Q: int,R: int] :
% 0.29/0.48 ( ( ( plus_plus_int @ ( times_times_int @ B_2 @ Q_1 ) @ R_1 )
% 0.29/0.48 = ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R ) )
% 0.29/0.48 => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R ) @ zero_zero_int )
% 0.29/0.48 => ( ( ord_less_int @ R_1 @ B_2 )
% 0.29/0.48 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 0.29/0.48 => ( ( ord_less_int @ zero_zero_int @ B_1 )
% 0.29/0.48 => ( ( ord_less_eq_int @ B_1 @ B_2 )
% 0.29/0.48 => ( ord_less_eq_int @ Q @ Q_1 ) ) ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1070_unique__quotient__lemma__neg,axiom,
% 0.29/0.48 ! [B_2: int,Q: int,R: int,Q_1: int,R_1: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B_2 @ Q ) @ R ) @ ( plus_plus_int @ ( times_times_int @ B_2 @ Q_1 ) @ R_1 ) )
% 0.29/0.48 => ( ( ord_less_eq_int @ R_1 @ zero_zero_int )
% 0.29/0.48 => ( ( ord_less_int @ B_2 @ R_1 )
% 0.29/0.48 => ( ( ord_less_int @ B_2 @ R )
% 0.29/0.48 => ( ord_less_eq_int @ Q_1 @ Q ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1071_number__of__is__id,axiom,
% 0.29/0.48 ! [K: int] :
% 0.29/0.48 ( ( number_number_of_int @ K )
% 0.29/0.48 = K ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1072_int__0__neq__1,axiom,
% 0.29/0.48 zero_zero_int != one_one_int ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1073_self__quotient__aux1,axiom,
% 0.29/0.48 ! [R_1: int,Q_1: int,A_1: int] :
% 0.29/0.48 ( ( ord_less_int @ zero_zero_int @ A_1 )
% 0.29/0.48 => ( ( A_1
% 0.29/0.48 = ( plus_plus_int @ R_1 @ ( times_times_int @ A_1 @ Q_1 ) ) )
% 0.29/0.48 => ( ( ord_less_int @ R_1 @ A_1 )
% 0.29/0.48 => ( ord_less_eq_int @ one_one_int @ Q_1 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1074_self__quotient__aux2,axiom,
% 0.29/0.48 ! [R_1: int,Q_1: int,A_1: int] :
% 0.29/0.48 ( ( ord_less_int @ zero_zero_int @ A_1 )
% 0.29/0.48 => ( ( A_1
% 0.29/0.48 = ( plus_plus_int @ R_1 @ ( times_times_int @ A_1 @ Q_1 ) ) )
% 0.29/0.48 => ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
% 0.29/0.48 => ( ord_less_eq_int @ Q_1 @ one_one_int ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1075_q__pos__lemma,axiom,
% 0.29/0.48 ! [B_1: int,Q: int,R: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R ) )
% 0.29/0.48 => ( ( ord_less_int @ R @ B_1 )
% 0.29/0.48 => ( ( ord_less_int @ zero_zero_int @ B_1 )
% 0.29/0.48 => ( ord_less_eq_int @ zero_zero_int @ Q ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1076_q__neg__lemma,axiom,
% 0.29/0.48 ! [B_1: int,Q: int,R: int] :
% 0.29/0.48 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R ) @ zero_zero_int )
% 0.29/0.48 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 0.29/0.48 => ( ( ord_less_int @ zero_zero_int @ B_1 )
% 0.29/0.48 => ( ord_less_eq_int @ Q @ zero_zero_int ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1077_unique__quotient__lemma,axiom,
% 0.29/0.48 ! [B_2: int,Q: int,R: int,Q_1: int,R_1: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B_2 @ Q ) @ R ) @ ( plus_plus_int @ ( times_times_int @ B_2 @ Q_1 ) @ R_1 ) )
% 0.29/0.48 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 0.29/0.48 => ( ( ord_less_int @ R @ B_2 )
% 0.29/0.48 => ( ( ord_less_int @ R_1 @ B_2 )
% 0.29/0.48 => ( ord_less_eq_int @ Q @ Q_1 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1078_zdiv__mono2__lemma,axiom,
% 0.29/0.48 ! [B_2: int,Q_1: int,R_1: int,B_1: int,Q: int,R: int] :
% 0.29/0.48 ( ( ( plus_plus_int @ ( times_times_int @ B_2 @ Q_1 ) @ R_1 )
% 0.29/0.48 = ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R ) )
% 0.29/0.48 => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R ) )
% 0.29/0.48 => ( ( ord_less_int @ R @ B_1 )
% 0.29/0.48 => ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
% 0.29/0.48 => ( ( ord_less_int @ zero_zero_int @ B_1 )
% 0.29/0.48 => ( ( ord_less_eq_int @ B_1 @ B_2 )
% 0.29/0.48 => ( ord_less_eq_int @ Q_1 @ Q ) ) ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1079_int__less__induct,axiom,
% 0.29/0.48 ! [P: int > $o,I_1: int,K: int] :
% 0.29/0.48 ( ( ord_less_int @ I_1 @ K )
% 0.29/0.48 => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 0.29/0.48 => ( ! [I: int] :
% 0.29/0.48 ( ( ord_less_int @ I @ K )
% 0.29/0.48 => ( ( P @ I )
% 0.29/0.48 => ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
% 0.29/0.48 => ( P @ I_1 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1080_int__le__induct,axiom,
% 0.29/0.48 ! [P: int > $o,I_1: int,K: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ I_1 @ K )
% 0.29/0.48 => ( ( P @ K )
% 0.29/0.48 => ( ! [I: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ I @ K )
% 0.29/0.48 => ( ( P @ I )
% 0.29/0.48 => ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
% 0.29/0.48 => ( P @ I_1 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1081_int__gr__induct,axiom,
% 0.29/0.48 ! [P: int > $o,K: int,I_1: int] :
% 0.29/0.48 ( ( ord_less_int @ K @ I_1 )
% 0.29/0.48 => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
% 0.29/0.48 => ( ! [I: int] :
% 0.29/0.48 ( ( ord_less_int @ K @ I )
% 0.29/0.48 => ( ( P @ I )
% 0.29/0.48 => ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
% 0.29/0.48 => ( P @ I_1 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1082_int__ge__induct,axiom,
% 0.29/0.48 ! [P: int > $o,K: int,I_1: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ K @ I_1 )
% 0.29/0.48 => ( ( P @ K )
% 0.29/0.48 => ( ! [I: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ K @ I )
% 0.29/0.48 => ( ( P @ I )
% 0.29/0.48 => ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
% 0.29/0.48 => ( P @ I_1 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1083_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
% 0.29/0.48 ! [Y: int,X: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.48 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.29/0.48 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ X @ Y ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1084_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
% 0.29/0.48 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1085_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
% 0.29/0.48 ! [Y: int,X: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.48 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.29/0.48 => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ X @ Y ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1086_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
% 0.29/0.48 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1087_transfer__nat__int__set__cong,axiom,
% 0.29/0.48 ! [P_1: int > $o,P: int > $o] :
% 0.29/0.48 ( ! [X_1: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ zero_zero_int @ X_1 )
% 0.29/0.48 => ( ( P @ X_1 )
% 0.29/0.48 <=> ( P_1 @ X_1 ) ) )
% 0.29/0.48 => ( ( collect_int
% 0.29/0.48 @ ^ [X_1: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ X_1 ) @ ( P @ X_1 ) ) )
% 0.29/0.48 = ( collect_int
% 0.29/0.48 @ ^ [X_1: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ X_1 ) @ ( P_1 @ X_1 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1088_decr__mult__lemma,axiom,
% 0.29/0.48 ! [K: int,P: int > $o,D: int] :
% 0.29/0.48 ( ( ord_less_int @ zero_zero_int @ D )
% 0.29/0.48 => ( ! [X_1: int] :
% 0.29/0.48 ( ( P @ X_1 )
% 0.29/0.48 => ( P @ ( minus_minus_int @ X_1 @ D ) ) )
% 0.29/0.48 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.29/0.48 => ! [X_1: int] :
% 0.29/0.48 ( ( P @ X_1 )
% 0.29/0.48 => ( P @ ( minus_minus_int @ X_1 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1089_conj__le__cong,axiom,
% 0.29/0.48 ! [P_1: $o,P: $o,X: int] :
% 0.29/0.48 ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.48 => ( P
% 0.29/0.48 <=> P_1 ) )
% 0.29/0.48 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.48 & P )
% 0.29/0.48 <=> ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.48 & P_1 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1090_imp__le__cong,axiom,
% 0.29/0.48 ! [P_1: $o,P: $o,X: int] :
% 0.29/0.48 ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.48 => ( P
% 0.29/0.48 <=> P_1 ) )
% 0.29/0.48 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.48 => P )
% 0.29/0.48 <=> ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.48 => P_1 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1091_incr__mult__lemma,axiom,
% 0.29/0.48 ! [K: int,P: int > $o,D: int] :
% 0.29/0.48 ( ( ord_less_int @ zero_zero_int @ D )
% 0.29/0.48 => ( ! [X_1: int] :
% 0.29/0.48 ( ( P @ X_1 )
% 0.29/0.48 => ( P @ ( plus_plus_int @ X_1 @ D ) ) )
% 0.29/0.48 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.29/0.48 => ! [X_1: int] :
% 0.29/0.48 ( ( P @ X_1 )
% 0.29/0.48 => ( P @ ( plus_plus_int @ X_1 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1092_int__induct,axiom,
% 0.29/0.48 ! [I_1: int,P: int > $o,K: int] :
% 0.29/0.48 ( ( P @ K )
% 0.29/0.48 => ( ! [I: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ K @ I )
% 0.29/0.48 => ( ( P @ I )
% 0.29/0.48 => ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
% 0.29/0.48 => ( ! [I: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ I @ K )
% 0.29/0.48 => ( ( P @ I )
% 0.29/0.48 => ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
% 0.29/0.48 => ( P @ I_1 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1093_minusinfinity,axiom,
% 0.29/0.48 ! [P: int > $o,P1: int > $o,D: int] :
% 0.29/0.48 ( ( ord_less_int @ zero_zero_int @ D )
% 0.29/0.48 => ( ! [X_1: int,K_1: int] :
% 0.29/0.48 ( ( P1 @ X_1 )
% 0.29/0.48 <=> ( P1 @ ( minus_minus_int @ X_1 @ ( times_times_int @ K_1 @ D ) ) ) )
% 0.29/0.48 => ( ? [Z_2: int] :
% 0.29/0.48 ! [X_1: int] :
% 0.29/0.48 ( ( ord_less_int @ X_1 @ Z_2 )
% 0.29/0.48 => ( ( P @ X_1 )
% 0.29/0.48 <=> ( P1 @ X_1 ) ) )
% 0.29/0.48 => ( ( ex @ P1 )
% 0.29/0.48 => ( ex @ P ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1094_plusinfinity,axiom,
% 0.29/0.48 ! [P: int > $o,P_1: int > $o,D: int] :
% 0.29/0.48 ( ( ord_less_int @ zero_zero_int @ D )
% 0.29/0.48 => ( ! [X_1: int,K_1: int] :
% 0.29/0.48 ( ( P_1 @ X_1 )
% 0.29/0.48 <=> ( P_1 @ ( minus_minus_int @ X_1 @ ( times_times_int @ K_1 @ D ) ) ) )
% 0.29/0.48 => ( ? [Z_2: int] :
% 0.29/0.48 ! [X_1: int] :
% 0.29/0.48 ( ( ord_less_int @ Z_2 @ X_1 )
% 0.29/0.48 => ( ( P @ X_1 )
% 0.29/0.48 <=> ( P_1 @ X_1 ) ) )
% 0.29/0.48 => ( ( ex @ P_1 )
% 0.29/0.48 => ( ex @ P ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1095_tsub__def,axiom,
% 0.29/0.48 ! [Y: int,X: int] :
% 0.29/0.48 ( ( ( ord_less_eq_int @ Y @ X )
% 0.29/0.48 => ( ( nat_tsub @ X @ Y )
% 0.29/0.48 = ( minus_minus_int @ X @ Y ) ) )
% 0.29/0.48 & ( ~ ( ord_less_eq_int @ Y @ X )
% 0.29/0.48 => ( ( nat_tsub @ X @ Y )
% 0.29/0.48 = zero_zero_int ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1096_zero__zle__int,axiom,
% 0.29/0.48 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1621563631at_int @ N ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1097_int__less__0__conv,axiom,
% 0.29/0.48 ! [K: nat] :
% 0.29/0.48 ~ ( ord_less_int @ ( semiri1621563631at_int @ K ) @ zero_zero_int ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1098_int__1,axiom,
% 0.29/0.48 ( ( semiri1621563631at_int @ one_one_nat )
% 0.29/0.48 = one_one_int ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1099_int__0,axiom,
% 0.29/0.48 ( ( semiri1621563631at_int @ zero_zero_nat )
% 0.29/0.48 = zero_zero_int ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1100_int__eq__0__conv,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ( ( ( semiri1621563631at_int @ N )
% 0.29/0.48 = zero_zero_int )
% 0.29/0.48 <=> ( N = zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1101_zless__int,axiom,
% 0.29/0.48 ! [M: nat,N: nat] :
% 0.29/0.48 ( ( ord_less_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
% 0.29/0.48 <=> ( ord_less_nat @ M @ N ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1102_zle__int,axiom,
% 0.29/0.48 ! [M: nat,N: nat] :
% 0.29/0.48 ( ( ord_less_eq_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
% 0.29/0.48 <=> ( ord_less_eq_nat @ M @ N ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1103_zadd__int,axiom,
% 0.29/0.48 ! [M: nat,N: nat] :
% 0.29/0.48 ( ( plus_plus_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
% 0.29/0.48 = ( semiri1621563631at_int @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1104_zadd__int__left,axiom,
% 0.29/0.48 ! [M: nat,N: nat,Z: int] :
% 0.29/0.48 ( ( plus_plus_int @ ( semiri1621563631at_int @ M ) @ ( plus_plus_int @ ( semiri1621563631at_int @ N ) @ Z ) )
% 0.29/0.48 = ( plus_plus_int @ ( semiri1621563631at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1105_zle__iff__zadd,axiom,
% 0.29/0.48 ! [W: int,Z: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ W @ Z )
% 0.29/0.48 <=> ? [N_1: nat] :
% 0.29/0.48 ( Z
% 0.29/0.48 = ( plus_plus_int @ W @ ( semiri1621563631at_int @ N_1 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1106_zmult__int,axiom,
% 0.29/0.48 ! [M: nat,N: nat] :
% 0.29/0.48 ( ( times_times_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
% 0.29/0.48 = ( semiri1621563631at_int @ ( times_times_nat @ M @ N ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1107_int__mult,axiom,
% 0.29/0.48 ! [M: nat,N: nat] :
% 0.29/0.48 ( ( semiri1621563631at_int @ ( times_times_nat @ M @ N ) )
% 0.29/0.48 = ( times_times_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1108_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,
% 0.29/0.48 ! [X: nat,Y: nat] :
% 0.29/0.48 ( ( times_times_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
% 0.29/0.48 = ( semiri1621563631at_int @ ( times_times_nat @ X @ Y ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1109_transfer__int__nat__relations_I2_J,axiom,
% 0.29/0.48 ! [X: nat,Y: nat] :
% 0.29/0.48 ( ( ord_less_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
% 0.29/0.48 <=> ( ord_less_nat @ X @ Y ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1110_transfer__int__nat__relations_I3_J,axiom,
% 0.29/0.48 ! [X: nat,Y: nat] :
% 0.29/0.48 ( ( ord_less_eq_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
% 0.29/0.48 <=> ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1111_Nat__Transfer_Otransfer__int__nat__functions_I3_J,axiom,
% 0.29/0.48 ! [X: nat,Y: nat] :
% 0.29/0.48 ( ( nat_tsub @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
% 0.29/0.48 = ( semiri1621563631at_int @ ( minus_minus_nat @ X @ Y ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1112_Nat__Transfer_Otransfer__int__nat__functions_I1_J,axiom,
% 0.29/0.48 ! [X: nat,Y: nat] :
% 0.29/0.48 ( ( plus_plus_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
% 0.29/0.48 = ( semiri1621563631at_int @ ( plus_plus_nat @ X @ Y ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1113_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J,axiom,
% 0.29/0.48 ! [A: nat > $o] :
% 0.29/0.48 ( ( finite_card_nat @ A )
% 0.29/0.48 = ( finite_card_int @ ( image_nat_int @ semiri1621563631at_int @ A ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1114_transfer__nat__int__set__relations_I1_J,axiom,
% 0.29/0.48 ! [A: nat > $o] :
% 0.29/0.48 ( ( finite_finite_nat @ A )
% 0.29/0.48 <=> ( finite_finite_int @ ( image_nat_int @ semiri1621563631at_int @ A ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1115_transfer__int__nat__numerals_I2_J,axiom,
% 0.29/0.48 ( one_one_int
% 0.29/0.48 = ( semiri1621563631at_int @ one_one_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1116_transfer__int__nat__numerals_I1_J,axiom,
% 0.29/0.48 ( zero_zero_int
% 0.29/0.48 = ( semiri1621563631at_int @ zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1117_Nat__Transfer_Otransfer__int__nat__set__functions_I5_J,axiom,
% 0.29/0.48 ! [P: int > $o] :
% 0.29/0.48 ( ( collect_int
% 0.29/0.48 @ ^ [X_1: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ X_1 ) @ ( P @ X_1 ) ) )
% 0.29/0.48 = ( image_nat_int @ semiri1621563631at_int
% 0.29/0.48 @ ( collect_nat
% 0.29/0.48 @ ^ [X_1: nat] : ( P @ ( semiri1621563631at_int @ X_1 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1118_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
% 0.29/0.48 ! [Z: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1621563631at_int @ Z ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1119_transfer__int__nat__quantifiers_I2_J,axiom,
% 0.29/0.48 ! [P: int > $o] :
% 0.29/0.48 ( ? [X_1: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ zero_zero_int @ X_1 )
% 0.29/0.48 & ( P @ X_1 ) )
% 0.29/0.48 <=> ? [X_1: nat] : ( P @ ( semiri1621563631at_int @ X_1 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1120_transfer__int__nat__quantifiers_I1_J,axiom,
% 0.29/0.48 ! [P: int > $o] :
% 0.29/0.48 ( ! [X_1: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ zero_zero_int @ X_1 )
% 0.29/0.48 => ( P @ X_1 ) )
% 0.29/0.48 <=> ! [X_1: nat] : ( P @ ( semiri1621563631at_int @ X_1 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1121_int__le__0__conv,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ( ( ord_less_eq_int @ ( semiri1621563631at_int @ N ) @ zero_zero_int )
% 0.29/0.48 <=> ( N = zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1122_int__Suc0__eq__1,axiom,
% 0.29/0.48 ( ( semiri1621563631at_int @ ( suc @ zero_zero_nat ) )
% 0.29/0.48 = one_one_int ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1123_zless__iff__Suc__zadd,axiom,
% 0.29/0.48 ! [W: int,Z: int] :
% 0.29/0.48 ( ( ord_less_int @ W @ Z )
% 0.29/0.48 <=> ? [N_1: nat] :
% 0.29/0.48 ( Z
% 0.29/0.48 = ( plus_plus_int @ W @ ( semiri1621563631at_int @ ( suc @ N_1 ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1124_int__Suc,axiom,
% 0.29/0.48 ! [M: nat] :
% 0.29/0.48 ( ( semiri1621563631at_int @ ( suc @ M ) )
% 0.29/0.48 = ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ M ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1125_zdiff__int,axiom,
% 0.29/0.48 ! [N: nat,M: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ N @ M )
% 0.29/0.48 => ( ( minus_minus_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
% 0.29/0.48 = ( semiri1621563631at_int @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1126_zero__less__int__conv,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ( ( ord_less_int @ zero_zero_int @ ( semiri1621563631at_int @ N ) )
% 0.29/0.48 <=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1127_zmult__zless__mono2__lemma,axiom,
% 0.29/0.48 ! [K: nat,I_1: int,J: int] :
% 0.29/0.48 ( ( ord_less_int @ I_1 @ J )
% 0.29/0.48 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.48 => ( ord_less_int @ ( times_times_int @ ( semiri1621563631at_int @ K ) @ I_1 ) @ ( times_times_int @ ( semiri1621563631at_int @ K ) @ J ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1128_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J,axiom,
% 0.29/0.48 ! [Y: int,X: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.29/0.48 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.29/0.48 => ( ord_less_eq_int @ zero_zero_int @ ( nat_tsub @ X @ Y ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1129_zdiff__int__split,axiom,
% 0.29/0.48 ! [P: int > $o,X: nat,Y: nat] :
% 0.29/0.48 ( ( P @ ( semiri1621563631at_int @ ( minus_minus_nat @ X @ Y ) ) )
% 0.29/0.48 <=> ( ( ( ord_less_eq_nat @ Y @ X )
% 0.29/0.48 => ( P @ ( minus_minus_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) ) ) )
% 0.29/0.48 & ( ( ord_less_nat @ X @ Y )
% 0.29/0.48 => ( P @ zero_zero_int ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1130_tsub__eq,axiom,
% 0.29/0.48 ! [Y: int,X: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ Y @ X )
% 0.29/0.48 => ( ( nat_tsub @ X @ Y )
% 0.29/0.48 = ( minus_minus_int @ X @ Y ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1131_zero__less__imp__eq__int,axiom,
% 0.29/0.48 ! [K: int] :
% 0.29/0.48 ( ( ord_less_int @ zero_zero_int @ K )
% 0.29/0.48 => ? [N_1: nat] :
% 0.29/0.48 ( ( ord_less_nat @ zero_zero_nat @ N_1 )
% 0.29/0.48 & ( K
% 0.29/0.48 = ( semiri1621563631at_int @ N_1 ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1132_int__int__eq,axiom,
% 0.29/0.48 ! [M: nat,N: nat] :
% 0.29/0.48 ( ( ( semiri1621563631at_int @ M )
% 0.29/0.48 = ( semiri1621563631at_int @ N ) )
% 0.29/0.48 <=> ( M = N ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1133_transfer__nat__int__set__relations_I4_J,axiom,
% 0.29/0.48 ! [A: nat > $o,B: nat > $o] :
% 0.29/0.48 ( ( ord_less_nat_o @ A @ B )
% 0.29/0.48 <=> ( ord_less_int_o @ ( image_nat_int @ semiri1621563631at_int @ A ) @ ( image_nat_int @ semiri1621563631at_int @ B ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1134_transfer__nat__int__set__relations_I5_J,axiom,
% 0.29/0.48 ! [A: nat > $o,B: nat > $o] :
% 0.29/0.48 ( ( ord_less_eq_nat_o @ A @ B )
% 0.29/0.48 <=> ( ord_less_eq_int_o @ ( image_nat_int @ semiri1621563631at_int @ A ) @ ( image_nat_int @ semiri1621563631at_int @ B ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1135_Nat__Transfer_Otransfer__int__nat__set__functions_I2_J,axiom,
% 0.29/0.48 ( bot_bot_int_o
% 0.29/0.48 = ( image_nat_int @ semiri1621563631at_int @ bot_bot_nat_o ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1136_transfer__nat__int__set__relations_I3_J,axiom,
% 0.29/0.48 ! [A: nat > $o,B: nat > $o] :
% 0.29/0.48 ( ( A = B )
% 0.29/0.48 <=> ( ( image_nat_int @ semiri1621563631at_int @ A )
% 0.29/0.48 = ( image_nat_int @ semiri1621563631at_int @ B ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1137_transfer__nat__int__set__relations_I2_J,axiom,
% 0.29/0.48 ! [X: nat,A: nat > $o] :
% 0.29/0.48 ( ( member_nat @ X @ A )
% 0.29/0.48 <=> ( member_int @ ( semiri1621563631at_int @ X ) @ ( image_nat_int @ semiri1621563631at_int @ A ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1138_int__if__cong,axiom,
% 0.29/0.48 ! [X: nat,Y: nat,P: $o] :
% 0.29/0.48 ( ( P
% 0.29/0.48 => ( ( semiri1621563631at_int @ X )
% 0.29/0.48 = ( semiri1621563631at_int @ ( if_nat @ P @ X @ Y ) ) ) )
% 0.29/0.48 & ( ~ P
% 0.29/0.48 => ( ( semiri1621563631at_int @ Y )
% 0.29/0.48 = ( semiri1621563631at_int @ ( if_nat @ P @ X @ Y ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1139_transfer__int__nat__relations_I1_J,axiom,
% 0.29/0.48 ! [X: nat,Y: nat] :
% 0.29/0.48 ( ( ( semiri1621563631at_int @ X )
% 0.29/0.48 = ( semiri1621563631at_int @ Y ) )
% 0.29/0.48 <=> ( X = Y ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1140_nonneg__int__cases,axiom,
% 0.29/0.48 ! [K: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.29/0.48 => ~ ! [N_1: nat] :
% 0.29/0.48 ( K
% 0.29/0.48 != ( semiri1621563631at_int @ N_1 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1141_nonneg__eq__int,axiom,
% 0.29/0.48 ! [Z: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.29/0.48 => ~ ! [M_1: nat] :
% 0.29/0.48 ( Z
% 0.29/0.48 != ( semiri1621563631at_int @ M_1 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1142_zero__le__imp__eq__int,axiom,
% 0.29/0.48 ! [K: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.29/0.48 => ? [N_1: nat] :
% 0.29/0.48 ( K
% 0.29/0.48 = ( semiri1621563631at_int @ N_1 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1143_int__diff__cases,axiom,
% 0.29/0.48 ! [Z: int] :
% 0.29/0.48 ~ ! [M_1: nat,N_1: nat] :
% 0.29/0.48 ( Z
% 0.29/0.48 != ( minus_minus_int @ ( semiri1621563631at_int @ M_1 ) @ ( semiri1621563631at_int @ N_1 ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1144_decr__lemma,axiom,
% 0.29/0.48 ! [X: int,Z: int,D: int] :
% 0.29/0.48 ( ( ord_less_int @ zero_zero_int @ D )
% 0.29/0.48 => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1145_zabs__less__one__iff,axiom,
% 0.29/0.48 ! [Z: int] :
% 0.29/0.48 ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
% 0.29/0.48 <=> ( Z = zero_zero_int ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1146_neg__def,axiom,
% 0.29/0.48 ! [Z_1: int] :
% 0.29/0.48 ( ( nat_neg @ Z_1 )
% 0.29/0.48 <=> ( ord_less_int @ Z_1 @ zero_zero_int ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1147_not__neg__eq__ge__0,axiom,
% 0.29/0.48 ! [X: int] :
% 0.29/0.48 ( ~ ( nat_neg @ X )
% 0.29/0.48 <=> ( ord_less_eq_int @ zero_zero_int @ X ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1148_not__neg__1,axiom,
% 0.29/0.48 ~ ( nat_neg @ one_one_int ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1149_not__neg__0,axiom,
% 0.29/0.48 ~ ( nat_neg @ zero_zero_int ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1150_abs__zmult__eq__1,axiom,
% 0.29/0.48 ! [M: int,N: int] :
% 0.29/0.48 ( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
% 0.29/0.48 = one_one_int )
% 0.29/0.48 => ( ( abs_abs_int @ M )
% 0.29/0.48 = one_one_int ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1151_not__neg__int,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ~ ( nat_neg @ ( semiri1621563631at_int @ N ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1152_abs__int__eq,axiom,
% 0.29/0.48 ! [M: nat] :
% 0.29/0.48 ( ( abs_abs_int @ ( semiri1621563631at_int @ M ) )
% 0.29/0.48 = ( semiri1621563631at_int @ M ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1153_neg__imp__number__of__eq__0,axiom,
% 0.29/0.48 ! [V: int] :
% 0.29/0.48 ( ( nat_neg @ ( number_number_of_int @ V ) )
% 0.29/0.48 => ( ( number_number_of_nat @ V )
% 0.29/0.48 = zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1154_eq__nat__number__of,axiom,
% 0.29/0.48 ! [V: int,V_1: int] :
% 0.29/0.48 ( ( ( number_number_of_nat @ V )
% 0.29/0.48 = ( number_number_of_nat @ V_1 ) )
% 0.29/0.48 <=> ( ( ( nat_neg @ ( number_number_of_int @ V ) )
% 0.29/0.48 => ( ord_less_eq_int @ ( number_number_of_int @ V_1 ) @ zero_zero_int ) )
% 0.29/0.48 & ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
% 0.29/0.48 => ( ( ( nat_neg @ ( number_number_of_int @ V_1 ) )
% 0.29/0.48 => ( ( number_number_of_int @ V )
% 0.29/0.48 = zero_zero_int ) )
% 0.29/0.48 & ( ~ ( nat_neg @ ( number_number_of_int @ V_1 ) )
% 0.29/0.48 => ( V = V_1 ) ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1155_nat__number__of__add__left,axiom,
% 0.29/0.48 ! [V_1: int,K: nat,V: int] :
% 0.29/0.48 ( ( ( nat_neg @ ( number_number_of_int @ V ) )
% 0.29/0.48 => ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K ) )
% 0.29/0.48 = ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K ) ) )
% 0.29/0.48 & ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
% 0.29/0.48 => ( ( ( nat_neg @ ( number_number_of_int @ V_1 ) )
% 0.29/0.48 => ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K ) )
% 0.29/0.48 = ( plus_plus_nat @ ( number_number_of_nat @ V ) @ K ) ) )
% 0.29/0.48 & ( ~ ( nat_neg @ ( number_number_of_int @ V_1 ) )
% 0.29/0.48 => ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K ) )
% 0.29/0.48 = ( plus_plus_nat @ ( number_number_of_nat @ ( plus_plus_int @ V @ V_1 ) ) @ K ) ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1156_int__nat__number__of,axiom,
% 0.29/0.48 ! [V: int] :
% 0.29/0.48 ( ( ( nat_neg @ ( number_number_of_int @ V ) )
% 0.29/0.48 => ( ( semiri1621563631at_int @ ( number_number_of_nat @ V ) )
% 0.29/0.48 = zero_zero_int ) )
% 0.29/0.48 & ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
% 0.29/0.48 => ( ( semiri1621563631at_int @ ( number_number_of_nat @ V ) )
% 0.29/0.48 = ( number_number_of_int @ V ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1157_incr__lemma,axiom,
% 0.29/0.48 ! [Z: int,X: int,D: int] :
% 0.29/0.48 ( ( ord_less_int @ zero_zero_int @ D )
% 0.29/0.48 => ( 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 ) @ D ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1158_int__val__lemma,axiom,
% 0.29/0.48 ! [K: int,F: nat > int,N: nat] :
% 0.29/0.48 ( ! [I: nat] :
% 0.29/0.48 ( ( ord_less_nat @ I @ N )
% 0.29/0.48 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( F @ I ) ) ) @ one_one_int ) )
% 0.29/0.48 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 0.29/0.48 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.29/0.48 => ? [I: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ I @ N )
% 0.29/0.48 & ( ( F @ I )
% 0.29/0.48 = K ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1159_nat0__intermed__int__val,axiom,
% 0.29/0.48 ! [K: int,F: nat > int,N: nat] :
% 0.29/0.48 ( ! [I: nat] :
% 0.29/0.48 ( ( ord_less_nat @ I @ N )
% 0.29/0.48 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( F @ I ) ) ) @ one_one_int ) )
% 0.29/0.48 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 0.29/0.48 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.29/0.48 => ? [I: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ I @ N )
% 0.29/0.48 & ( ( F @ I )
% 0.29/0.48 = K ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1160_nat__intermed__int__val,axiom,
% 0.29/0.48 ! [K: int,F: nat > int,N: nat,M: nat] :
% 0.29/0.48 ( ! [I: nat] :
% 0.29/0.48 ( ( ( ord_less_eq_nat @ M @ I )
% 0.29/0.48 & ( ord_less_nat @ I @ N ) )
% 0.29/0.48 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( F @ I ) ) ) @ one_one_int ) )
% 0.29/0.48 => ( ( ord_less_nat @ M @ N )
% 0.29/0.48 => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 0.29/0.48 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.29/0.48 => ? [I: nat] :
% 0.29/0.48 ( ( ord_less_eq_nat @ M @ I )
% 0.29/0.48 & ( ord_less_eq_nat @ I @ N )
% 0.29/0.48 & ( ( F @ I )
% 0.29/0.48 = K ) ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1161_Suc__nat__number__of__add,axiom,
% 0.29/0.48 ! [N: nat,V: int] :
% 0.29/0.48 ( ( ( nat_neg @ ( number_number_of_int @ V ) )
% 0.29/0.48 => ( ( suc @ ( plus_plus_nat @ ( number_number_of_nat @ V ) @ N ) )
% 0.29/0.48 = ( plus_plus_nat @ one_one_nat @ N ) ) )
% 0.29/0.48 & ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
% 0.29/0.48 => ( ( suc @ ( plus_plus_nat @ ( number_number_of_nat @ V ) @ N ) )
% 0.29/0.48 = ( plus_plus_nat @ ( number_number_of_nat @ ( succ @ V ) ) @ N ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1162_succ__def,axiom,
% 0.29/0.48 ! [K: int] :
% 0.29/0.48 ( ( succ @ K )
% 0.29/0.48 = ( plus_plus_int @ K @ one_one_int ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1163_Suc__nat__number__of,axiom,
% 0.29/0.48 ! [V: int] :
% 0.29/0.48 ( ( ( nat_neg @ ( number_number_of_int @ V ) )
% 0.29/0.48 => ( ( suc @ ( number_number_of_nat @ V ) )
% 0.29/0.48 = one_one_nat ) )
% 0.29/0.48 & ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
% 0.29/0.48 => ( ( suc @ ( number_number_of_nat @ V ) )
% 0.29/0.48 = ( number_number_of_nat @ ( succ @ V ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1164_nat__number__of__Bit1,axiom,
% 0.29/0.48 ! [W: int] :
% 0.29/0.48 ( ( ( nat_neg @ ( number_number_of_int @ W ) )
% 0.29/0.48 => ( ( number_number_of_nat @ ( bit1 @ W ) )
% 0.29/0.48 = zero_zero_nat ) )
% 0.29/0.48 & ( ~ ( nat_neg @ ( number_number_of_int @ W ) )
% 0.29/0.48 => ( ( number_number_of_nat @ ( bit1 @ W ) )
% 0.29/0.48 = ( suc @ ( plus_plus_nat @ ( number_number_of_nat @ W ) @ ( number_number_of_nat @ W ) ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1165_nat__1__add__number__of,axiom,
% 0.29/0.48 ! [V: int] :
% 0.29/0.48 ( ( ( ord_less_int @ V @ pls )
% 0.29/0.48 => ( ( plus_plus_nat @ one_one_nat @ ( number_number_of_nat @ V ) )
% 0.29/0.48 = one_one_nat ) )
% 0.29/0.48 & ( ~ ( ord_less_int @ V @ pls )
% 0.29/0.48 => ( ( plus_plus_nat @ one_one_nat @ ( number_number_of_nat @ V ) )
% 0.29/0.48 = ( number_number_of_nat @ ( succ @ V ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1166_succ__Pls,axiom,
% 0.29/0.48 ( ( succ @ pls )
% 0.29/0.48 = ( bit1 @ pls ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1167_neg__number__of__Bit1,axiom,
% 0.29/0.48 ! [W: int] :
% 0.29/0.48 ( ( nat_neg @ ( number_number_of_int @ ( bit1 @ W ) ) )
% 0.29/0.48 <=> ( nat_neg @ ( number_number_of_int @ W ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1168_not__neg__number__of__Pls,axiom,
% 0.29/0.48 ~ ( nat_neg @ ( number_number_of_int @ pls ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1169_transfer__int__nat__numerals_I4_J,axiom,
% 0.29/0.48 ( ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) )
% 0.29/0.48 = ( semiri1621563631at_int @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1170_numeral__1__eq__Suc__0,axiom,
% 0.29/0.48 ( ( number_number_of_nat @ ( bit1 @ pls ) )
% 0.29/0.48 = ( suc @ zero_zero_nat ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1171_numeral__3__eq__3,axiom,
% 0.29/0.48 ( ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) )
% 0.29/0.48 = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1172_Suc3__eq__add__3,axiom,
% 0.29/0.48 ! [N: nat] :
% 0.29/0.48 ( ( suc @ ( suc @ ( suc @ N ) ) )
% 0.29/0.48 = ( plus_plus_nat @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) @ N ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1173_Numeral1__eq1__nat,axiom,
% 0.29/0.48 ( one_one_nat
% 0.29/0.48 = ( number_number_of_nat @ ( bit1 @ pls ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1174_nat__numeral__1__eq__1,axiom,
% 0.29/0.48 ( ( number_number_of_nat @ ( bit1 @ pls ) )
% 0.29/0.48 = one_one_nat ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1175_one__is__num__one,axiom,
% 0.29/0.48 ( one_one_int
% 0.29/0.48 = ( number_number_of_int @ ( bit1 @ pls ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1176_zero__is__num__zero,axiom,
% 0.29/0.48 ( zero_zero_int
% 0.29/0.48 = ( number_number_of_int @ pls ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1177_Bit1__def,axiom,
% 0.29/0.48 ! [K: int] :
% 0.29/0.48 ( ( bit1 @ K )
% 0.29/0.48 = ( plus_plus_int @ ( plus_plus_int @ one_one_int @ K ) @ K ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1178_nat__number__of__Pls,axiom,
% 0.29/0.48 ( ( number_number_of_nat @ pls )
% 0.29/0.48 = zero_zero_nat ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1179_semiring__norm_I113_J,axiom,
% 0.29/0.48 ( zero_zero_nat
% 0.29/0.48 = ( number_number_of_nat @ pls ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1180_Pls__def,axiom,
% 0.29/0.48 pls = zero_zero_int ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1181_rel__simps_I51_J,axiom,
% 0.29/0.48 ! [K: int,L: int] :
% 0.29/0.48 ( ( ( bit1 @ K )
% 0.29/0.48 = ( bit1 @ L ) )
% 0.29/0.48 <=> ( K = L ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1182_rel__simps_I46_J,axiom,
% 0.29/0.48 ! [K: int] :
% 0.29/0.48 ( ( bit1 @ K )
% 0.29/0.48 != pls ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1183_rel__simps_I39_J,axiom,
% 0.29/0.48 ! [L: int] :
% 0.29/0.48 ( pls
% 0.29/0.48 != ( bit1 @ L ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1184_diff__bin__simps_I1_J,axiom,
% 0.29/0.48 ! [K: int] :
% 0.29/0.48 ( ( minus_minus_int @ K @ pls )
% 0.29/0.48 = K ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1185_add__Pls,axiom,
% 0.29/0.48 ! [K: int] :
% 0.29/0.48 ( ( plus_plus_int @ pls @ K )
% 0.29/0.48 = K ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1186_add__Pls__right,axiom,
% 0.29/0.48 ! [K: int] :
% 0.29/0.48 ( ( plus_plus_int @ K @ pls )
% 0.29/0.48 = K ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1187_mult__Pls,axiom,
% 0.29/0.48 ! [W: int] :
% 0.29/0.48 ( ( times_times_int @ pls @ W )
% 0.29/0.48 = pls ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1188_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J,axiom,
% 0.29/0.48 ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1189_rel__simps_I19_J,axiom,
% 0.29/0.48 ord_less_eq_int @ pls @ pls ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1190_rel__simps_I22_J,axiom,
% 0.29/0.48 ! [K: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ pls @ ( bit1 @ K ) )
% 0.29/0.48 <=> ( ord_less_eq_int @ pls @ K ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1191_rel__simps_I34_J,axiom,
% 0.29/0.48 ! [K: int,L: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ ( bit1 @ K ) @ ( bit1 @ L ) )
% 0.29/0.48 <=> ( ord_less_eq_int @ K @ L ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1192_less__eq__int__code_I16_J,axiom,
% 0.29/0.48 ! [K1: int,K2: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ ( bit1 @ K1 ) @ ( bit1 @ K2 ) )
% 0.29/0.48 <=> ( ord_less_eq_int @ K1 @ K2 ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1193_rel__simps_I2_J,axiom,
% 0.29/0.48 ~ ( ord_less_int @ pls @ pls ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1194_rel__simps_I12_J,axiom,
% 0.29/0.48 ! [K: int] :
% 0.29/0.48 ( ( ord_less_int @ ( bit1 @ K ) @ pls )
% 0.29/0.48 <=> ( ord_less_int @ K @ pls ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1195_rel__simps_I17_J,axiom,
% 0.29/0.48 ! [K: int,L: int] :
% 0.29/0.48 ( ( ord_less_int @ ( bit1 @ K ) @ ( bit1 @ L ) )
% 0.29/0.48 <=> ( ord_less_int @ K @ L ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1196_less__int__code_I16_J,axiom,
% 0.29/0.48 ! [K1: int,K2: int] :
% 0.29/0.48 ( ( ord_less_int @ ( bit1 @ K1 ) @ ( bit1 @ K2 ) )
% 0.29/0.48 <=> ( ord_less_int @ K1 @ K2 ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1197_rel__simps_I29_J,axiom,
% 0.29/0.48 ! [K: int] :
% 0.29/0.48 ( ( ord_less_eq_int @ ( bit1 @ K ) @ pls )
% 0.29/0.48 <=> ( ord_less_int @ K @ pls ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(fact_1198_rel__simps_I5_J,axiom,
% 0.29/0.48 ! [K: int] :
% 0.29/0.48 ( ( ord_less_int @ pls @ ( bit1 @ K ) )
% 0.29/0.48 <=> ( ord_less_eq_int @ pls @ K ) ) ).
% 0.29/0.48
% 0.29/0.48 %----Helper facts (9)
% 0.29/0.48 thf(help_fequal_1_1_fequal_000t__a_T,axiom,
% 0.29/0.48 ! [X: x_a,Y: x_a] :
% 0.29/0.48 ( ~ ( fequal_a @ X @ Y )
% 0.29/0.48 | ( X = Y ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(help_fequal_2_1_fequal_000t__a_T,axiom,
% 0.29/0.48 ! [X: x_a,Y: x_a] :
% 0.29/0.48 ( ( X != Y )
% 0.29/0.48 | ( fequal_a @ X @ Y ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(help_If_1_1_If_000tc__Nat__Onat_T,axiom,
% 0.29/0.48 ! [X: nat,Y: nat] :
% 0.29/0.48 ( ( if_nat @ $true @ X @ Y )
% 0.29/0.48 = X ) ).
% 0.29/0.48
% 0.29/0.48 thf(help_If_2_1_If_000tc__Nat__Onat_T,axiom,
% 0.29/0.48 ! [X: nat,Y: nat] :
% 0.29/0.48 ( ( if_nat @ $false @ X @ Y )
% 0.29/0.48 = Y ) ).
% 0.29/0.48
% 0.29/0.48 thf(help_If_3_1_If_000tc__Nat__Onat_T,axiom,
% 0.29/0.48 ! [P: $o] :
% 0.29/0.48 ( ( P = $true )
% 0.29/0.48 | ( P = $false ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(help_fequal_1_1_fequal_000tc__Int__Oint_T,axiom,
% 0.29/0.48 ! [X: int,Y: int] :
% 0.29/0.48 ( ~ ( fequal_int @ X @ Y )
% 0.29/0.48 | ( X = Y ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(help_fequal_2_1_fequal_000tc__Int__Oint_T,axiom,
% 0.29/0.48 ! [X: int,Y: int] :
% 0.29/0.48 ( ( X != Y )
% 0.29/0.48 | ( fequal_int @ X @ Y ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(help_fequal_1_1_fequal_000tc__Nat__Onat_T,axiom,
% 0.29/0.48 ! [X: nat,Y: nat] :
% 0.29/0.48 ( ~ ( fequal_nat @ X @ Y )
% 0.29/0.48 | ( X = Y ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(help_fequal_2_1_fequal_000tc__Nat__Onat_T,axiom,
% 0.29/0.48 ! [X: nat,Y: nat] :
% 0.29/0.48 ( ( X != Y )
% 0.29/0.48 | ( fequal_nat @ X @ Y ) ) ).
% 0.29/0.48
% 0.29/0.48 %----Conjectures (7)
% 0.29/0.48 thf(conj_0,hypothesis,
% 0.29/0.48 finite_finite_pname @ u ).
% 0.29/0.48
% 0.29/0.48 thf(conj_1,hypothesis,
% 0.29/0.48 ord_less_eq_a_o @ g @ ( image_pname_a @ mgt_call @ u ) ).
% 0.29/0.48
% 0.29/0.48 thf(conj_2,hypothesis,
% 0.29/0.48 ord_less_eq_nat @ ( suc @ na ) @ ( finite_card_a @ ( image_pname_a @ mgt_call @ u ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(conj_3,hypothesis,
% 0.29/0.48 ( ( finite_card_a @ g )
% 0.29/0.48 = ( minus_minus_nat @ ( finite_card_a @ ( image_pname_a @ mgt_call @ u ) ) @ ( suc @ na ) ) ) ).
% 0.29/0.48
% 0.29/0.48 thf(conj_4,hypothesis,
% 0.29/0.48 member_pname @ pn @ u ).
% 0.29/0.48
% 0.29/0.48 thf(conj_5,hypothesis,
% 0.29/0.48 ~ ( member_a @ ( mgt_call @ pn ) @ g ) ).
% 0.29/0.48
% 0.29/0.48 thf(conj_6,conjecture,
% 0.29/0.48 ord_less_eq_a_o @ ( insert_a @ ( mgt_call @ pn ) @ g ) @ ( image_pname_a @ mgt_call @ u ) ).
% 0.29/0.48
% 0.29/0.48 %------------------------------------------------------------------------------
% 0.29/0.48 ------------------------
% 3.29/1.63 ---- Embedded in THF ---
% 3.29/1.63 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.29/1.63 %%% Generated on Tue Feb 24 13:40:31 EST 2026
% 3.29/1.63 %%% using '$$dhol' embedding, version 1.3.0.
% 3.29/1.63 %%% Logic specification used:
% 3.29/1.63 %%% thf(spec, logic, $$dhol).
% 3.29/1.63
% 3.29/1.63 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.KmoHMpFwsY/DTF2DTF2194.p
% See solution above
% 3.29/1.64 ------------------------
% 3.36/1.71 ---- Cleaned THF ---
% 3.36/1.72 thf(ty_ty_t__a,type,
% 3.36/1.72 x_a: $tType ).
% 3.36/1.72
% 3.36/1.72 thf(ty_ty_tc__Com__Ocom,type,
% 3.36/1.72 com: $tType ).
% 3.36/1.72
% 3.36/1.72 thf(ty_ty_tc__Com__Opname,type,
% 3.36/1.72 pname: $tType ).
% 3.36/1.72
% 3.36/1.72 thf(ty_ty_tc__Int__Oint,type,
% 3.36/1.72 int: $tType ).
% 3.36/1.72
% 3.36/1.72 thf(ty_ty_tc__Nat__Onat,type,
% 3.36/1.72 nat: $tType ).
% 3.36/1.72
% 3.36/1.72 thf(ty_ty_tc__Option__Ooption_Itc__Com__Ocom_J,type,
% 3.36/1.72 option_com: $tType ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Com_Obody,type,
% 3.36/1.72 body: pname > option_com ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Ex,type,
% 3.36/1.72 ex: ( int > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ocard_000_062_It__a_M_Eo_J,type,
% 3.36/1.72 finite_card_a_o: ( ( x_a > $o ) > $o ) > nat ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ocard_000_062_Itc__Com__Opname_M_Eo_J,type,
% 3.36/1.72 finite_card_pname_o: ( ( pname > $o ) > $o ) > nat ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ocard_000_062_Itc__Int__Oint_M_Eo_J,type,
% 3.36/1.72 finite_card_int_o: ( ( int > $o ) > $o ) > nat ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ocard_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.36/1.72 finite_card_nat_o: ( ( nat > $o ) > $o ) > nat ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ocard_000t__a,type,
% 3.36/1.72 finite_card_a: ( x_a > $o ) > nat ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ocard_000tc__Com__Opname,type,
% 3.36/1.72 finite_card_pname: ( pname > $o ) > nat ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ocard_000tc__Int__Oint,type,
% 3.36/1.72 finite_card_int: ( int > $o ) > nat ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ocard_000tc__Nat__Onat,type,
% 3.36/1.72 finite_card_nat: ( nat > $o ) > nat ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ofinite_000_062_I_062_It__a_M_Eo_J_M_Eo_J,type,
% 3.36/1.72 finite_finite_a_o_o: ( ( ( x_a > $o ) > $o ) > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ofinite_000_062_I_062_Itc__Com__Opname_M_Eo_J_M_Eo_J,type,
% 3.36/1.72 finite1066544169me_o_o: ( ( ( pname > $o ) > $o ) > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ofinite_000_062_I_062_Itc__Int__Oint_M_Eo_J_M_Eo_J,type,
% 3.36/1.72 finite229719499nt_o_o: ( ( ( int > $o ) > $o ) > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ofinite_000_062_I_062_Itc__Nat__Onat_M_Eo_J_M_Eo_J,type,
% 3.36/1.72 finite1676163439at_o_o: ( ( ( nat > $o ) > $o ) > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ofinite_000_062_It__a_M_Eo_J,type,
% 3.36/1.72 finite_finite_a_o: ( ( x_a > $o ) > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ofinite_000_062_Itc__Com__Opname_M_Eo_J,type,
% 3.36/1.72 finite297249702name_o: ( ( pname > $o ) > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ofinite_000_062_Itc__Int__Oint_M_Eo_J,type,
% 3.36/1.72 finite_finite_int_o: ( ( int > $o ) > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ofinite_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.36/1.72 finite_finite_nat_o: ( ( nat > $o ) > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ofinite_000t__a,type,
% 3.36/1.72 finite_finite_a: ( x_a > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ofinite_000tc__Com__Opname,type,
% 3.36/1.72 finite_finite_pname: ( pname > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ofinite_000tc__Int__Oint,type,
% 3.36/1.72 finite_finite_int: ( int > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ofinite_000tc__Nat__Onat,type,
% 3.36/1.72 finite_finite_nat: ( nat > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ofolding__one_000t__a,type,
% 3.36/1.72 finite_folding_one_a: ( x_a > x_a > x_a ) > ( ( x_a > $o ) > x_a ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ofolding__one_000tc__Com__Opname,type,
% 3.36/1.72 finite1282449217_pname: ( pname > pname > pname ) > ( ( pname > $o ) > pname ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ofolding__one_000tc__Int__Oint,type,
% 3.36/1.72 finite1626084323ne_int: ( int > int > int ) > ( ( int > $o ) > int ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ofolding__one_000tc__Nat__Onat,type,
% 3.36/1.72 finite988810631ne_nat: ( nat > nat > nat ) > ( ( nat > $o ) > nat ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ofolding__one__idem_000t__a,type,
% 3.36/1.72 finite1819937229idem_a: ( x_a > x_a > x_a ) > ( ( x_a > $o ) > x_a ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Com__Opname,type,
% 3.36/1.72 finite89670078_pname: ( pname > pname > pname ) > ( ( pname > $o ) > pname ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Int__Oint,type,
% 3.36/1.72 finite1432773856em_int: ( int > int > int ) > ( ( int > $o ) > int ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Nat__Onat,type,
% 3.36/1.72 finite795500164em_nat: ( nat > nat > nat ) > ( ( nat > $o ) > nat ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Groups_Oabs__class_Oabs_000tc__Int__Oint,type,
% 3.36/1.72 abs_abs_int: int > int ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Groups_Ominus__class_Ominus_000_062_It__a_M_Eo_J,type,
% 3.36/1.72 minus_minus_a_o: ( x_a > $o ) > ( x_a > $o ) > x_a > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Com__Opname_M_Eo_J,type,
% 3.36/1.72 minus_minus_pname_o: ( pname > $o ) > ( pname > $o ) > pname > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Int__Oint_M_Eo_J,type,
% 3.36/1.72 minus_minus_int_o: ( int > $o ) > ( int > $o ) > int > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.36/1.72 minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Groups_Ominus__class_Ominus_000tc__Int__Oint,type,
% 3.36/1.72 minus_minus_int: int > int > int ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
% 3.36/1.72 minus_minus_nat: nat > nat > nat ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Groups_Oone__class_Oone_000tc__Int__Oint,type,
% 3.36/1.72 one_one_int: int ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
% 3.36/1.72 one_one_nat: nat ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Groups_Oplus__class_Oplus_000tc__Int__Oint,type,
% 3.36/1.72 plus_plus_int: int > int > int ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
% 3.36/1.72 plus_plus_nat: nat > nat > nat ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Groups_Otimes__class_Otimes_000tc__Int__Oint,type,
% 3.36/1.72 times_times_int: int > int > int ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
% 3.36/1.72 times_times_nat: nat > nat > nat ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Groups_Ozero__class_Ozero_000tc__Int__Oint,type,
% 3.36/1.72 zero_zero_int: int ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
% 3.36/1.72 zero_zero_nat: nat ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_HOL_OThe_000t__a,type,
% 3.36/1.72 the_a: ( x_a > $o ) > x_a ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_HOL_OThe_000tc__Int__Oint,type,
% 3.36/1.72 the_int: ( int > $o ) > int ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_HOL_OThe_000tc__Nat__Onat,type,
% 3.36/1.72 the_nat: ( nat > $o ) > nat ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_If_000tc__Nat__Onat,type,
% 3.36/1.72 if_nat: $o > nat > nat > nat ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Int_OBit1,type,
% 3.36/1.72 bit1: int > int ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Int_OPls,type,
% 3.36/1.72 pls: int ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Int_Onumber__class_Onumber__of_000tc__Int__Oint,type,
% 3.36/1.72 number_number_of_int: int > int ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Int_Onumber__class_Onumber__of_000tc__Nat__Onat,type,
% 3.36/1.72 number_number_of_nat: int > nat ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Int_Osucc,type,
% 3.36/1.72 succ: int > int ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Nat_OSuc,type,
% 3.36/1.72 suc: nat > nat ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Nat_Onat_Onat__case_000_Eo,type,
% 3.36/1.72 nat_case_o: $o > ( nat > $o ) > nat > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Nat_Onat_Onat__case_000tc__Nat__Onat,type,
% 3.36/1.72 nat_case_nat: nat > ( nat > nat ) > nat > nat ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Int__Oint,type,
% 3.36/1.72 semiri1621563631at_int: nat > int ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Nat__Numeral_Oneg,type,
% 3.36/1.72 nat_neg: int > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Nat__Transfer_Otsub,type,
% 3.36/1.72 nat_tsub: int > int > int ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Option_Othe_000tc__Com__Ocom,type,
% 3.36/1.72 the_com: option_com > com ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Orderings_Obot__class_Obot_000_062_It__a_M_Eo_J,type,
% 3.36/1.72 bot_bot_a_o: x_a > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Com__Opname_M_Eo_J,type,
% 3.36/1.72 bot_bot_pname_o: pname > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Int__Oint_M_Eo_J,type,
% 3.36/1.72 bot_bot_int_o: int > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.36/1.72 bot_bot_nat_o: nat > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Orderings_Obot__class_Obot_000_Eo,type,
% 3.36/1.72 bot_bot_o: $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Orderings_Obot__class_Obot_000tc__Nat__Onat,type,
% 3.36/1.72 bot_bot_nat: nat ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Orderings_Oord__class_Oless_000_062_Itc__Int__Oint_M_Eo_J,type,
% 3.36/1.72 ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Orderings_Oord__class_Oless_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.36/1.72 ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Orderings_Oord__class_Oless_000tc__Int__Oint,type,
% 3.36/1.72 ord_less_int: int > int > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,
% 3.36/1.72 ord_less_nat: nat > nat > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_I_062_It__a_M_Eo_J_M_Eo_J,type,
% 3.36/1.72 ord_less_eq_a_o_o: ( ( x_a > $o ) > $o ) > ( ( x_a > $o ) > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_I_062_Itc__Com__Opname_M_Eo_J_M_Eo_,type,
% 3.36/1.72 ord_le1205211808me_o_o: ( ( pname > $o ) > $o ) > ( ( pname > $o ) > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_I_062_Itc__Int__Oint_M_Eo_J_M_Eo_J,type,
% 3.36/1.72 ord_less_eq_int_o_o: ( ( int > $o ) > $o ) > ( ( int > $o ) > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_I_062_Itc__Nat__Onat_M_Eo_J_M_Eo_J,type,
% 3.36/1.72 ord_less_eq_nat_o_o: ( ( nat > $o ) > $o ) > ( ( nat > $o ) > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_It__a_M_Eo_J,type,
% 3.36/1.72 ord_less_eq_a_o: ( x_a > $o ) > ( x_a > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Com__Opname_M_Eo_J,type,
% 3.36/1.72 ord_less_eq_pname_o: ( pname > $o ) > ( pname > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Int__Oint_M_Eo_J,type,
% 3.36/1.72 ord_less_eq_int_o: ( int > $o ) > ( int > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.36/1.72 ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Orderings_Oord__class_Oless__eq_000_Eo,type,
% 3.36/1.72 ord_less_eq_o: $o > $o > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Int__Oint,type,
% 3.36/1.72 ord_less_eq_int: int > int > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
% 3.36/1.72 ord_less_eq_nat: nat > nat > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_OCollect_000_062_I_062_It__a_M_Eo_J_M_Eo_J,type,
% 3.36/1.72 collect_a_o_o: ( ( ( x_a > $o ) > $o ) > $o ) > ( ( x_a > $o ) > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_OCollect_000_062_I_062_Itc__Com__Opname_M_Eo_J_M_Eo_J,type,
% 3.36/1.72 collect_pname_o_o: ( ( ( pname > $o ) > $o ) > $o ) > ( ( pname > $o ) > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_OCollect_000_062_I_062_Itc__Int__Oint_M_Eo_J_M_Eo_J,type,
% 3.36/1.72 collect_int_o_o: ( ( ( int > $o ) > $o ) > $o ) > ( ( int > $o ) > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_OCollect_000_062_I_062_Itc__Nat__Onat_M_Eo_J_M_Eo_J,type,
% 3.36/1.72 collect_nat_o_o: ( ( ( nat > $o ) > $o ) > $o ) > ( ( nat > $o ) > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_OCollect_000_062_It__a_M_Eo_J,type,
% 3.36/1.72 collect_a_o: ( ( x_a > $o ) > $o ) > ( x_a > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_OCollect_000_062_Itc__Com__Opname_M_Eo_J,type,
% 3.36/1.72 collect_pname_o: ( ( pname > $o ) > $o ) > ( pname > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_OCollect_000_062_Itc__Int__Oint_M_Eo_J,type,
% 3.36/1.72 collect_int_o: ( ( int > $o ) > $o ) > ( int > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_OCollect_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.36/1.72 collect_nat_o: ( ( nat > $o ) > $o ) > ( nat > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_OCollect_000t__a,type,
% 3.36/1.72 collect_a: ( x_a > $o ) > x_a > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_OCollect_000tc__Com__Opname,type,
% 3.36/1.72 collect_pname: ( pname > $o ) > pname > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_OCollect_000tc__Int__Oint,type,
% 3.36/1.72 collect_int: ( int > $o ) > int > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_OCollect_000tc__Nat__Onat,type,
% 3.36/1.72 collect_nat: ( nat > $o ) > nat > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000_062_It__a_M_Eo_J_000t__a,type,
% 3.36/1.72 image_a_o_a: ( ( x_a > $o ) > x_a ) > ( ( x_a > $o ) > $o ) > x_a > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000_062_It__a_M_Eo_J_000tc__Com__Opname,type,
% 3.36/1.72 image_a_o_pname: ( ( x_a > $o ) > pname ) > ( ( x_a > $o ) > $o ) > pname > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000_062_It__a_M_Eo_J_000tc__Int__Oint,type,
% 3.36/1.72 image_a_o_int: ( ( x_a > $o ) > int ) > ( ( x_a > $o ) > $o ) > int > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000_062_It__a_M_Eo_J_000tc__Nat__Onat,type,
% 3.36/1.72 image_a_o_nat: ( ( x_a > $o ) > nat ) > ( ( x_a > $o ) > $o ) > nat > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000_062_Itc__Com__Opname_M_Eo_J_000t__a,type,
% 3.36/1.72 image_pname_o_a: ( ( pname > $o ) > x_a ) > ( ( pname > $o ) > $o ) > x_a > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000_062_Itc__Com__Opname_M_Eo_J_000tc__Com__Opname,type,
% 3.36/1.72 image_pname_o_pname: ( ( pname > $o ) > pname ) > ( ( pname > $o ) > $o ) > pname > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000_062_Itc__Com__Opname_M_Eo_J_000tc__Int__Oint,type,
% 3.36/1.72 image_pname_o_int: ( ( pname > $o ) > int ) > ( ( pname > $o ) > $o ) > int > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000_062_Itc__Com__Opname_M_Eo_J_000tc__Nat__Onat,type,
% 3.36/1.72 image_pname_o_nat: ( ( pname > $o ) > nat ) > ( ( pname > $o ) > $o ) > nat > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000_062_Itc__Int__Oint_M_Eo_J_000t__a,type,
% 3.36/1.72 image_int_o_a: ( ( int > $o ) > x_a ) > ( ( int > $o ) > $o ) > x_a > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000_062_Itc__Int__Oint_M_Eo_J_000tc__Com__Opname,type,
% 3.36/1.72 image_int_o_pname: ( ( int > $o ) > pname ) > ( ( int > $o ) > $o ) > pname > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000_062_Itc__Int__Oint_M_Eo_J_000tc__Int__Oint,type,
% 3.36/1.72 image_int_o_int: ( ( int > $o ) > int ) > ( ( int > $o ) > $o ) > int > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000_062_Itc__Int__Oint_M_Eo_J_000tc__Nat__Onat,type,
% 3.36/1.72 image_int_o_nat: ( ( int > $o ) > nat ) > ( ( int > $o ) > $o ) > nat > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000_062_Itc__Nat__Onat_M_Eo_J_000t__a,type,
% 3.36/1.72 image_nat_o_a: ( ( nat > $o ) > x_a ) > ( ( nat > $o ) > $o ) > x_a > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000_062_Itc__Nat__Onat_M_Eo_J_000tc__Com__Opname,type,
% 3.36/1.72 image_nat_o_pname: ( ( nat > $o ) > pname ) > ( ( nat > $o ) > $o ) > pname > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000_062_Itc__Nat__Onat_M_Eo_J_000tc__Int__Oint,type,
% 3.36/1.72 image_nat_o_int: ( ( nat > $o ) > int ) > ( ( nat > $o ) > $o ) > int > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000_062_Itc__Nat__Onat_M_Eo_J_000tc__Nat__Onat,type,
% 3.36/1.72 image_nat_o_nat: ( ( nat > $o ) > nat ) > ( ( nat > $o ) > $o ) > nat > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000t__a_000_062_It__a_M_Eo_J,type,
% 3.36/1.72 image_a_a_o: ( x_a > x_a > $o ) > ( x_a > $o ) > ( x_a > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000t__a_000_062_Itc__Com__Opname_M_Eo_J,type,
% 3.36/1.72 image_a_pname_o: ( x_a > pname > $o ) > ( x_a > $o ) > ( pname > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000t__a_000_062_Itc__Int__Oint_M_Eo_J,type,
% 3.36/1.72 image_a_int_o: ( x_a > int > $o ) > ( x_a > $o ) > ( int > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000t__a_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.36/1.72 image_a_nat_o: ( x_a > nat > $o ) > ( x_a > $o ) > ( nat > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000t__a_000t__a,type,
% 3.36/1.72 image_a_a: ( x_a > x_a ) > ( x_a > $o ) > x_a > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000t__a_000tc__Com__Opname,type,
% 3.36/1.72 image_a_pname: ( x_a > pname ) > ( x_a > $o ) > pname > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000t__a_000tc__Int__Oint,type,
% 3.36/1.72 image_a_int: ( x_a > int ) > ( x_a > $o ) > int > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000t__a_000tc__Nat__Onat,type,
% 3.36/1.72 image_a_nat: ( x_a > nat ) > ( x_a > $o ) > nat > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000tc__Com__Opname_000_062_It__a_M_Eo_J,type,
% 3.36/1.72 image_pname_a_o: ( pname > x_a > $o ) > ( pname > $o ) > ( x_a > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000tc__Com__Opname_000_062_Itc__Com__Opname_M_Eo_J,type,
% 3.36/1.72 image_pname_pname_o: ( pname > pname > $o ) > ( pname > $o ) > ( pname > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000tc__Com__Opname_000_062_Itc__Int__Oint_M_Eo_J,type,
% 3.36/1.72 image_pname_int_o: ( pname > int > $o ) > ( pname > $o ) > ( int > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000tc__Com__Opname_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.36/1.72 image_pname_nat_o: ( pname > nat > $o ) > ( pname > $o ) > ( nat > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000tc__Com__Opname_000t__a,type,
% 3.36/1.72 image_pname_a: ( pname > x_a ) > ( pname > $o ) > x_a > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000tc__Com__Opname_000tc__Com__Opname,type,
% 3.36/1.72 image_pname_pname: ( pname > pname ) > ( pname > $o ) > pname > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000tc__Com__Opname_000tc__Int__Oint,type,
% 3.36/1.72 image_pname_int: ( pname > int ) > ( pname > $o ) > int > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000tc__Com__Opname_000tc__Nat__Onat,type,
% 3.36/1.72 image_pname_nat: ( pname > nat ) > ( pname > $o ) > nat > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000tc__Int__Oint_000_062_It__a_M_Eo_J,type,
% 3.36/1.72 image_int_a_o: ( int > x_a > $o ) > ( int > $o ) > ( x_a > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000tc__Int__Oint_000_062_Itc__Com__Opname_M_Eo_J,type,
% 3.36/1.72 image_int_pname_o: ( int > pname > $o ) > ( int > $o ) > ( pname > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000tc__Int__Oint_000_062_Itc__Int__Oint_M_Eo_J,type,
% 3.36/1.72 image_int_int_o: ( int > int > $o ) > ( int > $o ) > ( int > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000tc__Int__Oint_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.36/1.72 image_int_nat_o: ( int > nat > $o ) > ( int > $o ) > ( nat > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000tc__Int__Oint_000t__a,type,
% 3.36/1.72 image_int_a: ( int > x_a ) > ( int > $o ) > x_a > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000tc__Int__Oint_000tc__Com__Opname,type,
% 3.36/1.72 image_int_pname: ( int > pname ) > ( int > $o ) > pname > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000tc__Nat__Onat_000_062_It__a_M_Eo_J,type,
% 3.36/1.72 image_nat_a_o: ( nat > x_a > $o ) > ( nat > $o ) > ( x_a > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000tc__Nat__Onat_000_062_Itc__Com__Opname_M_Eo_J,type,
% 3.36/1.72 image_nat_pname_o: ( nat > pname > $o ) > ( nat > $o ) > ( pname > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000tc__Nat__Onat_000_062_Itc__Int__Oint_M_Eo_J,type,
% 3.36/1.72 image_nat_int_o: ( nat > int > $o ) > ( nat > $o ) > ( int > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000tc__Nat__Onat_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.36/1.72 image_nat_nat_o: ( nat > nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000tc__Nat__Onat_000t__a,type,
% 3.36/1.72 image_nat_a: ( nat > x_a ) > ( nat > $o ) > x_a > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Com__Opname,type,
% 3.36/1.72 image_nat_pname: ( nat > pname ) > ( nat > $o ) > pname > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Int__Oint,type,
% 3.36/1.72 image_nat_int: ( nat > int ) > ( nat > $o ) > int > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oinsert_000_062_It__a_M_Eo_J,type,
% 3.36/1.72 insert_a_o: ( x_a > $o ) > ( ( x_a > $o ) > $o ) > ( x_a > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oinsert_000_062_Itc__Com__Opname_M_Eo_J,type,
% 3.36/1.72 insert_pname_o: ( pname > $o ) > ( ( pname > $o ) > $o ) > ( pname > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oinsert_000_062_Itc__Int__Oint_M_Eo_J,type,
% 3.36/1.72 insert_int_o: ( int > $o ) > ( ( int > $o ) > $o ) > ( int > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oinsert_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.36/1.72 insert_nat_o: ( nat > $o ) > ( ( nat > $o ) > $o ) > ( nat > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oinsert_000t__a,type,
% 3.36/1.72 insert_a: x_a > ( x_a > $o ) > x_a > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oinsert_000tc__Com__Opname,type,
% 3.36/1.72 insert_pname: pname > ( pname > $o ) > pname > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oinsert_000tc__Int__Oint,type,
% 3.36/1.72 insert_int: int > ( int > $o ) > int > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Oinsert_000tc__Nat__Onat,type,
% 3.36/1.72 insert_nat: nat > ( nat > $o ) > nat > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Othe__elem_000t__a,type,
% 3.36/1.72 the_elem_a: ( x_a > $o ) > x_a ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Othe__elem_000tc__Int__Oint,type,
% 3.36/1.72 the_elem_int: ( int > $o ) > int ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_Set_Othe__elem_000tc__Nat__Onat,type,
% 3.36/1.72 the_elem_nat: ( nat > $o ) > nat ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_fequal_000t__a,type,
% 3.36/1.72 fequal_a: x_a > x_a > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_fequal_000tc__Int__Oint,type,
% 3.36/1.72 fequal_int: int > int > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_fequal_000tc__Nat__Onat,type,
% 3.36/1.72 fequal_nat: nat > nat > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_member_000_062_It__a_M_Eo_J,type,
% 3.36/1.72 member_a_o: ( x_a > $o ) > ( ( x_a > $o ) > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_member_000_062_Itc__Com__Opname_M_Eo_J,type,
% 3.36/1.72 member_pname_o: ( pname > $o ) > ( ( pname > $o ) > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_member_000_062_Itc__Int__Oint_M_Eo_J,type,
% 3.36/1.72 member_int_o: ( int > $o ) > ( ( int > $o ) > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_member_000_062_Itc__Nat__Onat_M_Eo_J,type,
% 3.36/1.72 member_nat_o: ( nat > $o ) > ( ( nat > $o ) > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_member_000t__a,type,
% 3.36/1.72 member_a: x_a > ( x_a > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_member_000tc__Com__Opname,type,
% 3.36/1.72 member_pname: pname > ( pname > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_member_000tc__Int__Oint,type,
% 3.36/1.72 member_int: int > ( int > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_c_member_000tc__Nat__Onat,type,
% 3.36/1.72 member_nat: nat > ( nat > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_v_G,type,
% 3.36/1.72 g: x_a > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_v_P,type,
% 3.36/1.72 p: ( x_a > $o ) > ( x_a > $o ) > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_v_U,type,
% 3.36/1.72 u: pname > $o ).
% 3.36/1.72
% 3.36/1.72 thf(sy_v_mgt,type,
% 3.36/1.72 mgt: com > x_a ).
% 3.36/1.72
% 3.36/1.72 thf(sy_v_mgt__call,type,
% 3.36/1.72 mgt_call: pname > x_a ).
% 3.36/1.72
% 3.36/1.72 thf(sy_v_na,type,
% 3.36/1.72 na: nat ).
% 3.36/1.72
% 3.36/1.72 thf(sy_v_pn,type,
% 3.36/1.72 pn: pname ).
% 3.36/1.72
% 3.36/1.72 thf(sy_v_wt,type,
% 3.36/1.72 wt: com > $o ).
% 3.36/1.72
% 3.36/1.72 thf(fact_0_assms_I1_J,axiom,
% 3.36/1.72 ! [Ts: x_a > $o,G: x_a > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o @ Ts @ G )
% 3.36/1.72 => ( p @ G @ Ts ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_1_finite__Collect__subsets,axiom,
% 3.36/1.72 ! [A_161: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o @ A_161 )
% 3.36/1.72 => ( finite229719499nt_o_o
% 3.36/1.72 @ ( collect_int_o_o
% 3.36/1.72 @ ^ [B_26: ( int > $o ) > $o] : ( ord_less_eq_int_o_o @ B_26 @ A_161 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_2_finite__Collect__subsets,axiom,
% 3.36/1.72 ! [A_161: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o @ A_161 )
% 3.36/1.72 => ( finite1676163439at_o_o
% 3.36/1.72 @ ( collect_nat_o_o
% 3.36/1.72 @ ^ [B_26: ( nat > $o ) > $o] : ( ord_less_eq_nat_o_o @ B_26 @ A_161 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_3_finite__Collect__subsets,axiom,
% 3.36/1.72 ! [A_161: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o @ A_161 )
% 3.36/1.72 => ( finite1066544169me_o_o
% 3.36/1.72 @ ( collect_pname_o_o
% 3.36/1.72 @ ^ [B_26: ( pname > $o ) > $o] : ( ord_le1205211808me_o_o @ B_26 @ A_161 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_4_finite__Collect__subsets,axiom,
% 3.36/1.72 ! [A_161: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o @ A_161 )
% 3.36/1.72 => ( finite_finite_a_o_o
% 3.36/1.72 @ ( collect_a_o_o
% 3.36/1.72 @ ^ [B_26: ( x_a > $o ) > $o] : ( ord_less_eq_a_o_o @ B_26 @ A_161 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_5_finite__Collect__subsets,axiom,
% 3.36/1.72 ! [A_161: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ A_161 )
% 3.36/1.72 => ( finite_finite_a_o
% 3.36/1.72 @ ( collect_a_o
% 3.36/1.72 @ ^ [B_26: x_a > $o] : ( ord_less_eq_a_o @ B_26 @ A_161 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_6_finite__Collect__subsets,axiom,
% 3.36/1.72 ! [A_161: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ A_161 )
% 3.36/1.72 => ( finite297249702name_o
% 3.36/1.72 @ ( collect_pname_o
% 3.36/1.72 @ ^ [B_26: pname > $o] : ( ord_less_eq_pname_o @ B_26 @ A_161 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_7_finite__Collect__subsets,axiom,
% 3.36/1.72 ! [A_161: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ A_161 )
% 3.36/1.72 => ( finite_finite_nat_o
% 3.36/1.72 @ ( collect_nat_o
% 3.36/1.72 @ ^ [B_26: nat > $o] : ( ord_less_eq_nat_o @ B_26 @ A_161 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_8_finite__Collect__subsets,axiom,
% 3.36/1.72 ! [A_161: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ A_161 )
% 3.36/1.72 => ( finite_finite_int_o
% 3.36/1.72 @ ( collect_int_o
% 3.36/1.72 @ ^ [B_26: int > $o] : ( ord_less_eq_int_o @ B_26 @ A_161 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_9_finite__imageI,axiom,
% 3.36/1.72 ! [H: pname > int > $o,F_42: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ F_42 )
% 3.36/1.72 => ( finite_finite_int_o @ ( image_pname_int_o @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_10_finite__imageI,axiom,
% 3.36/1.72 ! [H: pname > nat > $o,F_42: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ F_42 )
% 3.36/1.72 => ( finite_finite_nat_o @ ( image_pname_nat_o @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_11_finite__imageI,axiom,
% 3.36/1.72 ! [H: pname > pname > $o,F_42: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ F_42 )
% 3.36/1.72 => ( finite297249702name_o @ ( image_pname_pname_o @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_12_finite__imageI,axiom,
% 3.36/1.72 ! [H: pname > x_a > $o,F_42: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ F_42 )
% 3.36/1.72 => ( finite_finite_a_o @ ( image_pname_a_o @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_13_finite__imageI,axiom,
% 3.36/1.72 ! [H: nat > x_a,F_42: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ F_42 )
% 3.36/1.72 => ( finite_finite_a @ ( image_nat_a @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_14_finite__imageI,axiom,
% 3.36/1.72 ! [H: nat > int > $o,F_42: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ F_42 )
% 3.36/1.72 => ( finite_finite_int_o @ ( image_nat_int_o @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_15_finite__imageI,axiom,
% 3.36/1.72 ! [H: nat > nat > $o,F_42: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ F_42 )
% 3.36/1.72 => ( finite_finite_nat_o @ ( image_nat_nat_o @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_16_finite__imageI,axiom,
% 3.36/1.72 ! [H: nat > pname > $o,F_42: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ F_42 )
% 3.36/1.72 => ( finite297249702name_o @ ( image_nat_pname_o @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_17_finite__imageI,axiom,
% 3.36/1.72 ! [H: nat > x_a > $o,F_42: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ F_42 )
% 3.36/1.72 => ( finite_finite_a_o @ ( image_nat_a_o @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_18_finite__imageI,axiom,
% 3.36/1.72 ! [H: int > x_a,F_42: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ F_42 )
% 3.36/1.72 => ( finite_finite_a @ ( image_int_a @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_19_finite__imageI,axiom,
% 3.36/1.72 ! [H: int > int > $o,F_42: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ F_42 )
% 3.36/1.72 => ( finite_finite_int_o @ ( image_int_int_o @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_20_finite__imageI,axiom,
% 3.36/1.72 ! [H: int > nat > $o,F_42: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ F_42 )
% 3.36/1.72 => ( finite_finite_nat_o @ ( image_int_nat_o @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_21_finite__imageI,axiom,
% 3.36/1.72 ! [H: int > pname > $o,F_42: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ F_42 )
% 3.36/1.72 => ( finite297249702name_o @ ( image_int_pname_o @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_22_finite__imageI,axiom,
% 3.36/1.72 ! [H: int > x_a > $o,F_42: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ F_42 )
% 3.36/1.72 => ( finite_finite_a_o @ ( image_int_a_o @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_23_finite__imageI,axiom,
% 3.36/1.72 ! [H: x_a > pname,F_42: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ F_42 )
% 3.36/1.72 => ( finite_finite_pname @ ( image_a_pname @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_24_finite__imageI,axiom,
% 3.36/1.72 ! [H: ( int > $o ) > pname,F_42: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o @ F_42 )
% 3.36/1.72 => ( finite_finite_pname @ ( image_int_o_pname @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_25_finite__imageI,axiom,
% 3.36/1.72 ! [H: ( nat > $o ) > pname,F_42: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o @ F_42 )
% 3.36/1.72 => ( finite_finite_pname @ ( image_nat_o_pname @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_26_finite__imageI,axiom,
% 3.36/1.72 ! [H: ( pname > $o ) > pname,F_42: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o @ F_42 )
% 3.36/1.72 => ( finite_finite_pname @ ( image_pname_o_pname @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_27_finite__imageI,axiom,
% 3.36/1.72 ! [H: ( x_a > $o ) > pname,F_42: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o @ F_42 )
% 3.36/1.72 => ( finite_finite_pname @ ( image_a_o_pname @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_28_finite__imageI,axiom,
% 3.36/1.72 ! [H: x_a > nat,F_42: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ F_42 )
% 3.36/1.72 => ( finite_finite_nat @ ( image_a_nat @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_29_finite__imageI,axiom,
% 3.36/1.72 ! [H: ( int > $o ) > nat,F_42: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o @ F_42 )
% 3.36/1.72 => ( finite_finite_nat @ ( image_int_o_nat @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_30_finite__imageI,axiom,
% 3.36/1.72 ! [H: ( nat > $o ) > nat,F_42: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o @ F_42 )
% 3.36/1.72 => ( finite_finite_nat @ ( image_nat_o_nat @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_31_finite__imageI,axiom,
% 3.36/1.72 ! [H: ( pname > $o ) > nat,F_42: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o @ F_42 )
% 3.36/1.72 => ( finite_finite_nat @ ( image_pname_o_nat @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_32_finite__imageI,axiom,
% 3.36/1.72 ! [H: ( x_a > $o ) > nat,F_42: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o @ F_42 )
% 3.36/1.72 => ( finite_finite_nat @ ( image_a_o_nat @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_33_finite__imageI,axiom,
% 3.36/1.72 ! [H: x_a > int,F_42: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ F_42 )
% 3.36/1.72 => ( finite_finite_int @ ( image_a_int @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_34_finite__imageI,axiom,
% 3.36/1.72 ! [H: ( int > $o ) > int,F_42: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o @ F_42 )
% 3.36/1.72 => ( finite_finite_int @ ( image_int_o_int @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_35_finite__imageI,axiom,
% 3.36/1.72 ! [H: ( nat > $o ) > int,F_42: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o @ F_42 )
% 3.36/1.72 => ( finite_finite_int @ ( image_nat_o_int @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_36_finite__imageI,axiom,
% 3.36/1.72 ! [H: ( pname > $o ) > int,F_42: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o @ F_42 )
% 3.36/1.72 => ( finite_finite_int @ ( image_pname_o_int @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_37_finite__imageI,axiom,
% 3.36/1.72 ! [H: ( x_a > $o ) > int,F_42: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o @ F_42 )
% 3.36/1.72 => ( finite_finite_int @ ( image_a_o_int @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_38_finite__imageI,axiom,
% 3.36/1.72 ! [H: pname > x_a,F_42: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ F_42 )
% 3.36/1.72 => ( finite_finite_a @ ( image_pname_a @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_39_finite__imageI,axiom,
% 3.36/1.72 ! [H: nat > int,F_42: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ F_42 )
% 3.36/1.72 => ( finite_finite_int @ ( image_nat_int @ H @ F_42 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_40_finite_OinsertI,axiom,
% 3.36/1.72 ! [A_160: int > $o,A_159: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o @ A_159 )
% 3.36/1.72 => ( finite_finite_int_o @ ( insert_int_o @ A_160 @ A_159 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_41_finite_OinsertI,axiom,
% 3.36/1.72 ! [A_160: nat > $o,A_159: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o @ A_159 )
% 3.36/1.72 => ( finite_finite_nat_o @ ( insert_nat_o @ A_160 @ A_159 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_42_finite_OinsertI,axiom,
% 3.36/1.72 ! [A_160: pname > $o,A_159: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o @ A_159 )
% 3.36/1.72 => ( finite297249702name_o @ ( insert_pname_o @ A_160 @ A_159 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_43_finite_OinsertI,axiom,
% 3.36/1.72 ! [A_160: x_a > $o,A_159: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o @ A_159 )
% 3.36/1.72 => ( finite_finite_a_o @ ( insert_a_o @ A_160 @ A_159 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_44_finite_OinsertI,axiom,
% 3.36/1.72 ! [A_160: pname,A_159: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ A_159 )
% 3.36/1.72 => ( finite_finite_pname @ ( insert_pname @ A_160 @ A_159 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_45_finite_OinsertI,axiom,
% 3.36/1.72 ! [A_160: nat,A_159: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ A_159 )
% 3.36/1.72 => ( finite_finite_nat @ ( insert_nat @ A_160 @ A_159 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_46_finite_OinsertI,axiom,
% 3.36/1.72 ! [A_160: int,A_159: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ A_159 )
% 3.36/1.72 => ( finite_finite_int @ ( insert_int @ A_160 @ A_159 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_47_finite_OinsertI,axiom,
% 3.36/1.72 ! [A_160: x_a,A_159: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ A_159 )
% 3.36/1.72 => ( finite_finite_a @ ( insert_a @ A_160 @ A_159 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_48_card__image__le,axiom,
% 3.36/1.72 ! [F_41: pname > pname,A_158: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ A_158 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_pname @ ( image_pname_pname @ F_41 @ A_158 ) ) @ ( finite_card_pname @ A_158 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_49_card__image__le,axiom,
% 3.36/1.72 ! [F_41: x_a > x_a,A_158: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ A_158 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_a @ ( image_a_a @ F_41 @ A_158 ) ) @ ( finite_card_a @ A_158 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_50_card__image__le,axiom,
% 3.36/1.72 ! [F_41: ( int > $o ) > x_a,A_158: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o @ A_158 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_a @ ( image_int_o_a @ F_41 @ A_158 ) ) @ ( finite_card_int_o @ A_158 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_51_card__image__le,axiom,
% 3.36/1.72 ! [F_41: ( nat > $o ) > x_a,A_158: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o @ A_158 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_a @ ( image_nat_o_a @ F_41 @ A_158 ) ) @ ( finite_card_nat_o @ A_158 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_52_card__image__le,axiom,
% 3.36/1.72 ! [F_41: ( pname > $o ) > x_a,A_158: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o @ A_158 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_a @ ( image_pname_o_a @ F_41 @ A_158 ) ) @ ( finite_card_pname_o @ A_158 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_53_card__image__le,axiom,
% 3.36/1.72 ! [F_41: ( x_a > $o ) > x_a,A_158: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o @ A_158 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_a @ ( image_a_o_a @ F_41 @ A_158 ) ) @ ( finite_card_a_o @ A_158 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_54_card__image__le,axiom,
% 3.36/1.72 ! [F_41: pname > nat,A_158: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ A_158 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_nat @ ( image_pname_nat @ F_41 @ A_158 ) ) @ ( finite_card_pname @ A_158 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_55_card__image__le,axiom,
% 3.36/1.72 ! [F_41: x_a > nat,A_158: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ A_158 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_nat @ ( image_a_nat @ F_41 @ A_158 ) ) @ ( finite_card_a @ A_158 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_56_card__image__le,axiom,
% 3.36/1.72 ! [F_41: ( int > $o ) > nat,A_158: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o @ A_158 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_nat @ ( image_int_o_nat @ F_41 @ A_158 ) ) @ ( finite_card_int_o @ A_158 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_57_card__image__le,axiom,
% 3.36/1.72 ! [F_41: ( nat > $o ) > nat,A_158: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o @ A_158 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_o_nat @ F_41 @ A_158 ) ) @ ( finite_card_nat_o @ A_158 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_58_card__image__le,axiom,
% 3.36/1.72 ! [F_41: ( pname > $o ) > nat,A_158: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o @ A_158 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_nat @ ( image_pname_o_nat @ F_41 @ A_158 ) ) @ ( finite_card_pname_o @ A_158 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_59_card__image__le,axiom,
% 3.36/1.72 ! [F_41: ( x_a > $o ) > nat,A_158: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o @ A_158 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_nat @ ( image_a_o_nat @ F_41 @ A_158 ) ) @ ( finite_card_a_o @ A_158 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_60_card__image__le,axiom,
% 3.36/1.72 ! [F_41: pname > int,A_158: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ A_158 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_int @ ( image_pname_int @ F_41 @ A_158 ) ) @ ( finite_card_pname @ A_158 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_61_card__image__le,axiom,
% 3.36/1.72 ! [F_41: x_a > int,A_158: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ A_158 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_int @ ( image_a_int @ F_41 @ A_158 ) ) @ ( finite_card_a @ A_158 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_62_card__image__le,axiom,
% 3.36/1.72 ! [F_41: ( int > $o ) > int,A_158: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o @ A_158 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_int @ ( image_int_o_int @ F_41 @ A_158 ) ) @ ( finite_card_int_o @ A_158 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_63_card__image__le,axiom,
% 3.36/1.72 ! [F_41: ( nat > $o ) > int,A_158: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o @ A_158 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_int @ ( image_nat_o_int @ F_41 @ A_158 ) ) @ ( finite_card_nat_o @ A_158 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_64_card__image__le,axiom,
% 3.36/1.72 ! [F_41: ( pname > $o ) > int,A_158: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o @ A_158 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_int @ ( image_pname_o_int @ F_41 @ A_158 ) ) @ ( finite_card_pname_o @ A_158 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_65_card__image__le,axiom,
% 3.36/1.72 ! [F_41: ( x_a > $o ) > int,A_158: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o @ A_158 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_int @ ( image_a_o_int @ F_41 @ A_158 ) ) @ ( finite_card_a_o @ A_158 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_66_card__image__le,axiom,
% 3.36/1.72 ! [F_41: x_a > pname,A_158: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ A_158 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_pname @ ( image_a_pname @ F_41 @ A_158 ) ) @ ( finite_card_a @ A_158 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_67_card__image__le,axiom,
% 3.36/1.72 ! [F_41: nat > pname,A_158: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ A_158 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_pname @ ( image_nat_pname @ F_41 @ A_158 ) ) @ ( finite_card_nat @ A_158 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_68_card__image__le,axiom,
% 3.36/1.72 ! [F_41: int > pname,A_158: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ A_158 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_pname @ ( image_int_pname @ F_41 @ A_158 ) ) @ ( finite_card_int @ A_158 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_69_card__image__le,axiom,
% 3.36/1.72 ! [F_41: pname > x_a,A_158: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ A_158 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_a @ ( image_pname_a @ F_41 @ A_158 ) ) @ ( finite_card_pname @ A_158 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_70_card__image__le,axiom,
% 3.36/1.72 ! [F_41: nat > int,A_158: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ A_158 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_int @ ( image_nat_int @ F_41 @ A_158 ) ) @ ( finite_card_nat @ A_158 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_71_card__mono,axiom,
% 3.36/1.72 ! [A_157: ( int > $o ) > $o,B_89: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o @ B_89 )
% 3.36/1.72 => ( ( ord_less_eq_int_o_o @ A_157 @ B_89 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_int_o @ A_157 ) @ ( finite_card_int_o @ B_89 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_72_card__mono,axiom,
% 3.36/1.72 ! [A_157: ( nat > $o ) > $o,B_89: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o @ B_89 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o_o @ A_157 @ B_89 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_nat_o @ A_157 ) @ ( finite_card_nat_o @ B_89 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_73_card__mono,axiom,
% 3.36/1.72 ! [A_157: ( pname > $o ) > $o,B_89: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o @ B_89 )
% 3.36/1.72 => ( ( ord_le1205211808me_o_o @ A_157 @ B_89 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_pname_o @ A_157 ) @ ( finite_card_pname_o @ B_89 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_74_card__mono,axiom,
% 3.36/1.72 ! [A_157: ( x_a > $o ) > $o,B_89: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o @ B_89 )
% 3.36/1.72 => ( ( ord_less_eq_a_o_o @ A_157 @ B_89 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_a_o @ A_157 ) @ ( finite_card_a_o @ B_89 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_75_card__mono,axiom,
% 3.36/1.72 ! [A_157: pname > $o,B_89: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ B_89 )
% 3.36/1.72 => ( ( ord_less_eq_pname_o @ A_157 @ B_89 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_pname @ A_157 ) @ ( finite_card_pname @ B_89 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_76_card__mono,axiom,
% 3.36/1.72 ! [A_157: x_a > $o,B_89: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ B_89 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ A_157 @ B_89 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_a @ A_157 ) @ ( finite_card_a @ B_89 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_77_card__mono,axiom,
% 3.36/1.72 ! [A_157: nat > $o,B_89: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ B_89 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o @ A_157 @ B_89 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_nat @ A_157 ) @ ( finite_card_nat @ B_89 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_78_card__mono,axiom,
% 3.36/1.72 ! [A_157: int > $o,B_89: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ B_89 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ A_157 @ B_89 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_int @ A_157 ) @ ( finite_card_int @ B_89 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_79_card__seteq,axiom,
% 3.36/1.72 ! [A_156: ( int > $o ) > $o,B_88: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o @ B_88 )
% 3.36/1.72 => ( ( ord_less_eq_int_o_o @ A_156 @ B_88 )
% 3.36/1.72 => ( ( ord_less_eq_nat @ ( finite_card_int_o @ B_88 ) @ ( finite_card_int_o @ A_156 ) )
% 3.36/1.72 => ( A_156 = B_88 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_80_card__seteq,axiom,
% 3.36/1.72 ! [A_156: ( nat > $o ) > $o,B_88: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o @ B_88 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o_o @ A_156 @ B_88 )
% 3.36/1.72 => ( ( ord_less_eq_nat @ ( finite_card_nat_o @ B_88 ) @ ( finite_card_nat_o @ A_156 ) )
% 3.36/1.72 => ( A_156 = B_88 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_81_card__seteq,axiom,
% 3.36/1.72 ! [A_156: ( pname > $o ) > $o,B_88: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o @ B_88 )
% 3.36/1.72 => ( ( ord_le1205211808me_o_o @ A_156 @ B_88 )
% 3.36/1.72 => ( ( ord_less_eq_nat @ ( finite_card_pname_o @ B_88 ) @ ( finite_card_pname_o @ A_156 ) )
% 3.36/1.72 => ( A_156 = B_88 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_82_card__seteq,axiom,
% 3.36/1.72 ! [A_156: ( x_a > $o ) > $o,B_88: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o @ B_88 )
% 3.36/1.72 => ( ( ord_less_eq_a_o_o @ A_156 @ B_88 )
% 3.36/1.72 => ( ( ord_less_eq_nat @ ( finite_card_a_o @ B_88 ) @ ( finite_card_a_o @ A_156 ) )
% 3.36/1.72 => ( A_156 = B_88 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_83_card__seteq,axiom,
% 3.36/1.72 ! [A_156: pname > $o,B_88: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ B_88 )
% 3.36/1.72 => ( ( ord_less_eq_pname_o @ A_156 @ B_88 )
% 3.36/1.72 => ( ( ord_less_eq_nat @ ( finite_card_pname @ B_88 ) @ ( finite_card_pname @ A_156 ) )
% 3.36/1.72 => ( A_156 = B_88 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_84_card__seteq,axiom,
% 3.36/1.72 ! [A_156: x_a > $o,B_88: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ B_88 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ A_156 @ B_88 )
% 3.36/1.72 => ( ( ord_less_eq_nat @ ( finite_card_a @ B_88 ) @ ( finite_card_a @ A_156 ) )
% 3.36/1.72 => ( A_156 = B_88 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_85_card__seteq,axiom,
% 3.36/1.72 ! [A_156: nat > $o,B_88: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ B_88 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o @ A_156 @ B_88 )
% 3.36/1.72 => ( ( ord_less_eq_nat @ ( finite_card_nat @ B_88 ) @ ( finite_card_nat @ A_156 ) )
% 3.36/1.72 => ( A_156 = B_88 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_86_card__seteq,axiom,
% 3.36/1.72 ! [A_156: int > $o,B_88: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ B_88 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ A_156 @ B_88 )
% 3.36/1.72 => ( ( ord_less_eq_nat @ ( finite_card_int @ B_88 ) @ ( finite_card_int @ A_156 ) )
% 3.36/1.72 => ( A_156 = B_88 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_87_card__insert__le,axiom,
% 3.36/1.72 ! [X_53: int > $o,A_155: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o @ A_155 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_int_o @ A_155 ) @ ( finite_card_int_o @ ( insert_int_o @ X_53 @ A_155 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_88_card__insert__le,axiom,
% 3.36/1.72 ! [X_53: nat > $o,A_155: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o @ A_155 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_nat_o @ A_155 ) @ ( finite_card_nat_o @ ( insert_nat_o @ X_53 @ A_155 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_89_card__insert__le,axiom,
% 3.36/1.72 ! [X_53: pname > $o,A_155: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o @ A_155 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_pname_o @ A_155 ) @ ( finite_card_pname_o @ ( insert_pname_o @ X_53 @ A_155 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_90_card__insert__le,axiom,
% 3.36/1.72 ! [X_53: x_a > $o,A_155: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o @ A_155 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_a_o @ A_155 ) @ ( finite_card_a_o @ ( insert_a_o @ X_53 @ A_155 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_91_card__insert__le,axiom,
% 3.36/1.72 ! [X_53: pname,A_155: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ A_155 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_pname @ A_155 ) @ ( finite_card_pname @ ( insert_pname @ X_53 @ A_155 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_92_card__insert__le,axiom,
% 3.36/1.72 ! [X_53: nat,A_155: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ A_155 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_nat @ A_155 ) @ ( finite_card_nat @ ( insert_nat @ X_53 @ A_155 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_93_card__insert__le,axiom,
% 3.36/1.72 ! [X_53: int,A_155: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ A_155 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_int @ A_155 ) @ ( finite_card_int @ ( insert_int @ X_53 @ A_155 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_94_card__insert__le,axiom,
% 3.36/1.72 ! [X_53: x_a,A_155: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ A_155 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( finite_card_a @ A_155 ) @ ( finite_card_a @ ( insert_a @ X_53 @ A_155 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_95_card__insert__if,axiom,
% 3.36/1.72 ! [X_52: int > $o,A_154: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o @ A_154 )
% 3.36/1.72 => ( ( ( member_int_o @ X_52 @ A_154 )
% 3.36/1.72 => ( ( finite_card_int_o @ ( insert_int_o @ X_52 @ A_154 ) )
% 3.36/1.72 = ( finite_card_int_o @ A_154 ) ) )
% 3.36/1.72 & ( ~ ( member_int_o @ X_52 @ A_154 )
% 3.36/1.72 => ( ( finite_card_int_o @ ( insert_int_o @ X_52 @ A_154 ) )
% 3.36/1.72 = ( suc @ ( finite_card_int_o @ A_154 ) ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_96_card__insert__if,axiom,
% 3.36/1.72 ! [X_52: nat > $o,A_154: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o @ A_154 )
% 3.36/1.72 => ( ( ( member_nat_o @ X_52 @ A_154 )
% 3.36/1.72 => ( ( finite_card_nat_o @ ( insert_nat_o @ X_52 @ A_154 ) )
% 3.36/1.72 = ( finite_card_nat_o @ A_154 ) ) )
% 3.36/1.72 & ( ~ ( member_nat_o @ X_52 @ A_154 )
% 3.36/1.72 => ( ( finite_card_nat_o @ ( insert_nat_o @ X_52 @ A_154 ) )
% 3.36/1.72 = ( suc @ ( finite_card_nat_o @ A_154 ) ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_97_card__insert__if,axiom,
% 3.36/1.72 ! [X_52: pname > $o,A_154: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o @ A_154 )
% 3.36/1.72 => ( ( ( member_pname_o @ X_52 @ A_154 )
% 3.36/1.72 => ( ( finite_card_pname_o @ ( insert_pname_o @ X_52 @ A_154 ) )
% 3.36/1.72 = ( finite_card_pname_o @ A_154 ) ) )
% 3.36/1.72 & ( ~ ( member_pname_o @ X_52 @ A_154 )
% 3.36/1.72 => ( ( finite_card_pname_o @ ( insert_pname_o @ X_52 @ A_154 ) )
% 3.36/1.72 = ( suc @ ( finite_card_pname_o @ A_154 ) ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_98_card__insert__if,axiom,
% 3.36/1.72 ! [X_52: x_a > $o,A_154: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o @ A_154 )
% 3.36/1.72 => ( ( ( member_a_o @ X_52 @ A_154 )
% 3.36/1.72 => ( ( finite_card_a_o @ ( insert_a_o @ X_52 @ A_154 ) )
% 3.36/1.72 = ( finite_card_a_o @ A_154 ) ) )
% 3.36/1.72 & ( ~ ( member_a_o @ X_52 @ A_154 )
% 3.36/1.72 => ( ( finite_card_a_o @ ( insert_a_o @ X_52 @ A_154 ) )
% 3.36/1.72 = ( suc @ ( finite_card_a_o @ A_154 ) ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_99_card__insert__if,axiom,
% 3.36/1.72 ! [X_52: pname,A_154: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ A_154 )
% 3.36/1.72 => ( ( ( member_pname @ X_52 @ A_154 )
% 3.36/1.72 => ( ( finite_card_pname @ ( insert_pname @ X_52 @ A_154 ) )
% 3.36/1.72 = ( finite_card_pname @ A_154 ) ) )
% 3.36/1.72 & ( ~ ( member_pname @ X_52 @ A_154 )
% 3.36/1.72 => ( ( finite_card_pname @ ( insert_pname @ X_52 @ A_154 ) )
% 3.36/1.72 = ( suc @ ( finite_card_pname @ A_154 ) ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_100_card__insert__if,axiom,
% 3.36/1.72 ! [X_52: nat,A_154: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ A_154 )
% 3.36/1.72 => ( ( ( member_nat @ X_52 @ A_154 )
% 3.36/1.72 => ( ( finite_card_nat @ ( insert_nat @ X_52 @ A_154 ) )
% 3.36/1.72 = ( finite_card_nat @ A_154 ) ) )
% 3.36/1.72 & ( ~ ( member_nat @ X_52 @ A_154 )
% 3.36/1.72 => ( ( finite_card_nat @ ( insert_nat @ X_52 @ A_154 ) )
% 3.36/1.72 = ( suc @ ( finite_card_nat @ A_154 ) ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_101_card__insert__if,axiom,
% 3.36/1.72 ! [X_52: int,A_154: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ A_154 )
% 3.36/1.72 => ( ( ( member_int @ X_52 @ A_154 )
% 3.36/1.72 => ( ( finite_card_int @ ( insert_int @ X_52 @ A_154 ) )
% 3.36/1.72 = ( finite_card_int @ A_154 ) ) )
% 3.36/1.72 & ( ~ ( member_int @ X_52 @ A_154 )
% 3.36/1.72 => ( ( finite_card_int @ ( insert_int @ X_52 @ A_154 ) )
% 3.36/1.72 = ( suc @ ( finite_card_int @ A_154 ) ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_102_card__insert__if,axiom,
% 3.36/1.72 ! [X_52: x_a,A_154: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ A_154 )
% 3.36/1.72 => ( ( ( member_a @ X_52 @ A_154 )
% 3.36/1.72 => ( ( finite_card_a @ ( insert_a @ X_52 @ A_154 ) )
% 3.36/1.72 = ( finite_card_a @ A_154 ) ) )
% 3.36/1.72 & ( ~ ( member_a @ X_52 @ A_154 )
% 3.36/1.72 => ( ( finite_card_a @ ( insert_a @ X_52 @ A_154 ) )
% 3.36/1.72 = ( suc @ ( finite_card_a @ A_154 ) ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_103_card__insert__disjoint,axiom,
% 3.36/1.72 ! [X_51: int > $o,A_153: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o @ A_153 )
% 3.36/1.72 => ( ~ ( member_int_o @ X_51 @ A_153 )
% 3.36/1.72 => ( ( finite_card_int_o @ ( insert_int_o @ X_51 @ A_153 ) )
% 3.36/1.72 = ( suc @ ( finite_card_int_o @ A_153 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_104_card__insert__disjoint,axiom,
% 3.36/1.72 ! [X_51: nat > $o,A_153: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o @ A_153 )
% 3.36/1.72 => ( ~ ( member_nat_o @ X_51 @ A_153 )
% 3.36/1.72 => ( ( finite_card_nat_o @ ( insert_nat_o @ X_51 @ A_153 ) )
% 3.36/1.72 = ( suc @ ( finite_card_nat_o @ A_153 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_105_card__insert__disjoint,axiom,
% 3.36/1.72 ! [X_51: pname > $o,A_153: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o @ A_153 )
% 3.36/1.72 => ( ~ ( member_pname_o @ X_51 @ A_153 )
% 3.36/1.72 => ( ( finite_card_pname_o @ ( insert_pname_o @ X_51 @ A_153 ) )
% 3.36/1.72 = ( suc @ ( finite_card_pname_o @ A_153 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_106_card__insert__disjoint,axiom,
% 3.36/1.72 ! [X_51: x_a > $o,A_153: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o @ A_153 )
% 3.36/1.72 => ( ~ ( member_a_o @ X_51 @ A_153 )
% 3.36/1.72 => ( ( finite_card_a_o @ ( insert_a_o @ X_51 @ A_153 ) )
% 3.36/1.72 = ( suc @ ( finite_card_a_o @ A_153 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_107_card__insert__disjoint,axiom,
% 3.36/1.72 ! [X_51: pname,A_153: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ A_153 )
% 3.36/1.72 => ( ~ ( member_pname @ X_51 @ A_153 )
% 3.36/1.72 => ( ( finite_card_pname @ ( insert_pname @ X_51 @ A_153 ) )
% 3.36/1.72 = ( suc @ ( finite_card_pname @ A_153 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_108_card__insert__disjoint,axiom,
% 3.36/1.72 ! [X_51: nat,A_153: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ A_153 )
% 3.36/1.72 => ( ~ ( member_nat @ X_51 @ A_153 )
% 3.36/1.72 => ( ( finite_card_nat @ ( insert_nat @ X_51 @ A_153 ) )
% 3.36/1.72 = ( suc @ ( finite_card_nat @ A_153 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_109_card__insert__disjoint,axiom,
% 3.36/1.72 ! [X_51: int,A_153: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ A_153 )
% 3.36/1.72 => ( ~ ( member_int @ X_51 @ A_153 )
% 3.36/1.72 => ( ( finite_card_int @ ( insert_int @ X_51 @ A_153 ) )
% 3.36/1.72 = ( suc @ ( finite_card_int @ A_153 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_110_card__insert__disjoint,axiom,
% 3.36/1.72 ! [X_51: x_a,A_153: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ A_153 )
% 3.36/1.72 => ( ~ ( member_a @ X_51 @ A_153 )
% 3.36/1.72 => ( ( finite_card_a @ ( insert_a @ X_51 @ A_153 ) )
% 3.36/1.72 = ( suc @ ( finite_card_a @ A_153 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_111_finite__Collect__conjI,axiom,
% 3.36/1.72 ! [Q_3: x_a > $o,P_13: x_a > $o] :
% 3.36/1.72 ( ( ( finite_finite_a @ ( collect_a @ P_13 ) )
% 3.36/1.72 | ( finite_finite_a @ ( collect_a @ Q_3 ) ) )
% 3.36/1.72 => ( finite_finite_a
% 3.36/1.72 @ ( collect_a
% 3.36/1.72 @ ^ [X_1: x_a] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_112_finite__Collect__conjI,axiom,
% 3.36/1.72 ! [Q_3: ( int > $o ) > $o,P_13: ( int > $o ) > $o] :
% 3.36/1.72 ( ( ( finite_finite_int_o @ ( collect_int_o @ P_13 ) )
% 3.36/1.72 | ( finite_finite_int_o @ ( collect_int_o @ Q_3 ) ) )
% 3.36/1.72 => ( finite_finite_int_o
% 3.36/1.72 @ ( collect_int_o
% 3.36/1.72 @ ^ [X_1: int > $o] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_113_finite__Collect__conjI,axiom,
% 3.36/1.72 ! [Q_3: ( nat > $o ) > $o,P_13: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( ( finite_finite_nat_o @ ( collect_nat_o @ P_13 ) )
% 3.36/1.72 | ( finite_finite_nat_o @ ( collect_nat_o @ Q_3 ) ) )
% 3.36/1.72 => ( finite_finite_nat_o
% 3.36/1.72 @ ( collect_nat_o
% 3.36/1.72 @ ^ [X_1: nat > $o] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_114_finite__Collect__conjI,axiom,
% 3.36/1.72 ! [Q_3: ( pname > $o ) > $o,P_13: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( ( finite297249702name_o @ ( collect_pname_o @ P_13 ) )
% 3.36/1.72 | ( finite297249702name_o @ ( collect_pname_o @ Q_3 ) ) )
% 3.36/1.72 => ( finite297249702name_o
% 3.36/1.72 @ ( collect_pname_o
% 3.36/1.72 @ ^ [X_1: pname > $o] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_115_finite__Collect__conjI,axiom,
% 3.36/1.72 ! [Q_3: ( x_a > $o ) > $o,P_13: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( ( finite_finite_a_o @ ( collect_a_o @ P_13 ) )
% 3.36/1.72 | ( finite_finite_a_o @ ( collect_a_o @ Q_3 ) ) )
% 3.36/1.72 => ( finite_finite_a_o
% 3.36/1.72 @ ( collect_a_o
% 3.36/1.72 @ ^ [X_1: x_a > $o] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_116_finite__Collect__conjI,axiom,
% 3.36/1.72 ! [Q_3: pname > $o,P_13: pname > $o] :
% 3.36/1.72 ( ( ( finite_finite_pname @ ( collect_pname @ P_13 ) )
% 3.36/1.72 | ( finite_finite_pname @ ( collect_pname @ Q_3 ) ) )
% 3.36/1.72 => ( finite_finite_pname
% 3.36/1.72 @ ( collect_pname
% 3.36/1.72 @ ^ [X_1: pname] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_117_finite__Collect__conjI,axiom,
% 3.36/1.72 ! [Q_3: nat > $o,P_13: nat > $o] :
% 3.36/1.72 ( ( ( finite_finite_nat @ ( collect_nat @ P_13 ) )
% 3.36/1.72 | ( finite_finite_nat @ ( collect_nat @ Q_3 ) ) )
% 3.36/1.72 => ( finite_finite_nat
% 3.36/1.72 @ ( collect_nat
% 3.36/1.72 @ ^ [X_1: nat] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_118_finite__Collect__conjI,axiom,
% 3.36/1.72 ! [Q_3: int > $o,P_13: int > $o] :
% 3.36/1.72 ( ( ( finite_finite_int @ ( collect_int @ P_13 ) )
% 3.36/1.72 | ( finite_finite_int @ ( collect_int @ Q_3 ) ) )
% 3.36/1.72 => ( finite_finite_int
% 3.36/1.72 @ ( collect_int
% 3.36/1.72 @ ^ [X_1: int] : ( (&) @ ( P_13 @ X_1 ) @ ( Q_3 @ X_1 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_119_Suc__diff__le,axiom,
% 3.36/1.72 ! [N: nat,M: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ N @ M )
% 3.36/1.72 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 3.36/1.72 = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_120_finite__Collect__le__nat,axiom,
% 3.36/1.72 ! [K: nat] :
% 3.36/1.72 ( finite_finite_nat
% 3.36/1.72 @ ( collect_nat
% 3.36/1.72 @ ^ [N_1: nat] : ( ord_less_eq_nat @ N_1 @ K ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_121_card__Collect__le__nat,axiom,
% 3.36/1.72 ! [N: nat] :
% 3.36/1.72 ( ( finite_card_nat
% 3.36/1.72 @ ( collect_nat
% 3.36/1.72 @ ^ [I: nat] : ( ord_less_eq_nat @ I @ N ) ) )
% 3.36/1.72 = ( suc @ N ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_122_Suc__inject,axiom,
% 3.36/1.72 ! [X: nat,Y: nat] :
% 3.36/1.72 ( ( ( suc @ X )
% 3.36/1.72 = ( suc @ Y ) )
% 3.36/1.72 => ( X = Y ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_123_nat_Oinject,axiom,
% 3.36/1.72 ! [Nat_4: nat,Nat_1: nat] :
% 3.36/1.72 ( ( ( suc @ Nat_4 )
% 3.36/1.72 = ( suc @ Nat_1 ) )
% 3.36/1.72 <=> ( Nat_4 = Nat_1 ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_124_Suc__n__not__n,axiom,
% 3.36/1.72 ! [N: nat] :
% 3.36/1.72 ( ( suc @ N )
% 3.36/1.72 != N ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_125_n__not__Suc__n,axiom,
% 3.36/1.72 ! [N: nat] :
% 3.36/1.72 ( N
% 3.36/1.72 != ( suc @ N ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_126_le__antisym,axiom,
% 3.36/1.72 ! [M: nat,N: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ M @ N )
% 3.36/1.72 => ( ( ord_less_eq_nat @ N @ M )
% 3.36/1.72 => ( M = N ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_127_le__trans,axiom,
% 3.36/1.72 ! [K: nat,I_1: nat,J: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ I_1 @ J )
% 3.36/1.72 => ( ( ord_less_eq_nat @ J @ K )
% 3.36/1.72 => ( ord_less_eq_nat @ I_1 @ K ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_128_eq__imp__le,axiom,
% 3.36/1.72 ! [M: nat,N: nat] :
% 3.36/1.72 ( ( M = N )
% 3.36/1.72 => ( ord_less_eq_nat @ M @ N ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_129_nat__le__linear,axiom,
% 3.36/1.72 ! [M: nat,N: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ M @ N )
% 3.36/1.72 | ( ord_less_eq_nat @ N @ M ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_130_le__refl,axiom,
% 3.36/1.72 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_131_diff__commute,axiom,
% 3.36/1.72 ! [I_1: nat,J: nat,K: nat] :
% 3.36/1.72 ( ( minus_minus_nat @ ( minus_minus_nat @ I_1 @ J ) @ K )
% 3.36/1.72 = ( minus_minus_nat @ ( minus_minus_nat @ I_1 @ K ) @ J ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_132_finite__Collect__disjI,axiom,
% 3.36/1.72 ! [P_12: x_a > $o,Q_2: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a
% 3.36/1.72 @ ( collect_a
% 3.36/1.72 @ ^ [X_1: x_a] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
% 3.36/1.72 <=> ( ( finite_finite_a @ ( collect_a @ P_12 ) )
% 3.36/1.72 & ( finite_finite_a @ ( collect_a @ Q_2 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_133_finite__Collect__disjI,axiom,
% 3.36/1.72 ! [P_12: ( int > $o ) > $o,Q_2: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o
% 3.36/1.72 @ ( collect_int_o
% 3.36/1.72 @ ^ [X_1: int > $o] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
% 3.36/1.72 <=> ( ( finite_finite_int_o @ ( collect_int_o @ P_12 ) )
% 3.36/1.72 & ( finite_finite_int_o @ ( collect_int_o @ Q_2 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_134_finite__Collect__disjI,axiom,
% 3.36/1.72 ! [P_12: ( nat > $o ) > $o,Q_2: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o
% 3.36/1.72 @ ( collect_nat_o
% 3.36/1.72 @ ^ [X_1: nat > $o] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
% 3.36/1.72 <=> ( ( finite_finite_nat_o @ ( collect_nat_o @ P_12 ) )
% 3.36/1.72 & ( finite_finite_nat_o @ ( collect_nat_o @ Q_2 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_135_finite__Collect__disjI,axiom,
% 3.36/1.72 ! [P_12: ( pname > $o ) > $o,Q_2: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o
% 3.36/1.72 @ ( collect_pname_o
% 3.36/1.72 @ ^ [X_1: pname > $o] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
% 3.36/1.72 <=> ( ( finite297249702name_o @ ( collect_pname_o @ P_12 ) )
% 3.36/1.72 & ( finite297249702name_o @ ( collect_pname_o @ Q_2 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_136_finite__Collect__disjI,axiom,
% 3.36/1.72 ! [P_12: ( x_a > $o ) > $o,Q_2: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o
% 3.36/1.72 @ ( collect_a_o
% 3.36/1.72 @ ^ [X_1: x_a > $o] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
% 3.36/1.72 <=> ( ( finite_finite_a_o @ ( collect_a_o @ P_12 ) )
% 3.36/1.72 & ( finite_finite_a_o @ ( collect_a_o @ Q_2 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_137_finite__Collect__disjI,axiom,
% 3.36/1.72 ! [P_12: pname > $o,Q_2: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname
% 3.36/1.72 @ ( collect_pname
% 3.36/1.72 @ ^ [X_1: pname] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
% 3.36/1.72 <=> ( ( finite_finite_pname @ ( collect_pname @ P_12 ) )
% 3.36/1.72 & ( finite_finite_pname @ ( collect_pname @ Q_2 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_138_finite__Collect__disjI,axiom,
% 3.36/1.72 ! [P_12: nat > $o,Q_2: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat
% 3.36/1.72 @ ( collect_nat
% 3.36/1.72 @ ^ [X_1: nat] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
% 3.36/1.72 <=> ( ( finite_finite_nat @ ( collect_nat @ P_12 ) )
% 3.36/1.72 & ( finite_finite_nat @ ( collect_nat @ Q_2 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_139_finite__Collect__disjI,axiom,
% 3.36/1.72 ! [P_12: int > $o,Q_2: int > $o] :
% 3.36/1.72 ( ( finite_finite_int
% 3.36/1.72 @ ( collect_int
% 3.36/1.72 @ ^ [X_1: int] : ( (|) @ ( P_12 @ X_1 ) @ ( Q_2 @ X_1 ) ) ) )
% 3.36/1.72 <=> ( ( finite_finite_int @ ( collect_int @ P_12 ) )
% 3.36/1.72 & ( finite_finite_int @ ( collect_int @ Q_2 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_140_finite__insert,axiom,
% 3.36/1.72 ! [A_152: int > $o,A_151: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o @ ( insert_int_o @ A_152 @ A_151 ) )
% 3.36/1.72 <=> ( finite_finite_int_o @ A_151 ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_141_finite__insert,axiom,
% 3.36/1.72 ! [A_152: nat > $o,A_151: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o @ ( insert_nat_o @ A_152 @ A_151 ) )
% 3.36/1.72 <=> ( finite_finite_nat_o @ A_151 ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_142_finite__insert,axiom,
% 3.36/1.72 ! [A_152: pname > $o,A_151: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o @ ( insert_pname_o @ A_152 @ A_151 ) )
% 3.36/1.72 <=> ( finite297249702name_o @ A_151 ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_143_finite__insert,axiom,
% 3.36/1.72 ! [A_152: x_a > $o,A_151: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o @ ( insert_a_o @ A_152 @ A_151 ) )
% 3.36/1.72 <=> ( finite_finite_a_o @ A_151 ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_144_finite__insert,axiom,
% 3.36/1.72 ! [A_152: pname,A_151: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ ( insert_pname @ A_152 @ A_151 ) )
% 3.36/1.72 <=> ( finite_finite_pname @ A_151 ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_145_finite__insert,axiom,
% 3.36/1.72 ! [A_152: nat,A_151: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ ( insert_nat @ A_152 @ A_151 ) )
% 3.36/1.72 <=> ( finite_finite_nat @ A_151 ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_146_finite__insert,axiom,
% 3.36/1.72 ! [A_152: int,A_151: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ ( insert_int @ A_152 @ A_151 ) )
% 3.36/1.72 <=> ( finite_finite_int @ A_151 ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_147_finite__insert,axiom,
% 3.36/1.72 ! [A_152: x_a,A_151: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ ( insert_a @ A_152 @ A_151 ) )
% 3.36/1.72 <=> ( finite_finite_a @ A_151 ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_148_finite__subset,axiom,
% 3.36/1.72 ! [A_150: ( int > $o ) > $o,B_87: ( int > $o ) > $o] :
% 3.36/1.72 ( ( ord_less_eq_int_o_o @ A_150 @ B_87 )
% 3.36/1.72 => ( ( finite_finite_int_o @ B_87 )
% 3.36/1.72 => ( finite_finite_int_o @ A_150 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_149_finite__subset,axiom,
% 3.36/1.72 ! [A_150: ( nat > $o ) > $o,B_87: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( ord_less_eq_nat_o_o @ A_150 @ B_87 )
% 3.36/1.72 => ( ( finite_finite_nat_o @ B_87 )
% 3.36/1.72 => ( finite_finite_nat_o @ A_150 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_150_finite__subset,axiom,
% 3.36/1.72 ! [A_150: ( pname > $o ) > $o,B_87: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( ord_le1205211808me_o_o @ A_150 @ B_87 )
% 3.36/1.72 => ( ( finite297249702name_o @ B_87 )
% 3.36/1.72 => ( finite297249702name_o @ A_150 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_151_finite__subset,axiom,
% 3.36/1.72 ! [A_150: ( x_a > $o ) > $o,B_87: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o_o @ A_150 @ B_87 )
% 3.36/1.72 => ( ( finite_finite_a_o @ B_87 )
% 3.36/1.72 => ( finite_finite_a_o @ A_150 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_152_finite__subset,axiom,
% 3.36/1.72 ! [A_150: x_a > $o,B_87: x_a > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o @ A_150 @ B_87 )
% 3.36/1.72 => ( ( finite_finite_a @ B_87 )
% 3.36/1.72 => ( finite_finite_a @ A_150 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_153_finite__subset,axiom,
% 3.36/1.72 ! [A_150: pname > $o,B_87: pname > $o] :
% 3.36/1.72 ( ( ord_less_eq_pname_o @ A_150 @ B_87 )
% 3.36/1.72 => ( ( finite_finite_pname @ B_87 )
% 3.36/1.72 => ( finite_finite_pname @ A_150 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_154_finite__subset,axiom,
% 3.36/1.72 ! [A_150: nat > $o,B_87: nat > $o] :
% 3.36/1.72 ( ( ord_less_eq_nat_o @ A_150 @ B_87 )
% 3.36/1.72 => ( ( finite_finite_nat @ B_87 )
% 3.36/1.72 => ( finite_finite_nat @ A_150 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_155_finite__subset,axiom,
% 3.36/1.72 ! [A_150: int > $o,B_87: int > $o] :
% 3.36/1.72 ( ( ord_less_eq_int_o @ A_150 @ B_87 )
% 3.36/1.72 => ( ( finite_finite_int @ B_87 )
% 3.36/1.72 => ( finite_finite_int @ A_150 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_156_rev__finite__subset,axiom,
% 3.36/1.72 ! [A_149: ( int > $o ) > $o,B_86: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o @ B_86 )
% 3.36/1.72 => ( ( ord_less_eq_int_o_o @ A_149 @ B_86 )
% 3.36/1.72 => ( finite_finite_int_o @ A_149 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_157_rev__finite__subset,axiom,
% 3.36/1.72 ! [A_149: ( nat > $o ) > $o,B_86: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o @ B_86 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o_o @ A_149 @ B_86 )
% 3.36/1.72 => ( finite_finite_nat_o @ A_149 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_158_rev__finite__subset,axiom,
% 3.36/1.72 ! [A_149: ( pname > $o ) > $o,B_86: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o @ B_86 )
% 3.36/1.72 => ( ( ord_le1205211808me_o_o @ A_149 @ B_86 )
% 3.36/1.72 => ( finite297249702name_o @ A_149 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_159_rev__finite__subset,axiom,
% 3.36/1.72 ! [A_149: ( x_a > $o ) > $o,B_86: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o @ B_86 )
% 3.36/1.72 => ( ( ord_less_eq_a_o_o @ A_149 @ B_86 )
% 3.36/1.72 => ( finite_finite_a_o @ A_149 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_160_rev__finite__subset,axiom,
% 3.36/1.72 ! [A_149: x_a > $o,B_86: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ B_86 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ A_149 @ B_86 )
% 3.36/1.72 => ( finite_finite_a @ A_149 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_161_rev__finite__subset,axiom,
% 3.36/1.72 ! [A_149: pname > $o,B_86: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ B_86 )
% 3.36/1.72 => ( ( ord_less_eq_pname_o @ A_149 @ B_86 )
% 3.36/1.72 => ( finite_finite_pname @ A_149 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_162_rev__finite__subset,axiom,
% 3.36/1.72 ! [A_149: nat > $o,B_86: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ B_86 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o @ A_149 @ B_86 )
% 3.36/1.72 => ( finite_finite_nat @ A_149 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_163_rev__finite__subset,axiom,
% 3.36/1.72 ! [A_149: int > $o,B_86: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ B_86 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ A_149 @ B_86 )
% 3.36/1.72 => ( finite_finite_int @ A_149 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_164_Suc__leD,axiom,
% 3.36/1.72 ! [M: nat,N: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 3.36/1.72 => ( ord_less_eq_nat @ M @ N ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_165_le__SucE,axiom,
% 3.36/1.72 ! [M: nat,N: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 3.36/1.72 => ( ~ ( ord_less_eq_nat @ M @ N )
% 3.36/1.72 => ( M
% 3.36/1.72 = ( suc @ N ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_166_le__SucI,axiom,
% 3.36/1.72 ! [M: nat,N: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ M @ N )
% 3.36/1.72 => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_167_Suc__le__mono,axiom,
% 3.36/1.72 ! [N: nat,M: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 3.36/1.72 <=> ( ord_less_eq_nat @ N @ M ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_168_le__Suc__eq,axiom,
% 3.36/1.72 ! [M: nat,N: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 3.36/1.72 <=> ( ( ord_less_eq_nat @ M @ N )
% 3.36/1.72 | ( M
% 3.36/1.72 = ( suc @ N ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_169_not__less__eq__eq,axiom,
% 3.36/1.72 ! [M: nat,N: nat] :
% 3.36/1.72 ( ~ ( ord_less_eq_nat @ M @ N )
% 3.36/1.72 <=> ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_170_Suc__n__not__le__n,axiom,
% 3.36/1.72 ! [N: nat] :
% 3.36/1.72 ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_171_Suc__diff__diff,axiom,
% 3.36/1.72 ! [M: nat,N: nat,K: nat] :
% 3.36/1.72 ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 3.36/1.72 = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_172_diff__Suc__Suc,axiom,
% 3.36/1.72 ! [M: nat,N: nat] :
% 3.36/1.72 ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 3.36/1.72 = ( minus_minus_nat @ M @ N ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_173_le__diff__iff,axiom,
% 3.36/1.72 ! [N: nat,K: nat,M: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ K @ M )
% 3.36/1.72 => ( ( ord_less_eq_nat @ K @ N )
% 3.36/1.72 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 3.36/1.72 <=> ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_174_Nat_Odiff__diff__eq,axiom,
% 3.36/1.72 ! [N: nat,K: nat,M: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ K @ M )
% 3.36/1.72 => ( ( ord_less_eq_nat @ K @ N )
% 3.36/1.72 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 3.36/1.72 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_175_eq__diff__iff,axiom,
% 3.36/1.72 ! [N: nat,K: nat,M: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ K @ M )
% 3.36/1.72 => ( ( ord_less_eq_nat @ K @ N )
% 3.36/1.72 => ( ( ( minus_minus_nat @ M @ K )
% 3.36/1.72 = ( minus_minus_nat @ N @ K ) )
% 3.36/1.72 <=> ( M = N ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_176_diff__diff__cancel,axiom,
% 3.36/1.72 ! [I_1: nat,N: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ I_1 @ N )
% 3.36/1.72 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I_1 ) )
% 3.36/1.72 = I_1 ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_177_diff__le__mono,axiom,
% 3.36/1.72 ! [L: nat,M: nat,N: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ M @ N )
% 3.36/1.72 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_178_diff__le__mono2,axiom,
% 3.36/1.72 ! [L: nat,M: nat,N: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ M @ N )
% 3.36/1.72 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_179_diff__le__self,axiom,
% 3.36/1.72 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_180_finite__surj,axiom,
% 3.36/1.72 ! [B_85: x_a > $o,F_40: x_a > x_a,A_148: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ B_85 @ ( image_a_a @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_a @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_181_finite__surj,axiom,
% 3.36/1.72 ! [B_85: x_a > $o,F_40: ( int > $o ) > x_a,A_148: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ B_85 @ ( image_int_o_a @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_a @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_182_finite__surj,axiom,
% 3.36/1.72 ! [B_85: x_a > $o,F_40: ( nat > $o ) > x_a,A_148: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ B_85 @ ( image_nat_o_a @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_a @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_183_finite__surj,axiom,
% 3.36/1.72 ! [B_85: x_a > $o,F_40: ( pname > $o ) > x_a,A_148: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ B_85 @ ( image_pname_o_a @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_a @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_184_finite__surj,axiom,
% 3.36/1.72 ! [B_85: x_a > $o,F_40: ( x_a > $o ) > x_a,A_148: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ B_85 @ ( image_a_o_a @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_a @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_185_finite__surj,axiom,
% 3.36/1.72 ! [B_85: ( int > $o ) > $o,F_40: pname > int > $o,A_148: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_int_o_o @ B_85 @ ( image_pname_int_o @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_int_o @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_186_finite__surj,axiom,
% 3.36/1.72 ! [B_85: ( nat > $o ) > $o,F_40: pname > nat > $o,A_148: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o_o @ B_85 @ ( image_pname_nat_o @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_nat_o @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_187_finite__surj,axiom,
% 3.36/1.72 ! [B_85: ( pname > $o ) > $o,F_40: pname > pname > $o,A_148: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ A_148 )
% 3.36/1.72 => ( ( ord_le1205211808me_o_o @ B_85 @ ( image_pname_pname_o @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite297249702name_o @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_188_finite__surj,axiom,
% 3.36/1.72 ! [B_85: ( x_a > $o ) > $o,F_40: pname > x_a > $o,A_148: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_a_o_o @ B_85 @ ( image_pname_a_o @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_a_o @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_189_finite__surj,axiom,
% 3.36/1.72 ! [B_85: pname > $o,F_40: pname > pname,A_148: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_pname_o @ B_85 @ ( image_pname_pname @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_pname @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_190_finite__surj,axiom,
% 3.36/1.72 ! [B_85: x_a > $o,F_40: nat > x_a,A_148: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ B_85 @ ( image_nat_a @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_a @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_191_finite__surj,axiom,
% 3.36/1.72 ! [B_85: ( int > $o ) > $o,F_40: nat > int > $o,A_148: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_int_o_o @ B_85 @ ( image_nat_int_o @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_int_o @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_192_finite__surj,axiom,
% 3.36/1.72 ! [B_85: ( nat > $o ) > $o,F_40: nat > nat > $o,A_148: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o_o @ B_85 @ ( image_nat_nat_o @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_nat_o @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_193_finite__surj,axiom,
% 3.36/1.72 ! [B_85: ( pname > $o ) > $o,F_40: nat > pname > $o,A_148: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ A_148 )
% 3.36/1.72 => ( ( ord_le1205211808me_o_o @ B_85 @ ( image_nat_pname_o @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite297249702name_o @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_194_finite__surj,axiom,
% 3.36/1.72 ! [B_85: ( x_a > $o ) > $o,F_40: nat > x_a > $o,A_148: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_a_o_o @ B_85 @ ( image_nat_a_o @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_a_o @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_195_finite__surj,axiom,
% 3.36/1.72 ! [B_85: pname > $o,F_40: nat > pname,A_148: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_pname_o @ B_85 @ ( image_nat_pname @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_pname @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_196_finite__surj,axiom,
% 3.36/1.72 ! [B_85: x_a > $o,F_40: int > x_a,A_148: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ B_85 @ ( image_int_a @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_a @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_197_finite__surj,axiom,
% 3.36/1.72 ! [B_85: ( int > $o ) > $o,F_40: int > int > $o,A_148: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_int_o_o @ B_85 @ ( image_int_int_o @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_int_o @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_198_finite__surj,axiom,
% 3.36/1.72 ! [B_85: ( nat > $o ) > $o,F_40: int > nat > $o,A_148: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o_o @ B_85 @ ( image_int_nat_o @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_nat_o @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_199_finite__surj,axiom,
% 3.36/1.72 ! [B_85: ( pname > $o ) > $o,F_40: int > pname > $o,A_148: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ A_148 )
% 3.36/1.72 => ( ( ord_le1205211808me_o_o @ B_85 @ ( image_int_pname_o @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite297249702name_o @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_200_finite__surj,axiom,
% 3.36/1.72 ! [B_85: ( x_a > $o ) > $o,F_40: int > x_a > $o,A_148: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_a_o_o @ B_85 @ ( image_int_a_o @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_a_o @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_201_finite__surj,axiom,
% 3.36/1.72 ! [B_85: pname > $o,F_40: int > pname,A_148: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_pname_o @ B_85 @ ( image_int_pname @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_pname @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_202_finite__surj,axiom,
% 3.36/1.72 ! [B_85: pname > $o,F_40: x_a > pname,A_148: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_pname_o @ B_85 @ ( image_a_pname @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_pname @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_203_finite__surj,axiom,
% 3.36/1.72 ! [B_85: pname > $o,F_40: ( int > $o ) > pname,A_148: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_pname_o @ B_85 @ ( image_int_o_pname @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_pname @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_204_finite__surj,axiom,
% 3.36/1.72 ! [B_85: pname > $o,F_40: ( nat > $o ) > pname,A_148: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_pname_o @ B_85 @ ( image_nat_o_pname @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_pname @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_205_finite__surj,axiom,
% 3.36/1.72 ! [B_85: pname > $o,F_40: ( pname > $o ) > pname,A_148: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_pname_o @ B_85 @ ( image_pname_o_pname @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_pname @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_206_finite__surj,axiom,
% 3.36/1.72 ! [B_85: pname > $o,F_40: ( x_a > $o ) > pname,A_148: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_pname_o @ B_85 @ ( image_a_o_pname @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_pname @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_207_finite__surj,axiom,
% 3.36/1.72 ! [B_85: nat > $o,F_40: x_a > nat,A_148: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o @ B_85 @ ( image_a_nat @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_nat @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_208_finite__surj,axiom,
% 3.36/1.72 ! [B_85: nat > $o,F_40: ( int > $o ) > nat,A_148: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o @ B_85 @ ( image_int_o_nat @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_nat @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_209_finite__surj,axiom,
% 3.36/1.72 ! [B_85: nat > $o,F_40: ( nat > $o ) > nat,A_148: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o @ B_85 @ ( image_nat_o_nat @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_nat @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_210_finite__surj,axiom,
% 3.36/1.72 ! [B_85: nat > $o,F_40: ( pname > $o ) > nat,A_148: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o @ B_85 @ ( image_pname_o_nat @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_nat @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_211_finite__surj,axiom,
% 3.36/1.72 ! [B_85: nat > $o,F_40: ( x_a > $o ) > nat,A_148: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o @ B_85 @ ( image_a_o_nat @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_nat @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_212_finite__surj,axiom,
% 3.36/1.72 ! [B_85: int > $o,F_40: x_a > int,A_148: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ B_85 @ ( image_a_int @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_int @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_213_finite__surj,axiom,
% 3.36/1.72 ! [B_85: int > $o,F_40: ( int > $o ) > int,A_148: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ B_85 @ ( image_int_o_int @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_int @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_214_finite__surj,axiom,
% 3.36/1.72 ! [B_85: int > $o,F_40: ( nat > $o ) > int,A_148: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ B_85 @ ( image_nat_o_int @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_int @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_215_finite__surj,axiom,
% 3.36/1.72 ! [B_85: int > $o,F_40: ( pname > $o ) > int,A_148: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ B_85 @ ( image_pname_o_int @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_int @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_216_finite__surj,axiom,
% 3.36/1.72 ! [B_85: int > $o,F_40: ( x_a > $o ) > int,A_148: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ B_85 @ ( image_a_o_int @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_int @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_217_finite__surj,axiom,
% 3.36/1.72 ! [B_85: x_a > $o,F_40: pname > x_a,A_148: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ B_85 @ ( image_pname_a @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_a @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_218_finite__surj,axiom,
% 3.36/1.72 ! [B_85: int > $o,F_40: nat > int,A_148: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ A_148 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ B_85 @ ( image_nat_int @ F_40 @ A_148 ) )
% 3.36/1.72 => ( finite_finite_int @ B_85 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_219_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: ( int > $o ) > x_a,A_147: ( int > $o ) > $o,B_84: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ B_84 @ ( image_int_o_a @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: ( int > $o ) > $o] :
% 3.36/1.72 ( ( ord_less_eq_int_o_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_int_o @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_int_o_a @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_220_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: ( nat > $o ) > x_a,A_147: ( nat > $o ) > $o,B_84: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ B_84 @ ( image_nat_o_a @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( ord_less_eq_nat_o_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_nat_o @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_nat_o_a @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_221_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: ( pname > $o ) > x_a,A_147: ( pname > $o ) > $o,B_84: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ B_84 @ ( image_pname_o_a @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( ord_le1205211808me_o_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite297249702name_o @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_pname_o_a @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_222_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: ( x_a > $o ) > x_a,A_147: ( x_a > $o ) > $o,B_84: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ B_84 @ ( image_a_o_a @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_a_o @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_a_o_a @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_223_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: x_a > x_a,A_147: x_a > $o,B_84: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ B_84 @ ( image_a_a @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: x_a > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_a @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_a_a @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_224_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: x_a > int > $o,A_147: x_a > $o,B_84: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_int_o_o @ B_84 @ ( image_a_int_o @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: x_a > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_a @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_a_int_o @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_225_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: x_a > nat > $o,A_147: x_a > $o,B_84: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o_o @ B_84 @ ( image_a_nat_o @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: x_a > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_a @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_a_nat_o @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_226_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: x_a > pname > $o,A_147: x_a > $o,B_84: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o @ B_84 )
% 3.36/1.72 => ( ( ord_le1205211808me_o_o @ B_84 @ ( image_a_pname_o @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: x_a > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_a @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_a_pname_o @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_227_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: x_a > x_a > $o,A_147: x_a > $o,B_84: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_a_o_o @ B_84 @ ( image_a_a_o @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: x_a > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_a @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_a_a_o @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_228_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: x_a > pname,A_147: x_a > $o,B_84: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_pname_o @ B_84 @ ( image_a_pname @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: x_a > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_a @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_a_pname @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_229_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: ( int > $o ) > pname,A_147: ( int > $o ) > $o,B_84: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_pname_o @ B_84 @ ( image_int_o_pname @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: ( int > $o ) > $o] :
% 3.36/1.72 ( ( ord_less_eq_int_o_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_int_o @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_int_o_pname @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_230_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: ( nat > $o ) > pname,A_147: ( nat > $o ) > $o,B_84: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_pname_o @ B_84 @ ( image_nat_o_pname @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( ord_less_eq_nat_o_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_nat_o @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_nat_o_pname @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_231_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: ( pname > $o ) > pname,A_147: ( pname > $o ) > $o,B_84: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_pname_o @ B_84 @ ( image_pname_o_pname @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( ord_le1205211808me_o_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite297249702name_o @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_pname_o_pname @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_232_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: ( x_a > $o ) > pname,A_147: ( x_a > $o ) > $o,B_84: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_pname_o @ B_84 @ ( image_a_o_pname @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_a_o @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_a_o_pname @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_233_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: x_a > nat,A_147: x_a > $o,B_84: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o @ B_84 @ ( image_a_nat @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: x_a > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_a @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_a_nat @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_234_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: ( int > $o ) > nat,A_147: ( int > $o ) > $o,B_84: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o @ B_84 @ ( image_int_o_nat @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: ( int > $o ) > $o] :
% 3.36/1.72 ( ( ord_less_eq_int_o_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_int_o @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_int_o_nat @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_235_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: ( nat > $o ) > nat,A_147: ( nat > $o ) > $o,B_84: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o @ B_84 @ ( image_nat_o_nat @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( ord_less_eq_nat_o_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_nat_o @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_nat_o_nat @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_236_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: ( pname > $o ) > nat,A_147: ( pname > $o ) > $o,B_84: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o @ B_84 @ ( image_pname_o_nat @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( ord_le1205211808me_o_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite297249702name_o @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_pname_o_nat @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_237_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: ( x_a > $o ) > nat,A_147: ( x_a > $o ) > $o,B_84: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o @ B_84 @ ( image_a_o_nat @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_a_o @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_a_o_nat @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_238_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: pname > nat,A_147: pname > $o,B_84: nat > $o] :
% 3.36/1.72 ( ( finite_finite_nat @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o @ B_84 @ ( image_pname_nat @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: pname > $o] :
% 3.36/1.72 ( ( ord_less_eq_pname_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_pname @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_pname_nat @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_239_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: x_a > int,A_147: x_a > $o,B_84: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ B_84 @ ( image_a_int @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: x_a > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_a @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_a_int @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_240_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: ( int > $o ) > int,A_147: ( int > $o ) > $o,B_84: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ B_84 @ ( image_int_o_int @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: ( int > $o ) > $o] :
% 3.36/1.72 ( ( ord_less_eq_int_o_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_int_o @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_int_o_int @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_241_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: ( nat > $o ) > int,A_147: ( nat > $o ) > $o,B_84: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ B_84 @ ( image_nat_o_int @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( ord_less_eq_nat_o_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_nat_o @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_nat_o_int @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_242_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: ( pname > $o ) > int,A_147: ( pname > $o ) > $o,B_84: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ B_84 @ ( image_pname_o_int @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( ord_le1205211808me_o_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite297249702name_o @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_pname_o_int @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_243_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: ( x_a > $o ) > int,A_147: ( x_a > $o ) > $o,B_84: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ B_84 @ ( image_a_o_int @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_a_o @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_a_o_int @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_244_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: pname > int,A_147: pname > $o,B_84: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ B_84 @ ( image_pname_int @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: pname > $o] :
% 3.36/1.72 ( ( ord_less_eq_pname_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_pname @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_pname_int @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_245_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: pname > int > $o,A_147: pname > $o,B_84: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_int_o_o @ B_84 @ ( image_pname_int_o @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: pname > $o] :
% 3.36/1.72 ( ( ord_less_eq_pname_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_pname @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_pname_int_o @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_246_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: pname > nat > $o,A_147: pname > $o,B_84: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o_o @ B_84 @ ( image_pname_nat_o @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: pname > $o] :
% 3.36/1.72 ( ( ord_less_eq_pname_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_pname @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_pname_nat_o @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_247_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: pname > pname > $o,A_147: pname > $o,B_84: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o @ B_84 )
% 3.36/1.72 => ( ( ord_le1205211808me_o_o @ B_84 @ ( image_pname_pname_o @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: pname > $o] :
% 3.36/1.72 ( ( ord_less_eq_pname_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_pname @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_pname_pname_o @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_248_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: pname > x_a > $o,A_147: pname > $o,B_84: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_a_o_o @ B_84 @ ( image_pname_a_o @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: pname > $o] :
% 3.36/1.72 ( ( ord_less_eq_pname_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_pname @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_pname_a_o @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_249_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: pname > pname,A_147: pname > $o,B_84: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_pname_o @ B_84 @ ( image_pname_pname @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: pname > $o] :
% 3.36/1.72 ( ( ord_less_eq_pname_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_pname @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_pname_pname @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_250_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: nat > x_a,A_147: nat > $o,B_84: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ B_84 @ ( image_nat_a @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: nat > $o] :
% 3.36/1.72 ( ( ord_less_eq_nat_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_nat @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_nat_a @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_251_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: nat > int > $o,A_147: nat > $o,B_84: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_int_o_o @ B_84 @ ( image_nat_int_o @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: nat > $o] :
% 3.36/1.72 ( ( ord_less_eq_nat_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_nat @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_nat_int_o @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_252_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: nat > nat > $o,A_147: nat > $o,B_84: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o_o @ B_84 @ ( image_nat_nat_o @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: nat > $o] :
% 3.36/1.72 ( ( ord_less_eq_nat_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_nat @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_nat_nat_o @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_253_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: nat > pname > $o,A_147: nat > $o,B_84: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o @ B_84 )
% 3.36/1.72 => ( ( ord_le1205211808me_o_o @ B_84 @ ( image_nat_pname_o @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: nat > $o] :
% 3.36/1.72 ( ( ord_less_eq_nat_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_nat @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_nat_pname_o @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_254_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: nat > x_a > $o,A_147: nat > $o,B_84: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_a_o_o @ B_84 @ ( image_nat_a_o @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: nat > $o] :
% 3.36/1.72 ( ( ord_less_eq_nat_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_nat @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_nat_a_o @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_255_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: nat > pname,A_147: nat > $o,B_84: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_pname_o @ B_84 @ ( image_nat_pname @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: nat > $o] :
% 3.36/1.72 ( ( ord_less_eq_nat_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_nat @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_nat_pname @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_256_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: int > x_a,A_147: int > $o,B_84: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ B_84 @ ( image_int_a @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: int > $o] :
% 3.36/1.72 ( ( ord_less_eq_int_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_int @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_int_a @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_257_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: int > int > $o,A_147: int > $o,B_84: ( int > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_int_o @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_int_o_o @ B_84 @ ( image_int_int_o @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: int > $o] :
% 3.36/1.72 ( ( ord_less_eq_int_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_int @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_int_int_o @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_258_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: int > nat > $o,A_147: int > $o,B_84: ( nat > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_nat_o @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o_o @ B_84 @ ( image_int_nat_o @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: int > $o] :
% 3.36/1.72 ( ( ord_less_eq_int_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_int @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_int_nat_o @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_259_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: int > pname > $o,A_147: int > $o,B_84: ( pname > $o ) > $o] :
% 3.36/1.72 ( ( finite297249702name_o @ B_84 )
% 3.36/1.72 => ( ( ord_le1205211808me_o_o @ B_84 @ ( image_int_pname_o @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: int > $o] :
% 3.36/1.72 ( ( ord_less_eq_int_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_int @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_int_pname_o @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_260_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: int > x_a > $o,A_147: int > $o,B_84: ( x_a > $o ) > $o] :
% 3.36/1.72 ( ( finite_finite_a_o @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_a_o_o @ B_84 @ ( image_int_a_o @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: int > $o] :
% 3.36/1.72 ( ( ord_less_eq_int_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_int @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_int_a_o @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_261_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: int > pname,A_147: int > $o,B_84: pname > $o] :
% 3.36/1.72 ( ( finite_finite_pname @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_pname_o @ B_84 @ ( image_int_pname @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: int > $o] :
% 3.36/1.72 ( ( ord_less_eq_int_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_int @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_int_pname @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_262_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: pname > x_a,A_147: pname > $o,B_84: x_a > $o] :
% 3.36/1.72 ( ( finite_finite_a @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ B_84 @ ( image_pname_a @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: pname > $o] :
% 3.36/1.72 ( ( ord_less_eq_pname_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_pname @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_pname_a @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_263_finite__subset__image,axiom,
% 3.36/1.72 ! [F_39: nat > int,A_147: nat > $o,B_84: int > $o] :
% 3.36/1.72 ( ( finite_finite_int @ B_84 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ B_84 @ ( image_nat_int @ F_39 @ A_147 ) )
% 3.36/1.72 => ? [C_34: nat > $o] :
% 3.36/1.72 ( ( ord_less_eq_nat_o @ C_34 @ A_147 )
% 3.36/1.72 & ( finite_finite_nat @ C_34 )
% 3.36/1.72 & ( B_84
% 3.36/1.72 = ( image_nat_int @ F_39 @ C_34 ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_264_lift__Suc__mono__le,axiom,
% 3.36/1.72 ! [N_4: nat,N_3: nat,F_38: nat > pname > $o] :
% 3.36/1.72 ( ! [N_1: nat] : ( ord_less_eq_pname_o @ ( F_38 @ N_1 ) @ ( F_38 @ ( suc @ N_1 ) ) )
% 3.36/1.72 => ( ( ord_less_eq_nat @ N_4 @ N_3 )
% 3.36/1.72 => ( ord_less_eq_pname_o @ ( F_38 @ N_4 ) @ ( F_38 @ N_3 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_265_lift__Suc__mono__le,axiom,
% 3.36/1.72 ! [N_4: nat,N_3: nat,F_38: nat > $o] :
% 3.36/1.72 ( ! [N_1: nat] : ( ord_less_eq_o @ ( F_38 @ N_1 ) @ ( F_38 @ ( suc @ N_1 ) ) )
% 3.36/1.72 => ( ( ord_less_eq_nat @ N_4 @ N_3 )
% 3.36/1.72 => ( ord_less_eq_o @ ( F_38 @ N_4 ) @ ( F_38 @ N_3 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_266_lift__Suc__mono__le,axiom,
% 3.36/1.72 ! [N_4: nat,N_3: nat,F_38: nat > x_a > $o] :
% 3.36/1.72 ( ! [N_1: nat] : ( ord_less_eq_a_o @ ( F_38 @ N_1 ) @ ( F_38 @ ( suc @ N_1 ) ) )
% 3.36/1.72 => ( ( ord_less_eq_nat @ N_4 @ N_3 )
% 3.36/1.72 => ( ord_less_eq_a_o @ ( F_38 @ N_4 ) @ ( F_38 @ N_3 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_267_lift__Suc__mono__le,axiom,
% 3.36/1.72 ! [N_4: nat,N_3: nat,F_38: nat > nat] :
% 3.36/1.72 ( ! [N_1: nat] : ( ord_less_eq_nat @ ( F_38 @ N_1 ) @ ( F_38 @ ( suc @ N_1 ) ) )
% 3.36/1.72 => ( ( ord_less_eq_nat @ N_4 @ N_3 )
% 3.36/1.72 => ( ord_less_eq_nat @ ( F_38 @ N_4 ) @ ( F_38 @ N_3 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_268_lift__Suc__mono__le,axiom,
% 3.36/1.72 ! [N_4: nat,N_3: nat,F_38: nat > int] :
% 3.36/1.72 ( ! [N_1: nat] : ( ord_less_eq_int @ ( F_38 @ N_1 ) @ ( F_38 @ ( suc @ N_1 ) ) )
% 3.36/1.72 => ( ( ord_less_eq_nat @ N_4 @ N_3 )
% 3.36/1.72 => ( ord_less_eq_int @ ( F_38 @ N_4 ) @ ( F_38 @ N_3 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_269_lift__Suc__mono__le,axiom,
% 3.36/1.72 ! [N_4: nat,N_3: nat,F_38: nat > nat > $o] :
% 3.36/1.72 ( ! [N_1: nat] : ( ord_less_eq_nat_o @ ( F_38 @ N_1 ) @ ( F_38 @ ( suc @ N_1 ) ) )
% 3.36/1.72 => ( ( ord_less_eq_nat @ N_4 @ N_3 )
% 3.36/1.72 => ( ord_less_eq_nat_o @ ( F_38 @ N_4 ) @ ( F_38 @ N_3 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_270_lift__Suc__mono__le,axiom,
% 3.36/1.72 ! [N_4: nat,N_3: nat,F_38: nat > int > $o] :
% 3.36/1.72 ( ! [N_1: nat] : ( ord_less_eq_int_o @ ( F_38 @ N_1 ) @ ( F_38 @ ( suc @ N_1 ) ) )
% 3.36/1.72 => ( ( ord_less_eq_nat @ N_4 @ N_3 )
% 3.36/1.72 => ( ord_less_eq_int_o @ ( F_38 @ N_4 ) @ ( F_38 @ N_3 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_271_pigeonhole__infinite,axiom,
% 3.36/1.72 ! [F_37: nat > int > $o,A_146: nat > $o] :
% 3.36/1.72 ( ~ ( finite_finite_nat @ A_146 )
% 3.36/1.72 => ( ( finite_finite_int_o @ ( image_nat_int_o @ F_37 @ A_146 ) )
% 3.36/1.72 => ? [X_1: nat] :
% 3.36/1.72 ( ( member_nat @ X_1 @ A_146 )
% 3.36/1.72 & ~ ( finite_finite_nat
% 3.36/1.72 @ ( collect_nat
% 3.36/1.72 @ ^ [A_37: nat] :
% 3.36/1.72 ( (&) @ ( member_nat @ A_37 @ A_146 )
% 3.36/1.72 @ ( ( F_37 @ A_37 )
% 3.36/1.72 = ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_272_pigeonhole__infinite,axiom,
% 3.36/1.72 ! [F_37: nat > nat > $o,A_146: nat > $o] :
% 3.36/1.72 ( ~ ( finite_finite_nat @ A_146 )
% 3.36/1.72 => ( ( finite_finite_nat_o @ ( image_nat_nat_o @ F_37 @ A_146 ) )
% 3.36/1.72 => ? [X_1: nat] :
% 3.36/1.72 ( ( member_nat @ X_1 @ A_146 )
% 3.36/1.72 & ~ ( finite_finite_nat
% 3.36/1.72 @ ( collect_nat
% 3.36/1.72 @ ^ [A_37: nat] :
% 3.36/1.72 ( (&) @ ( member_nat @ A_37 @ A_146 )
% 3.36/1.72 @ ( ( F_37 @ A_37 )
% 3.36/1.72 = ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_273_pigeonhole__infinite,axiom,
% 3.36/1.72 ! [F_37: nat > pname > $o,A_146: nat > $o] :
% 3.36/1.72 ( ~ ( finite_finite_nat @ A_146 )
% 3.36/1.72 => ( ( finite297249702name_o @ ( image_nat_pname_o @ F_37 @ A_146 ) )
% 3.36/1.72 => ? [X_1: nat] :
% 3.36/1.72 ( ( member_nat @ X_1 @ A_146 )
% 3.36/1.72 & ~ ( finite_finite_nat
% 3.36/1.72 @ ( collect_nat
% 3.36/1.72 @ ^ [A_37: nat] :
% 3.36/1.72 ( (&) @ ( member_nat @ A_37 @ A_146 )
% 3.36/1.72 @ ( ( F_37 @ A_37 )
% 3.36/1.72 = ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_274_pigeonhole__infinite,axiom,
% 3.36/1.72 ! [F_37: nat > x_a > $o,A_146: nat > $o] :
% 3.36/1.72 ( ~ ( finite_finite_nat @ A_146 )
% 3.36/1.72 => ( ( finite_finite_a_o @ ( image_nat_a_o @ F_37 @ A_146 ) )
% 3.36/1.72 => ? [X_1: nat] :
% 3.36/1.72 ( ( member_nat @ X_1 @ A_146 )
% 3.36/1.72 & ~ ( finite_finite_nat
% 3.36/1.72 @ ( collect_nat
% 3.36/1.72 @ ^ [A_37: nat] :
% 3.36/1.72 ( (&) @ ( member_nat @ A_37 @ A_146 )
% 3.36/1.72 @ ( ( F_37 @ A_37 )
% 3.36/1.72 = ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_275_pigeonhole__infinite,axiom,
% 3.36/1.72 ! [F_37: int > x_a,A_146: int > $o] :
% 3.36/1.72 ( ~ ( finite_finite_int @ A_146 )
% 3.36/1.72 => ( ( finite_finite_a @ ( image_int_a @ F_37 @ A_146 ) )
% 3.36/1.72 => ? [X_1: int] :
% 3.36/1.72 ( ( member_int @ X_1 @ A_146 )
% 3.36/1.72 & ~ ( finite_finite_int
% 3.36/1.72 @ ( collect_int
% 3.36/1.72 @ ^ [A_37: int] :
% 3.36/1.72 ( (&) @ ( member_int @ A_37 @ A_146 )
% 3.36/1.72 @ ( ( F_37 @ A_37 )
% 3.36/1.72 = ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_276_pigeonhole__infinite,axiom,
% 3.36/1.72 ! [F_37: int > int > $o,A_146: int > $o] :
% 3.36/1.72 ( ~ ( finite_finite_int @ A_146 )
% 3.36/1.72 => ( ( finite_finite_int_o @ ( image_int_int_o @ F_37 @ A_146 ) )
% 3.36/1.72 => ? [X_1: int] :
% 3.36/1.72 ( ( member_int @ X_1 @ A_146 )
% 3.36/1.72 & ~ ( finite_finite_int
% 3.36/1.72 @ ( collect_int
% 3.36/1.72 @ ^ [A_37: int] :
% 3.36/1.72 ( (&) @ ( member_int @ A_37 @ A_146 )
% 3.36/1.72 @ ( ( F_37 @ A_37 )
% 3.36/1.72 = ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_277_pigeonhole__infinite,axiom,
% 3.36/1.72 ! [F_37: int > nat > $o,A_146: int > $o] :
% 3.36/1.72 ( ~ ( finite_finite_int @ A_146 )
% 3.36/1.72 => ( ( finite_finite_nat_o @ ( image_int_nat_o @ F_37 @ A_146 ) )
% 3.36/1.72 => ? [X_1: int] :
% 3.36/1.72 ( ( member_int @ X_1 @ A_146 )
% 3.36/1.72 & ~ ( finite_finite_int
% 3.36/1.72 @ ( collect_int
% 3.36/1.72 @ ^ [A_37: int] :
% 3.36/1.72 ( (&) @ ( member_int @ A_37 @ A_146 )
% 3.36/1.72 @ ( ( F_37 @ A_37 )
% 3.36/1.72 = ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_278_pigeonhole__infinite,axiom,
% 3.36/1.72 ! [F_37: int > pname > $o,A_146: int > $o] :
% 3.36/1.72 ( ~ ( finite_finite_int @ A_146 )
% 3.36/1.72 => ( ( finite297249702name_o @ ( image_int_pname_o @ F_37 @ A_146 ) )
% 3.36/1.72 => ? [X_1: int] :
% 3.36/1.72 ( ( member_int @ X_1 @ A_146 )
% 3.36/1.72 & ~ ( finite_finite_int
% 3.36/1.72 @ ( collect_int
% 3.36/1.72 @ ^ [A_37: int] :
% 3.36/1.72 ( (&) @ ( member_int @ A_37 @ A_146 )
% 3.36/1.72 @ ( ( F_37 @ A_37 )
% 3.36/1.72 = ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_279_pigeonhole__infinite,axiom,
% 3.36/1.72 ! [F_37: int > x_a > $o,A_146: int > $o] :
% 3.36/1.72 ( ~ ( finite_finite_int @ A_146 )
% 3.36/1.72 => ( ( finite_finite_a_o @ ( image_int_a_o @ F_37 @ A_146 ) )
% 3.36/1.72 => ? [X_1: int] :
% 3.36/1.72 ( ( member_int @ X_1 @ A_146 )
% 3.36/1.72 & ~ ( finite_finite_int
% 3.36/1.72 @ ( collect_int
% 3.36/1.72 @ ^ [A_37: int] :
% 3.36/1.72 ( (&) @ ( member_int @ A_37 @ A_146 )
% 3.36/1.72 @ ( ( F_37 @ A_37 )
% 3.36/1.72 = ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_280_pigeonhole__infinite,axiom,
% 3.36/1.72 ! [F_37: pname > x_a,A_146: pname > $o] :
% 3.36/1.72 ( ~ ( finite_finite_pname @ A_146 )
% 3.36/1.72 => ( ( finite_finite_a @ ( image_pname_a @ F_37 @ A_146 ) )
% 3.36/1.72 => ? [X_1: pname] :
% 3.36/1.72 ( ( member_pname @ X_1 @ A_146 )
% 3.36/1.72 & ~ ( finite_finite_pname
% 3.36/1.72 @ ( collect_pname
% 3.36/1.72 @ ^ [A_37: pname] :
% 3.36/1.72 ( (&) @ ( member_pname @ A_37 @ A_146 )
% 3.36/1.72 @ ( ( F_37 @ A_37 )
% 3.36/1.72 = ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_281_pigeonhole__infinite,axiom,
% 3.36/1.72 ! [F_37: nat > int,A_146: nat > $o] :
% 3.36/1.72 ( ~ ( finite_finite_nat @ A_146 )
% 3.36/1.72 => ( ( finite_finite_int @ ( image_nat_int @ F_37 @ A_146 ) )
% 3.36/1.72 => ? [X_1: nat] :
% 3.36/1.72 ( ( member_nat @ X_1 @ A_146 )
% 3.36/1.72 & ~ ( finite_finite_nat
% 3.36/1.72 @ ( collect_nat
% 3.36/1.72 @ ^ [A_37: nat] :
% 3.36/1.72 ( (&) @ ( member_nat @ A_37 @ A_146 )
% 3.36/1.72 @ ( ( F_37 @ A_37 )
% 3.36/1.72 = ( F_37 @ X_1 ) ) ) ) ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_282_image__eqI,axiom,
% 3.36/1.72 ! [A_145: nat > $o,B_83: int,F_36: nat > int,X_50: nat] :
% 3.36/1.72 ( ( B_83
% 3.36/1.72 = ( F_36 @ X_50 ) )
% 3.36/1.72 => ( ( member_nat @ X_50 @ A_145 )
% 3.36/1.72 => ( member_int @ B_83 @ ( image_nat_int @ F_36 @ A_145 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_283_image__eqI,axiom,
% 3.36/1.72 ! [A_145: pname > $o,B_83: x_a,F_36: pname > x_a,X_50: pname] :
% 3.36/1.72 ( ( B_83
% 3.36/1.72 = ( F_36 @ X_50 ) )
% 3.36/1.72 => ( ( member_pname @ X_50 @ A_145 )
% 3.36/1.72 => ( member_a @ B_83 @ ( image_pname_a @ F_36 @ A_145 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_284_equalityI,axiom,
% 3.36/1.72 ! [A_144: int > $o,B_82: int > $o] :
% 3.36/1.72 ( ( ord_less_eq_int_o @ A_144 @ B_82 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ B_82 @ A_144 )
% 3.36/1.72 => ( A_144 = B_82 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_285_equalityI,axiom,
% 3.36/1.72 ! [A_144: nat > $o,B_82: nat > $o] :
% 3.36/1.72 ( ( ord_less_eq_nat_o @ A_144 @ B_82 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o @ B_82 @ A_144 )
% 3.36/1.72 => ( A_144 = B_82 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_286_equalityI,axiom,
% 3.36/1.72 ! [A_144: x_a > $o,B_82: x_a > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o @ A_144 @ B_82 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ B_82 @ A_144 )
% 3.36/1.72 => ( A_144 = B_82 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_287_subsetD,axiom,
% 3.36/1.72 ! [C_33: int,A_143: int > $o,B_81: int > $o] :
% 3.36/1.72 ( ( ord_less_eq_int_o @ A_143 @ B_81 )
% 3.36/1.72 => ( ( member_int @ C_33 @ A_143 )
% 3.36/1.72 => ( member_int @ C_33 @ B_81 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_288_subsetD,axiom,
% 3.36/1.72 ! [C_33: nat,A_143: nat > $o,B_81: nat > $o] :
% 3.36/1.72 ( ( ord_less_eq_nat_o @ A_143 @ B_81 )
% 3.36/1.72 => ( ( member_nat @ C_33 @ A_143 )
% 3.36/1.72 => ( member_nat @ C_33 @ B_81 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_289_subsetD,axiom,
% 3.36/1.72 ! [C_33: x_a,A_143: x_a > $o,B_81: x_a > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o @ A_143 @ B_81 )
% 3.36/1.72 => ( ( member_a @ C_33 @ A_143 )
% 3.36/1.72 => ( member_a @ C_33 @ B_81 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_290_subsetD,axiom,
% 3.36/1.72 ! [C_33: pname,A_143: pname > $o,B_81: pname > $o] :
% 3.36/1.72 ( ( ord_less_eq_pname_o @ A_143 @ B_81 )
% 3.36/1.72 => ( ( member_pname @ C_33 @ A_143 )
% 3.36/1.72 => ( member_pname @ C_33 @ B_81 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_291_insertCI,axiom,
% 3.36/1.72 ! [B_80: x_a,A_142: x_a,B_79: x_a > $o] :
% 3.36/1.72 ( ( ~ ( member_a @ A_142 @ B_79 )
% 3.36/1.72 => ( A_142 = B_80 ) )
% 3.36/1.72 => ( member_a @ A_142 @ ( insert_a @ B_80 @ B_79 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_292_insertCI,axiom,
% 3.36/1.72 ! [B_80: int,A_142: int,B_79: int > $o] :
% 3.36/1.72 ( ( ~ ( member_int @ A_142 @ B_79 )
% 3.36/1.72 => ( A_142 = B_80 ) )
% 3.36/1.72 => ( member_int @ A_142 @ ( insert_int @ B_80 @ B_79 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_293_insertCI,axiom,
% 3.36/1.72 ! [B_80: nat,A_142: nat,B_79: nat > $o] :
% 3.36/1.72 ( ( ~ ( member_nat @ A_142 @ B_79 )
% 3.36/1.72 => ( A_142 = B_80 ) )
% 3.36/1.72 => ( member_nat @ A_142 @ ( insert_nat @ B_80 @ B_79 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_294_insertCI,axiom,
% 3.36/1.72 ! [B_80: pname,A_142: pname,B_79: pname > $o] :
% 3.36/1.72 ( ( ~ ( member_pname @ A_142 @ B_79 )
% 3.36/1.72 => ( A_142 = B_80 ) )
% 3.36/1.72 => ( member_pname @ A_142 @ ( insert_pname @ B_80 @ B_79 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_295_insertE,axiom,
% 3.36/1.72 ! [A_141: x_a,B_78: x_a,A_140: x_a > $o] :
% 3.36/1.72 ( ( member_a @ A_141 @ ( insert_a @ B_78 @ A_140 ) )
% 3.36/1.72 => ( ( A_141 != B_78 )
% 3.36/1.72 => ( member_a @ A_141 @ A_140 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_296_insertE,axiom,
% 3.36/1.72 ! [A_141: int,B_78: int,A_140: int > $o] :
% 3.36/1.72 ( ( member_int @ A_141 @ ( insert_int @ B_78 @ A_140 ) )
% 3.36/1.72 => ( ( A_141 != B_78 )
% 3.36/1.72 => ( member_int @ A_141 @ A_140 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_297_insertE,axiom,
% 3.36/1.72 ! [A_141: nat,B_78: nat,A_140: nat > $o] :
% 3.36/1.72 ( ( member_nat @ A_141 @ ( insert_nat @ B_78 @ A_140 ) )
% 3.36/1.72 => ( ( A_141 != B_78 )
% 3.36/1.72 => ( member_nat @ A_141 @ A_140 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_298_insertE,axiom,
% 3.36/1.72 ! [A_141: pname,B_78: pname,A_140: pname > $o] :
% 3.36/1.72 ( ( member_pname @ A_141 @ ( insert_pname @ B_78 @ A_140 ) )
% 3.36/1.72 => ( ( A_141 != B_78 )
% 3.36/1.72 => ( member_pname @ A_141 @ A_140 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_299_zero__induct__lemma,axiom,
% 3.36/1.72 ! [I_1: nat,P: nat > $o,K: nat] :
% 3.36/1.72 ( ( P @ K )
% 3.36/1.72 => ( ! [N_1: nat] :
% 3.36/1.72 ( ( P @ ( suc @ N_1 ) )
% 3.36/1.72 => ( P @ N_1 ) )
% 3.36/1.72 => ( P @ ( minus_minus_nat @ K @ I_1 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_300_Suc__le__D,axiom,
% 3.36/1.72 ! [N: nat,M_3: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ ( suc @ N ) @ M_3 )
% 3.36/1.72 => ? [M_1: nat] :
% 3.36/1.72 ( M_3
% 3.36/1.72 = ( suc @ M_1 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_301_order__refl,axiom,
% 3.36/1.72 ! [X_49: int > $o] : ( ord_less_eq_int_o @ X_49 @ X_49 ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_302_order__refl,axiom,
% 3.36/1.72 ! [X_49: nat > $o] : ( ord_less_eq_nat_o @ X_49 @ X_49 ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_303_order__refl,axiom,
% 3.36/1.72 ! [X_49: int] : ( ord_less_eq_int @ X_49 @ X_49 ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_304_order__refl,axiom,
% 3.36/1.72 ! [X_49: nat] : ( ord_less_eq_nat @ X_49 @ X_49 ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_305_order__refl,axiom,
% 3.36/1.72 ! [X_49: x_a > $o] : ( ord_less_eq_a_o @ X_49 @ X_49 ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_306_linorder__linear,axiom,
% 3.36/1.72 ! [X_48: int,Y_12: int] :
% 3.36/1.72 ( ( ord_less_eq_int @ X_48 @ Y_12 )
% 3.36/1.72 | ( ord_less_eq_int @ Y_12 @ X_48 ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_307_linorder__linear,axiom,
% 3.36/1.72 ! [X_48: nat,Y_12: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ X_48 @ Y_12 )
% 3.36/1.72 | ( ord_less_eq_nat @ Y_12 @ X_48 ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_308_order__eq__iff,axiom,
% 3.36/1.72 ! [X_47: int > $o,Y_11: int > $o] :
% 3.36/1.72 ( ( X_47 = Y_11 )
% 3.36/1.72 <=> ( ( ord_less_eq_int_o @ X_47 @ Y_11 )
% 3.36/1.72 & ( ord_less_eq_int_o @ Y_11 @ X_47 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_309_order__eq__iff,axiom,
% 3.36/1.72 ! [X_47: nat > $o,Y_11: nat > $o] :
% 3.36/1.72 ( ( X_47 = Y_11 )
% 3.36/1.72 <=> ( ( ord_less_eq_nat_o @ X_47 @ Y_11 )
% 3.36/1.72 & ( ord_less_eq_nat_o @ Y_11 @ X_47 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_310_order__eq__iff,axiom,
% 3.36/1.72 ! [X_47: int,Y_11: int] :
% 3.36/1.72 ( ( X_47 = Y_11 )
% 3.36/1.72 <=> ( ( ord_less_eq_int @ X_47 @ Y_11 )
% 3.36/1.72 & ( ord_less_eq_int @ Y_11 @ X_47 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_311_order__eq__iff,axiom,
% 3.36/1.72 ! [X_47: nat,Y_11: nat] :
% 3.36/1.72 ( ( X_47 = Y_11 )
% 3.36/1.72 <=> ( ( ord_less_eq_nat @ X_47 @ Y_11 )
% 3.36/1.72 & ( ord_less_eq_nat @ Y_11 @ X_47 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_312_order__eq__iff,axiom,
% 3.36/1.72 ! [X_47: x_a > $o,Y_11: x_a > $o] :
% 3.36/1.72 ( ( X_47 = Y_11 )
% 3.36/1.72 <=> ( ( ord_less_eq_a_o @ X_47 @ Y_11 )
% 3.36/1.72 & ( ord_less_eq_a_o @ Y_11 @ X_47 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_313_order__eq__refl,axiom,
% 3.36/1.72 ! [X_46: int > $o,Y_10: int > $o] :
% 3.36/1.72 ( ( X_46 = Y_10 )
% 3.36/1.72 => ( ord_less_eq_int_o @ X_46 @ Y_10 ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_314_order__eq__refl,axiom,
% 3.36/1.72 ! [X_46: nat > $o,Y_10: nat > $o] :
% 3.36/1.72 ( ( X_46 = Y_10 )
% 3.36/1.72 => ( ord_less_eq_nat_o @ X_46 @ Y_10 ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_315_order__eq__refl,axiom,
% 3.36/1.72 ! [X_46: int,Y_10: int] :
% 3.36/1.72 ( ( X_46 = Y_10 )
% 3.36/1.72 => ( ord_less_eq_int @ X_46 @ Y_10 ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_316_order__eq__refl,axiom,
% 3.36/1.72 ! [X_46: nat,Y_10: nat] :
% 3.36/1.72 ( ( X_46 = Y_10 )
% 3.36/1.72 => ( ord_less_eq_nat @ X_46 @ Y_10 ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_317_order__eq__refl,axiom,
% 3.36/1.72 ! [X_46: x_a > $o,Y_10: x_a > $o] :
% 3.36/1.72 ( ( X_46 = Y_10 )
% 3.36/1.72 => ( ord_less_eq_a_o @ X_46 @ Y_10 ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_318_order__antisym__conv,axiom,
% 3.36/1.72 ! [Y_9: int > $o,X_45: int > $o] :
% 3.36/1.72 ( ( ord_less_eq_int_o @ Y_9 @ X_45 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ X_45 @ Y_9 )
% 3.36/1.72 <=> ( X_45 = Y_9 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_319_order__antisym__conv,axiom,
% 3.36/1.72 ! [Y_9: nat > $o,X_45: nat > $o] :
% 3.36/1.72 ( ( ord_less_eq_nat_o @ Y_9 @ X_45 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o @ X_45 @ Y_9 )
% 3.36/1.72 <=> ( X_45 = Y_9 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_320_order__antisym__conv,axiom,
% 3.36/1.72 ! [Y_9: int,X_45: int] :
% 3.36/1.72 ( ( ord_less_eq_int @ Y_9 @ X_45 )
% 3.36/1.72 => ( ( ord_less_eq_int @ X_45 @ Y_9 )
% 3.36/1.72 <=> ( X_45 = Y_9 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_321_order__antisym__conv,axiom,
% 3.36/1.72 ! [Y_9: nat,X_45: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ Y_9 @ X_45 )
% 3.36/1.72 => ( ( ord_less_eq_nat @ X_45 @ Y_9 )
% 3.36/1.72 <=> ( X_45 = Y_9 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_322_order__antisym__conv,axiom,
% 3.36/1.72 ! [Y_9: x_a > $o,X_45: x_a > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o @ Y_9 @ X_45 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ X_45 @ Y_9 )
% 3.36/1.72 <=> ( X_45 = Y_9 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_323_ord__eq__le__trans,axiom,
% 3.36/1.72 ! [C_32: int > $o,A_139: int > $o,B_77: int > $o] :
% 3.36/1.72 ( ( A_139 = B_77 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ B_77 @ C_32 )
% 3.36/1.72 => ( ord_less_eq_int_o @ A_139 @ C_32 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_324_ord__eq__le__trans,axiom,
% 3.36/1.72 ! [C_32: nat > $o,A_139: nat > $o,B_77: nat > $o] :
% 3.36/1.72 ( ( A_139 = B_77 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o @ B_77 @ C_32 )
% 3.36/1.72 => ( ord_less_eq_nat_o @ A_139 @ C_32 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_325_ord__eq__le__trans,axiom,
% 3.36/1.72 ! [C_32: int,A_139: int,B_77: int] :
% 3.36/1.72 ( ( A_139 = B_77 )
% 3.36/1.72 => ( ( ord_less_eq_int @ B_77 @ C_32 )
% 3.36/1.72 => ( ord_less_eq_int @ A_139 @ C_32 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_326_ord__eq__le__trans,axiom,
% 3.36/1.72 ! [C_32: nat,A_139: nat,B_77: nat] :
% 3.36/1.72 ( ( A_139 = B_77 )
% 3.36/1.72 => ( ( ord_less_eq_nat @ B_77 @ C_32 )
% 3.36/1.72 => ( ord_less_eq_nat @ A_139 @ C_32 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_327_ord__eq__le__trans,axiom,
% 3.36/1.72 ! [C_32: x_a > $o,A_139: x_a > $o,B_77: x_a > $o] :
% 3.36/1.72 ( ( A_139 = B_77 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ B_77 @ C_32 )
% 3.36/1.72 => ( ord_less_eq_a_o @ A_139 @ C_32 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_328_xt1_I3_J,axiom,
% 3.36/1.72 ! [C_31: int > $o,A_138: int > $o,B_76: int > $o] :
% 3.36/1.72 ( ( A_138 = B_76 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ C_31 @ B_76 )
% 3.36/1.72 => ( ord_less_eq_int_o @ C_31 @ A_138 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_329_xt1_I3_J,axiom,
% 3.36/1.72 ! [C_31: nat > $o,A_138: nat > $o,B_76: nat > $o] :
% 3.36/1.72 ( ( A_138 = B_76 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o @ C_31 @ B_76 )
% 3.36/1.72 => ( ord_less_eq_nat_o @ C_31 @ A_138 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_330_xt1_I3_J,axiom,
% 3.36/1.72 ! [C_31: int,A_138: int,B_76: int] :
% 3.36/1.72 ( ( A_138 = B_76 )
% 3.36/1.72 => ( ( ord_less_eq_int @ C_31 @ B_76 )
% 3.36/1.72 => ( ord_less_eq_int @ C_31 @ A_138 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_331_xt1_I3_J,axiom,
% 3.36/1.72 ! [C_31: nat,A_138: nat,B_76: nat] :
% 3.36/1.72 ( ( A_138 = B_76 )
% 3.36/1.72 => ( ( ord_less_eq_nat @ C_31 @ B_76 )
% 3.36/1.72 => ( ord_less_eq_nat @ C_31 @ A_138 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_332_xt1_I3_J,axiom,
% 3.36/1.72 ! [C_31: x_a > $o,A_138: x_a > $o,B_76: x_a > $o] :
% 3.36/1.72 ( ( A_138 = B_76 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ C_31 @ B_76 )
% 3.36/1.72 => ( ord_less_eq_a_o @ C_31 @ A_138 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_333_ord__le__eq__trans,axiom,
% 3.36/1.72 ! [C_30: int > $o,A_137: int > $o,B_75: int > $o] :
% 3.36/1.72 ( ( ord_less_eq_int_o @ A_137 @ B_75 )
% 3.36/1.72 => ( ( B_75 = C_30 )
% 3.36/1.72 => ( ord_less_eq_int_o @ A_137 @ C_30 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_334_ord__le__eq__trans,axiom,
% 3.36/1.72 ! [C_30: nat > $o,A_137: nat > $o,B_75: nat > $o] :
% 3.36/1.72 ( ( ord_less_eq_nat_o @ A_137 @ B_75 )
% 3.36/1.72 => ( ( B_75 = C_30 )
% 3.36/1.72 => ( ord_less_eq_nat_o @ A_137 @ C_30 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_335_ord__le__eq__trans,axiom,
% 3.36/1.72 ! [C_30: int,A_137: int,B_75: int] :
% 3.36/1.72 ( ( ord_less_eq_int @ A_137 @ B_75 )
% 3.36/1.72 => ( ( B_75 = C_30 )
% 3.36/1.72 => ( ord_less_eq_int @ A_137 @ C_30 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_336_ord__le__eq__trans,axiom,
% 3.36/1.72 ! [C_30: nat,A_137: nat,B_75: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ A_137 @ B_75 )
% 3.36/1.72 => ( ( B_75 = C_30 )
% 3.36/1.72 => ( ord_less_eq_nat @ A_137 @ C_30 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_337_ord__le__eq__trans,axiom,
% 3.36/1.72 ! [C_30: x_a > $o,A_137: x_a > $o,B_75: x_a > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o @ A_137 @ B_75 )
% 3.36/1.72 => ( ( B_75 = C_30 )
% 3.36/1.72 => ( ord_less_eq_a_o @ A_137 @ C_30 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_338_xt1_I4_J,axiom,
% 3.36/1.72 ! [C_29: int > $o,B_74: int > $o,A_136: int > $o] :
% 3.36/1.72 ( ( ord_less_eq_int_o @ B_74 @ A_136 )
% 3.36/1.72 => ( ( B_74 = C_29 )
% 3.36/1.72 => ( ord_less_eq_int_o @ C_29 @ A_136 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_339_xt1_I4_J,axiom,
% 3.36/1.72 ! [C_29: nat > $o,B_74: nat > $o,A_136: nat > $o] :
% 3.36/1.72 ( ( ord_less_eq_nat_o @ B_74 @ A_136 )
% 3.36/1.72 => ( ( B_74 = C_29 )
% 3.36/1.72 => ( ord_less_eq_nat_o @ C_29 @ A_136 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_340_xt1_I4_J,axiom,
% 3.36/1.72 ! [C_29: int,B_74: int,A_136: int] :
% 3.36/1.72 ( ( ord_less_eq_int @ B_74 @ A_136 )
% 3.36/1.72 => ( ( B_74 = C_29 )
% 3.36/1.72 => ( ord_less_eq_int @ C_29 @ A_136 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_341_xt1_I4_J,axiom,
% 3.36/1.72 ! [C_29: nat,B_74: nat,A_136: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ B_74 @ A_136 )
% 3.36/1.72 => ( ( B_74 = C_29 )
% 3.36/1.72 => ( ord_less_eq_nat @ C_29 @ A_136 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_342_xt1_I4_J,axiom,
% 3.36/1.72 ! [C_29: x_a > $o,B_74: x_a > $o,A_136: x_a > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o @ B_74 @ A_136 )
% 3.36/1.72 => ( ( B_74 = C_29 )
% 3.36/1.72 => ( ord_less_eq_a_o @ C_29 @ A_136 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_343_order__antisym,axiom,
% 3.36/1.72 ! [X_44: int > $o,Y_8: int > $o] :
% 3.36/1.72 ( ( ord_less_eq_int_o @ X_44 @ Y_8 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ Y_8 @ X_44 )
% 3.36/1.72 => ( X_44 = Y_8 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_344_order__antisym,axiom,
% 3.36/1.72 ! [X_44: nat > $o,Y_8: nat > $o] :
% 3.36/1.72 ( ( ord_less_eq_nat_o @ X_44 @ Y_8 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o @ Y_8 @ X_44 )
% 3.36/1.72 => ( X_44 = Y_8 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_345_order__antisym,axiom,
% 3.36/1.72 ! [X_44: int,Y_8: int] :
% 3.36/1.72 ( ( ord_less_eq_int @ X_44 @ Y_8 )
% 3.36/1.72 => ( ( ord_less_eq_int @ Y_8 @ X_44 )
% 3.36/1.72 => ( X_44 = Y_8 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_346_order__antisym,axiom,
% 3.36/1.72 ! [X_44: nat,Y_8: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ X_44 @ Y_8 )
% 3.36/1.72 => ( ( ord_less_eq_nat @ Y_8 @ X_44 )
% 3.36/1.72 => ( X_44 = Y_8 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_347_order__antisym,axiom,
% 3.36/1.72 ! [X_44: x_a > $o,Y_8: x_a > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o @ X_44 @ Y_8 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ Y_8 @ X_44 )
% 3.36/1.72 => ( X_44 = Y_8 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_348_order__trans,axiom,
% 3.36/1.72 ! [Z_6: int > $o,X_43: int > $o,Y_7: int > $o] :
% 3.36/1.72 ( ( ord_less_eq_int_o @ X_43 @ Y_7 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ Y_7 @ Z_6 )
% 3.36/1.72 => ( ord_less_eq_int_o @ X_43 @ Z_6 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_349_order__trans,axiom,
% 3.36/1.72 ! [Z_6: nat > $o,X_43: nat > $o,Y_7: nat > $o] :
% 3.36/1.72 ( ( ord_less_eq_nat_o @ X_43 @ Y_7 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o @ Y_7 @ Z_6 )
% 3.36/1.72 => ( ord_less_eq_nat_o @ X_43 @ Z_6 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_350_order__trans,axiom,
% 3.36/1.72 ! [Z_6: int,X_43: int,Y_7: int] :
% 3.36/1.72 ( ( ord_less_eq_int @ X_43 @ Y_7 )
% 3.36/1.72 => ( ( ord_less_eq_int @ Y_7 @ Z_6 )
% 3.36/1.72 => ( ord_less_eq_int @ X_43 @ Z_6 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_351_order__trans,axiom,
% 3.36/1.72 ! [Z_6: nat,X_43: nat,Y_7: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ X_43 @ Y_7 )
% 3.36/1.72 => ( ( ord_less_eq_nat @ Y_7 @ Z_6 )
% 3.36/1.72 => ( ord_less_eq_nat @ X_43 @ Z_6 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_352_order__trans,axiom,
% 3.36/1.72 ! [Z_6: x_a > $o,X_43: x_a > $o,Y_7: x_a > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o @ X_43 @ Y_7 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ Y_7 @ Z_6 )
% 3.36/1.72 => ( ord_less_eq_a_o @ X_43 @ Z_6 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_353_xt1_I5_J,axiom,
% 3.36/1.72 ! [Y_6: int > $o,X_42: int > $o] :
% 3.36/1.72 ( ( ord_less_eq_int_o @ Y_6 @ X_42 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ X_42 @ Y_6 )
% 3.36/1.72 => ( X_42 = Y_6 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_354_xt1_I5_J,axiom,
% 3.36/1.72 ! [Y_6: nat > $o,X_42: nat > $o] :
% 3.36/1.72 ( ( ord_less_eq_nat_o @ Y_6 @ X_42 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o @ X_42 @ Y_6 )
% 3.36/1.72 => ( X_42 = Y_6 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_355_xt1_I5_J,axiom,
% 3.36/1.72 ! [Y_6: int,X_42: int] :
% 3.36/1.72 ( ( ord_less_eq_int @ Y_6 @ X_42 )
% 3.36/1.72 => ( ( ord_less_eq_int @ X_42 @ Y_6 )
% 3.36/1.72 => ( X_42 = Y_6 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_356_xt1_I5_J,axiom,
% 3.36/1.72 ! [Y_6: nat,X_42: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ Y_6 @ X_42 )
% 3.36/1.72 => ( ( ord_less_eq_nat @ X_42 @ Y_6 )
% 3.36/1.72 => ( X_42 = Y_6 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_357_xt1_I5_J,axiom,
% 3.36/1.72 ! [Y_6: x_a > $o,X_42: x_a > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o @ Y_6 @ X_42 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ X_42 @ Y_6 )
% 3.36/1.72 => ( X_42 = Y_6 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_358_xt1_I6_J,axiom,
% 3.36/1.72 ! [Z_5: int > $o,Y_5: int > $o,X_41: int > $o] :
% 3.36/1.72 ( ( ord_less_eq_int_o @ Y_5 @ X_41 )
% 3.36/1.72 => ( ( ord_less_eq_int_o @ Z_5 @ Y_5 )
% 3.36/1.72 => ( ord_less_eq_int_o @ Z_5 @ X_41 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_359_xt1_I6_J,axiom,
% 3.36/1.72 ! [Z_5: nat > $o,Y_5: nat > $o,X_41: nat > $o] :
% 3.36/1.72 ( ( ord_less_eq_nat_o @ Y_5 @ X_41 )
% 3.36/1.72 => ( ( ord_less_eq_nat_o @ Z_5 @ Y_5 )
% 3.36/1.72 => ( ord_less_eq_nat_o @ Z_5 @ X_41 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_360_xt1_I6_J,axiom,
% 3.36/1.72 ! [Z_5: int,Y_5: int,X_41: int] :
% 3.36/1.72 ( ( ord_less_eq_int @ Y_5 @ X_41 )
% 3.36/1.72 => ( ( ord_less_eq_int @ Z_5 @ Y_5 )
% 3.36/1.72 => ( ord_less_eq_int @ Z_5 @ X_41 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_361_xt1_I6_J,axiom,
% 3.36/1.72 ! [Z_5: nat,Y_5: nat,X_41: nat] :
% 3.36/1.72 ( ( ord_less_eq_nat @ Y_5 @ X_41 )
% 3.36/1.72 => ( ( ord_less_eq_nat @ Z_5 @ Y_5 )
% 3.36/1.72 => ( ord_less_eq_nat @ Z_5 @ X_41 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_362_xt1_I6_J,axiom,
% 3.36/1.72 ! [Z_5: x_a > $o,Y_5: x_a > $o,X_41: x_a > $o] :
% 3.36/1.72 ( ( ord_less_eq_a_o @ Y_5 @ X_41 )
% 3.36/1.72 => ( ( ord_less_eq_a_o @ Z_5 @ Y_5 )
% 3.36/1.72 => ( ord_less_eq_a_o @ Z_5 @ X_41 ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_363_linorder__le__cases,axiom,
% 3.36/1.72 ! [X_40: int,Y_4: int] :
% 3.36/1.72 ( ~ ( ord_less_eq_int @ X_40 @ Y_4 )
% 3.36/1.72 => ( ord_less_eq_int @ Y_4 @ X_40 ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_364_linorder__le__cases,axiom,
% 3.36/1.72 ! [X_40: nat,Y_4: nat] :
% 3.36/1.72 ( ~ ( ord_less_eq_nat @ X_40 @ Y_4 )
% 3.36/1.72 => ( ord_less_eq_nat @ Y_4 @ X_40 ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_365_insertI1,axiom,
% 3.36/1.72 ! [A_135: x_a,B_73: x_a > $o] : ( member_a @ A_135 @ ( insert_a @ A_135 @ B_73 ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_366_insertI1,axiom,
% 3.36/1.72 ! [A_135: int,B_73: int > $o] : ( member_int @ A_135 @ ( insert_int @ A_135 @ B_73 ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_367_insertI1,axiom,
% 3.36/1.72 ! [A_135: nat,B_73: nat > $o] : ( member_nat @ A_135 @ ( insert_nat @ A_135 @ B_73 ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_368_insertI1,axiom,
% 3.36/1.72 ! [A_135: pname,B_73: pname > $o] : ( member_pname @ A_135 @ ( insert_pname @ A_135 @ B_73 ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_369_insert__compr,axiom,
% 3.36/1.72 ! [A_134: x_a,B_72: x_a > $o] :
% 3.36/1.72 ( ( insert_a @ A_134 @ B_72 )
% 3.36/1.72 = ( collect_a
% 3.36/1.72 @ ^ [X_1: x_a] : ( (|) @ ( X_1 = A_134 ) @ ( member_a @ X_1 @ B_72 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_370_insert__compr,axiom,
% 3.36/1.72 ! [A_134: int,B_72: int > $o] :
% 3.36/1.72 ( ( insert_int @ A_134 @ B_72 )
% 3.36/1.72 = ( collect_int
% 3.36/1.72 @ ^ [X_1: int] : ( (|) @ ( X_1 = A_134 ) @ ( member_int @ X_1 @ B_72 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_371_insert__compr,axiom,
% 3.36/1.72 ! [A_134: nat,B_72: nat > $o] :
% 3.36/1.72 ( ( insert_nat @ A_134 @ B_72 )
% 3.36/1.72 = ( collect_nat
% 3.36/1.72 @ ^ [X_1: nat] : ( (|) @ ( X_1 = A_134 ) @ ( member_nat @ X_1 @ B_72 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_372_insert__compr,axiom,
% 3.36/1.72 ! [A_134: pname,B_72: pname > $o] :
% 3.36/1.72 ( ( insert_pname @ A_134 @ B_72 )
% 3.36/1.72 = ( collect_pname
% 3.36/1.72 @ ^ [X_1: pname] : ( (|) @ ( X_1 = A_134 ) @ ( member_pname @ X_1 @ B_72 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_373_insert__Collect,axiom,
% 3.36/1.72 ! [A_133: x_a,P_11: x_a > $o] :
% 3.36/1.72 ( ( insert_a @ A_133 @ ( collect_a @ P_11 ) )
% 3.36/1.72 = ( collect_a
% 3.36/1.72 @ ^ [U_1: x_a] : ( (=>) @ ( (~) @ ( U_1 = A_133 ) ) @ ( P_11 @ U_1 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_374_insert__Collect,axiom,
% 3.36/1.72 ! [A_133: int,P_11: int > $o] :
% 3.36/1.72 ( ( insert_int @ A_133 @ ( collect_int @ P_11 ) )
% 3.36/1.72 = ( collect_int
% 3.36/1.72 @ ^ [U_1: int] : ( (=>) @ ( (~) @ ( U_1 = A_133 ) ) @ ( P_11 @ U_1 ) ) ) ) ).
% 3.36/1.72
% 3.36/1.72 thf(fact_375_insert__Collect,axiom,
% 3.36/1.73 ! [A_133: nat,P_11: nat > $o] :
% 3.36/1.73 ( ( insert_nat @ A_133 @ ( collect_nat @ P_11 ) )
% 3.36/1.73 = ( collect_nat
% 3.36/1.73 @ ^ [U_1: nat] : ( (=>) @ ( (~) @ ( U_1 = A_133 ) ) @ ( P_11 @ U_1 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_376_mem__def,axiom,
% 3.36/1.73 ! [X_39: int,A_132: int > $o] :
% 3.36/1.73 ( ( member_int @ X_39 @ A_132 )
% 3.36/1.73 <=> ( A_132 @ X_39 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_377_mem__def,axiom,
% 3.36/1.73 ! [X_39: nat,A_132: nat > $o] :
% 3.36/1.73 ( ( member_nat @ X_39 @ A_132 )
% 3.36/1.73 <=> ( A_132 @ X_39 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_378_mem__def,axiom,
% 3.36/1.73 ! [X_39: x_a,A_132: x_a > $o] :
% 3.36/1.73 ( ( member_a @ X_39 @ A_132 )
% 3.36/1.73 <=> ( A_132 @ X_39 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_379_mem__def,axiom,
% 3.36/1.73 ! [X_39: pname,A_132: pname > $o] :
% 3.36/1.73 ( ( member_pname @ X_39 @ A_132 )
% 3.36/1.73 <=> ( A_132 @ X_39 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_380_Collect__def,axiom,
% 3.36/1.73 ! [P_10: int > $o] :
% 3.36/1.73 ( ( collect_int @ P_10 )
% 3.36/1.73 = P_10 ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_381_Collect__def,axiom,
% 3.36/1.73 ! [P_10: nat > $o] :
% 3.36/1.73 ( ( collect_nat @ P_10 )
% 3.36/1.73 = P_10 ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_382_insert__absorb2,axiom,
% 3.36/1.73 ! [X_38: x_a,A_131: x_a > $o] :
% 3.36/1.73 ( ( insert_a @ X_38 @ ( insert_a @ X_38 @ A_131 ) )
% 3.36/1.73 = ( insert_a @ X_38 @ A_131 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_383_insert__commute,axiom,
% 3.36/1.73 ! [X_37: x_a,Y_3: x_a,A_130: x_a > $o] :
% 3.36/1.73 ( ( insert_a @ X_37 @ ( insert_a @ Y_3 @ A_130 ) )
% 3.36/1.73 = ( insert_a @ Y_3 @ ( insert_a @ X_37 @ A_130 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_384_insert__iff,axiom,
% 3.36/1.73 ! [A_129: x_a,B_71: x_a,A_128: x_a > $o] :
% 3.36/1.73 ( ( member_a @ A_129 @ ( insert_a @ B_71 @ A_128 ) )
% 3.36/1.73 <=> ( ( A_129 = B_71 )
% 3.36/1.73 | ( member_a @ A_129 @ A_128 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_385_insert__iff,axiom,
% 3.36/1.73 ! [A_129: int,B_71: int,A_128: int > $o] :
% 3.36/1.73 ( ( member_int @ A_129 @ ( insert_int @ B_71 @ A_128 ) )
% 3.36/1.73 <=> ( ( A_129 = B_71 )
% 3.36/1.73 | ( member_int @ A_129 @ A_128 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_386_insert__iff,axiom,
% 3.36/1.73 ! [A_129: nat,B_71: nat,A_128: nat > $o] :
% 3.36/1.73 ( ( member_nat @ A_129 @ ( insert_nat @ B_71 @ A_128 ) )
% 3.36/1.73 <=> ( ( A_129 = B_71 )
% 3.36/1.73 | ( member_nat @ A_129 @ A_128 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_387_insert__iff,axiom,
% 3.36/1.73 ! [A_129: pname,B_71: pname,A_128: pname > $o] :
% 3.36/1.73 ( ( member_pname @ A_129 @ ( insert_pname @ B_71 @ A_128 ) )
% 3.36/1.73 <=> ( ( A_129 = B_71 )
% 3.36/1.73 | ( member_pname @ A_129 @ A_128 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_388_insert__code,axiom,
% 3.36/1.73 ! [Y_2: x_a,A_127: x_a > $o,X_36: x_a] :
% 3.36/1.73 ( ( insert_a @ Y_2 @ A_127 @ X_36 )
% 3.36/1.73 <=> ( ( Y_2 = X_36 )
% 3.36/1.73 | ( A_127 @ X_36 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_389_insert__ident,axiom,
% 3.36/1.73 ! [B_70: x_a > $o,X_35: x_a,A_126: x_a > $o] :
% 3.36/1.73 ( ~ ( member_a @ X_35 @ A_126 )
% 3.36/1.73 => ( ~ ( member_a @ X_35 @ B_70 )
% 3.36/1.73 => ( ( ( insert_a @ X_35 @ A_126 )
% 3.36/1.73 = ( insert_a @ X_35 @ B_70 ) )
% 3.36/1.73 <=> ( A_126 = B_70 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_390_insert__ident,axiom,
% 3.36/1.73 ! [B_70: int > $o,X_35: int,A_126: int > $o] :
% 3.36/1.73 ( ~ ( member_int @ X_35 @ A_126 )
% 3.36/1.73 => ( ~ ( member_int @ X_35 @ B_70 )
% 3.36/1.73 => ( ( ( insert_int @ X_35 @ A_126 )
% 3.36/1.73 = ( insert_int @ X_35 @ B_70 ) )
% 3.36/1.73 <=> ( A_126 = B_70 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_391_insert__ident,axiom,
% 3.36/1.73 ! [B_70: nat > $o,X_35: nat,A_126: nat > $o] :
% 3.36/1.73 ( ~ ( member_nat @ X_35 @ A_126 )
% 3.36/1.73 => ( ~ ( member_nat @ X_35 @ B_70 )
% 3.36/1.73 => ( ( ( insert_nat @ X_35 @ A_126 )
% 3.36/1.73 = ( insert_nat @ X_35 @ B_70 ) )
% 3.36/1.73 <=> ( A_126 = B_70 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_392_insert__ident,axiom,
% 3.36/1.73 ! [B_70: pname > $o,X_35: pname,A_126: pname > $o] :
% 3.36/1.73 ( ~ ( member_pname @ X_35 @ A_126 )
% 3.36/1.73 => ( ~ ( member_pname @ X_35 @ B_70 )
% 3.36/1.73 => ( ( ( insert_pname @ X_35 @ A_126 )
% 3.36/1.73 = ( insert_pname @ X_35 @ B_70 ) )
% 3.36/1.73 <=> ( A_126 = B_70 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_393_insertI2,axiom,
% 3.36/1.73 ! [B_69: x_a,A_125: x_a,B_68: x_a > $o] :
% 3.36/1.73 ( ( member_a @ A_125 @ B_68 )
% 3.36/1.73 => ( member_a @ A_125 @ ( insert_a @ B_69 @ B_68 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_394_insertI2,axiom,
% 3.36/1.73 ! [B_69: int,A_125: int,B_68: int > $o] :
% 3.36/1.73 ( ( member_int @ A_125 @ B_68 )
% 3.36/1.73 => ( member_int @ A_125 @ ( insert_int @ B_69 @ B_68 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_395_insertI2,axiom,
% 3.36/1.73 ! [B_69: nat,A_125: nat,B_68: nat > $o] :
% 3.36/1.73 ( ( member_nat @ A_125 @ B_68 )
% 3.36/1.73 => ( member_nat @ A_125 @ ( insert_nat @ B_69 @ B_68 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_396_insertI2,axiom,
% 3.36/1.73 ! [B_69: pname,A_125: pname,B_68: pname > $o] :
% 3.36/1.73 ( ( member_pname @ A_125 @ B_68 )
% 3.36/1.73 => ( member_pname @ A_125 @ ( insert_pname @ B_69 @ B_68 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_397_insert__absorb,axiom,
% 3.36/1.73 ! [A_124: x_a,A_123: x_a > $o] :
% 3.36/1.73 ( ( member_a @ A_124 @ A_123 )
% 3.36/1.73 => ( ( insert_a @ A_124 @ A_123 )
% 3.36/1.73 = A_123 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_398_insert__absorb,axiom,
% 3.36/1.73 ! [A_124: int,A_123: int > $o] :
% 3.36/1.73 ( ( member_int @ A_124 @ A_123 )
% 3.36/1.73 => ( ( insert_int @ A_124 @ A_123 )
% 3.36/1.73 = A_123 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_399_insert__absorb,axiom,
% 3.36/1.73 ! [A_124: nat,A_123: nat > $o] :
% 3.36/1.73 ( ( member_nat @ A_124 @ A_123 )
% 3.36/1.73 => ( ( insert_nat @ A_124 @ A_123 )
% 3.36/1.73 = A_123 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_400_insert__absorb,axiom,
% 3.36/1.73 ! [A_124: pname,A_123: pname > $o] :
% 3.36/1.73 ( ( member_pname @ A_124 @ A_123 )
% 3.36/1.73 => ( ( insert_pname @ A_124 @ A_123 )
% 3.36/1.73 = A_123 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_401_subset__refl,axiom,
% 3.36/1.73 ! [A_122: int > $o] : ( ord_less_eq_int_o @ A_122 @ A_122 ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_402_subset__refl,axiom,
% 3.36/1.73 ! [A_122: nat > $o] : ( ord_less_eq_nat_o @ A_122 @ A_122 ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_403_subset__refl,axiom,
% 3.36/1.73 ! [A_122: x_a > $o] : ( ord_less_eq_a_o @ A_122 @ A_122 ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_404_set__eq__subset,axiom,
% 3.36/1.73 ! [A_121: int > $o,B_67: int > $o] :
% 3.36/1.73 ( ( A_121 = B_67 )
% 3.36/1.73 <=> ( ( ord_less_eq_int_o @ A_121 @ B_67 )
% 3.36/1.73 & ( ord_less_eq_int_o @ B_67 @ A_121 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_405_set__eq__subset,axiom,
% 3.36/1.73 ! [A_121: nat > $o,B_67: nat > $o] :
% 3.36/1.73 ( ( A_121 = B_67 )
% 3.36/1.73 <=> ( ( ord_less_eq_nat_o @ A_121 @ B_67 )
% 3.36/1.73 & ( ord_less_eq_nat_o @ B_67 @ A_121 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_406_set__eq__subset,axiom,
% 3.36/1.73 ! [A_121: x_a > $o,B_67: x_a > $o] :
% 3.36/1.73 ( ( A_121 = B_67 )
% 3.36/1.73 <=> ( ( ord_less_eq_a_o @ A_121 @ B_67 )
% 3.36/1.73 & ( ord_less_eq_a_o @ B_67 @ A_121 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_407_equalityD1,axiom,
% 3.36/1.73 ! [A_120: int > $o,B_66: int > $o] :
% 3.36/1.73 ( ( A_120 = B_66 )
% 3.36/1.73 => ( ord_less_eq_int_o @ A_120 @ B_66 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_408_equalityD1,axiom,
% 3.36/1.73 ! [A_120: nat > $o,B_66: nat > $o] :
% 3.36/1.73 ( ( A_120 = B_66 )
% 3.36/1.73 => ( ord_less_eq_nat_o @ A_120 @ B_66 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_409_equalityD1,axiom,
% 3.36/1.73 ! [A_120: x_a > $o,B_66: x_a > $o] :
% 3.36/1.73 ( ( A_120 = B_66 )
% 3.36/1.73 => ( ord_less_eq_a_o @ A_120 @ B_66 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_410_equalityD2,axiom,
% 3.36/1.73 ! [A_119: int > $o,B_65: int > $o] :
% 3.36/1.73 ( ( A_119 = B_65 )
% 3.36/1.73 => ( ord_less_eq_int_o @ B_65 @ A_119 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_411_equalityD2,axiom,
% 3.36/1.73 ! [A_119: nat > $o,B_65: nat > $o] :
% 3.36/1.73 ( ( A_119 = B_65 )
% 3.36/1.73 => ( ord_less_eq_nat_o @ B_65 @ A_119 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_412_equalityD2,axiom,
% 3.36/1.73 ! [A_119: x_a > $o,B_65: x_a > $o] :
% 3.36/1.73 ( ( A_119 = B_65 )
% 3.36/1.73 => ( ord_less_eq_a_o @ B_65 @ A_119 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_413_in__mono,axiom,
% 3.36/1.73 ! [X_34: int,A_118: int > $o,B_64: int > $o] :
% 3.36/1.73 ( ( ord_less_eq_int_o @ A_118 @ B_64 )
% 3.36/1.73 => ( ( member_int @ X_34 @ A_118 )
% 3.36/1.73 => ( member_int @ X_34 @ B_64 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_414_in__mono,axiom,
% 3.36/1.73 ! [X_34: nat,A_118: nat > $o,B_64: nat > $o] :
% 3.36/1.73 ( ( ord_less_eq_nat_o @ A_118 @ B_64 )
% 3.36/1.73 => ( ( member_nat @ X_34 @ A_118 )
% 3.36/1.73 => ( member_nat @ X_34 @ B_64 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_415_in__mono,axiom,
% 3.36/1.73 ! [X_34: x_a,A_118: x_a > $o,B_64: x_a > $o] :
% 3.36/1.73 ( ( ord_less_eq_a_o @ A_118 @ B_64 )
% 3.36/1.73 => ( ( member_a @ X_34 @ A_118 )
% 3.36/1.73 => ( member_a @ X_34 @ B_64 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_416_in__mono,axiom,
% 3.36/1.73 ! [X_34: pname,A_118: pname > $o,B_64: pname > $o] :
% 3.36/1.73 ( ( ord_less_eq_pname_o @ A_118 @ B_64 )
% 3.36/1.73 => ( ( member_pname @ X_34 @ A_118 )
% 3.36/1.73 => ( member_pname @ X_34 @ B_64 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_417_set__rev__mp,axiom,
% 3.36/1.73 ! [B_63: int > $o,X_33: int,A_117: int > $o] :
% 3.36/1.73 ( ( member_int @ X_33 @ A_117 )
% 3.36/1.73 => ( ( ord_less_eq_int_o @ A_117 @ B_63 )
% 3.36/1.73 => ( member_int @ X_33 @ B_63 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_418_set__rev__mp,axiom,
% 3.36/1.73 ! [B_63: nat > $o,X_33: nat,A_117: nat > $o] :
% 3.36/1.73 ( ( member_nat @ X_33 @ A_117 )
% 3.36/1.73 => ( ( ord_less_eq_nat_o @ A_117 @ B_63 )
% 3.36/1.73 => ( member_nat @ X_33 @ B_63 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_419_set__rev__mp,axiom,
% 3.36/1.73 ! [B_63: x_a > $o,X_33: x_a,A_117: x_a > $o] :
% 3.36/1.73 ( ( member_a @ X_33 @ A_117 )
% 3.36/1.73 => ( ( ord_less_eq_a_o @ A_117 @ B_63 )
% 3.36/1.73 => ( member_a @ X_33 @ B_63 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_420_set__rev__mp,axiom,
% 3.36/1.73 ! [B_63: pname > $o,X_33: pname,A_117: pname > $o] :
% 3.36/1.73 ( ( member_pname @ X_33 @ A_117 )
% 3.36/1.73 => ( ( ord_less_eq_pname_o @ A_117 @ B_63 )
% 3.36/1.73 => ( member_pname @ X_33 @ B_63 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_421_set__mp,axiom,
% 3.36/1.73 ! [X_32: int,A_116: int > $o,B_62: int > $o] :
% 3.36/1.73 ( ( ord_less_eq_int_o @ A_116 @ B_62 )
% 3.36/1.73 => ( ( member_int @ X_32 @ A_116 )
% 3.36/1.73 => ( member_int @ X_32 @ B_62 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_422_set__mp,axiom,
% 3.36/1.73 ! [X_32: nat,A_116: nat > $o,B_62: nat > $o] :
% 3.36/1.73 ( ( ord_less_eq_nat_o @ A_116 @ B_62 )
% 3.36/1.73 => ( ( member_nat @ X_32 @ A_116 )
% 3.36/1.73 => ( member_nat @ X_32 @ B_62 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_423_set__mp,axiom,
% 3.36/1.73 ! [X_32: x_a,A_116: x_a > $o,B_62: x_a > $o] :
% 3.36/1.73 ( ( ord_less_eq_a_o @ A_116 @ B_62 )
% 3.36/1.73 => ( ( member_a @ X_32 @ A_116 )
% 3.36/1.73 => ( member_a @ X_32 @ B_62 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_424_set__mp,axiom,
% 3.36/1.73 ! [X_32: pname,A_116: pname > $o,B_62: pname > $o] :
% 3.36/1.73 ( ( ord_less_eq_pname_o @ A_116 @ B_62 )
% 3.36/1.73 => ( ( member_pname @ X_32 @ A_116 )
% 3.36/1.73 => ( member_pname @ X_32 @ B_62 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_425_subset__trans,axiom,
% 3.36/1.73 ! [C_28: int > $o,A_115: int > $o,B_61: int > $o] :
% 3.36/1.73 ( ( ord_less_eq_int_o @ A_115 @ B_61 )
% 3.36/1.73 => ( ( ord_less_eq_int_o @ B_61 @ C_28 )
% 3.36/1.73 => ( ord_less_eq_int_o @ A_115 @ C_28 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_426_subset__trans,axiom,
% 3.36/1.73 ! [C_28: nat > $o,A_115: nat > $o,B_61: nat > $o] :
% 3.36/1.73 ( ( ord_less_eq_nat_o @ A_115 @ B_61 )
% 3.36/1.73 => ( ( ord_less_eq_nat_o @ B_61 @ C_28 )
% 3.36/1.73 => ( ord_less_eq_nat_o @ A_115 @ C_28 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_427_subset__trans,axiom,
% 3.36/1.73 ! [C_28: x_a > $o,A_115: x_a > $o,B_61: x_a > $o] :
% 3.36/1.73 ( ( ord_less_eq_a_o @ A_115 @ B_61 )
% 3.36/1.73 => ( ( ord_less_eq_a_o @ B_61 @ C_28 )
% 3.36/1.73 => ( ord_less_eq_a_o @ A_115 @ C_28 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_428_equalityE,axiom,
% 3.36/1.73 ! [A_114: int > $o,B_60: int > $o] :
% 3.36/1.73 ( ( A_114 = B_60 )
% 3.36/1.73 => ~ ( ( ord_less_eq_int_o @ A_114 @ B_60 )
% 3.36/1.73 => ~ ( ord_less_eq_int_o @ B_60 @ A_114 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_429_equalityE,axiom,
% 3.36/1.73 ! [A_114: nat > $o,B_60: nat > $o] :
% 3.36/1.73 ( ( A_114 = B_60 )
% 3.36/1.73 => ~ ( ( ord_less_eq_nat_o @ A_114 @ B_60 )
% 3.36/1.73 => ~ ( ord_less_eq_nat_o @ B_60 @ A_114 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_430_equalityE,axiom,
% 3.36/1.73 ! [A_114: x_a > $o,B_60: x_a > $o] :
% 3.36/1.73 ( ( A_114 = B_60 )
% 3.36/1.73 => ~ ( ( ord_less_eq_a_o @ A_114 @ B_60 )
% 3.36/1.73 => ~ ( ord_less_eq_a_o @ B_60 @ A_114 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_431_image__iff,axiom,
% 3.36/1.73 ! [Z_4: int,F_35: nat > int,A_113: nat > $o] :
% 3.36/1.73 ( ( member_int @ Z_4 @ ( image_nat_int @ F_35 @ A_113 ) )
% 3.36/1.73 <=> ? [X_1: nat] :
% 3.36/1.73 ( ( member_nat @ X_1 @ A_113 )
% 3.36/1.73 & ( Z_4
% 3.36/1.73 = ( F_35 @ X_1 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_432_image__iff,axiom,
% 3.36/1.73 ! [Z_4: x_a,F_35: pname > x_a,A_113: pname > $o] :
% 3.36/1.73 ( ( member_a @ Z_4 @ ( image_pname_a @ F_35 @ A_113 ) )
% 3.36/1.73 <=> ? [X_1: pname] :
% 3.36/1.73 ( ( member_pname @ X_1 @ A_113 )
% 3.36/1.73 & ( Z_4
% 3.36/1.73 = ( F_35 @ X_1 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_433_imageI,axiom,
% 3.36/1.73 ! [F_34: nat > int,X_31: nat,A_112: nat > $o] :
% 3.36/1.73 ( ( member_nat @ X_31 @ A_112 )
% 3.36/1.73 => ( member_int @ ( F_34 @ X_31 ) @ ( image_nat_int @ F_34 @ A_112 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_434_imageI,axiom,
% 3.36/1.73 ! [F_34: pname > x_a,X_31: pname,A_112: pname > $o] :
% 3.36/1.73 ( ( member_pname @ X_31 @ A_112 )
% 3.36/1.73 => ( member_a @ ( F_34 @ X_31 ) @ ( image_pname_a @ F_34 @ A_112 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_435_rev__image__eqI,axiom,
% 3.36/1.73 ! [B_59: int,F_33: nat > int,X_30: nat,A_111: nat > $o] :
% 3.36/1.73 ( ( member_nat @ X_30 @ A_111 )
% 3.36/1.73 => ( ( B_59
% 3.36/1.73 = ( F_33 @ X_30 ) )
% 3.36/1.73 => ( member_int @ B_59 @ ( image_nat_int @ F_33 @ A_111 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_436_rev__image__eqI,axiom,
% 3.36/1.73 ! [B_59: x_a,F_33: pname > x_a,X_30: pname,A_111: pname > $o] :
% 3.36/1.73 ( ( member_pname @ X_30 @ A_111 )
% 3.36/1.73 => ( ( B_59
% 3.36/1.73 = ( F_33 @ X_30 ) )
% 3.36/1.73 => ( member_a @ B_59 @ ( image_pname_a @ F_33 @ A_111 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_437_insert__compr__raw,axiom,
% 3.36/1.73 ! [X_1: x_a,Xa: x_a > $o] :
% 3.36/1.73 ( ( insert_a @ X_1 @ Xa )
% 3.36/1.73 = ( collect_a
% 3.36/1.73 @ ^ [Y_1: x_a] : ( (|) @ ( Y_1 = X_1 ) @ ( member_a @ Y_1 @ Xa ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_438_insert__compr__raw,axiom,
% 3.36/1.73 ! [X_1: int,Xa: int > $o] :
% 3.36/1.73 ( ( insert_int @ X_1 @ Xa )
% 3.36/1.73 = ( collect_int
% 3.36/1.73 @ ^ [Y_1: int] : ( (|) @ ( Y_1 = X_1 ) @ ( member_int @ Y_1 @ Xa ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_439_insert__compr__raw,axiom,
% 3.36/1.73 ! [X_1: nat,Xa: nat > $o] :
% 3.36/1.73 ( ( insert_nat @ X_1 @ Xa )
% 3.36/1.73 = ( collect_nat
% 3.36/1.73 @ ^ [Y_1: nat] : ( (|) @ ( Y_1 = X_1 ) @ ( member_nat @ Y_1 @ Xa ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_440_insert__compr__raw,axiom,
% 3.36/1.73 ! [X_1: pname,Xa: pname > $o] :
% 3.36/1.73 ( ( insert_pname @ X_1 @ Xa )
% 3.36/1.73 = ( collect_pname
% 3.36/1.73 @ ^ [Y_1: pname] : ( (|) @ ( Y_1 = X_1 ) @ ( member_pname @ Y_1 @ Xa ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_441_le__fun__def,axiom,
% 3.36/1.73 ! [F_32: int > $o,G_4: int > $o] :
% 3.36/1.73 ( ( ord_less_eq_int_o @ F_32 @ G_4 )
% 3.36/1.73 <=> ! [X_1: int] : ( ord_less_eq_o @ ( F_32 @ X_1 ) @ ( G_4 @ X_1 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_442_le__fun__def,axiom,
% 3.36/1.73 ! [F_32: nat > $o,G_4: nat > $o] :
% 3.36/1.73 ( ( ord_less_eq_nat_o @ F_32 @ G_4 )
% 3.36/1.73 <=> ! [X_1: nat] : ( ord_less_eq_o @ ( F_32 @ X_1 ) @ ( G_4 @ X_1 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_443_le__fun__def,axiom,
% 3.36/1.73 ! [F_32: x_a > $o,G_4: x_a > $o] :
% 3.36/1.73 ( ( ord_less_eq_a_o @ F_32 @ G_4 )
% 3.36/1.73 <=> ! [X_1: x_a] : ( ord_less_eq_o @ ( F_32 @ X_1 ) @ ( G_4 @ X_1 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_444_le__funD,axiom,
% 3.36/1.73 ! [X_29: int,F_31: int > $o,G_3: int > $o] :
% 3.36/1.73 ( ( ord_less_eq_int_o @ F_31 @ G_3 )
% 3.36/1.73 => ( ord_less_eq_o @ ( F_31 @ X_29 ) @ ( G_3 @ X_29 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_445_le__funD,axiom,
% 3.36/1.73 ! [X_29: nat,F_31: nat > $o,G_3: nat > $o] :
% 3.36/1.73 ( ( ord_less_eq_nat_o @ F_31 @ G_3 )
% 3.36/1.73 => ( ord_less_eq_o @ ( F_31 @ X_29 ) @ ( G_3 @ X_29 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_446_le__funD,axiom,
% 3.36/1.73 ! [X_29: x_a,F_31: x_a > $o,G_3: x_a > $o] :
% 3.36/1.73 ( ( ord_less_eq_a_o @ F_31 @ G_3 )
% 3.36/1.73 => ( ord_less_eq_o @ ( F_31 @ X_29 ) @ ( G_3 @ X_29 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_447_le__funE,axiom,
% 3.36/1.73 ! [X_28: int,F_30: int > $o,G_2: int > $o] :
% 3.36/1.73 ( ( ord_less_eq_int_o @ F_30 @ G_2 )
% 3.36/1.73 => ( ord_less_eq_o @ ( F_30 @ X_28 ) @ ( G_2 @ X_28 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_448_le__funE,axiom,
% 3.36/1.73 ! [X_28: nat,F_30: nat > $o,G_2: nat > $o] :
% 3.36/1.73 ( ( ord_less_eq_nat_o @ F_30 @ G_2 )
% 3.36/1.73 => ( ord_less_eq_o @ ( F_30 @ X_28 ) @ ( G_2 @ X_28 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_449_le__funE,axiom,
% 3.36/1.73 ! [X_28: x_a,F_30: x_a > $o,G_2: x_a > $o] :
% 3.36/1.73 ( ( ord_less_eq_a_o @ F_30 @ G_2 )
% 3.36/1.73 => ( ord_less_eq_o @ ( F_30 @ X_28 ) @ ( G_2 @ X_28 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_450_subset__insertI,axiom,
% 3.36/1.73 ! [B_58: x_a > $o,A_110: x_a] : ( ord_less_eq_a_o @ B_58 @ ( insert_a @ A_110 @ B_58 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_451_subset__insertI,axiom,
% 3.36/1.73 ! [B_58: int > $o,A_110: int] : ( ord_less_eq_int_o @ B_58 @ ( insert_int @ A_110 @ B_58 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_452_subset__insertI,axiom,
% 3.36/1.73 ! [B_58: nat > $o,A_110: nat] : ( ord_less_eq_nat_o @ B_58 @ ( insert_nat @ A_110 @ B_58 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_453_insert__subset,axiom,
% 3.36/1.73 ! [X_27: x_a,A_109: x_a > $o,B_57: x_a > $o] :
% 3.36/1.73 ( ( ord_less_eq_a_o @ ( insert_a @ X_27 @ A_109 ) @ B_57 )
% 3.36/1.73 <=> ( ( member_a @ X_27 @ B_57 )
% 3.36/1.73 & ( ord_less_eq_a_o @ A_109 @ B_57 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_454_insert__subset,axiom,
% 3.36/1.73 ! [X_27: int,A_109: int > $o,B_57: int > $o] :
% 3.36/1.73 ( ( ord_less_eq_int_o @ ( insert_int @ X_27 @ A_109 ) @ B_57 )
% 3.36/1.73 <=> ( ( member_int @ X_27 @ B_57 )
% 3.36/1.73 & ( ord_less_eq_int_o @ A_109 @ B_57 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_455_insert__subset,axiom,
% 3.36/1.73 ! [X_27: nat,A_109: nat > $o,B_57: nat > $o] :
% 3.36/1.73 ( ( ord_less_eq_nat_o @ ( insert_nat @ X_27 @ A_109 ) @ B_57 )
% 3.36/1.73 <=> ( ( member_nat @ X_27 @ B_57 )
% 3.36/1.73 & ( ord_less_eq_nat_o @ A_109 @ B_57 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_456_insert__subset,axiom,
% 3.36/1.73 ! [X_27: pname,A_109: pname > $o,B_57: pname > $o] :
% 3.36/1.73 ( ( ord_less_eq_pname_o @ ( insert_pname @ X_27 @ A_109 ) @ B_57 )
% 3.36/1.73 <=> ( ( member_pname @ X_27 @ B_57 )
% 3.36/1.73 & ( ord_less_eq_pname_o @ A_109 @ B_57 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_457_subset__insert,axiom,
% 3.36/1.73 ! [B_56: x_a > $o,X_26: x_a,A_108: x_a > $o] :
% 3.36/1.73 ( ~ ( member_a @ X_26 @ A_108 )
% 3.36/1.73 => ( ( ord_less_eq_a_o @ A_108 @ ( insert_a @ X_26 @ B_56 ) )
% 3.36/1.73 <=> ( ord_less_eq_a_o @ A_108 @ B_56 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_458_subset__insert,axiom,
% 3.36/1.73 ! [B_56: int > $o,X_26: int,A_108: int > $o] :
% 3.36/1.73 ( ~ ( member_int @ X_26 @ A_108 )
% 3.36/1.73 => ( ( ord_less_eq_int_o @ A_108 @ ( insert_int @ X_26 @ B_56 ) )
% 3.36/1.73 <=> ( ord_less_eq_int_o @ A_108 @ B_56 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_459_subset__insert,axiom,
% 3.36/1.73 ! [B_56: nat > $o,X_26: nat,A_108: nat > $o] :
% 3.36/1.73 ( ~ ( member_nat @ X_26 @ A_108 )
% 3.36/1.73 => ( ( ord_less_eq_nat_o @ A_108 @ ( insert_nat @ X_26 @ B_56 ) )
% 3.36/1.73 <=> ( ord_less_eq_nat_o @ A_108 @ B_56 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_460_subset__insert,axiom,
% 3.36/1.73 ! [B_56: pname > $o,X_26: pname,A_108: pname > $o] :
% 3.36/1.73 ( ~ ( member_pname @ X_26 @ A_108 )
% 3.36/1.73 => ( ( ord_less_eq_pname_o @ A_108 @ ( insert_pname @ X_26 @ B_56 ) )
% 3.36/1.73 <=> ( ord_less_eq_pname_o @ A_108 @ B_56 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_461_subset__insertI2,axiom,
% 3.36/1.73 ! [B_55: x_a,A_107: x_a > $o,B_54: x_a > $o] :
% 3.36/1.73 ( ( ord_less_eq_a_o @ A_107 @ B_54 )
% 3.36/1.73 => ( ord_less_eq_a_o @ A_107 @ ( insert_a @ B_55 @ B_54 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_462_subset__insertI2,axiom,
% 3.36/1.73 ! [B_55: int,A_107: int > $o,B_54: int > $o] :
% 3.36/1.73 ( ( ord_less_eq_int_o @ A_107 @ B_54 )
% 3.36/1.73 => ( ord_less_eq_int_o @ A_107 @ ( insert_int @ B_55 @ B_54 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_463_subset__insertI2,axiom,
% 3.36/1.73 ! [B_55: nat,A_107: nat > $o,B_54: nat > $o] :
% 3.36/1.73 ( ( ord_less_eq_nat_o @ A_107 @ B_54 )
% 3.36/1.73 => ( ord_less_eq_nat_o @ A_107 @ ( insert_nat @ B_55 @ B_54 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_464_insert__mono,axiom,
% 3.36/1.73 ! [A_106: x_a,C_27: x_a > $o,D_7: x_a > $o] :
% 3.36/1.73 ( ( ord_less_eq_a_o @ C_27 @ D_7 )
% 3.36/1.73 => ( ord_less_eq_a_o @ ( insert_a @ A_106 @ C_27 ) @ ( insert_a @ A_106 @ D_7 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_465_insert__mono,axiom,
% 3.36/1.73 ! [A_106: int,C_27: int > $o,D_7: int > $o] :
% 3.36/1.73 ( ( ord_less_eq_int_o @ C_27 @ D_7 )
% 3.36/1.73 => ( ord_less_eq_int_o @ ( insert_int @ A_106 @ C_27 ) @ ( insert_int @ A_106 @ D_7 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_466_insert__mono,axiom,
% 3.36/1.73 ! [A_106: nat,C_27: nat > $o,D_7: nat > $o] :
% 3.36/1.73 ( ( ord_less_eq_nat_o @ C_27 @ D_7 )
% 3.36/1.73 => ( ord_less_eq_nat_o @ ( insert_nat @ A_106 @ C_27 ) @ ( insert_nat @ A_106 @ D_7 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_467_image__insert,axiom,
% 3.36/1.73 ! [F_29: nat > int,A_105: nat,B_53: nat > $o] :
% 3.36/1.73 ( ( image_nat_int @ F_29 @ ( insert_nat @ A_105 @ B_53 ) )
% 3.36/1.73 = ( insert_int @ ( F_29 @ A_105 ) @ ( image_nat_int @ F_29 @ B_53 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_468_image__insert,axiom,
% 3.36/1.73 ! [F_29: pname > x_a,A_105: pname,B_53: pname > $o] :
% 3.36/1.73 ( ( image_pname_a @ F_29 @ ( insert_pname @ A_105 @ B_53 ) )
% 3.36/1.73 = ( insert_a @ ( F_29 @ A_105 ) @ ( image_pname_a @ F_29 @ B_53 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_469_insert__image,axiom,
% 3.36/1.73 ! [F_28: nat > int,X_25: nat,A_104: nat > $o] :
% 3.36/1.73 ( ( member_nat @ X_25 @ A_104 )
% 3.36/1.73 => ( ( insert_int @ ( F_28 @ X_25 ) @ ( image_nat_int @ F_28 @ A_104 ) )
% 3.36/1.73 = ( image_nat_int @ F_28 @ A_104 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_470_insert__image,axiom,
% 3.36/1.73 ! [F_28: pname > x_a,X_25: pname,A_104: pname > $o] :
% 3.36/1.73 ( ( member_pname @ X_25 @ A_104 )
% 3.36/1.73 => ( ( insert_a @ ( F_28 @ X_25 ) @ ( image_pname_a @ F_28 @ A_104 ) )
% 3.36/1.73 = ( image_pname_a @ F_28 @ A_104 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_471_subset__image__iff,axiom,
% 3.36/1.73 ! [B_52: int > $o,F_27: nat > int,A_103: nat > $o] :
% 3.36/1.73 ( ( ord_less_eq_int_o @ B_52 @ ( image_nat_int @ F_27 @ A_103 ) )
% 3.36/1.73 <=> ? [AA: nat > $o] :
% 3.36/1.73 ( ( ord_less_eq_nat_o @ AA @ A_103 )
% 3.36/1.73 & ( B_52
% 3.36/1.73 = ( image_nat_int @ F_27 @ AA ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_472_subset__image__iff,axiom,
% 3.36/1.73 ! [B_52: x_a > $o,F_27: pname > x_a,A_103: pname > $o] :
% 3.36/1.73 ( ( ord_less_eq_a_o @ B_52 @ ( image_pname_a @ F_27 @ A_103 ) )
% 3.36/1.73 <=> ? [AA: pname > $o] :
% 3.36/1.73 ( ( ord_less_eq_pname_o @ AA @ A_103 )
% 3.36/1.73 & ( B_52
% 3.36/1.73 = ( image_pname_a @ F_27 @ AA ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_473_image__mono,axiom,
% 3.36/1.73 ! [F_26: nat > int,A_102: nat > $o,B_51: nat > $o] :
% 3.36/1.73 ( ( ord_less_eq_nat_o @ A_102 @ B_51 )
% 3.36/1.73 => ( ord_less_eq_int_o @ ( image_nat_int @ F_26 @ A_102 ) @ ( image_nat_int @ F_26 @ B_51 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_474_image__mono,axiom,
% 3.36/1.73 ! [F_26: pname > x_a,A_102: pname > $o,B_51: pname > $o] :
% 3.36/1.73 ( ( ord_less_eq_pname_o @ A_102 @ B_51 )
% 3.36/1.73 => ( ord_less_eq_a_o @ ( image_pname_a @ F_26 @ A_102 ) @ ( image_pname_a @ F_26 @ B_51 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_475_imageE,axiom,
% 3.36/1.73 ! [B_50: int,F_25: nat > int,A_101: nat > $o] :
% 3.36/1.73 ( ( member_int @ B_50 @ ( image_nat_int @ F_25 @ A_101 ) )
% 3.36/1.73 => ~ ! [X_1: nat] :
% 3.36/1.73 ( ( B_50
% 3.36/1.73 = ( F_25 @ X_1 ) )
% 3.36/1.73 => ~ ( member_nat @ X_1 @ A_101 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_476_imageE,axiom,
% 3.36/1.73 ! [B_50: x_a,F_25: pname > x_a,A_101: pname > $o] :
% 3.36/1.73 ( ( member_a @ B_50 @ ( image_pname_a @ F_25 @ A_101 ) )
% 3.36/1.73 => ~ ! [X_1: pname] :
% 3.36/1.73 ( ( B_50
% 3.36/1.73 = ( F_25 @ X_1 ) )
% 3.36/1.73 => ~ ( member_pname @ X_1 @ A_101 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_477_subsetI,axiom,
% 3.36/1.73 ! [B_49: int > $o,A_100: int > $o] :
% 3.36/1.73 ( ! [X_1: int] :
% 3.36/1.73 ( ( member_int @ X_1 @ A_100 )
% 3.36/1.73 => ( member_int @ X_1 @ B_49 ) )
% 3.36/1.73 => ( ord_less_eq_int_o @ A_100 @ B_49 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_478_subsetI,axiom,
% 3.36/1.73 ! [B_49: nat > $o,A_100: nat > $o] :
% 3.36/1.73 ( ! [X_1: nat] :
% 3.36/1.73 ( ( member_nat @ X_1 @ A_100 )
% 3.36/1.73 => ( member_nat @ X_1 @ B_49 ) )
% 3.36/1.73 => ( ord_less_eq_nat_o @ A_100 @ B_49 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_479_subsetI,axiom,
% 3.36/1.73 ! [B_49: x_a > $o,A_100: x_a > $o] :
% 3.36/1.73 ( ! [X_1: x_a] :
% 3.36/1.73 ( ( member_a @ X_1 @ A_100 )
% 3.36/1.73 => ( member_a @ X_1 @ B_49 ) )
% 3.36/1.73 => ( ord_less_eq_a_o @ A_100 @ B_49 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_480_subsetI,axiom,
% 3.36/1.73 ! [B_49: pname > $o,A_100: pname > $o] :
% 3.36/1.73 ( ! [X_1: pname] :
% 3.36/1.73 ( ( member_pname @ X_1 @ A_100 )
% 3.36/1.73 => ( member_pname @ X_1 @ B_49 ) )
% 3.36/1.73 => ( ord_less_eq_pname_o @ A_100 @ B_49 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_481_image__subsetI,axiom,
% 3.36/1.73 ! [F_24: nat > int,B_48: int > $o,A_99: nat > $o] :
% 3.36/1.73 ( ! [X_1: nat] :
% 3.36/1.73 ( ( member_nat @ X_1 @ A_99 )
% 3.36/1.73 => ( member_int @ ( F_24 @ X_1 ) @ B_48 ) )
% 3.36/1.73 => ( ord_less_eq_int_o @ ( image_nat_int @ F_24 @ A_99 ) @ B_48 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_482_image__subsetI,axiom,
% 3.36/1.73 ! [F_24: pname > x_a,B_48: x_a > $o,A_99: pname > $o] :
% 3.36/1.73 ( ! [X_1: pname] :
% 3.36/1.73 ( ( member_pname @ X_1 @ A_99 )
% 3.36/1.73 => ( member_a @ ( F_24 @ X_1 ) @ B_48 ) )
% 3.36/1.73 => ( ord_less_eq_a_o @ ( image_pname_a @ F_24 @ A_99 ) @ B_48 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_483_le__funI,axiom,
% 3.36/1.73 ! [F_23: int > $o,G_1: int > $o] :
% 3.36/1.73 ( ! [X_1: int] : ( ord_less_eq_o @ ( F_23 @ X_1 ) @ ( G_1 @ X_1 ) )
% 3.36/1.73 => ( ord_less_eq_int_o @ F_23 @ G_1 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_484_le__funI,axiom,
% 3.36/1.73 ! [F_23: nat > $o,G_1: nat > $o] :
% 3.36/1.73 ( ! [X_1: nat] : ( ord_less_eq_o @ ( F_23 @ X_1 ) @ ( G_1 @ X_1 ) )
% 3.36/1.73 => ( ord_less_eq_nat_o @ F_23 @ G_1 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_485_le__funI,axiom,
% 3.36/1.73 ! [F_23: x_a > $o,G_1: x_a > $o] :
% 3.36/1.73 ( ! [X_1: x_a] : ( ord_less_eq_o @ ( F_23 @ X_1 ) @ ( G_1 @ X_1 ) )
% 3.36/1.73 => ( ord_less_eq_a_o @ F_23 @ G_1 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_486_finite__nat__set__iff__bounded__le,axiom,
% 3.36/1.73 ! [N_2: nat > $o] :
% 3.36/1.73 ( ( finite_finite_nat @ N_2 )
% 3.36/1.73 <=> ? [M_1: nat] :
% 3.36/1.73 ! [X_1: nat] :
% 3.36/1.73 ( ( member_nat @ X_1 @ N_2 )
% 3.36/1.73 => ( ord_less_eq_nat @ X_1 @ M_1 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_487_assms_I3_J,axiom,
% 3.36/1.73 ! [G: x_a > $o,C: com] :
% 3.36/1.73 ( ( wt @ C )
% 3.36/1.73 => ( ! [X_1: pname] :
% 3.36/1.73 ( ( member_pname @ X_1 @ u )
% 3.36/1.73 => ( p @ G @ ( insert_a @ ( mgt_call @ X_1 ) @ bot_bot_a_o ) ) )
% 3.36/1.73 => ( p @ G @ ( insert_a @ ( mgt @ C ) @ bot_bot_a_o ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_488_diff__eq__diff__less__eq,axiom,
% 3.36/1.73 ! [A_98: int,B_47: int,C_26: int,D_6: int] :
% 3.36/1.73 ( ( ( minus_minus_int @ A_98 @ B_47 )
% 3.36/1.73 = ( minus_minus_int @ C_26 @ D_6 ) )
% 3.36/1.73 => ( ( ord_less_eq_int @ A_98 @ B_47 )
% 3.36/1.73 <=> ( ord_less_eq_int @ C_26 @ D_6 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_489_less__eq__nat_Osimps_I2_J,axiom,
% 3.36/1.73 ! [M: nat,N: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 3.36/1.73 <=> ( nat_case_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_490_emptyE,axiom,
% 3.36/1.73 ! [A_97: int] :
% 3.36/1.73 ~ ( member_int @ A_97 @ bot_bot_int_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_491_emptyE,axiom,
% 3.36/1.73 ! [A_97: nat] :
% 3.36/1.73 ~ ( member_nat @ A_97 @ bot_bot_nat_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_492_emptyE,axiom,
% 3.36/1.73 ! [A_97: x_a] :
% 3.36/1.73 ~ ( member_a @ A_97 @ bot_bot_a_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_493_emptyE,axiom,
% 3.36/1.73 ! [A_97: pname] :
% 3.36/1.73 ~ ( member_pname @ A_97 @ bot_bot_pname_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_494_finite__Diff,axiom,
% 3.36/1.73 ! [B_46: int > $o,A_96: int > $o] :
% 3.36/1.73 ( ( finite_finite_int @ A_96 )
% 3.36/1.73 => ( finite_finite_int @ ( minus_minus_int_o @ A_96 @ B_46 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_495_finite__Diff,axiom,
% 3.36/1.73 ! [B_46: nat > $o,A_96: nat > $o] :
% 3.36/1.73 ( ( finite_finite_nat @ A_96 )
% 3.36/1.73 => ( finite_finite_nat @ ( minus_minus_nat_o @ A_96 @ B_46 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_496_finite__Diff,axiom,
% 3.36/1.73 ! [B_46: pname > $o,A_96: pname > $o] :
% 3.36/1.73 ( ( finite_finite_pname @ A_96 )
% 3.36/1.73 => ( finite_finite_pname @ ( minus_minus_pname_o @ A_96 @ B_46 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_497_finite_OemptyI,axiom,
% 3.36/1.73 finite_finite_int @ bot_bot_int_o ).
% 3.36/1.73
% 3.36/1.73 thf(fact_498_finite_OemptyI,axiom,
% 3.36/1.73 finite_finite_nat @ bot_bot_nat_o ).
% 3.36/1.73
% 3.36/1.73 thf(fact_499_finite_OemptyI,axiom,
% 3.36/1.73 finite_finite_pname @ bot_bot_pname_o ).
% 3.36/1.73
% 3.36/1.73 thf(fact_500_finite_OemptyI,axiom,
% 3.36/1.73 finite_finite_a @ bot_bot_a_o ).
% 3.36/1.73
% 3.36/1.73 thf(fact_501_empty__subsetI,axiom,
% 3.36/1.73 ! [A_95: nat > $o] : ( ord_less_eq_nat_o @ bot_bot_nat_o @ A_95 ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_502_empty__subsetI,axiom,
% 3.36/1.73 ! [A_95: int > $o] : ( ord_less_eq_int_o @ bot_bot_int_o @ A_95 ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_503_empty__subsetI,axiom,
% 3.36/1.73 ! [A_95: x_a > $o] : ( ord_less_eq_a_o @ bot_bot_a_o @ A_95 ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_504_equals0D,axiom,
% 3.36/1.73 ! [A_94: int,A_93: int > $o] :
% 3.36/1.73 ( ( A_93 = bot_bot_int_o )
% 3.36/1.73 => ~ ( member_int @ A_94 @ A_93 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_505_equals0D,axiom,
% 3.36/1.73 ! [A_94: nat,A_93: nat > $o] :
% 3.36/1.73 ( ( A_93 = bot_bot_nat_o )
% 3.36/1.73 => ~ ( member_nat @ A_94 @ A_93 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_506_equals0D,axiom,
% 3.36/1.73 ! [A_94: x_a,A_93: x_a > $o] :
% 3.36/1.73 ( ( A_93 = bot_bot_a_o )
% 3.36/1.73 => ~ ( member_a @ A_94 @ A_93 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_507_equals0D,axiom,
% 3.36/1.73 ! [A_94: pname,A_93: pname > $o] :
% 3.36/1.73 ( ( A_93 = bot_bot_pname_o )
% 3.36/1.73 => ~ ( member_pname @ A_94 @ A_93 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_508_Collect__empty__eq,axiom,
% 3.36/1.73 ! [P_9: int > $o] :
% 3.36/1.73 ( ( ( collect_int @ P_9 )
% 3.36/1.73 = bot_bot_int_o )
% 3.36/1.73 <=> ! [X_1: int] :
% 3.36/1.73 ~ ( P_9 @ X_1 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_509_Collect__empty__eq,axiom,
% 3.36/1.73 ! [P_9: nat > $o] :
% 3.36/1.73 ( ( ( collect_nat @ P_9 )
% 3.36/1.73 = bot_bot_nat_o )
% 3.36/1.73 <=> ! [X_1: nat] :
% 3.36/1.73 ~ ( P_9 @ X_1 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_510_Collect__empty__eq,axiom,
% 3.36/1.73 ! [P_9: x_a > $o] :
% 3.36/1.73 ( ( ( collect_a @ P_9 )
% 3.36/1.73 = bot_bot_a_o )
% 3.36/1.73 <=> ! [X_1: x_a] :
% 3.36/1.73 ~ ( P_9 @ X_1 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_511_Diff__cancel,axiom,
% 3.36/1.73 ! [A_92: nat > $o] :
% 3.36/1.73 ( ( minus_minus_nat_o @ A_92 @ A_92 )
% 3.36/1.73 = bot_bot_nat_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_512_Diff__cancel,axiom,
% 3.36/1.73 ! [A_92: int > $o] :
% 3.36/1.73 ( ( minus_minus_int_o @ A_92 @ A_92 )
% 3.36/1.73 = bot_bot_int_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_513_Diff__cancel,axiom,
% 3.36/1.73 ! [A_92: x_a > $o] :
% 3.36/1.73 ( ( minus_minus_a_o @ A_92 @ A_92 )
% 3.36/1.73 = bot_bot_a_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_514_Diff__empty,axiom,
% 3.36/1.73 ! [A_91: nat > $o] :
% 3.36/1.73 ( ( minus_minus_nat_o @ A_91 @ bot_bot_nat_o )
% 3.36/1.73 = A_91 ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_515_Diff__empty,axiom,
% 3.36/1.73 ! [A_91: int > $o] :
% 3.36/1.73 ( ( minus_minus_int_o @ A_91 @ bot_bot_int_o )
% 3.36/1.73 = A_91 ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_516_Diff__empty,axiom,
% 3.36/1.73 ! [A_91: x_a > $o] :
% 3.36/1.73 ( ( minus_minus_a_o @ A_91 @ bot_bot_a_o )
% 3.36/1.73 = A_91 ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_517_empty__iff,axiom,
% 3.36/1.73 ! [C_25: int] :
% 3.36/1.73 ~ ( member_int @ C_25 @ bot_bot_int_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_518_empty__iff,axiom,
% 3.36/1.73 ! [C_25: nat] :
% 3.36/1.73 ~ ( member_nat @ C_25 @ bot_bot_nat_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_519_empty__iff,axiom,
% 3.36/1.73 ! [C_25: x_a] :
% 3.36/1.73 ~ ( member_a @ C_25 @ bot_bot_a_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_520_empty__iff,axiom,
% 3.36/1.73 ! [C_25: pname] :
% 3.36/1.73 ~ ( member_pname @ C_25 @ bot_bot_pname_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_521_empty__Collect__eq,axiom,
% 3.36/1.73 ! [P_8: int > $o] :
% 3.36/1.73 ( ( bot_bot_int_o
% 3.36/1.73 = ( collect_int @ P_8 ) )
% 3.36/1.73 <=> ! [X_1: int] :
% 3.36/1.73 ~ ( P_8 @ X_1 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_522_empty__Collect__eq,axiom,
% 3.36/1.73 ! [P_8: nat > $o] :
% 3.36/1.73 ( ( bot_bot_nat_o
% 3.36/1.73 = ( collect_nat @ P_8 ) )
% 3.36/1.73 <=> ! [X_1: nat] :
% 3.36/1.73 ~ ( P_8 @ X_1 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_523_empty__Collect__eq,axiom,
% 3.36/1.73 ! [P_8: x_a > $o] :
% 3.36/1.73 ( ( bot_bot_a_o
% 3.36/1.73 = ( collect_a @ P_8 ) )
% 3.36/1.73 <=> ! [X_1: x_a] :
% 3.36/1.73 ~ ( P_8 @ X_1 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_524_empty__Diff,axiom,
% 3.36/1.73 ! [A_90: nat > $o] :
% 3.36/1.73 ( ( minus_minus_nat_o @ bot_bot_nat_o @ A_90 )
% 3.36/1.73 = bot_bot_nat_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_525_empty__Diff,axiom,
% 3.36/1.73 ! [A_90: int > $o] :
% 3.36/1.73 ( ( minus_minus_int_o @ bot_bot_int_o @ A_90 )
% 3.36/1.73 = bot_bot_int_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_526_empty__Diff,axiom,
% 3.36/1.73 ! [A_90: x_a > $o] :
% 3.36/1.73 ( ( minus_minus_a_o @ bot_bot_a_o @ A_90 )
% 3.36/1.73 = bot_bot_a_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_527_ex__in__conv,axiom,
% 3.36/1.73 ! [A_89: int > $o] :
% 3.36/1.73 ( ? [X_1: int] : ( member_int @ X_1 @ A_89 )
% 3.36/1.73 <=> ( A_89 != bot_bot_int_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_528_ex__in__conv,axiom,
% 3.36/1.73 ! [A_89: nat > $o] :
% 3.36/1.73 ( ? [X_1: nat] : ( member_nat @ X_1 @ A_89 )
% 3.36/1.73 <=> ( A_89 != bot_bot_nat_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_529_ex__in__conv,axiom,
% 3.36/1.73 ! [A_89: x_a > $o] :
% 3.36/1.73 ( ? [X_1: x_a] : ( member_a @ X_1 @ A_89 )
% 3.36/1.73 <=> ( A_89 != bot_bot_a_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_530_ex__in__conv,axiom,
% 3.36/1.73 ! [A_89: pname > $o] :
% 3.36/1.73 ( ? [X_1: pname] : ( member_pname @ X_1 @ A_89 )
% 3.36/1.73 <=> ( A_89 != bot_bot_pname_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_531_all__not__in__conv,axiom,
% 3.36/1.73 ! [A_88: int > $o] :
% 3.36/1.73 ( ! [X_1: int] :
% 3.36/1.73 ~ ( member_int @ X_1 @ A_88 )
% 3.36/1.73 <=> ( A_88 = bot_bot_int_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_532_all__not__in__conv,axiom,
% 3.36/1.73 ! [A_88: nat > $o] :
% 3.36/1.73 ( ! [X_1: nat] :
% 3.36/1.73 ~ ( member_nat @ X_1 @ A_88 )
% 3.36/1.73 <=> ( A_88 = bot_bot_nat_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_533_all__not__in__conv,axiom,
% 3.36/1.73 ! [A_88: x_a > $o] :
% 3.36/1.73 ( ! [X_1: x_a] :
% 3.36/1.73 ~ ( member_a @ X_1 @ A_88 )
% 3.36/1.73 <=> ( A_88 = bot_bot_a_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_534_all__not__in__conv,axiom,
% 3.36/1.73 ! [A_88: pname > $o] :
% 3.36/1.73 ( ! [X_1: pname] :
% 3.36/1.73 ~ ( member_pname @ X_1 @ A_88 )
% 3.36/1.73 <=> ( A_88 = bot_bot_pname_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_535_bot__apply,axiom,
% 3.36/1.73 ! [X_24: nat] :
% 3.36/1.73 ( ( bot_bot_nat_o @ X_24 )
% 3.36/1.73 <=> bot_bot_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_536_bot__apply,axiom,
% 3.36/1.73 ! [X_24: int] :
% 3.36/1.73 ( ( bot_bot_int_o @ X_24 )
% 3.36/1.73 <=> bot_bot_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_537_bot__apply,axiom,
% 3.36/1.73 ! [X_24: x_a] :
% 3.36/1.73 ( ( bot_bot_a_o @ X_24 )
% 3.36/1.73 <=> bot_bot_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_538_empty__def,axiom,
% 3.36/1.73 ( bot_bot_int_o
% 3.36/1.73 = ( collect_int
% 3.36/1.73 @ ^ [X_1: int] : $false ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_539_empty__def,axiom,
% 3.36/1.73 ( bot_bot_nat_o
% 3.36/1.73 = ( collect_nat
% 3.36/1.73 @ ^ [X_1: nat] : $false ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_540_empty__def,axiom,
% 3.36/1.73 ( bot_bot_a_o
% 3.36/1.73 = ( collect_a
% 3.36/1.73 @ ^ [X_1: x_a] : $false ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_541_bot__fun__def,axiom,
% 3.36/1.73 ! [X_1: nat] :
% 3.36/1.73 ( ( bot_bot_nat_o @ X_1 )
% 3.36/1.73 <=> bot_bot_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_542_bot__fun__def,axiom,
% 3.36/1.73 ! [X_1: int] :
% 3.36/1.73 ( ( bot_bot_int_o @ X_1 )
% 3.36/1.73 <=> bot_bot_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_543_bot__fun__def,axiom,
% 3.36/1.73 ! [X_1: x_a] :
% 3.36/1.73 ( ( bot_bot_a_o @ X_1 )
% 3.36/1.73 <=> bot_bot_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_544_insert__Diff,axiom,
% 3.36/1.73 ! [A_87: x_a,A_86: x_a > $o] :
% 3.36/1.73 ( ( member_a @ A_87 @ A_86 )
% 3.36/1.73 => ( ( insert_a @ A_87 @ ( minus_minus_a_o @ A_86 @ ( insert_a @ A_87 @ bot_bot_a_o ) ) )
% 3.36/1.73 = A_86 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_545_insert__Diff,axiom,
% 3.36/1.73 ! [A_87: int,A_86: int > $o] :
% 3.36/1.73 ( ( member_int @ A_87 @ A_86 )
% 3.36/1.73 => ( ( insert_int @ A_87 @ ( minus_minus_int_o @ A_86 @ ( insert_int @ A_87 @ bot_bot_int_o ) ) )
% 3.36/1.73 = A_86 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_546_insert__Diff,axiom,
% 3.36/1.73 ! [A_87: nat,A_86: nat > $o] :
% 3.36/1.73 ( ( member_nat @ A_87 @ A_86 )
% 3.36/1.73 => ( ( insert_nat @ A_87 @ ( minus_minus_nat_o @ A_86 @ ( insert_nat @ A_87 @ bot_bot_nat_o ) ) )
% 3.36/1.73 = A_86 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_547_insert__Diff,axiom,
% 3.36/1.73 ! [A_87: pname,A_86: pname > $o] :
% 3.36/1.73 ( ( member_pname @ A_87 @ A_86 )
% 3.36/1.73 => ( ( insert_pname @ A_87 @ ( minus_minus_pname_o @ A_86 @ ( insert_pname @ A_87 @ bot_bot_pname_o ) ) )
% 3.36/1.73 = A_86 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_548_Diff__insert__absorb,axiom,
% 3.36/1.73 ! [X_23: x_a,A_85: x_a > $o] :
% 3.36/1.73 ( ~ ( member_a @ X_23 @ A_85 )
% 3.36/1.73 => ( ( minus_minus_a_o @ ( insert_a @ X_23 @ A_85 ) @ ( insert_a @ X_23 @ bot_bot_a_o ) )
% 3.36/1.73 = A_85 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_549_Diff__insert__absorb,axiom,
% 3.36/1.73 ! [X_23: int,A_85: int > $o] :
% 3.36/1.73 ( ~ ( member_int @ X_23 @ A_85 )
% 3.36/1.73 => ( ( minus_minus_int_o @ ( insert_int @ X_23 @ A_85 ) @ ( insert_int @ X_23 @ bot_bot_int_o ) )
% 3.36/1.73 = A_85 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_550_Diff__insert__absorb,axiom,
% 3.36/1.73 ! [X_23: nat,A_85: nat > $o] :
% 3.36/1.73 ( ~ ( member_nat @ X_23 @ A_85 )
% 3.36/1.73 => ( ( minus_minus_nat_o @ ( insert_nat @ X_23 @ A_85 ) @ ( insert_nat @ X_23 @ bot_bot_nat_o ) )
% 3.36/1.73 = A_85 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_551_Diff__insert__absorb,axiom,
% 3.36/1.73 ! [X_23: pname,A_85: pname > $o] :
% 3.36/1.73 ( ~ ( member_pname @ X_23 @ A_85 )
% 3.36/1.73 => ( ( minus_minus_pname_o @ ( insert_pname @ X_23 @ A_85 ) @ ( insert_pname @ X_23 @ bot_bot_pname_o ) )
% 3.36/1.73 = A_85 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_552_insert__Diff__single,axiom,
% 3.36/1.73 ! [A_84: x_a,A_83: x_a > $o] :
% 3.36/1.73 ( ( insert_a @ A_84 @ ( minus_minus_a_o @ A_83 @ ( insert_a @ A_84 @ bot_bot_a_o ) ) )
% 3.36/1.73 = ( insert_a @ A_84 @ A_83 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_553_insert__Diff__single,axiom,
% 3.36/1.73 ! [A_84: nat,A_83: nat > $o] :
% 3.36/1.73 ( ( insert_nat @ A_84 @ ( minus_minus_nat_o @ A_83 @ ( insert_nat @ A_84 @ bot_bot_nat_o ) ) )
% 3.36/1.73 = ( insert_nat @ A_84 @ A_83 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_554_insert__Diff__single,axiom,
% 3.36/1.73 ! [A_84: int,A_83: int > $o] :
% 3.36/1.73 ( ( insert_int @ A_84 @ ( minus_minus_int_o @ A_83 @ ( insert_int @ A_84 @ bot_bot_int_o ) ) )
% 3.36/1.73 = ( insert_int @ A_84 @ A_83 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_555_Diff__insert2,axiom,
% 3.36/1.73 ! [A_82: x_a > $o,A_81: x_a,B_45: x_a > $o] :
% 3.36/1.73 ( ( minus_minus_a_o @ A_82 @ ( insert_a @ A_81 @ B_45 ) )
% 3.36/1.73 = ( minus_minus_a_o @ ( minus_minus_a_o @ A_82 @ ( insert_a @ A_81 @ bot_bot_a_o ) ) @ B_45 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_556_Diff__insert2,axiom,
% 3.36/1.73 ! [A_82: nat > $o,A_81: nat,B_45: nat > $o] :
% 3.36/1.73 ( ( minus_minus_nat_o @ A_82 @ ( insert_nat @ A_81 @ B_45 ) )
% 3.36/1.73 = ( minus_minus_nat_o @ ( minus_minus_nat_o @ A_82 @ ( insert_nat @ A_81 @ bot_bot_nat_o ) ) @ B_45 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_557_Diff__insert2,axiom,
% 3.36/1.73 ! [A_82: int > $o,A_81: int,B_45: int > $o] :
% 3.36/1.73 ( ( minus_minus_int_o @ A_82 @ ( insert_int @ A_81 @ B_45 ) )
% 3.36/1.73 = ( minus_minus_int_o @ ( minus_minus_int_o @ A_82 @ ( insert_int @ A_81 @ bot_bot_int_o ) ) @ B_45 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_558_Diff__insert,axiom,
% 3.36/1.73 ! [A_80: x_a > $o,A_79: x_a,B_44: x_a > $o] :
% 3.36/1.73 ( ( minus_minus_a_o @ A_80 @ ( insert_a @ A_79 @ B_44 ) )
% 3.36/1.73 = ( minus_minus_a_o @ ( minus_minus_a_o @ A_80 @ B_44 ) @ ( insert_a @ A_79 @ bot_bot_a_o ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_559_Diff__insert,axiom,
% 3.36/1.73 ! [A_80: nat > $o,A_79: nat,B_44: nat > $o] :
% 3.36/1.73 ( ( minus_minus_nat_o @ A_80 @ ( insert_nat @ A_79 @ B_44 ) )
% 3.36/1.73 = ( minus_minus_nat_o @ ( minus_minus_nat_o @ A_80 @ B_44 ) @ ( insert_nat @ A_79 @ bot_bot_nat_o ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_560_Diff__insert,axiom,
% 3.36/1.73 ! [A_80: int > $o,A_79: int,B_44: int > $o] :
% 3.36/1.73 ( ( minus_minus_int_o @ A_80 @ ( insert_int @ A_79 @ B_44 ) )
% 3.36/1.73 = ( minus_minus_int_o @ ( minus_minus_int_o @ A_80 @ B_44 ) @ ( insert_int @ A_79 @ bot_bot_int_o ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_561_diff__single__insert,axiom,
% 3.36/1.73 ! [A_78: x_a > $o,X_22: x_a,B_43: x_a > $o] :
% 3.36/1.73 ( ( ord_less_eq_a_o @ ( minus_minus_a_o @ A_78 @ ( insert_a @ X_22 @ bot_bot_a_o ) ) @ B_43 )
% 3.36/1.73 => ( ( member_a @ X_22 @ A_78 )
% 3.36/1.73 => ( ord_less_eq_a_o @ A_78 @ ( insert_a @ X_22 @ B_43 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_562_diff__single__insert,axiom,
% 3.36/1.73 ! [A_78: int > $o,X_22: int,B_43: int > $o] :
% 3.36/1.73 ( ( ord_less_eq_int_o @ ( minus_minus_int_o @ A_78 @ ( insert_int @ X_22 @ bot_bot_int_o ) ) @ B_43 )
% 3.36/1.73 => ( ( member_int @ X_22 @ A_78 )
% 3.36/1.73 => ( ord_less_eq_int_o @ A_78 @ ( insert_int @ X_22 @ B_43 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_563_diff__single__insert,axiom,
% 3.36/1.73 ! [A_78: nat > $o,X_22: nat,B_43: nat > $o] :
% 3.36/1.73 ( ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_78 @ ( insert_nat @ X_22 @ bot_bot_nat_o ) ) @ B_43 )
% 3.36/1.73 => ( ( member_nat @ X_22 @ A_78 )
% 3.36/1.73 => ( ord_less_eq_nat_o @ A_78 @ ( insert_nat @ X_22 @ B_43 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_564_diff__single__insert,axiom,
% 3.36/1.73 ! [A_78: pname > $o,X_22: pname,B_43: pname > $o] :
% 3.36/1.73 ( ( ord_less_eq_pname_o @ ( minus_minus_pname_o @ A_78 @ ( insert_pname @ X_22 @ bot_bot_pname_o ) ) @ B_43 )
% 3.36/1.73 => ( ( member_pname @ X_22 @ A_78 )
% 3.36/1.73 => ( ord_less_eq_pname_o @ A_78 @ ( insert_pname @ X_22 @ B_43 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_565_subset__insert__iff,axiom,
% 3.36/1.73 ! [A_77: x_a > $o,X_21: x_a,B_42: x_a > $o] :
% 3.36/1.73 ( ( ord_less_eq_a_o @ A_77 @ ( insert_a @ X_21 @ B_42 ) )
% 3.36/1.73 <=> ( ( ( member_a @ X_21 @ A_77 )
% 3.36/1.73 => ( ord_less_eq_a_o @ ( minus_minus_a_o @ A_77 @ ( insert_a @ X_21 @ bot_bot_a_o ) ) @ B_42 ) )
% 3.36/1.73 & ( ~ ( member_a @ X_21 @ A_77 )
% 3.36/1.73 => ( ord_less_eq_a_o @ A_77 @ B_42 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_566_subset__insert__iff,axiom,
% 3.36/1.73 ! [A_77: int > $o,X_21: int,B_42: int > $o] :
% 3.36/1.73 ( ( ord_less_eq_int_o @ A_77 @ ( insert_int @ X_21 @ B_42 ) )
% 3.36/1.73 <=> ( ( ( member_int @ X_21 @ A_77 )
% 3.36/1.73 => ( ord_less_eq_int_o @ ( minus_minus_int_o @ A_77 @ ( insert_int @ X_21 @ bot_bot_int_o ) ) @ B_42 ) )
% 3.36/1.73 & ( ~ ( member_int @ X_21 @ A_77 )
% 3.36/1.73 => ( ord_less_eq_int_o @ A_77 @ B_42 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_567_subset__insert__iff,axiom,
% 3.36/1.73 ! [A_77: nat > $o,X_21: nat,B_42: nat > $o] :
% 3.36/1.73 ( ( ord_less_eq_nat_o @ A_77 @ ( insert_nat @ X_21 @ B_42 ) )
% 3.36/1.73 <=> ( ( ( member_nat @ X_21 @ A_77 )
% 3.36/1.73 => ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_77 @ ( insert_nat @ X_21 @ bot_bot_nat_o ) ) @ B_42 ) )
% 3.36/1.73 & ( ~ ( member_nat @ X_21 @ A_77 )
% 3.36/1.73 => ( ord_less_eq_nat_o @ A_77 @ B_42 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_568_subset__insert__iff,axiom,
% 3.36/1.73 ! [A_77: pname > $o,X_21: pname,B_42: pname > $o] :
% 3.36/1.73 ( ( ord_less_eq_pname_o @ A_77 @ ( insert_pname @ X_21 @ B_42 ) )
% 3.36/1.73 <=> ( ( ( member_pname @ X_21 @ A_77 )
% 3.36/1.73 => ( ord_less_eq_pname_o @ ( minus_minus_pname_o @ A_77 @ ( insert_pname @ X_21 @ bot_bot_pname_o ) ) @ B_42 ) )
% 3.36/1.73 & ( ~ ( member_pname @ X_21 @ A_77 )
% 3.36/1.73 => ( ord_less_eq_pname_o @ A_77 @ B_42 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_569_finite__Diff2,axiom,
% 3.36/1.73 ! [A_76: int > $o,B_41: int > $o] :
% 3.36/1.73 ( ( finite_finite_int @ B_41 )
% 3.36/1.73 => ( ( finite_finite_int @ ( minus_minus_int_o @ A_76 @ B_41 ) )
% 3.36/1.73 <=> ( finite_finite_int @ A_76 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_570_finite__Diff2,axiom,
% 3.36/1.73 ! [A_76: nat > $o,B_41: nat > $o] :
% 3.36/1.73 ( ( finite_finite_nat @ B_41 )
% 3.36/1.73 => ( ( finite_finite_nat @ ( minus_minus_nat_o @ A_76 @ B_41 ) )
% 3.36/1.73 <=> ( finite_finite_nat @ A_76 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_571_finite__Diff2,axiom,
% 3.36/1.73 ! [A_76: pname > $o,B_41: pname > $o] :
% 3.36/1.73 ( ( finite_finite_pname @ B_41 )
% 3.36/1.73 => ( ( finite_finite_pname @ ( minus_minus_pname_o @ A_76 @ B_41 ) )
% 3.36/1.73 <=> ( finite_finite_pname @ A_76 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_572_insert__Diff1,axiom,
% 3.36/1.73 ! [A_75: x_a > $o,X_20: x_a,B_40: x_a > $o] :
% 3.36/1.73 ( ( member_a @ X_20 @ B_40 )
% 3.36/1.73 => ( ( minus_minus_a_o @ ( insert_a @ X_20 @ A_75 ) @ B_40 )
% 3.36/1.73 = ( minus_minus_a_o @ A_75 @ B_40 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_573_insert__Diff1,axiom,
% 3.36/1.73 ! [A_75: int > $o,X_20: int,B_40: int > $o] :
% 3.36/1.73 ( ( member_int @ X_20 @ B_40 )
% 3.36/1.73 => ( ( minus_minus_int_o @ ( insert_int @ X_20 @ A_75 ) @ B_40 )
% 3.36/1.73 = ( minus_minus_int_o @ A_75 @ B_40 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_574_insert__Diff1,axiom,
% 3.36/1.73 ! [A_75: nat > $o,X_20: nat,B_40: nat > $o] :
% 3.36/1.73 ( ( member_nat @ X_20 @ B_40 )
% 3.36/1.73 => ( ( minus_minus_nat_o @ ( insert_nat @ X_20 @ A_75 ) @ B_40 )
% 3.36/1.73 = ( minus_minus_nat_o @ A_75 @ B_40 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_575_insert__Diff1,axiom,
% 3.36/1.73 ! [A_75: pname > $o,X_20: pname,B_40: pname > $o] :
% 3.36/1.73 ( ( member_pname @ X_20 @ B_40 )
% 3.36/1.73 => ( ( minus_minus_pname_o @ ( insert_pname @ X_20 @ A_75 ) @ B_40 )
% 3.36/1.73 = ( minus_minus_pname_o @ A_75 @ B_40 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_576_insert__Diff__if,axiom,
% 3.36/1.73 ! [A_74: x_a > $o,X_19: x_a,B_39: x_a > $o] :
% 3.36/1.73 ( ( ( member_a @ X_19 @ B_39 )
% 3.36/1.73 => ( ( minus_minus_a_o @ ( insert_a @ X_19 @ A_74 ) @ B_39 )
% 3.36/1.73 = ( minus_minus_a_o @ A_74 @ B_39 ) ) )
% 3.36/1.73 & ( ~ ( member_a @ X_19 @ B_39 )
% 3.36/1.73 => ( ( minus_minus_a_o @ ( insert_a @ X_19 @ A_74 ) @ B_39 )
% 3.36/1.73 = ( insert_a @ X_19 @ ( minus_minus_a_o @ A_74 @ B_39 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_577_insert__Diff__if,axiom,
% 3.36/1.73 ! [A_74: int > $o,X_19: int,B_39: int > $o] :
% 3.36/1.73 ( ( ( member_int @ X_19 @ B_39 )
% 3.36/1.73 => ( ( minus_minus_int_o @ ( insert_int @ X_19 @ A_74 ) @ B_39 )
% 3.36/1.73 = ( minus_minus_int_o @ A_74 @ B_39 ) ) )
% 3.36/1.73 & ( ~ ( member_int @ X_19 @ B_39 )
% 3.36/1.73 => ( ( minus_minus_int_o @ ( insert_int @ X_19 @ A_74 ) @ B_39 )
% 3.36/1.73 = ( insert_int @ X_19 @ ( minus_minus_int_o @ A_74 @ B_39 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_578_insert__Diff__if,axiom,
% 3.36/1.73 ! [A_74: nat > $o,X_19: nat,B_39: nat > $o] :
% 3.36/1.73 ( ( ( member_nat @ X_19 @ B_39 )
% 3.36/1.73 => ( ( minus_minus_nat_o @ ( insert_nat @ X_19 @ A_74 ) @ B_39 )
% 3.36/1.73 = ( minus_minus_nat_o @ A_74 @ B_39 ) ) )
% 3.36/1.73 & ( ~ ( member_nat @ X_19 @ B_39 )
% 3.36/1.73 => ( ( minus_minus_nat_o @ ( insert_nat @ X_19 @ A_74 ) @ B_39 )
% 3.36/1.73 = ( insert_nat @ X_19 @ ( minus_minus_nat_o @ A_74 @ B_39 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_579_insert__Diff__if,axiom,
% 3.36/1.73 ! [A_74: pname > $o,X_19: pname,B_39: pname > $o] :
% 3.36/1.73 ( ( ( member_pname @ X_19 @ B_39 )
% 3.36/1.73 => ( ( minus_minus_pname_o @ ( insert_pname @ X_19 @ A_74 ) @ B_39 )
% 3.36/1.73 = ( minus_minus_pname_o @ A_74 @ B_39 ) ) )
% 3.36/1.73 & ( ~ ( member_pname @ X_19 @ B_39 )
% 3.36/1.73 => ( ( minus_minus_pname_o @ ( insert_pname @ X_19 @ A_74 ) @ B_39 )
% 3.36/1.73 = ( insert_pname @ X_19 @ ( minus_minus_pname_o @ A_74 @ B_39 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_580_double__diff,axiom,
% 3.36/1.73 ! [C_24: int > $o,A_73: int > $o,B_38: int > $o] :
% 3.36/1.73 ( ( ord_less_eq_int_o @ A_73 @ B_38 )
% 3.36/1.73 => ( ( ord_less_eq_int_o @ B_38 @ C_24 )
% 3.36/1.73 => ( ( minus_minus_int_o @ B_38 @ ( minus_minus_int_o @ C_24 @ A_73 ) )
% 3.36/1.73 = A_73 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_581_double__diff,axiom,
% 3.36/1.73 ! [C_24: nat > $o,A_73: nat > $o,B_38: nat > $o] :
% 3.36/1.73 ( ( ord_less_eq_nat_o @ A_73 @ B_38 )
% 3.36/1.73 => ( ( ord_less_eq_nat_o @ B_38 @ C_24 )
% 3.36/1.73 => ( ( minus_minus_nat_o @ B_38 @ ( minus_minus_nat_o @ C_24 @ A_73 ) )
% 3.36/1.73 = A_73 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_582_double__diff,axiom,
% 3.36/1.73 ! [C_24: x_a > $o,A_73: x_a > $o,B_38: x_a > $o] :
% 3.36/1.73 ( ( ord_less_eq_a_o @ A_73 @ B_38 )
% 3.36/1.73 => ( ( ord_less_eq_a_o @ B_38 @ C_24 )
% 3.36/1.73 => ( ( minus_minus_a_o @ B_38 @ ( minus_minus_a_o @ C_24 @ A_73 ) )
% 3.36/1.73 = A_73 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_583_Diff__mono,axiom,
% 3.36/1.73 ! [D_5: int > $o,B_37: int > $o,A_72: int > $o,C_23: int > $o] :
% 3.36/1.73 ( ( ord_less_eq_int_o @ A_72 @ C_23 )
% 3.36/1.73 => ( ( ord_less_eq_int_o @ D_5 @ B_37 )
% 3.36/1.73 => ( ord_less_eq_int_o @ ( minus_minus_int_o @ A_72 @ B_37 ) @ ( minus_minus_int_o @ C_23 @ D_5 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_584_Diff__mono,axiom,
% 3.36/1.73 ! [D_5: nat > $o,B_37: nat > $o,A_72: nat > $o,C_23: nat > $o] :
% 3.36/1.73 ( ( ord_less_eq_nat_o @ A_72 @ C_23 )
% 3.36/1.73 => ( ( ord_less_eq_nat_o @ D_5 @ B_37 )
% 3.36/1.73 => ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_72 @ B_37 ) @ ( minus_minus_nat_o @ C_23 @ D_5 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_585_Diff__mono,axiom,
% 3.36/1.73 ! [D_5: x_a > $o,B_37: x_a > $o,A_72: x_a > $o,C_23: x_a > $o] :
% 3.36/1.73 ( ( ord_less_eq_a_o @ A_72 @ C_23 )
% 3.36/1.73 => ( ( ord_less_eq_a_o @ D_5 @ B_37 )
% 3.36/1.73 => ( ord_less_eq_a_o @ ( minus_minus_a_o @ A_72 @ B_37 ) @ ( minus_minus_a_o @ C_23 @ D_5 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_586_Diff__subset,axiom,
% 3.36/1.73 ! [A_71: int > $o,B_36: int > $o] : ( ord_less_eq_int_o @ ( minus_minus_int_o @ A_71 @ B_36 ) @ A_71 ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_587_Diff__subset,axiom,
% 3.36/1.73 ! [A_71: nat > $o,B_36: nat > $o] : ( ord_less_eq_nat_o @ ( minus_minus_nat_o @ A_71 @ B_36 ) @ A_71 ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_588_Diff__subset,axiom,
% 3.36/1.73 ! [A_71: x_a > $o,B_36: x_a > $o] : ( ord_less_eq_a_o @ ( minus_minus_a_o @ A_71 @ B_36 ) @ A_71 ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_589_singleton__inject,axiom,
% 3.36/1.73 ! [A_70: x_a,B_35: x_a] :
% 3.36/1.73 ( ( ( insert_a @ A_70 @ bot_bot_a_o )
% 3.36/1.73 = ( insert_a @ B_35 @ bot_bot_a_o ) )
% 3.36/1.73 => ( A_70 = B_35 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_590_singleton__inject,axiom,
% 3.36/1.73 ! [A_70: nat,B_35: nat] :
% 3.36/1.73 ( ( ( insert_nat @ A_70 @ bot_bot_nat_o )
% 3.36/1.73 = ( insert_nat @ B_35 @ bot_bot_nat_o ) )
% 3.36/1.73 => ( A_70 = B_35 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_591_singleton__inject,axiom,
% 3.36/1.73 ! [A_70: int,B_35: int] :
% 3.36/1.73 ( ( ( insert_int @ A_70 @ bot_bot_int_o )
% 3.36/1.73 = ( insert_int @ B_35 @ bot_bot_int_o ) )
% 3.36/1.73 => ( A_70 = B_35 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_592_singletonE,axiom,
% 3.36/1.73 ! [B_34: x_a,A_69: x_a] :
% 3.36/1.73 ( ( member_a @ B_34 @ ( insert_a @ A_69 @ bot_bot_a_o ) )
% 3.36/1.73 => ( B_34 = A_69 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_593_singletonE,axiom,
% 3.36/1.73 ! [B_34: int,A_69: int] :
% 3.36/1.73 ( ( member_int @ B_34 @ ( insert_int @ A_69 @ bot_bot_int_o ) )
% 3.36/1.73 => ( B_34 = A_69 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_594_singletonE,axiom,
% 3.36/1.73 ! [B_34: nat,A_69: nat] :
% 3.36/1.73 ( ( member_nat @ B_34 @ ( insert_nat @ A_69 @ bot_bot_nat_o ) )
% 3.36/1.73 => ( B_34 = A_69 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_595_singletonE,axiom,
% 3.36/1.73 ! [B_34: pname,A_69: pname] :
% 3.36/1.73 ( ( member_pname @ B_34 @ ( insert_pname @ A_69 @ bot_bot_pname_o ) )
% 3.36/1.73 => ( B_34 = A_69 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_596_doubleton__eq__iff,axiom,
% 3.36/1.73 ! [A_68: x_a,B_33: x_a,C_22: x_a,D_4: x_a] :
% 3.36/1.73 ( ( ( insert_a @ A_68 @ ( insert_a @ B_33 @ bot_bot_a_o ) )
% 3.36/1.73 = ( insert_a @ C_22 @ ( insert_a @ D_4 @ bot_bot_a_o ) ) )
% 3.36/1.73 <=> ( ( ( A_68 = C_22 )
% 3.36/1.73 & ( B_33 = D_4 ) )
% 3.36/1.73 | ( ( A_68 = D_4 )
% 3.36/1.73 & ( B_33 = C_22 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_597_doubleton__eq__iff,axiom,
% 3.36/1.73 ! [A_68: nat,B_33: nat,C_22: nat,D_4: nat] :
% 3.36/1.73 ( ( ( insert_nat @ A_68 @ ( insert_nat @ B_33 @ bot_bot_nat_o ) )
% 3.36/1.73 = ( insert_nat @ C_22 @ ( insert_nat @ D_4 @ bot_bot_nat_o ) ) )
% 3.36/1.73 <=> ( ( ( A_68 = C_22 )
% 3.36/1.73 & ( B_33 = D_4 ) )
% 3.36/1.73 | ( ( A_68 = D_4 )
% 3.36/1.73 & ( B_33 = C_22 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_598_doubleton__eq__iff,axiom,
% 3.36/1.73 ! [A_68: int,B_33: int,C_22: int,D_4: int] :
% 3.36/1.73 ( ( ( insert_int @ A_68 @ ( insert_int @ B_33 @ bot_bot_int_o ) )
% 3.36/1.73 = ( insert_int @ C_22 @ ( insert_int @ D_4 @ bot_bot_int_o ) ) )
% 3.36/1.73 <=> ( ( ( A_68 = C_22 )
% 3.36/1.73 & ( B_33 = D_4 ) )
% 3.36/1.73 | ( ( A_68 = D_4 )
% 3.36/1.73 & ( B_33 = C_22 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_599_singleton__iff,axiom,
% 3.36/1.73 ! [B_32: x_a,A_67: x_a] :
% 3.36/1.73 ( ( member_a @ B_32 @ ( insert_a @ A_67 @ bot_bot_a_o ) )
% 3.36/1.73 <=> ( B_32 = A_67 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_600_singleton__iff,axiom,
% 3.36/1.73 ! [B_32: int,A_67: int] :
% 3.36/1.73 ( ( member_int @ B_32 @ ( insert_int @ A_67 @ bot_bot_int_o ) )
% 3.36/1.73 <=> ( B_32 = A_67 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_601_singleton__iff,axiom,
% 3.36/1.73 ! [B_32: nat,A_67: nat] :
% 3.36/1.73 ( ( member_nat @ B_32 @ ( insert_nat @ A_67 @ bot_bot_nat_o ) )
% 3.36/1.73 <=> ( B_32 = A_67 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_602_singleton__iff,axiom,
% 3.36/1.73 ! [B_32: pname,A_67: pname] :
% 3.36/1.73 ( ( member_pname @ B_32 @ ( insert_pname @ A_67 @ bot_bot_pname_o ) )
% 3.36/1.73 <=> ( B_32 = A_67 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_603_insert__not__empty,axiom,
% 3.36/1.73 ! [A_66: x_a,A_65: x_a > $o] :
% 3.36/1.73 ( ( insert_a @ A_66 @ A_65 )
% 3.36/1.73 != bot_bot_a_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_604_insert__not__empty,axiom,
% 3.36/1.73 ! [A_66: nat,A_65: nat > $o] :
% 3.36/1.73 ( ( insert_nat @ A_66 @ A_65 )
% 3.36/1.73 != bot_bot_nat_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_605_insert__not__empty,axiom,
% 3.36/1.73 ! [A_66: int,A_65: int > $o] :
% 3.36/1.73 ( ( insert_int @ A_66 @ A_65 )
% 3.36/1.73 != bot_bot_int_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_606_empty__not__insert,axiom,
% 3.36/1.73 ! [A_64: x_a,A_63: x_a > $o] :
% 3.36/1.73 ( bot_bot_a_o
% 3.36/1.73 != ( insert_a @ A_64 @ A_63 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_607_empty__not__insert,axiom,
% 3.36/1.73 ! [A_64: nat,A_63: nat > $o] :
% 3.36/1.73 ( bot_bot_nat_o
% 3.36/1.73 != ( insert_nat @ A_64 @ A_63 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_608_empty__not__insert,axiom,
% 3.36/1.73 ! [A_64: int,A_63: int > $o] :
% 3.36/1.73 ( bot_bot_int_o
% 3.36/1.73 != ( insert_int @ A_64 @ A_63 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_609_subset__empty,axiom,
% 3.36/1.73 ! [A_62: nat > $o] :
% 3.36/1.73 ( ( ord_less_eq_nat_o @ A_62 @ bot_bot_nat_o )
% 3.36/1.73 <=> ( A_62 = bot_bot_nat_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_610_subset__empty,axiom,
% 3.36/1.73 ! [A_62: int > $o] :
% 3.36/1.73 ( ( ord_less_eq_int_o @ A_62 @ bot_bot_int_o )
% 3.36/1.73 <=> ( A_62 = bot_bot_int_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_611_subset__empty,axiom,
% 3.36/1.73 ! [A_62: x_a > $o] :
% 3.36/1.73 ( ( ord_less_eq_a_o @ A_62 @ bot_bot_a_o )
% 3.36/1.73 <=> ( A_62 = bot_bot_a_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_612_image__is__empty,axiom,
% 3.36/1.73 ! [F_22: nat > int,A_61: nat > $o] :
% 3.36/1.73 ( ( ( image_nat_int @ F_22 @ A_61 )
% 3.36/1.73 = bot_bot_int_o )
% 3.36/1.73 <=> ( A_61 = bot_bot_nat_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_613_image__is__empty,axiom,
% 3.36/1.73 ! [F_22: pname > x_a,A_61: pname > $o] :
% 3.36/1.73 ( ( ( image_pname_a @ F_22 @ A_61 )
% 3.36/1.73 = bot_bot_a_o )
% 3.36/1.73 <=> ( A_61 = bot_bot_pname_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_614_image__empty,axiom,
% 3.36/1.73 ! [F_21: nat > int] :
% 3.36/1.73 ( ( image_nat_int @ F_21 @ bot_bot_nat_o )
% 3.36/1.73 = bot_bot_int_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_615_image__empty,axiom,
% 3.36/1.73 ! [F_21: pname > x_a] :
% 3.36/1.73 ( ( image_pname_a @ F_21 @ bot_bot_pname_o )
% 3.36/1.73 = bot_bot_a_o ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_616_empty__is__image,axiom,
% 3.36/1.73 ! [F_20: nat > int,A_60: nat > $o] :
% 3.36/1.73 ( ( bot_bot_int_o
% 3.36/1.73 = ( image_nat_int @ F_20 @ A_60 ) )
% 3.36/1.73 <=> ( A_60 = bot_bot_nat_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_617_empty__is__image,axiom,
% 3.36/1.73 ! [F_20: pname > x_a,A_60: pname > $o] :
% 3.36/1.73 ( ( bot_bot_a_o
% 3.36/1.73 = ( image_pname_a @ F_20 @ A_60 ) )
% 3.36/1.73 <=> ( A_60 = bot_bot_pname_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_618_le__bot,axiom,
% 3.36/1.73 ! [A_59: nat > $o] :
% 3.36/1.73 ( ( ord_less_eq_nat_o @ A_59 @ bot_bot_nat_o )
% 3.36/1.73 => ( A_59 = bot_bot_nat_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_619_le__bot,axiom,
% 3.36/1.73 ! [A_59: int > $o] :
% 3.36/1.73 ( ( ord_less_eq_int_o @ A_59 @ bot_bot_int_o )
% 3.36/1.73 => ( A_59 = bot_bot_int_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_620_le__bot,axiom,
% 3.36/1.73 ! [A_59: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ A_59 @ bot_bot_nat )
% 3.36/1.73 => ( A_59 = bot_bot_nat ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_621_le__bot,axiom,
% 3.36/1.73 ! [A_59: x_a > $o] :
% 3.36/1.73 ( ( ord_less_eq_a_o @ A_59 @ bot_bot_a_o )
% 3.36/1.73 => ( A_59 = bot_bot_a_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_622_bot__unique,axiom,
% 3.36/1.73 ! [A_58: nat > $o] :
% 3.36/1.73 ( ( ord_less_eq_nat_o @ A_58 @ bot_bot_nat_o )
% 3.36/1.73 <=> ( A_58 = bot_bot_nat_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_623_bot__unique,axiom,
% 3.36/1.73 ! [A_58: int > $o] :
% 3.36/1.73 ( ( ord_less_eq_int_o @ A_58 @ bot_bot_int_o )
% 3.36/1.73 <=> ( A_58 = bot_bot_int_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_624_bot__unique,axiom,
% 3.36/1.73 ! [A_58: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ A_58 @ bot_bot_nat )
% 3.36/1.73 <=> ( A_58 = bot_bot_nat ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_625_bot__unique,axiom,
% 3.36/1.73 ! [A_58: x_a > $o] :
% 3.36/1.73 ( ( ord_less_eq_a_o @ A_58 @ bot_bot_a_o )
% 3.36/1.73 <=> ( A_58 = bot_bot_a_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_626_bot__least,axiom,
% 3.36/1.73 ! [A_57: nat > $o] : ( ord_less_eq_nat_o @ bot_bot_nat_o @ A_57 ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_627_bot__least,axiom,
% 3.36/1.73 ! [A_57: int > $o] : ( ord_less_eq_int_o @ bot_bot_int_o @ A_57 ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_628_bot__least,axiom,
% 3.36/1.73 ! [A_57: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A_57 ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_629_bot__least,axiom,
% 3.36/1.73 ! [A_57: x_a > $o] : ( ord_less_eq_a_o @ bot_bot_a_o @ A_57 ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_630_Collect__conv__if,axiom,
% 3.36/1.73 ! [P_7: x_a > $o,A_56: x_a] :
% 3.36/1.73 ( ( ( P_7 @ A_56 )
% 3.36/1.73 => ( ( collect_a
% 3.36/1.73 @ ^ [X_1: x_a] : ( (&) @ ( X_1 = A_56 ) @ ( P_7 @ X_1 ) ) )
% 3.36/1.73 = ( insert_a @ A_56 @ bot_bot_a_o ) ) )
% 3.36/1.73 & ( ~ ( P_7 @ A_56 )
% 3.36/1.73 => ( ( collect_a
% 3.36/1.73 @ ^ [X_1: x_a] : ( (&) @ ( X_1 = A_56 ) @ ( P_7 @ X_1 ) ) )
% 3.36/1.73 = bot_bot_a_o ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_631_Collect__conv__if,axiom,
% 3.36/1.73 ! [P_7: int > $o,A_56: int] :
% 3.36/1.73 ( ( ( P_7 @ A_56 )
% 3.36/1.73 => ( ( collect_int
% 3.36/1.73 @ ^ [X_1: int] : ( (&) @ ( X_1 = A_56 ) @ ( P_7 @ X_1 ) ) )
% 3.36/1.73 = ( insert_int @ A_56 @ bot_bot_int_o ) ) )
% 3.36/1.73 & ( ~ ( P_7 @ A_56 )
% 3.36/1.73 => ( ( collect_int
% 3.36/1.73 @ ^ [X_1: int] : ( (&) @ ( X_1 = A_56 ) @ ( P_7 @ X_1 ) ) )
% 3.36/1.73 = bot_bot_int_o ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_632_Collect__conv__if,axiom,
% 3.36/1.73 ! [P_7: nat > $o,A_56: nat] :
% 3.36/1.73 ( ( ( P_7 @ A_56 )
% 3.36/1.73 => ( ( collect_nat
% 3.36/1.73 @ ^ [X_1: nat] : ( (&) @ ( X_1 = A_56 ) @ ( P_7 @ X_1 ) ) )
% 3.36/1.73 = ( insert_nat @ A_56 @ bot_bot_nat_o ) ) )
% 3.36/1.73 & ( ~ ( P_7 @ A_56 )
% 3.36/1.73 => ( ( collect_nat
% 3.36/1.73 @ ^ [X_1: nat] : ( (&) @ ( X_1 = A_56 ) @ ( P_7 @ X_1 ) ) )
% 3.36/1.73 = bot_bot_nat_o ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_633_Collect__conv__if2,axiom,
% 3.36/1.73 ! [P_6: x_a > $o,A_55: x_a] :
% 3.36/1.73 ( ( ( P_6 @ A_55 )
% 3.36/1.73 => ( ( collect_a
% 3.36/1.73 @ ^ [X_1: x_a] : ( (&) @ ( A_55 = X_1 ) @ ( P_6 @ X_1 ) ) )
% 3.36/1.73 = ( insert_a @ A_55 @ bot_bot_a_o ) ) )
% 3.36/1.73 & ( ~ ( P_6 @ A_55 )
% 3.36/1.73 => ( ( collect_a
% 3.36/1.73 @ ^ [X_1: x_a] : ( (&) @ ( A_55 = X_1 ) @ ( P_6 @ X_1 ) ) )
% 3.36/1.73 = bot_bot_a_o ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_634_Collect__conv__if2,axiom,
% 3.36/1.73 ! [P_6: int > $o,A_55: int] :
% 3.36/1.73 ( ( ( P_6 @ A_55 )
% 3.36/1.73 => ( ( collect_int
% 3.36/1.73 @ ^ [X_1: int] : ( (&) @ ( A_55 = X_1 ) @ ( P_6 @ X_1 ) ) )
% 3.36/1.73 = ( insert_int @ A_55 @ bot_bot_int_o ) ) )
% 3.36/1.73 & ( ~ ( P_6 @ A_55 )
% 3.36/1.73 => ( ( collect_int
% 3.36/1.73 @ ^ [X_1: int] : ( (&) @ ( A_55 = X_1 ) @ ( P_6 @ X_1 ) ) )
% 3.36/1.73 = bot_bot_int_o ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_635_Collect__conv__if2,axiom,
% 3.36/1.73 ! [P_6: nat > $o,A_55: nat] :
% 3.36/1.73 ( ( ( P_6 @ A_55 )
% 3.36/1.73 => ( ( collect_nat
% 3.36/1.73 @ ^ [X_1: nat] : ( (&) @ ( A_55 = X_1 ) @ ( P_6 @ X_1 ) ) )
% 3.36/1.73 = ( insert_nat @ A_55 @ bot_bot_nat_o ) ) )
% 3.36/1.73 & ( ~ ( P_6 @ A_55 )
% 3.36/1.73 => ( ( collect_nat
% 3.36/1.73 @ ^ [X_1: nat] : ( (&) @ ( A_55 = X_1 ) @ ( P_6 @ X_1 ) ) )
% 3.36/1.73 = bot_bot_nat_o ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_636_singleton__conv,axiom,
% 3.36/1.73 ! [A_54: x_a] :
% 3.36/1.73 ( ( collect_a
% 3.36/1.73 @ ^ [X_1: x_a] : ( X_1 = A_54 ) )
% 3.36/1.73 = ( insert_a @ A_54 @ bot_bot_a_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_637_singleton__conv,axiom,
% 3.36/1.73 ! [A_54: int] :
% 3.36/1.73 ( ( collect_int
% 3.36/1.73 @ ^ [X_1: int] : ( X_1 = A_54 ) )
% 3.36/1.73 = ( insert_int @ A_54 @ bot_bot_int_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_638_singleton__conv,axiom,
% 3.36/1.73 ! [A_54: nat] :
% 3.36/1.73 ( ( collect_nat
% 3.36/1.73 @ ^ [X_1: nat] : ( X_1 = A_54 ) )
% 3.36/1.73 = ( insert_nat @ A_54 @ bot_bot_nat_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_639_singleton__conv2,axiom,
% 3.36/1.73 ! [A_53: x_a] :
% 3.36/1.73 ( ( collect_a @ ( fequal_a @ A_53 ) )
% 3.36/1.73 = ( insert_a @ A_53 @ bot_bot_a_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_640_singleton__conv2,axiom,
% 3.36/1.73 ! [A_53: int] :
% 3.36/1.73 ( ( collect_int @ ( fequal_int @ A_53 ) )
% 3.36/1.73 = ( insert_int @ A_53 @ bot_bot_int_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_641_singleton__conv2,axiom,
% 3.36/1.73 ! [A_53: nat] :
% 3.36/1.73 ( ( collect_nat @ ( fequal_nat @ A_53 ) )
% 3.36/1.73 = ( insert_nat @ A_53 @ bot_bot_nat_o ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_642_card__Suc__Diff1,axiom,
% 3.36/1.73 ! [X_18: x_a,A_52: x_a > $o] :
% 3.36/1.73 ( ( finite_finite_a @ A_52 )
% 3.36/1.73 => ( ( member_a @ X_18 @ A_52 )
% 3.36/1.73 => ( ( suc @ ( finite_card_a @ ( minus_minus_a_o @ A_52 @ ( insert_a @ X_18 @ bot_bot_a_o ) ) ) )
% 3.36/1.73 = ( finite_card_a @ A_52 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_643_card__Suc__Diff1,axiom,
% 3.36/1.73 ! [X_18: int,A_52: int > $o] :
% 3.36/1.73 ( ( finite_finite_int @ A_52 )
% 3.36/1.73 => ( ( member_int @ X_18 @ A_52 )
% 3.36/1.73 => ( ( suc @ ( finite_card_int @ ( minus_minus_int_o @ A_52 @ ( insert_int @ X_18 @ bot_bot_int_o ) ) ) )
% 3.36/1.73 = ( finite_card_int @ A_52 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_644_card__Suc__Diff1,axiom,
% 3.36/1.73 ! [X_18: nat,A_52: nat > $o] :
% 3.36/1.73 ( ( finite_finite_nat @ A_52 )
% 3.36/1.73 => ( ( member_nat @ X_18 @ A_52 )
% 3.36/1.73 => ( ( suc @ ( finite_card_nat @ ( minus_minus_nat_o @ A_52 @ ( insert_nat @ X_18 @ bot_bot_nat_o ) ) ) )
% 3.36/1.73 = ( finite_card_nat @ A_52 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_645_card__Suc__Diff1,axiom,
% 3.36/1.73 ! [X_18: pname,A_52: pname > $o] :
% 3.36/1.73 ( ( finite_finite_pname @ A_52 )
% 3.36/1.73 => ( ( member_pname @ X_18 @ A_52 )
% 3.36/1.73 => ( ( suc @ ( finite_card_pname @ ( minus_minus_pname_o @ A_52 @ ( insert_pname @ X_18 @ bot_bot_pname_o ) ) ) )
% 3.36/1.73 = ( finite_card_pname @ A_52 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_646_card__insert,axiom,
% 3.36/1.73 ! [X_17: x_a,A_51: x_a > $o] :
% 3.36/1.73 ( ( finite_finite_a @ A_51 )
% 3.36/1.73 => ( ( finite_card_a @ ( insert_a @ X_17 @ A_51 ) )
% 3.36/1.73 = ( suc @ ( finite_card_a @ ( minus_minus_a_o @ A_51 @ ( insert_a @ X_17 @ bot_bot_a_o ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_647_card__insert,axiom,
% 3.36/1.73 ! [X_17: int,A_51: int > $o] :
% 3.36/1.73 ( ( finite_finite_int @ A_51 )
% 3.36/1.73 => ( ( finite_card_int @ ( insert_int @ X_17 @ A_51 ) )
% 3.36/1.73 = ( suc @ ( finite_card_int @ ( minus_minus_int_o @ A_51 @ ( insert_int @ X_17 @ bot_bot_int_o ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_648_card__insert,axiom,
% 3.36/1.73 ! [X_17: nat,A_51: nat > $o] :
% 3.36/1.73 ( ( finite_finite_nat @ A_51 )
% 3.36/1.73 => ( ( finite_card_nat @ ( insert_nat @ X_17 @ A_51 ) )
% 3.36/1.73 = ( suc @ ( finite_card_nat @ ( minus_minus_nat_o @ A_51 @ ( insert_nat @ X_17 @ bot_bot_nat_o ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_649_card__insert,axiom,
% 3.36/1.73 ! [X_17: pname,A_51: pname > $o] :
% 3.36/1.73 ( ( finite_finite_pname @ A_51 )
% 3.36/1.73 => ( ( finite_card_pname @ ( insert_pname @ X_17 @ A_51 ) )
% 3.36/1.73 = ( suc @ ( finite_card_pname @ ( minus_minus_pname_o @ A_51 @ ( insert_pname @ X_17 @ bot_bot_pname_o ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_650_card__Diff1__le,axiom,
% 3.36/1.73 ! [X_16: x_a,A_50: x_a > $o] :
% 3.36/1.73 ( ( finite_finite_a @ A_50 )
% 3.36/1.73 => ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_a_o @ A_50 @ ( insert_a @ X_16 @ bot_bot_a_o ) ) ) @ ( finite_card_a @ A_50 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_651_card__Diff1__le,axiom,
% 3.36/1.73 ! [X_16: int,A_50: int > $o] :
% 3.36/1.73 ( ( finite_finite_int @ A_50 )
% 3.36/1.73 => ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_int_o @ A_50 @ ( insert_int @ X_16 @ bot_bot_int_o ) ) ) @ ( finite_card_int @ A_50 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_652_card__Diff1__le,axiom,
% 3.36/1.73 ! [X_16: nat,A_50: nat > $o] :
% 3.36/1.73 ( ( finite_finite_nat @ A_50 )
% 3.36/1.73 => ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_nat_o @ A_50 @ ( insert_nat @ X_16 @ bot_bot_nat_o ) ) ) @ ( finite_card_nat @ A_50 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_653_card__Diff1__le,axiom,
% 3.36/1.73 ! [X_16: pname,A_50: pname > $o] :
% 3.36/1.73 ( ( finite_finite_pname @ A_50 )
% 3.36/1.73 => ( ord_less_eq_nat @ ( finite_card_pname @ ( minus_minus_pname_o @ A_50 @ ( insert_pname @ X_16 @ bot_bot_pname_o ) ) ) @ ( finite_card_pname @ A_50 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_654_finite__Diff__insert,axiom,
% 3.36/1.73 ! [A_49: x_a > $o,A_48: x_a,B_31: x_a > $o] :
% 3.36/1.73 ( ( finite_finite_a @ ( minus_minus_a_o @ A_49 @ ( insert_a @ A_48 @ B_31 ) ) )
% 3.36/1.73 <=> ( finite_finite_a @ ( minus_minus_a_o @ A_49 @ B_31 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_655_finite__Diff__insert,axiom,
% 3.36/1.73 ! [A_49: int > $o,A_48: int,B_31: int > $o] :
% 3.36/1.73 ( ( finite_finite_int @ ( minus_minus_int_o @ A_49 @ ( insert_int @ A_48 @ B_31 ) ) )
% 3.36/1.73 <=> ( finite_finite_int @ ( minus_minus_int_o @ A_49 @ B_31 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_656_finite__Diff__insert,axiom,
% 3.36/1.73 ! [A_49: nat > $o,A_48: nat,B_31: nat > $o] :
% 3.36/1.73 ( ( finite_finite_nat @ ( minus_minus_nat_o @ A_49 @ ( insert_nat @ A_48 @ B_31 ) ) )
% 3.36/1.73 <=> ( finite_finite_nat @ ( minus_minus_nat_o @ A_49 @ B_31 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_657_finite__Diff__insert,axiom,
% 3.36/1.73 ! [A_49: pname > $o,A_48: pname,B_31: pname > $o] :
% 3.36/1.73 ( ( finite_finite_pname @ ( minus_minus_pname_o @ A_49 @ ( insert_pname @ A_48 @ B_31 ) ) )
% 3.36/1.73 <=> ( finite_finite_pname @ ( minus_minus_pname_o @ A_49 @ B_31 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_658_image__diff__subset,axiom,
% 3.36/1.73 ! [F_19: nat > int,A_47: nat > $o,B_30: nat > $o] : ( ord_less_eq_int_o @ ( minus_minus_int_o @ ( image_nat_int @ F_19 @ A_47 ) @ ( image_nat_int @ F_19 @ B_30 ) ) @ ( image_nat_int @ F_19 @ ( minus_minus_nat_o @ A_47 @ B_30 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_659_image__diff__subset,axiom,
% 3.36/1.73 ! [F_19: pname > x_a,A_47: pname > $o,B_30: pname > $o] : ( ord_less_eq_a_o @ ( minus_minus_a_o @ ( image_pname_a @ F_19 @ A_47 ) @ ( image_pname_a @ F_19 @ B_30 ) ) @ ( image_pname_a @ F_19 @ ( minus_minus_pname_o @ A_47 @ B_30 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_660_subset__singletonD,axiom,
% 3.36/1.73 ! [A_46: x_a > $o,X_15: x_a] :
% 3.36/1.73 ( ( ord_less_eq_a_o @ A_46 @ ( insert_a @ X_15 @ bot_bot_a_o ) )
% 3.36/1.73 => ( ( A_46 = bot_bot_a_o )
% 3.36/1.73 | ( A_46
% 3.36/1.73 = ( insert_a @ X_15 @ bot_bot_a_o ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_661_subset__singletonD,axiom,
% 3.36/1.73 ! [A_46: nat > $o,X_15: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat_o @ A_46 @ ( insert_nat @ X_15 @ bot_bot_nat_o ) )
% 3.36/1.73 => ( ( A_46 = bot_bot_nat_o )
% 3.36/1.73 | ( A_46
% 3.36/1.73 = ( insert_nat @ X_15 @ bot_bot_nat_o ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_662_subset__singletonD,axiom,
% 3.36/1.73 ! [A_46: int > $o,X_15: int] :
% 3.36/1.73 ( ( ord_less_eq_int_o @ A_46 @ ( insert_int @ X_15 @ bot_bot_int_o ) )
% 3.36/1.73 => ( ( A_46 = bot_bot_int_o )
% 3.36/1.73 | ( A_46
% 3.36/1.73 = ( insert_int @ X_15 @ bot_bot_int_o ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_663_nat__case__Suc,axiom,
% 3.36/1.73 ! [F1: nat,F2: nat > nat,Nat_3: nat] :
% 3.36/1.73 ( ( nat_case_nat @ F1 @ F2 @ ( suc @ Nat_3 ) )
% 3.36/1.73 = ( F2 @ Nat_3 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_664_nat__case__Suc,axiom,
% 3.36/1.73 ! [F1: $o,F2: nat > $o,Nat_3: nat] :
% 3.36/1.73 ( ( nat_case_o @ F1 @ F2 @ ( suc @ Nat_3 ) )
% 3.36/1.73 <=> ( F2 @ Nat_3 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_665_image__constant,axiom,
% 3.36/1.73 ! [C_21: int,X_14: nat,A_45: nat > $o] :
% 3.36/1.73 ( ( member_nat @ X_14 @ A_45 )
% 3.36/1.73 => ( ( image_nat_int
% 3.36/1.73 @ ^ [X_1: nat] : C_21
% 3.36/1.73 @ A_45 )
% 3.36/1.73 = ( insert_int @ C_21 @ bot_bot_int_o ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_666_image__constant,axiom,
% 3.36/1.73 ! [C_21: x_a,X_14: pname,A_45: pname > $o] :
% 3.36/1.73 ( ( member_pname @ X_14 @ A_45 )
% 3.36/1.73 => ( ( image_pname_a
% 3.36/1.73 @ ^ [X_1: pname] : C_21
% 3.36/1.73 @ A_45 )
% 3.36/1.73 = ( insert_a @ C_21 @ bot_bot_a_o ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_667_image__constant__conv,axiom,
% 3.36/1.73 ! [C_20: int,A_44: nat > $o] :
% 3.36/1.73 ( ( ( A_44 = bot_bot_nat_o )
% 3.36/1.73 => ( ( image_nat_int
% 3.36/1.73 @ ^ [X_1: nat] : C_20
% 3.36/1.73 @ A_44 )
% 3.36/1.73 = bot_bot_int_o ) )
% 3.36/1.73 & ( ( A_44 != bot_bot_nat_o )
% 3.36/1.73 => ( ( image_nat_int
% 3.36/1.73 @ ^ [X_1: nat] : C_20
% 3.36/1.73 @ A_44 )
% 3.36/1.73 = ( insert_int @ C_20 @ bot_bot_int_o ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_668_image__constant__conv,axiom,
% 3.36/1.73 ! [C_20: x_a,A_44: pname > $o] :
% 3.36/1.73 ( ( ( A_44 = bot_bot_pname_o )
% 3.36/1.73 => ( ( image_pname_a
% 3.36/1.73 @ ^ [X_1: pname] : C_20
% 3.36/1.73 @ A_44 )
% 3.36/1.73 = bot_bot_a_o ) )
% 3.36/1.73 & ( ( A_44 != bot_bot_pname_o )
% 3.36/1.73 => ( ( image_pname_a
% 3.36/1.73 @ ^ [X_1: pname] : C_20
% 3.36/1.73 @ A_44 )
% 3.36/1.73 = ( insert_a @ C_20 @ bot_bot_a_o ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_669_diff__eq__diff__eq,axiom,
% 3.36/1.73 ! [A_43: int,B_29: int,C_19: int,D_3: int] :
% 3.36/1.73 ( ( ( minus_minus_int @ A_43 @ B_29 )
% 3.36/1.73 = ( minus_minus_int @ C_19 @ D_3 ) )
% 3.36/1.73 => ( ( A_43 = B_29 )
% 3.36/1.73 <=> ( C_19 = D_3 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_670_card__Diff__subset,axiom,
% 3.36/1.73 ! [A_42: int > $o,B_28: int > $o] :
% 3.36/1.73 ( ( finite_finite_int @ B_28 )
% 3.36/1.73 => ( ( ord_less_eq_int_o @ B_28 @ A_42 )
% 3.36/1.73 => ( ( finite_card_int @ ( minus_minus_int_o @ A_42 @ B_28 ) )
% 3.36/1.73 = ( minus_minus_nat @ ( finite_card_int @ A_42 ) @ ( finite_card_int @ B_28 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_671_card__Diff__subset,axiom,
% 3.36/1.73 ! [A_42: nat > $o,B_28: nat > $o] :
% 3.36/1.73 ( ( finite_finite_nat @ B_28 )
% 3.36/1.73 => ( ( ord_less_eq_nat_o @ B_28 @ A_42 )
% 3.36/1.73 => ( ( finite_card_nat @ ( minus_minus_nat_o @ A_42 @ B_28 ) )
% 3.36/1.73 = ( minus_minus_nat @ ( finite_card_nat @ A_42 ) @ ( finite_card_nat @ B_28 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_672_card__Diff__subset,axiom,
% 3.36/1.73 ! [A_42: x_a > $o,B_28: x_a > $o] :
% 3.36/1.73 ( ( finite_finite_a @ B_28 )
% 3.36/1.73 => ( ( ord_less_eq_a_o @ B_28 @ A_42 )
% 3.36/1.73 => ( ( finite_card_a @ ( minus_minus_a_o @ A_42 @ B_28 ) )
% 3.36/1.73 = ( minus_minus_nat @ ( finite_card_a @ A_42 ) @ ( finite_card_a @ B_28 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_673_card__Diff__subset,axiom,
% 3.36/1.73 ! [A_42: pname > $o,B_28: pname > $o] :
% 3.36/1.73 ( ( finite_finite_pname @ B_28 )
% 3.36/1.73 => ( ( ord_less_eq_pname_o @ B_28 @ A_42 )
% 3.36/1.73 => ( ( finite_card_pname @ ( minus_minus_pname_o @ A_42 @ B_28 ) )
% 3.36/1.73 = ( minus_minus_nat @ ( finite_card_pname @ A_42 ) @ ( finite_card_pname @ B_28 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_674_diff__card__le__card__Diff,axiom,
% 3.36/1.73 ! [A_41: int > $o,B_27: int > $o] :
% 3.36/1.73 ( ( finite_finite_int @ B_27 )
% 3.36/1.73 => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_int @ A_41 ) @ ( finite_card_int @ B_27 ) ) @ ( finite_card_int @ ( minus_minus_int_o @ A_41 @ B_27 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_675_diff__card__le__card__Diff,axiom,
% 3.36/1.73 ! [A_41: nat > $o,B_27: nat > $o] :
% 3.36/1.73 ( ( finite_finite_nat @ B_27 )
% 3.36/1.73 => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A_41 ) @ ( finite_card_nat @ B_27 ) ) @ ( finite_card_nat @ ( minus_minus_nat_o @ A_41 @ B_27 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_676_diff__card__le__card__Diff,axiom,
% 3.36/1.73 ! [A_41: x_a > $o,B_27: x_a > $o] :
% 3.36/1.73 ( ( finite_finite_a @ B_27 )
% 3.36/1.73 => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_a @ A_41 ) @ ( finite_card_a @ B_27 ) ) @ ( finite_card_a @ ( minus_minus_a_o @ A_41 @ B_27 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_677_diff__card__le__card__Diff,axiom,
% 3.36/1.73 ! [A_41: pname > $o,B_27: pname > $o] :
% 3.36/1.73 ( ( finite_finite_pname @ B_27 )
% 3.36/1.73 => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_pname @ A_41 ) @ ( finite_card_pname @ B_27 ) ) @ ( finite_card_pname @ ( minus_minus_pname_o @ A_41 @ B_27 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_678_finite__subset__induct,axiom,
% 3.36/1.73 ! [P_5: ( x_a > $o ) > $o,A_40: x_a > $o,F_18: x_a > $o] :
% 3.36/1.73 ( ( finite_finite_a @ F_18 )
% 3.36/1.73 => ( ( ord_less_eq_a_o @ F_18 @ A_40 )
% 3.36/1.73 => ( ( P_5 @ bot_bot_a_o )
% 3.36/1.73 => ( ! [A_37: x_a,F_2: x_a > $o] :
% 3.36/1.73 ( ( finite_finite_a @ F_2 )
% 3.36/1.73 => ( ( member_a @ A_37 @ A_40 )
% 3.36/1.73 => ( ~ ( member_a @ A_37 @ F_2 )
% 3.36/1.73 => ( ( P_5 @ F_2 )
% 3.36/1.73 => ( P_5 @ ( insert_a @ A_37 @ F_2 ) ) ) ) ) )
% 3.36/1.73 => ( P_5 @ F_18 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_679_finite__subset__induct,axiom,
% 3.36/1.73 ! [P_5: ( int > $o ) > $o,A_40: int > $o,F_18: int > $o] :
% 3.36/1.73 ( ( finite_finite_int @ F_18 )
% 3.36/1.73 => ( ( ord_less_eq_int_o @ F_18 @ A_40 )
% 3.36/1.73 => ( ( P_5 @ bot_bot_int_o )
% 3.36/1.73 => ( ! [A_37: int,F_2: int > $o] :
% 3.36/1.73 ( ( finite_finite_int @ F_2 )
% 3.36/1.73 => ( ( member_int @ A_37 @ A_40 )
% 3.36/1.73 => ( ~ ( member_int @ A_37 @ F_2 )
% 3.36/1.73 => ( ( P_5 @ F_2 )
% 3.36/1.73 => ( P_5 @ ( insert_int @ A_37 @ F_2 ) ) ) ) ) )
% 3.36/1.73 => ( P_5 @ F_18 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_680_finite__subset__induct,axiom,
% 3.36/1.73 ! [P_5: ( nat > $o ) > $o,A_40: nat > $o,F_18: nat > $o] :
% 3.36/1.73 ( ( finite_finite_nat @ F_18 )
% 3.36/1.73 => ( ( ord_less_eq_nat_o @ F_18 @ A_40 )
% 3.36/1.73 => ( ( P_5 @ bot_bot_nat_o )
% 3.36/1.73 => ( ! [A_37: nat,F_2: nat > $o] :
% 3.36/1.73 ( ( finite_finite_nat @ F_2 )
% 3.36/1.73 => ( ( member_nat @ A_37 @ A_40 )
% 3.36/1.73 => ( ~ ( member_nat @ A_37 @ F_2 )
% 3.36/1.73 => ( ( P_5 @ F_2 )
% 3.36/1.73 => ( P_5 @ ( insert_nat @ A_37 @ F_2 ) ) ) ) ) )
% 3.36/1.73 => ( P_5 @ F_18 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_681_finite__subset__induct,axiom,
% 3.36/1.73 ! [P_5: ( pname > $o ) > $o,A_40: pname > $o,F_18: pname > $o] :
% 3.36/1.73 ( ( finite_finite_pname @ F_18 )
% 3.36/1.73 => ( ( ord_less_eq_pname_o @ F_18 @ A_40 )
% 3.36/1.73 => ( ( P_5 @ bot_bot_pname_o )
% 3.36/1.73 => ( ! [A_37: pname,F_2: pname > $o] :
% 3.36/1.73 ( ( finite_finite_pname @ F_2 )
% 3.36/1.73 => ( ( member_pname @ A_37 @ A_40 )
% 3.36/1.73 => ( ~ ( member_pname @ A_37 @ F_2 )
% 3.36/1.73 => ( ( P_5 @ F_2 )
% 3.36/1.73 => ( P_5 @ ( insert_pname @ A_37 @ F_2 ) ) ) ) ) )
% 3.36/1.73 => ( P_5 @ F_18 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_682_assms_I2_J,axiom,
% 3.36/1.73 ! [Pn: pname,G: x_a > $o] :
% 3.36/1.73 ( ( p @ ( insert_a @ ( mgt_call @ Pn ) @ G ) @ ( insert_a @ ( mgt @ ( the_com @ ( body @ Pn ) ) ) @ bot_bot_a_o ) )
% 3.36/1.73 => ( p @ G @ ( insert_a @ ( mgt_call @ Pn ) @ bot_bot_a_o ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_683_finite__empty__induct,axiom,
% 3.36/1.73 ! [P_4: ( x_a > $o ) > $o,A_39: x_a > $o] :
% 3.36/1.73 ( ( finite_finite_a @ A_39 )
% 3.36/1.73 => ( ( P_4 @ A_39 )
% 3.36/1.73 => ( ! [A_37: x_a,A_38: x_a > $o] :
% 3.36/1.73 ( ( finite_finite_a @ A_38 )
% 3.36/1.73 => ( ( member_a @ A_37 @ A_38 )
% 3.36/1.73 => ( ( P_4 @ A_38 )
% 3.36/1.73 => ( P_4 @ ( minus_minus_a_o @ A_38 @ ( insert_a @ A_37 @ bot_bot_a_o ) ) ) ) ) )
% 3.36/1.73 => ( P_4 @ bot_bot_a_o ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_684_finite__empty__induct,axiom,
% 3.36/1.73 ! [P_4: ( int > $o ) > $o,A_39: int > $o] :
% 3.36/1.73 ( ( finite_finite_int @ A_39 )
% 3.36/1.73 => ( ( P_4 @ A_39 )
% 3.36/1.73 => ( ! [A_37: int,A_38: int > $o] :
% 3.36/1.73 ( ( finite_finite_int @ A_38 )
% 3.36/1.73 => ( ( member_int @ A_37 @ A_38 )
% 3.36/1.73 => ( ( P_4 @ A_38 )
% 3.36/1.73 => ( P_4 @ ( minus_minus_int_o @ A_38 @ ( insert_int @ A_37 @ bot_bot_int_o ) ) ) ) ) )
% 3.36/1.73 => ( P_4 @ bot_bot_int_o ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_685_finite__empty__induct,axiom,
% 3.36/1.73 ! [P_4: ( nat > $o ) > $o,A_39: nat > $o] :
% 3.36/1.73 ( ( finite_finite_nat @ A_39 )
% 3.36/1.73 => ( ( P_4 @ A_39 )
% 3.36/1.73 => ( ! [A_37: nat,A_38: nat > $o] :
% 3.36/1.73 ( ( finite_finite_nat @ A_38 )
% 3.36/1.73 => ( ( member_nat @ A_37 @ A_38 )
% 3.36/1.73 => ( ( P_4 @ A_38 )
% 3.36/1.73 => ( P_4 @ ( minus_minus_nat_o @ A_38 @ ( insert_nat @ A_37 @ bot_bot_nat_o ) ) ) ) ) )
% 3.36/1.73 => ( P_4 @ bot_bot_nat_o ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_686_finite__empty__induct,axiom,
% 3.36/1.73 ! [P_4: ( pname > $o ) > $o,A_39: pname > $o] :
% 3.36/1.73 ( ( finite_finite_pname @ A_39 )
% 3.36/1.73 => ( ( P_4 @ A_39 )
% 3.36/1.73 => ( ! [A_37: pname,A_38: pname > $o] :
% 3.36/1.73 ( ( finite_finite_pname @ A_38 )
% 3.36/1.73 => ( ( member_pname @ A_37 @ A_38 )
% 3.36/1.73 => ( ( P_4 @ A_38 )
% 3.36/1.73 => ( P_4 @ ( minus_minus_pname_o @ A_38 @ ( insert_pname @ A_37 @ bot_bot_pname_o ) ) ) ) ) )
% 3.36/1.73 => ( P_4 @ bot_bot_pname_o ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_687_finite__induct,axiom,
% 3.36/1.73 ! [P_3: ( x_a > $o ) > $o,F_17: x_a > $o] :
% 3.36/1.73 ( ( finite_finite_a @ F_17 )
% 3.36/1.73 => ( ( P_3 @ bot_bot_a_o )
% 3.36/1.73 => ( ! [X_1: x_a,F_2: x_a > $o] :
% 3.36/1.73 ( ( finite_finite_a @ F_2 )
% 3.36/1.73 => ( ~ ( member_a @ X_1 @ F_2 )
% 3.36/1.73 => ( ( P_3 @ F_2 )
% 3.36/1.73 => ( P_3 @ ( insert_a @ X_1 @ F_2 ) ) ) ) )
% 3.36/1.73 => ( P_3 @ F_17 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_688_finite__induct,axiom,
% 3.36/1.73 ! [P_3: ( int > $o ) > $o,F_17: int > $o] :
% 3.36/1.73 ( ( finite_finite_int @ F_17 )
% 3.36/1.73 => ( ( P_3 @ bot_bot_int_o )
% 3.36/1.73 => ( ! [X_1: int,F_2: int > $o] :
% 3.36/1.73 ( ( finite_finite_int @ F_2 )
% 3.36/1.73 => ( ~ ( member_int @ X_1 @ F_2 )
% 3.36/1.73 => ( ( P_3 @ F_2 )
% 3.36/1.73 => ( P_3 @ ( insert_int @ X_1 @ F_2 ) ) ) ) )
% 3.36/1.73 => ( P_3 @ F_17 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_689_finite__induct,axiom,
% 3.36/1.73 ! [P_3: ( nat > $o ) > $o,F_17: nat > $o] :
% 3.36/1.73 ( ( finite_finite_nat @ F_17 )
% 3.36/1.73 => ( ( P_3 @ bot_bot_nat_o )
% 3.36/1.73 => ( ! [X_1: nat,F_2: nat > $o] :
% 3.36/1.73 ( ( finite_finite_nat @ F_2 )
% 3.36/1.73 => ( ~ ( member_nat @ X_1 @ F_2 )
% 3.36/1.73 => ( ( P_3 @ F_2 )
% 3.36/1.73 => ( P_3 @ ( insert_nat @ X_1 @ F_2 ) ) ) ) )
% 3.36/1.73 => ( P_3 @ F_17 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_690_finite__induct,axiom,
% 3.36/1.73 ! [P_3: ( pname > $o ) > $o,F_17: pname > $o] :
% 3.36/1.73 ( ( finite_finite_pname @ F_17 )
% 3.36/1.73 => ( ( P_3 @ bot_bot_pname_o )
% 3.36/1.73 => ( ! [X_1: pname,F_2: pname > $o] :
% 3.36/1.73 ( ( finite_finite_pname @ F_2 )
% 3.36/1.73 => ( ~ ( member_pname @ X_1 @ F_2 )
% 3.36/1.73 => ( ( P_3 @ F_2 )
% 3.36/1.73 => ( P_3 @ ( insert_pname @ X_1 @ F_2 ) ) ) ) )
% 3.36/1.73 => ( P_3 @ F_17 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_691_finite_Osimps,axiom,
% 3.36/1.73 ! [A_36: x_a > $o] :
% 3.36/1.73 ( ( finite_finite_a @ A_36 )
% 3.36/1.73 <=> ( ( A_36 = bot_bot_a_o )
% 3.36/1.73 | ? [A_38: x_a > $o,A_37: x_a] :
% 3.36/1.73 ( ( A_36
% 3.36/1.73 = ( insert_a @ A_37 @ A_38 ) )
% 3.36/1.73 & ( finite_finite_a @ A_38 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_692_finite_Osimps,axiom,
% 3.36/1.73 ! [A_36: int > $o] :
% 3.36/1.73 ( ( finite_finite_int @ A_36 )
% 3.36/1.73 <=> ( ( A_36 = bot_bot_int_o )
% 3.36/1.73 | ? [A_38: int > $o,A_37: int] :
% 3.36/1.73 ( ( A_36
% 3.36/1.73 = ( insert_int @ A_37 @ A_38 ) )
% 3.36/1.73 & ( finite_finite_int @ A_38 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_693_finite_Osimps,axiom,
% 3.36/1.73 ! [A_36: nat > $o] :
% 3.36/1.73 ( ( finite_finite_nat @ A_36 )
% 3.36/1.73 <=> ( ( A_36 = bot_bot_nat_o )
% 3.36/1.73 | ? [A_38: nat > $o,A_37: nat] :
% 3.36/1.73 ( ( A_36
% 3.36/1.73 = ( insert_nat @ A_37 @ A_38 ) )
% 3.36/1.73 & ( finite_finite_nat @ A_38 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_694_finite_Osimps,axiom,
% 3.36/1.73 ! [A_36: pname > $o] :
% 3.36/1.73 ( ( finite_finite_pname @ A_36 )
% 3.36/1.73 <=> ( ( A_36 = bot_bot_pname_o )
% 3.36/1.73 | ? [A_38: pname > $o,A_37: pname] :
% 3.36/1.73 ( ( A_36
% 3.36/1.73 = ( insert_pname @ A_37 @ A_38 ) )
% 3.36/1.73 & ( finite_finite_pname @ A_38 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_695_finite__less__ub,axiom,
% 3.36/1.73 ! [U: nat,F: nat > nat] :
% 3.36/1.73 ( ! [N_1: nat] : ( ord_less_eq_nat @ N_1 @ ( F @ N_1 ) )
% 3.36/1.73 => ( finite_finite_nat
% 3.36/1.73 @ ( collect_nat
% 3.36/1.73 @ ^ [N_1: nat] : ( ord_less_eq_nat @ ( F @ N_1 ) @ U ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_696_the__elem__eq,axiom,
% 3.36/1.73 ! [X_13: x_a] :
% 3.36/1.73 ( ( the_elem_a @ ( insert_a @ X_13 @ bot_bot_a_o ) )
% 3.36/1.73 = X_13 ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_697_the__elem__eq,axiom,
% 3.36/1.73 ! [X_13: nat] :
% 3.36/1.73 ( ( the_elem_nat @ ( insert_nat @ X_13 @ bot_bot_nat_o ) )
% 3.36/1.73 = X_13 ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_698_the__elem__eq,axiom,
% 3.36/1.73 ! [X_13: int] :
% 3.36/1.73 ( ( the_elem_int @ ( insert_int @ X_13 @ bot_bot_int_o ) )
% 3.36/1.73 = X_13 ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_699_nonempty__iff,axiom,
% 3.36/1.73 ! [A_35: x_a > $o] :
% 3.36/1.73 ( ( A_35 != bot_bot_a_o )
% 3.36/1.73 <=> ? [X_1: x_a,B_26: x_a > $o] :
% 3.36/1.73 ( ( A_35
% 3.36/1.73 = ( insert_a @ X_1 @ B_26 ) )
% 3.36/1.73 & ~ ( member_a @ X_1 @ B_26 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_700_nonempty__iff,axiom,
% 3.36/1.73 ! [A_35: int > $o] :
% 3.36/1.73 ( ( A_35 != bot_bot_int_o )
% 3.36/1.73 <=> ? [X_1: int,B_26: int > $o] :
% 3.36/1.73 ( ( A_35
% 3.36/1.73 = ( insert_int @ X_1 @ B_26 ) )
% 3.36/1.73 & ~ ( member_int @ X_1 @ B_26 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_701_nonempty__iff,axiom,
% 3.36/1.73 ! [A_35: nat > $o] :
% 3.36/1.73 ( ( A_35 != bot_bot_nat_o )
% 3.36/1.73 <=> ? [X_1: nat,B_26: nat > $o] :
% 3.36/1.73 ( ( A_35
% 3.36/1.73 = ( insert_nat @ X_1 @ B_26 ) )
% 3.36/1.73 & ~ ( member_nat @ X_1 @ B_26 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_702_nonempty__iff,axiom,
% 3.36/1.73 ! [A_35: pname > $o] :
% 3.36/1.73 ( ( A_35 != bot_bot_pname_o )
% 3.36/1.73 <=> ? [X_1: pname,B_26: pname > $o] :
% 3.36/1.73 ( ( A_35
% 3.36/1.73 = ( insert_pname @ X_1 @ B_26 ) )
% 3.36/1.73 & ~ ( member_pname @ X_1 @ B_26 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_703_assms_I4_J,axiom,
% 3.36/1.73 ! [Pn: pname] :
% 3.36/1.73 ( ( member_pname @ Pn @ u )
% 3.36/1.73 => ( wt @ ( the_com @ ( body @ Pn ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_704_DiffE,axiom,
% 3.36/1.73 ! [C_18: int,A_34: int > $o,B_25: int > $o] :
% 3.36/1.73 ( ( member_int @ C_18 @ ( minus_minus_int_o @ A_34 @ B_25 ) )
% 3.36/1.73 => ~ ( ( member_int @ C_18 @ A_34 )
% 3.36/1.73 => ( member_int @ C_18 @ B_25 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_705_DiffE,axiom,
% 3.36/1.73 ! [C_18: nat,A_34: nat > $o,B_25: nat > $o] :
% 3.36/1.73 ( ( member_nat @ C_18 @ ( minus_minus_nat_o @ A_34 @ B_25 ) )
% 3.36/1.73 => ~ ( ( member_nat @ C_18 @ A_34 )
% 3.36/1.73 => ( member_nat @ C_18 @ B_25 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_706_DiffE,axiom,
% 3.36/1.73 ! [C_18: x_a,A_34: x_a > $o,B_25: x_a > $o] :
% 3.36/1.73 ( ( member_a @ C_18 @ ( minus_minus_a_o @ A_34 @ B_25 ) )
% 3.36/1.73 => ~ ( ( member_a @ C_18 @ A_34 )
% 3.36/1.73 => ( member_a @ C_18 @ B_25 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_707_DiffE,axiom,
% 3.36/1.73 ! [C_18: pname,A_34: pname > $o,B_25: pname > $o] :
% 3.36/1.73 ( ( member_pname @ C_18 @ ( minus_minus_pname_o @ A_34 @ B_25 ) )
% 3.36/1.73 => ~ ( ( member_pname @ C_18 @ A_34 )
% 3.36/1.73 => ( member_pname @ C_18 @ B_25 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_708_DiffI,axiom,
% 3.36/1.73 ! [B_24: int > $o,C_17: int,A_33: int > $o] :
% 3.36/1.73 ( ( member_int @ C_17 @ A_33 )
% 3.36/1.73 => ( ~ ( member_int @ C_17 @ B_24 )
% 3.36/1.73 => ( member_int @ C_17 @ ( minus_minus_int_o @ A_33 @ B_24 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_709_DiffI,axiom,
% 3.36/1.73 ! [B_24: nat > $o,C_17: nat,A_33: nat > $o] :
% 3.36/1.73 ( ( member_nat @ C_17 @ A_33 )
% 3.36/1.73 => ( ~ ( member_nat @ C_17 @ B_24 )
% 3.36/1.73 => ( member_nat @ C_17 @ ( minus_minus_nat_o @ A_33 @ B_24 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_710_DiffI,axiom,
% 3.36/1.73 ! [B_24: x_a > $o,C_17: x_a,A_33: x_a > $o] :
% 3.36/1.73 ( ( member_a @ C_17 @ A_33 )
% 3.36/1.73 => ( ~ ( member_a @ C_17 @ B_24 )
% 3.36/1.73 => ( member_a @ C_17 @ ( minus_minus_a_o @ A_33 @ B_24 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_711_DiffI,axiom,
% 3.36/1.73 ! [B_24: pname > $o,C_17: pname,A_33: pname > $o] :
% 3.36/1.73 ( ( member_pname @ C_17 @ A_33 )
% 3.36/1.73 => ( ~ ( member_pname @ C_17 @ B_24 )
% 3.36/1.73 => ( member_pname @ C_17 @ ( minus_minus_pname_o @ A_33 @ B_24 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_712_DiffD2,axiom,
% 3.36/1.73 ! [C_16: int,A_32: int > $o,B_23: int > $o] :
% 3.36/1.73 ( ( member_int @ C_16 @ ( minus_minus_int_o @ A_32 @ B_23 ) )
% 3.36/1.73 => ~ ( member_int @ C_16 @ B_23 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_713_DiffD2,axiom,
% 3.36/1.73 ! [C_16: nat,A_32: nat > $o,B_23: nat > $o] :
% 3.36/1.73 ( ( member_nat @ C_16 @ ( minus_minus_nat_o @ A_32 @ B_23 ) )
% 3.36/1.73 => ~ ( member_nat @ C_16 @ B_23 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_714_DiffD2,axiom,
% 3.36/1.73 ! [C_16: x_a,A_32: x_a > $o,B_23: x_a > $o] :
% 3.36/1.73 ( ( member_a @ C_16 @ ( minus_minus_a_o @ A_32 @ B_23 ) )
% 3.36/1.73 => ~ ( member_a @ C_16 @ B_23 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_715_DiffD2,axiom,
% 3.36/1.73 ! [C_16: pname,A_32: pname > $o,B_23: pname > $o] :
% 3.36/1.73 ( ( member_pname @ C_16 @ ( minus_minus_pname_o @ A_32 @ B_23 ) )
% 3.36/1.73 => ~ ( member_pname @ C_16 @ B_23 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_716_DiffD1,axiom,
% 3.36/1.73 ! [C_15: int,A_31: int > $o,B_22: int > $o] :
% 3.36/1.73 ( ( member_int @ C_15 @ ( minus_minus_int_o @ A_31 @ B_22 ) )
% 3.36/1.73 => ( member_int @ C_15 @ A_31 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_717_DiffD1,axiom,
% 3.36/1.73 ! [C_15: nat,A_31: nat > $o,B_22: nat > $o] :
% 3.36/1.73 ( ( member_nat @ C_15 @ ( minus_minus_nat_o @ A_31 @ B_22 ) )
% 3.36/1.73 => ( member_nat @ C_15 @ A_31 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_718_DiffD1,axiom,
% 3.36/1.73 ! [C_15: x_a,A_31: x_a > $o,B_22: x_a > $o] :
% 3.36/1.73 ( ( member_a @ C_15 @ ( minus_minus_a_o @ A_31 @ B_22 ) )
% 3.36/1.73 => ( member_a @ C_15 @ A_31 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_719_DiffD1,axiom,
% 3.36/1.73 ! [C_15: pname,A_31: pname > $o,B_22: pname > $o] :
% 3.36/1.73 ( ( member_pname @ C_15 @ ( minus_minus_pname_o @ A_31 @ B_22 ) )
% 3.36/1.73 => ( member_pname @ C_15 @ A_31 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_720_Diff__iff,axiom,
% 3.36/1.73 ! [C_14: int,A_30: int > $o,B_21: int > $o] :
% 3.36/1.73 ( ( member_int @ C_14 @ ( minus_minus_int_o @ A_30 @ B_21 ) )
% 3.36/1.73 <=> ( ( member_int @ C_14 @ A_30 )
% 3.36/1.73 & ~ ( member_int @ C_14 @ B_21 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_721_Diff__iff,axiom,
% 3.36/1.73 ! [C_14: nat,A_30: nat > $o,B_21: nat > $o] :
% 3.36/1.73 ( ( member_nat @ C_14 @ ( minus_minus_nat_o @ A_30 @ B_21 ) )
% 3.36/1.73 <=> ( ( member_nat @ C_14 @ A_30 )
% 3.36/1.73 & ~ ( member_nat @ C_14 @ B_21 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_722_Diff__iff,axiom,
% 3.36/1.73 ! [C_14: x_a,A_30: x_a > $o,B_21: x_a > $o] :
% 3.36/1.73 ( ( member_a @ C_14 @ ( minus_minus_a_o @ A_30 @ B_21 ) )
% 3.36/1.73 <=> ( ( member_a @ C_14 @ A_30 )
% 3.36/1.73 & ~ ( member_a @ C_14 @ B_21 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_723_Diff__iff,axiom,
% 3.36/1.73 ! [C_14: pname,A_30: pname > $o,B_21: pname > $o] :
% 3.36/1.73 ( ( member_pname @ C_14 @ ( minus_minus_pname_o @ A_30 @ B_21 ) )
% 3.36/1.73 <=> ( ( member_pname @ C_14 @ A_30 )
% 3.36/1.73 & ~ ( member_pname @ C_14 @ B_21 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_724_set__diff__eq,axiom,
% 3.36/1.73 ! [A_29: int > $o,B_20: int > $o] :
% 3.36/1.73 ( ( minus_minus_int_o @ A_29 @ B_20 )
% 3.36/1.73 = ( collect_int
% 3.36/1.73 @ ^ [X_1: int] : ( (&) @ ( member_int @ X_1 @ A_29 ) @ ( (~) @ ( member_int @ X_1 @ B_20 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_725_set__diff__eq,axiom,
% 3.36/1.73 ! [A_29: nat > $o,B_20: nat > $o] :
% 3.36/1.73 ( ( minus_minus_nat_o @ A_29 @ B_20 )
% 3.36/1.73 = ( collect_nat
% 3.36/1.73 @ ^ [X_1: nat] : ( (&) @ ( member_nat @ X_1 @ A_29 ) @ ( (~) @ ( member_nat @ X_1 @ B_20 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_726_set__diff__eq,axiom,
% 3.36/1.73 ! [A_29: x_a > $o,B_20: x_a > $o] :
% 3.36/1.73 ( ( minus_minus_a_o @ A_29 @ B_20 )
% 3.36/1.73 = ( collect_a
% 3.36/1.73 @ ^ [X_1: x_a] : ( (&) @ ( member_a @ X_1 @ A_29 ) @ ( (~) @ ( member_a @ X_1 @ B_20 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_727_set__diff__eq,axiom,
% 3.36/1.73 ! [A_29: pname > $o,B_20: pname > $o] :
% 3.36/1.73 ( ( minus_minus_pname_o @ A_29 @ B_20 )
% 3.36/1.73 = ( collect_pname
% 3.36/1.73 @ ^ [X_1: pname] : ( (&) @ ( member_pname @ X_1 @ A_29 ) @ ( (~) @ ( member_pname @ X_1 @ B_20 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_728_folding__one_Oinsert__remove,axiom,
% 3.36/1.73 ! [X_12: x_a,A_28: x_a > $o,F_16: x_a > x_a > x_a,F_15: ( x_a > $o ) > x_a] :
% 3.36/1.73 ( ( finite_folding_one_a @ F_16 @ F_15 )
% 3.36/1.73 => ( ( finite_finite_a @ A_28 )
% 3.36/1.73 => ( ( ( ( minus_minus_a_o @ A_28 @ ( insert_a @ X_12 @ bot_bot_a_o ) )
% 3.36/1.73 = bot_bot_a_o )
% 3.36/1.73 => ( ( F_15 @ ( insert_a @ X_12 @ A_28 ) )
% 3.36/1.73 = X_12 ) )
% 3.36/1.73 & ( ( ( minus_minus_a_o @ A_28 @ ( insert_a @ X_12 @ bot_bot_a_o ) )
% 3.36/1.73 != bot_bot_a_o )
% 3.36/1.73 => ( ( F_15 @ ( insert_a @ X_12 @ A_28 ) )
% 3.36/1.73 = ( F_16 @ X_12 @ ( F_15 @ ( minus_minus_a_o @ A_28 @ ( insert_a @ X_12 @ bot_bot_a_o ) ) ) ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_729_folding__one_Oinsert__remove,axiom,
% 3.36/1.73 ! [X_12: int,A_28: int > $o,F_16: int > int > int,F_15: ( int > $o ) > int] :
% 3.36/1.73 ( ( finite1626084323ne_int @ F_16 @ F_15 )
% 3.36/1.73 => ( ( finite_finite_int @ A_28 )
% 3.36/1.73 => ( ( ( ( minus_minus_int_o @ A_28 @ ( insert_int @ X_12 @ bot_bot_int_o ) )
% 3.36/1.73 = bot_bot_int_o )
% 3.36/1.73 => ( ( F_15 @ ( insert_int @ X_12 @ A_28 ) )
% 3.36/1.73 = X_12 ) )
% 3.36/1.73 & ( ( ( minus_minus_int_o @ A_28 @ ( insert_int @ X_12 @ bot_bot_int_o ) )
% 3.36/1.73 != bot_bot_int_o )
% 3.36/1.73 => ( ( F_15 @ ( insert_int @ X_12 @ A_28 ) )
% 3.36/1.73 = ( F_16 @ X_12 @ ( F_15 @ ( minus_minus_int_o @ A_28 @ ( insert_int @ X_12 @ bot_bot_int_o ) ) ) ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_730_folding__one_Oinsert__remove,axiom,
% 3.36/1.73 ! [X_12: nat,A_28: nat > $o,F_16: nat > nat > nat,F_15: ( nat > $o ) > nat] :
% 3.36/1.73 ( ( finite988810631ne_nat @ F_16 @ F_15 )
% 3.36/1.73 => ( ( finite_finite_nat @ A_28 )
% 3.36/1.73 => ( ( ( ( minus_minus_nat_o @ A_28 @ ( insert_nat @ X_12 @ bot_bot_nat_o ) )
% 3.36/1.73 = bot_bot_nat_o )
% 3.36/1.73 => ( ( F_15 @ ( insert_nat @ X_12 @ A_28 ) )
% 3.36/1.73 = X_12 ) )
% 3.36/1.73 & ( ( ( minus_minus_nat_o @ A_28 @ ( insert_nat @ X_12 @ bot_bot_nat_o ) )
% 3.36/1.73 != bot_bot_nat_o )
% 3.36/1.73 => ( ( F_15 @ ( insert_nat @ X_12 @ A_28 ) )
% 3.36/1.73 = ( F_16 @ X_12 @ ( F_15 @ ( minus_minus_nat_o @ A_28 @ ( insert_nat @ X_12 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_731_folding__one_Oinsert__remove,axiom,
% 3.36/1.73 ! [X_12: pname,A_28: pname > $o,F_16: pname > pname > pname,F_15: ( pname > $o ) > pname] :
% 3.36/1.73 ( ( finite1282449217_pname @ F_16 @ F_15 )
% 3.36/1.73 => ( ( finite_finite_pname @ A_28 )
% 3.36/1.73 => ( ( ( ( minus_minus_pname_o @ A_28 @ ( insert_pname @ X_12 @ bot_bot_pname_o ) )
% 3.36/1.73 = bot_bot_pname_o )
% 3.36/1.73 => ( ( F_15 @ ( insert_pname @ X_12 @ A_28 ) )
% 3.36/1.73 = X_12 ) )
% 3.36/1.73 & ( ( ( minus_minus_pname_o @ A_28 @ ( insert_pname @ X_12 @ bot_bot_pname_o ) )
% 3.36/1.73 != bot_bot_pname_o )
% 3.36/1.73 => ( ( F_15 @ ( insert_pname @ X_12 @ A_28 ) )
% 3.36/1.73 = ( F_16 @ X_12 @ ( F_15 @ ( minus_minus_pname_o @ A_28 @ ( insert_pname @ X_12 @ bot_bot_pname_o ) ) ) ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_732_folding__one_Oremove,axiom,
% 3.36/1.73 ! [X_11: x_a,A_27: x_a > $o,F_14: x_a > x_a > x_a,F_13: ( x_a > $o ) > x_a] :
% 3.36/1.73 ( ( finite_folding_one_a @ F_14 @ F_13 )
% 3.36/1.73 => ( ( finite_finite_a @ A_27 )
% 3.36/1.73 => ( ( member_a @ X_11 @ A_27 )
% 3.36/1.73 => ( ( ( ( minus_minus_a_o @ A_27 @ ( insert_a @ X_11 @ bot_bot_a_o ) )
% 3.36/1.73 = bot_bot_a_o )
% 3.36/1.73 => ( ( F_13 @ A_27 )
% 3.36/1.73 = X_11 ) )
% 3.36/1.73 & ( ( ( minus_minus_a_o @ A_27 @ ( insert_a @ X_11 @ bot_bot_a_o ) )
% 3.36/1.73 != bot_bot_a_o )
% 3.36/1.73 => ( ( F_13 @ A_27 )
% 3.36/1.73 = ( F_14 @ X_11 @ ( F_13 @ ( minus_minus_a_o @ A_27 @ ( insert_a @ X_11 @ bot_bot_a_o ) ) ) ) ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_733_folding__one_Oremove,axiom,
% 3.36/1.73 ! [X_11: int,A_27: int > $o,F_14: int > int > int,F_13: ( int > $o ) > int] :
% 3.36/1.73 ( ( finite1626084323ne_int @ F_14 @ F_13 )
% 3.36/1.73 => ( ( finite_finite_int @ A_27 )
% 3.36/1.73 => ( ( member_int @ X_11 @ A_27 )
% 3.36/1.73 => ( ( ( ( minus_minus_int_o @ A_27 @ ( insert_int @ X_11 @ bot_bot_int_o ) )
% 3.36/1.73 = bot_bot_int_o )
% 3.36/1.73 => ( ( F_13 @ A_27 )
% 3.36/1.73 = X_11 ) )
% 3.36/1.73 & ( ( ( minus_minus_int_o @ A_27 @ ( insert_int @ X_11 @ bot_bot_int_o ) )
% 3.36/1.73 != bot_bot_int_o )
% 3.36/1.73 => ( ( F_13 @ A_27 )
% 3.36/1.73 = ( F_14 @ X_11 @ ( F_13 @ ( minus_minus_int_o @ A_27 @ ( insert_int @ X_11 @ bot_bot_int_o ) ) ) ) ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_734_folding__one_Oremove,axiom,
% 3.36/1.73 ! [X_11: nat,A_27: nat > $o,F_14: nat > nat > nat,F_13: ( nat > $o ) > nat] :
% 3.36/1.73 ( ( finite988810631ne_nat @ F_14 @ F_13 )
% 3.36/1.73 => ( ( finite_finite_nat @ A_27 )
% 3.36/1.73 => ( ( member_nat @ X_11 @ A_27 )
% 3.36/1.73 => ( ( ( ( minus_minus_nat_o @ A_27 @ ( insert_nat @ X_11 @ bot_bot_nat_o ) )
% 3.36/1.73 = bot_bot_nat_o )
% 3.36/1.73 => ( ( F_13 @ A_27 )
% 3.36/1.73 = X_11 ) )
% 3.36/1.73 & ( ( ( minus_minus_nat_o @ A_27 @ ( insert_nat @ X_11 @ bot_bot_nat_o ) )
% 3.36/1.73 != bot_bot_nat_o )
% 3.36/1.73 => ( ( F_13 @ A_27 )
% 3.36/1.73 = ( F_14 @ X_11 @ ( F_13 @ ( minus_minus_nat_o @ A_27 @ ( insert_nat @ X_11 @ bot_bot_nat_o ) ) ) ) ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_735_folding__one_Oremove,axiom,
% 3.36/1.73 ! [X_11: pname,A_27: pname > $o,F_14: pname > pname > pname,F_13: ( pname > $o ) > pname] :
% 3.36/1.73 ( ( finite1282449217_pname @ F_14 @ F_13 )
% 3.36/1.73 => ( ( finite_finite_pname @ A_27 )
% 3.36/1.73 => ( ( member_pname @ X_11 @ A_27 )
% 3.36/1.73 => ( ( ( ( minus_minus_pname_o @ A_27 @ ( insert_pname @ X_11 @ bot_bot_pname_o ) )
% 3.36/1.73 = bot_bot_pname_o )
% 3.36/1.73 => ( ( F_13 @ A_27 )
% 3.36/1.73 = X_11 ) )
% 3.36/1.73 & ( ( ( minus_minus_pname_o @ A_27 @ ( insert_pname @ X_11 @ bot_bot_pname_o ) )
% 3.36/1.73 != bot_bot_pname_o )
% 3.36/1.73 => ( ( F_13 @ A_27 )
% 3.36/1.73 = ( F_14 @ X_11 @ ( F_13 @ ( minus_minus_pname_o @ A_27 @ ( insert_pname @ X_11 @ bot_bot_pname_o ) ) ) ) ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_736_card__Diff__singleton__if,axiom,
% 3.36/1.73 ! [X_10: x_a,A_26: x_a > $o] :
% 3.36/1.73 ( ( finite_finite_a @ A_26 )
% 3.36/1.73 => ( ( ( member_a @ X_10 @ A_26 )
% 3.36/1.73 => ( ( finite_card_a @ ( minus_minus_a_o @ A_26 @ ( insert_a @ X_10 @ bot_bot_a_o ) ) )
% 3.36/1.73 = ( minus_minus_nat @ ( finite_card_a @ A_26 ) @ one_one_nat ) ) )
% 3.36/1.73 & ( ~ ( member_a @ X_10 @ A_26 )
% 3.36/1.73 => ( ( finite_card_a @ ( minus_minus_a_o @ A_26 @ ( insert_a @ X_10 @ bot_bot_a_o ) ) )
% 3.36/1.73 = ( finite_card_a @ A_26 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_737_card__Diff__singleton__if,axiom,
% 3.36/1.73 ! [X_10: int,A_26: int > $o] :
% 3.36/1.73 ( ( finite_finite_int @ A_26 )
% 3.36/1.73 => ( ( ( member_int @ X_10 @ A_26 )
% 3.36/1.73 => ( ( finite_card_int @ ( minus_minus_int_o @ A_26 @ ( insert_int @ X_10 @ bot_bot_int_o ) ) )
% 3.36/1.73 = ( minus_minus_nat @ ( finite_card_int @ A_26 ) @ one_one_nat ) ) )
% 3.36/1.73 & ( ~ ( member_int @ X_10 @ A_26 )
% 3.36/1.73 => ( ( finite_card_int @ ( minus_minus_int_o @ A_26 @ ( insert_int @ X_10 @ bot_bot_int_o ) ) )
% 3.36/1.73 = ( finite_card_int @ A_26 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_738_card__Diff__singleton__if,axiom,
% 3.36/1.73 ! [X_10: nat,A_26: nat > $o] :
% 3.36/1.73 ( ( finite_finite_nat @ A_26 )
% 3.36/1.73 => ( ( ( member_nat @ X_10 @ A_26 )
% 3.36/1.73 => ( ( finite_card_nat @ ( minus_minus_nat_o @ A_26 @ ( insert_nat @ X_10 @ bot_bot_nat_o ) ) )
% 3.36/1.73 = ( minus_minus_nat @ ( finite_card_nat @ A_26 ) @ one_one_nat ) ) )
% 3.36/1.73 & ( ~ ( member_nat @ X_10 @ A_26 )
% 3.36/1.73 => ( ( finite_card_nat @ ( minus_minus_nat_o @ A_26 @ ( insert_nat @ X_10 @ bot_bot_nat_o ) ) )
% 3.36/1.73 = ( finite_card_nat @ A_26 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_739_card__Diff__singleton__if,axiom,
% 3.36/1.73 ! [X_10: pname,A_26: pname > $o] :
% 3.36/1.73 ( ( finite_finite_pname @ A_26 )
% 3.36/1.73 => ( ( ( member_pname @ X_10 @ A_26 )
% 3.36/1.73 => ( ( finite_card_pname @ ( minus_minus_pname_o @ A_26 @ ( insert_pname @ X_10 @ bot_bot_pname_o ) ) )
% 3.36/1.73 = ( minus_minus_nat @ ( finite_card_pname @ A_26 ) @ one_one_nat ) ) )
% 3.36/1.73 & ( ~ ( member_pname @ X_10 @ A_26 )
% 3.36/1.73 => ( ( finite_card_pname @ ( minus_minus_pname_o @ A_26 @ ( insert_pname @ X_10 @ bot_bot_pname_o ) ) )
% 3.36/1.73 = ( finite_card_pname @ A_26 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_740_card__Diff__singleton,axiom,
% 3.36/1.73 ! [X_9: x_a,A_25: x_a > $o] :
% 3.36/1.73 ( ( finite_finite_a @ A_25 )
% 3.36/1.73 => ( ( member_a @ X_9 @ A_25 )
% 3.36/1.73 => ( ( finite_card_a @ ( minus_minus_a_o @ A_25 @ ( insert_a @ X_9 @ bot_bot_a_o ) ) )
% 3.36/1.73 = ( minus_minus_nat @ ( finite_card_a @ A_25 ) @ one_one_nat ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_741_card__Diff__singleton,axiom,
% 3.36/1.73 ! [X_9: int,A_25: int > $o] :
% 3.36/1.73 ( ( finite_finite_int @ A_25 )
% 3.36/1.73 => ( ( member_int @ X_9 @ A_25 )
% 3.36/1.73 => ( ( finite_card_int @ ( minus_minus_int_o @ A_25 @ ( insert_int @ X_9 @ bot_bot_int_o ) ) )
% 3.36/1.73 = ( minus_minus_nat @ ( finite_card_int @ A_25 ) @ one_one_nat ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_742_card__Diff__singleton,axiom,
% 3.36/1.73 ! [X_9: nat,A_25: nat > $o] :
% 3.36/1.73 ( ( finite_finite_nat @ A_25 )
% 3.36/1.73 => ( ( member_nat @ X_9 @ A_25 )
% 3.36/1.73 => ( ( finite_card_nat @ ( minus_minus_nat_o @ A_25 @ ( insert_nat @ X_9 @ bot_bot_nat_o ) ) )
% 3.36/1.73 = ( minus_minus_nat @ ( finite_card_nat @ A_25 ) @ one_one_nat ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_743_card__Diff__singleton,axiom,
% 3.36/1.73 ! [X_9: pname,A_25: pname > $o] :
% 3.36/1.73 ( ( finite_finite_pname @ A_25 )
% 3.36/1.73 => ( ( member_pname @ X_9 @ A_25 )
% 3.36/1.73 => ( ( finite_card_pname @ ( minus_minus_pname_o @ A_25 @ ( insert_pname @ X_9 @ bot_bot_pname_o ) ) )
% 3.36/1.73 = ( minus_minus_nat @ ( finite_card_pname @ A_25 ) @ one_one_nat ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_744_one__reorient,axiom,
% 3.36/1.73 ! [X_8: int] :
% 3.36/1.73 ( ( one_one_int = X_8 )
% 3.36/1.73 <=> ( X_8 = one_one_int ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_745_one__reorient,axiom,
% 3.36/1.73 ! [X_8: nat] :
% 3.36/1.73 ( ( one_one_nat = X_8 )
% 3.36/1.73 <=> ( X_8 = one_one_nat ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_746_diff__Suc__1,axiom,
% 3.36/1.73 ! [N: nat] :
% 3.36/1.73 ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
% 3.36/1.73 = N ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_747_diff__Suc__eq__diff__pred,axiom,
% 3.36/1.73 ! [M: nat,N: nat] :
% 3.36/1.73 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 3.36/1.73 = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_748_folding__one_Osingleton,axiom,
% 3.36/1.73 ! [X_7: x_a,F_12: x_a > x_a > x_a,F_11: ( x_a > $o ) > x_a] :
% 3.36/1.73 ( ( finite_folding_one_a @ F_12 @ F_11 )
% 3.36/1.73 => ( ( F_11 @ ( insert_a @ X_7 @ bot_bot_a_o ) )
% 3.36/1.73 = X_7 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_749_folding__one_Osingleton,axiom,
% 3.36/1.73 ! [X_7: nat,F_12: nat > nat > nat,F_11: ( nat > $o ) > nat] :
% 3.36/1.73 ( ( finite988810631ne_nat @ F_12 @ F_11 )
% 3.36/1.73 => ( ( F_11 @ ( insert_nat @ X_7 @ bot_bot_nat_o ) )
% 3.36/1.73 = X_7 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_750_folding__one_Osingleton,axiom,
% 3.36/1.73 ! [X_7: int,F_12: int > int > int,F_11: ( int > $o ) > int] :
% 3.36/1.73 ( ( finite1626084323ne_int @ F_12 @ F_11 )
% 3.36/1.73 => ( ( F_11 @ ( insert_int @ X_7 @ bot_bot_int_o ) )
% 3.36/1.73 = X_7 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_751_card__Diff__insert,axiom,
% 3.36/1.73 ! [B_19: x_a > $o,A_24: x_a,A_23: x_a > $o] :
% 3.36/1.73 ( ( finite_finite_a @ A_23 )
% 3.36/1.73 => ( ( member_a @ A_24 @ A_23 )
% 3.36/1.73 => ( ~ ( member_a @ A_24 @ B_19 )
% 3.36/1.73 => ( ( finite_card_a @ ( minus_minus_a_o @ A_23 @ ( insert_a @ A_24 @ B_19 ) ) )
% 3.36/1.73 = ( minus_minus_nat @ ( finite_card_a @ ( minus_minus_a_o @ A_23 @ B_19 ) ) @ one_one_nat ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_752_card__Diff__insert,axiom,
% 3.36/1.73 ! [B_19: int > $o,A_24: int,A_23: int > $o] :
% 3.36/1.73 ( ( finite_finite_int @ A_23 )
% 3.36/1.73 => ( ( member_int @ A_24 @ A_23 )
% 3.36/1.73 => ( ~ ( member_int @ A_24 @ B_19 )
% 3.36/1.73 => ( ( finite_card_int @ ( minus_minus_int_o @ A_23 @ ( insert_int @ A_24 @ B_19 ) ) )
% 3.36/1.73 = ( minus_minus_nat @ ( finite_card_int @ ( minus_minus_int_o @ A_23 @ B_19 ) ) @ one_one_nat ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_753_card__Diff__insert,axiom,
% 3.36/1.73 ! [B_19: nat > $o,A_24: nat,A_23: nat > $o] :
% 3.36/1.73 ( ( finite_finite_nat @ A_23 )
% 3.36/1.73 => ( ( member_nat @ A_24 @ A_23 )
% 3.36/1.73 => ( ~ ( member_nat @ A_24 @ B_19 )
% 3.36/1.73 => ( ( finite_card_nat @ ( minus_minus_nat_o @ A_23 @ ( insert_nat @ A_24 @ B_19 ) ) )
% 3.36/1.73 = ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_nat_o @ A_23 @ B_19 ) ) @ one_one_nat ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_754_card__Diff__insert,axiom,
% 3.36/1.73 ! [B_19: pname > $o,A_24: pname,A_23: pname > $o] :
% 3.36/1.73 ( ( finite_finite_pname @ A_23 )
% 3.36/1.73 => ( ( member_pname @ A_24 @ A_23 )
% 3.36/1.73 => ( ~ ( member_pname @ A_24 @ B_19 )
% 3.36/1.73 => ( ( finite_card_pname @ ( minus_minus_pname_o @ A_23 @ ( insert_pname @ A_24 @ B_19 ) ) )
% 3.36/1.73 = ( minus_minus_nat @ ( finite_card_pname @ ( minus_minus_pname_o @ A_23 @ B_19 ) ) @ one_one_nat ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_755_folding__one_Oinsert,axiom,
% 3.36/1.73 ! [X_6: x_a,A_22: x_a > $o,F_10: x_a > x_a > x_a,F_9: ( x_a > $o ) > x_a] :
% 3.36/1.73 ( ( finite_folding_one_a @ F_10 @ F_9 )
% 3.36/1.73 => ( ( finite_finite_a @ A_22 )
% 3.36/1.73 => ( ~ ( member_a @ X_6 @ A_22 )
% 3.36/1.73 => ( ( A_22 != bot_bot_a_o )
% 3.36/1.73 => ( ( F_9 @ ( insert_a @ X_6 @ A_22 ) )
% 3.36/1.73 = ( F_10 @ X_6 @ ( F_9 @ A_22 ) ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_756_folding__one_Oinsert,axiom,
% 3.36/1.73 ! [X_6: int,A_22: int > $o,F_10: int > int > int,F_9: ( int > $o ) > int] :
% 3.36/1.73 ( ( finite1626084323ne_int @ F_10 @ F_9 )
% 3.36/1.73 => ( ( finite_finite_int @ A_22 )
% 3.36/1.73 => ( ~ ( member_int @ X_6 @ A_22 )
% 3.36/1.73 => ( ( A_22 != bot_bot_int_o )
% 3.36/1.73 => ( ( F_9 @ ( insert_int @ X_6 @ A_22 ) )
% 3.36/1.73 = ( F_10 @ X_6 @ ( F_9 @ A_22 ) ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_757_folding__one_Oinsert,axiom,
% 3.36/1.73 ! [X_6: nat,A_22: nat > $o,F_10: nat > nat > nat,F_9: ( nat > $o ) > nat] :
% 3.36/1.73 ( ( finite988810631ne_nat @ F_10 @ F_9 )
% 3.36/1.73 => ( ( finite_finite_nat @ A_22 )
% 3.36/1.73 => ( ~ ( member_nat @ X_6 @ A_22 )
% 3.36/1.73 => ( ( A_22 != bot_bot_nat_o )
% 3.36/1.73 => ( ( F_9 @ ( insert_nat @ X_6 @ A_22 ) )
% 3.36/1.73 = ( F_10 @ X_6 @ ( F_9 @ A_22 ) ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_758_folding__one_Oinsert,axiom,
% 3.36/1.73 ! [X_6: pname,A_22: pname > $o,F_10: pname > pname > pname,F_9: ( pname > $o ) > pname] :
% 3.36/1.73 ( ( finite1282449217_pname @ F_10 @ F_9 )
% 3.36/1.73 => ( ( finite_finite_pname @ A_22 )
% 3.36/1.73 => ( ~ ( member_pname @ X_6 @ A_22 )
% 3.36/1.73 => ( ( A_22 != bot_bot_pname_o )
% 3.36/1.73 => ( ( F_9 @ ( insert_pname @ X_6 @ A_22 ) )
% 3.36/1.73 = ( F_10 @ X_6 @ ( F_9 @ A_22 ) ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_759_folding__one_Oclosed,axiom,
% 3.36/1.73 ! [A_21: x_a > $o,F_8: x_a > x_a > x_a,F_7: ( x_a > $o ) > x_a] :
% 3.36/1.73 ( ( finite_folding_one_a @ F_8 @ F_7 )
% 3.36/1.73 => ( ( finite_finite_a @ A_21 )
% 3.36/1.73 => ( ( A_21 != bot_bot_a_o )
% 3.36/1.73 => ( ! [X_1: x_a,Y_1: x_a] : ( member_a @ ( F_8 @ X_1 @ Y_1 ) @ ( insert_a @ X_1 @ ( insert_a @ Y_1 @ bot_bot_a_o ) ) )
% 3.36/1.73 => ( member_a @ ( F_7 @ A_21 ) @ A_21 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_760_folding__one_Oclosed,axiom,
% 3.36/1.73 ! [A_21: int > $o,F_8: int > int > int,F_7: ( int > $o ) > int] :
% 3.36/1.73 ( ( finite1626084323ne_int @ F_8 @ F_7 )
% 3.36/1.73 => ( ( finite_finite_int @ A_21 )
% 3.36/1.73 => ( ( A_21 != bot_bot_int_o )
% 3.36/1.73 => ( ! [X_1: int,Y_1: int] : ( member_int @ ( F_8 @ X_1 @ Y_1 ) @ ( insert_int @ X_1 @ ( insert_int @ Y_1 @ bot_bot_int_o ) ) )
% 3.36/1.73 => ( member_int @ ( F_7 @ A_21 ) @ A_21 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_761_folding__one_Oclosed,axiom,
% 3.36/1.73 ! [A_21: nat > $o,F_8: nat > nat > nat,F_7: ( nat > $o ) > nat] :
% 3.36/1.73 ( ( finite988810631ne_nat @ F_8 @ F_7 )
% 3.36/1.73 => ( ( finite_finite_nat @ A_21 )
% 3.36/1.73 => ( ( A_21 != bot_bot_nat_o )
% 3.36/1.73 => ( ! [X_1: nat,Y_1: nat] : ( member_nat @ ( F_8 @ X_1 @ Y_1 ) @ ( insert_nat @ X_1 @ ( insert_nat @ Y_1 @ bot_bot_nat_o ) ) )
% 3.36/1.73 => ( member_nat @ ( F_7 @ A_21 ) @ A_21 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_762_folding__one_Oclosed,axiom,
% 3.36/1.73 ! [A_21: pname > $o,F_8: pname > pname > pname,F_7: ( pname > $o ) > pname] :
% 3.36/1.73 ( ( finite1282449217_pname @ F_8 @ F_7 )
% 3.36/1.73 => ( ( finite_finite_pname @ A_21 )
% 3.36/1.73 => ( ( A_21 != bot_bot_pname_o )
% 3.36/1.73 => ( ! [X_1: pname,Y_1: pname] : ( member_pname @ ( F_8 @ X_1 @ Y_1 ) @ ( insert_pname @ X_1 @ ( insert_pname @ Y_1 @ bot_bot_pname_o ) ) )
% 3.36/1.73 => ( member_pname @ ( F_7 @ A_21 ) @ A_21 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_763_card_Oremove,axiom,
% 3.36/1.73 ! [X_5: x_a,A_20: x_a > $o] :
% 3.36/1.73 ( ( finite_finite_a @ A_20 )
% 3.36/1.73 => ( ( member_a @ X_5 @ A_20 )
% 3.36/1.73 => ( ( finite_card_a @ A_20 )
% 3.36/1.73 = ( plus_plus_nat @ one_one_nat @ ( finite_card_a @ ( minus_minus_a_o @ A_20 @ ( insert_a @ X_5 @ bot_bot_a_o ) ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_764_card_Oremove,axiom,
% 3.36/1.73 ! [X_5: int,A_20: int > $o] :
% 3.36/1.73 ( ( finite_finite_int @ A_20 )
% 3.36/1.73 => ( ( member_int @ X_5 @ A_20 )
% 3.36/1.73 => ( ( finite_card_int @ A_20 )
% 3.36/1.73 = ( plus_plus_nat @ one_one_nat @ ( finite_card_int @ ( minus_minus_int_o @ A_20 @ ( insert_int @ X_5 @ bot_bot_int_o ) ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_765_card_Oremove,axiom,
% 3.36/1.73 ! [X_5: nat,A_20: nat > $o] :
% 3.36/1.73 ( ( finite_finite_nat @ A_20 )
% 3.36/1.73 => ( ( member_nat @ X_5 @ A_20 )
% 3.36/1.73 => ( ( finite_card_nat @ A_20 )
% 3.36/1.73 = ( plus_plus_nat @ one_one_nat @ ( finite_card_nat @ ( minus_minus_nat_o @ A_20 @ ( insert_nat @ X_5 @ bot_bot_nat_o ) ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_766_card_Oremove,axiom,
% 3.36/1.73 ! [X_5: pname,A_20: pname > $o] :
% 3.36/1.73 ( ( finite_finite_pname @ A_20 )
% 3.36/1.73 => ( ( member_pname @ X_5 @ A_20 )
% 3.36/1.73 => ( ( finite_card_pname @ A_20 )
% 3.36/1.73 = ( plus_plus_nat @ one_one_nat @ ( finite_card_pname @ ( minus_minus_pname_o @ A_20 @ ( insert_pname @ X_5 @ bot_bot_pname_o ) ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_767_card_Oinsert__remove,axiom,
% 3.36/1.73 ! [X_4: x_a,A_19: x_a > $o] :
% 3.36/1.73 ( ( finite_finite_a @ A_19 )
% 3.36/1.73 => ( ( finite_card_a @ ( insert_a @ X_4 @ A_19 ) )
% 3.36/1.73 = ( plus_plus_nat @ one_one_nat @ ( finite_card_a @ ( minus_minus_a_o @ A_19 @ ( insert_a @ X_4 @ bot_bot_a_o ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_768_card_Oinsert__remove,axiom,
% 3.36/1.73 ! [X_4: int,A_19: int > $o] :
% 3.36/1.73 ( ( finite_finite_int @ A_19 )
% 3.36/1.73 => ( ( finite_card_int @ ( insert_int @ X_4 @ A_19 ) )
% 3.36/1.73 = ( plus_plus_nat @ one_one_nat @ ( finite_card_int @ ( minus_minus_int_o @ A_19 @ ( insert_int @ X_4 @ bot_bot_int_o ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_769_card_Oinsert__remove,axiom,
% 3.36/1.73 ! [X_4: nat,A_19: nat > $o] :
% 3.36/1.73 ( ( finite_finite_nat @ A_19 )
% 3.36/1.73 => ( ( finite_card_nat @ ( insert_nat @ X_4 @ A_19 ) )
% 3.36/1.73 = ( plus_plus_nat @ one_one_nat @ ( finite_card_nat @ ( minus_minus_nat_o @ A_19 @ ( insert_nat @ X_4 @ bot_bot_nat_o ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_770_card_Oinsert__remove,axiom,
% 3.36/1.73 ! [X_4: pname,A_19: pname > $o] :
% 3.36/1.73 ( ( finite_finite_pname @ A_19 )
% 3.36/1.73 => ( ( finite_card_pname @ ( insert_pname @ X_4 @ A_19 ) )
% 3.36/1.73 = ( plus_plus_nat @ one_one_nat @ ( finite_card_pname @ ( minus_minus_pname_o @ A_19 @ ( insert_pname @ X_4 @ bot_bot_pname_o ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_771_folding__one__idem_Osubset__idem,axiom,
% 3.36/1.73 ! [B_18: int > $o,A_18: int > $o,F_6: int > int > int,F_5: ( int > $o ) > int] :
% 3.36/1.73 ( ( finite1432773856em_int @ F_6 @ F_5 )
% 3.36/1.73 => ( ( finite_finite_int @ A_18 )
% 3.36/1.73 => ( ( B_18 != bot_bot_int_o )
% 3.36/1.73 => ( ( ord_less_eq_int_o @ B_18 @ A_18 )
% 3.36/1.73 => ( ( F_6 @ ( F_5 @ B_18 ) @ ( F_5 @ A_18 ) )
% 3.36/1.73 = ( F_5 @ A_18 ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_772_folding__one__idem_Osubset__idem,axiom,
% 3.36/1.73 ! [B_18: nat > $o,A_18: nat > $o,F_6: nat > nat > nat,F_5: ( nat > $o ) > nat] :
% 3.36/1.73 ( ( finite795500164em_nat @ F_6 @ F_5 )
% 3.36/1.73 => ( ( finite_finite_nat @ A_18 )
% 3.36/1.73 => ( ( B_18 != bot_bot_nat_o )
% 3.36/1.73 => ( ( ord_less_eq_nat_o @ B_18 @ A_18 )
% 3.36/1.73 => ( ( F_6 @ ( F_5 @ B_18 ) @ ( F_5 @ A_18 ) )
% 3.36/1.73 = ( F_5 @ A_18 ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_773_folding__one__idem_Osubset__idem,axiom,
% 3.36/1.73 ! [B_18: pname > $o,A_18: pname > $o,F_6: pname > pname > pname,F_5: ( pname > $o ) > pname] :
% 3.36/1.73 ( ( finite89670078_pname @ F_6 @ F_5 )
% 3.36/1.73 => ( ( finite_finite_pname @ A_18 )
% 3.36/1.73 => ( ( B_18 != bot_bot_pname_o )
% 3.36/1.73 => ( ( ord_less_eq_pname_o @ B_18 @ A_18 )
% 3.36/1.73 => ( ( F_6 @ ( F_5 @ B_18 ) @ ( F_5 @ A_18 ) )
% 3.36/1.73 = ( F_5 @ A_18 ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_774_folding__one__idem_Osubset__idem,axiom,
% 3.36/1.73 ! [B_18: x_a > $o,A_18: x_a > $o,F_6: x_a > x_a > x_a,F_5: ( x_a > $o ) > x_a] :
% 3.36/1.73 ( ( finite1819937229idem_a @ F_6 @ F_5 )
% 3.36/1.73 => ( ( finite_finite_a @ A_18 )
% 3.36/1.73 => ( ( B_18 != bot_bot_a_o )
% 3.36/1.73 => ( ( ord_less_eq_a_o @ B_18 @ A_18 )
% 3.36/1.73 => ( ( F_6 @ ( F_5 @ B_18 ) @ ( F_5 @ A_18 ) )
% 3.36/1.73 = ( F_5 @ A_18 ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_775_folding__one__idem_Oinsert__idem,axiom,
% 3.36/1.73 ! [X_3: x_a,A_17: x_a > $o,F_4: x_a > x_a > x_a,F_3: ( x_a > $o ) > x_a] :
% 3.36/1.73 ( ( finite1819937229idem_a @ F_4 @ F_3 )
% 3.36/1.73 => ( ( finite_finite_a @ A_17 )
% 3.36/1.73 => ( ( A_17 != bot_bot_a_o )
% 3.36/1.73 => ( ( F_3 @ ( insert_a @ X_3 @ A_17 ) )
% 3.36/1.73 = ( F_4 @ X_3 @ ( F_3 @ A_17 ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_776_folding__one__idem_Oinsert__idem,axiom,
% 3.36/1.73 ! [X_3: int,A_17: int > $o,F_4: int > int > int,F_3: ( int > $o ) > int] :
% 3.36/1.73 ( ( finite1432773856em_int @ F_4 @ F_3 )
% 3.36/1.73 => ( ( finite_finite_int @ A_17 )
% 3.36/1.73 => ( ( A_17 != bot_bot_int_o )
% 3.36/1.73 => ( ( F_3 @ ( insert_int @ X_3 @ A_17 ) )
% 3.36/1.73 = ( F_4 @ X_3 @ ( F_3 @ A_17 ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_777_folding__one__idem_Oinsert__idem,axiom,
% 3.36/1.73 ! [X_3: nat,A_17: nat > $o,F_4: nat > nat > nat,F_3: ( nat > $o ) > nat] :
% 3.36/1.73 ( ( finite795500164em_nat @ F_4 @ F_3 )
% 3.36/1.73 => ( ( finite_finite_nat @ A_17 )
% 3.36/1.73 => ( ( A_17 != bot_bot_nat_o )
% 3.36/1.73 => ( ( F_3 @ ( insert_nat @ X_3 @ A_17 ) )
% 3.36/1.73 = ( F_4 @ X_3 @ ( F_3 @ A_17 ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_778_folding__one__idem_Oinsert__idem,axiom,
% 3.36/1.73 ! [X_3: pname,A_17: pname > $o,F_4: pname > pname > pname,F_3: ( pname > $o ) > pname] :
% 3.36/1.73 ( ( finite89670078_pname @ F_4 @ F_3 )
% 3.36/1.73 => ( ( finite_finite_pname @ A_17 )
% 3.36/1.73 => ( ( A_17 != bot_bot_pname_o )
% 3.36/1.73 => ( ( F_3 @ ( insert_pname @ X_3 @ A_17 ) )
% 3.36/1.73 = ( F_4 @ X_3 @ ( F_3 @ A_17 ) ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_779_finite__ne__induct,axiom,
% 3.36/1.73 ! [P_2: ( x_a > $o ) > $o,F_1: x_a > $o] :
% 3.36/1.73 ( ( finite_finite_a @ F_1 )
% 3.36/1.73 => ( ( F_1 != bot_bot_a_o )
% 3.36/1.73 => ( ! [X_1: x_a] : ( P_2 @ ( insert_a @ X_1 @ bot_bot_a_o ) )
% 3.36/1.73 => ( ! [X_1: x_a,F_2: x_a > $o] :
% 3.36/1.73 ( ( finite_finite_a @ F_2 )
% 3.36/1.73 => ( ( F_2 != bot_bot_a_o )
% 3.36/1.73 => ( ~ ( member_a @ X_1 @ F_2 )
% 3.36/1.73 => ( ( P_2 @ F_2 )
% 3.36/1.73 => ( P_2 @ ( insert_a @ X_1 @ F_2 ) ) ) ) ) )
% 3.36/1.73 => ( P_2 @ F_1 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_780_finite__ne__induct,axiom,
% 3.36/1.73 ! [P_2: ( int > $o ) > $o,F_1: int > $o] :
% 3.36/1.73 ( ( finite_finite_int @ F_1 )
% 3.36/1.73 => ( ( F_1 != bot_bot_int_o )
% 3.36/1.73 => ( ! [X_1: int] : ( P_2 @ ( insert_int @ X_1 @ bot_bot_int_o ) )
% 3.36/1.73 => ( ! [X_1: int,F_2: int > $o] :
% 3.36/1.73 ( ( finite_finite_int @ F_2 )
% 3.36/1.73 => ( ( F_2 != bot_bot_int_o )
% 3.36/1.73 => ( ~ ( member_int @ X_1 @ F_2 )
% 3.36/1.73 => ( ( P_2 @ F_2 )
% 3.36/1.73 => ( P_2 @ ( insert_int @ X_1 @ F_2 ) ) ) ) ) )
% 3.36/1.73 => ( P_2 @ F_1 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_781_finite__ne__induct,axiom,
% 3.36/1.73 ! [P_2: ( nat > $o ) > $o,F_1: nat > $o] :
% 3.36/1.73 ( ( finite_finite_nat @ F_1 )
% 3.36/1.73 => ( ( F_1 != bot_bot_nat_o )
% 3.36/1.73 => ( ! [X_1: nat] : ( P_2 @ ( insert_nat @ X_1 @ bot_bot_nat_o ) )
% 3.36/1.73 => ( ! [X_1: nat,F_2: nat > $o] :
% 3.36/1.73 ( ( finite_finite_nat @ F_2 )
% 3.36/1.73 => ( ( F_2 != bot_bot_nat_o )
% 3.36/1.73 => ( ~ ( member_nat @ X_1 @ F_2 )
% 3.36/1.73 => ( ( P_2 @ F_2 )
% 3.36/1.73 => ( P_2 @ ( insert_nat @ X_1 @ F_2 ) ) ) ) ) )
% 3.36/1.73 => ( P_2 @ F_1 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_782_finite__ne__induct,axiom,
% 3.36/1.73 ! [P_2: ( pname > $o ) > $o,F_1: pname > $o] :
% 3.36/1.73 ( ( finite_finite_pname @ F_1 )
% 3.36/1.73 => ( ( F_1 != bot_bot_pname_o )
% 3.36/1.73 => ( ! [X_1: pname] : ( P_2 @ ( insert_pname @ X_1 @ bot_bot_pname_o ) )
% 3.36/1.73 => ( ! [X_1: pname,F_2: pname > $o] :
% 3.36/1.73 ( ( finite_finite_pname @ F_2 )
% 3.36/1.73 => ( ( F_2 != bot_bot_pname_o )
% 3.36/1.73 => ( ~ ( member_pname @ X_1 @ F_2 )
% 3.36/1.73 => ( ( P_2 @ F_2 )
% 3.36/1.73 => ( P_2 @ ( insert_pname @ X_1 @ F_2 ) ) ) ) ) )
% 3.36/1.73 => ( P_2 @ F_1 ) ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_783_the__elem__def,axiom,
% 3.36/1.73 ! [X_2: x_a > $o] :
% 3.36/1.73 ( ( the_elem_a @ X_2 )
% 3.36/1.73 = ( the_a
% 3.36/1.73 @ ^ [X_1: x_a] :
% 3.36/1.73 ( X_2
% 3.36/1.73 = ( insert_a @ X_1 @ bot_bot_a_o ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_784_the__elem__def,axiom,
% 3.36/1.73 ! [X_2: nat > $o] :
% 3.36/1.73 ( ( the_elem_nat @ X_2 )
% 3.36/1.73 = ( the_nat
% 3.36/1.73 @ ^ [X_1: nat] :
% 3.36/1.73 ( X_2
% 3.36/1.73 = ( insert_nat @ X_1 @ bot_bot_nat_o ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_785_the__elem__def,axiom,
% 3.36/1.73 ! [X_2: int > $o] :
% 3.36/1.73 ( ( the_elem_int @ X_2 )
% 3.36/1.73 = ( the_int
% 3.36/1.73 @ ^ [X_1: int] :
% 3.36/1.73 ( X_2
% 3.36/1.73 = ( insert_int @ X_1 @ bot_bot_int_o ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_786_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 3.36/1.73 ! [A_16: int,B_17: int,C_13: int] :
% 3.36/1.73 ( ( plus_plus_int @ ( plus_plus_int @ A_16 @ B_17 ) @ C_13 )
% 3.36/1.73 = ( plus_plus_int @ A_16 @ ( plus_plus_int @ B_17 @ C_13 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_787_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 3.36/1.73 ! [A_16: nat,B_17: nat,C_13: nat] :
% 3.36/1.73 ( ( plus_plus_nat @ ( plus_plus_nat @ A_16 @ B_17 ) @ C_13 )
% 3.36/1.73 = ( plus_plus_nat @ A_16 @ ( plus_plus_nat @ B_17 @ C_13 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_788_add__left__cancel,axiom,
% 3.36/1.73 ! [A_15: int,B_16: int,C_12: int] :
% 3.36/1.73 ( ( ( plus_plus_int @ A_15 @ B_16 )
% 3.36/1.73 = ( plus_plus_int @ A_15 @ C_12 ) )
% 3.36/1.73 <=> ( B_16 = C_12 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_789_add__left__cancel,axiom,
% 3.36/1.73 ! [A_15: nat,B_16: nat,C_12: nat] :
% 3.36/1.73 ( ( ( plus_plus_nat @ A_15 @ B_16 )
% 3.36/1.73 = ( plus_plus_nat @ A_15 @ C_12 ) )
% 3.36/1.73 <=> ( B_16 = C_12 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_790_add__right__cancel,axiom,
% 3.36/1.73 ! [B_15: int,A_14: int,C_11: int] :
% 3.36/1.73 ( ( ( plus_plus_int @ B_15 @ A_14 )
% 3.36/1.73 = ( plus_plus_int @ C_11 @ A_14 ) )
% 3.36/1.73 <=> ( B_15 = C_11 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_791_add__right__cancel,axiom,
% 3.36/1.73 ! [B_15: nat,A_14: nat,C_11: nat] :
% 3.36/1.73 ( ( ( plus_plus_nat @ B_15 @ A_14 )
% 3.36/1.73 = ( plus_plus_nat @ C_11 @ A_14 ) )
% 3.36/1.73 <=> ( B_15 = C_11 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_792_add__left__imp__eq,axiom,
% 3.36/1.73 ! [A_13: int,B_14: int,C_10: int] :
% 3.36/1.73 ( ( ( plus_plus_int @ A_13 @ B_14 )
% 3.36/1.73 = ( plus_plus_int @ A_13 @ C_10 ) )
% 3.36/1.73 => ( B_14 = C_10 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_793_add__left__imp__eq,axiom,
% 3.36/1.73 ! [A_13: nat,B_14: nat,C_10: nat] :
% 3.36/1.73 ( ( ( plus_plus_nat @ A_13 @ B_14 )
% 3.36/1.73 = ( plus_plus_nat @ A_13 @ C_10 ) )
% 3.36/1.73 => ( B_14 = C_10 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_794_add__imp__eq,axiom,
% 3.36/1.73 ! [A_12: int,B_13: int,C_9: int] :
% 3.36/1.73 ( ( ( plus_plus_int @ A_12 @ B_13 )
% 3.36/1.73 = ( plus_plus_int @ A_12 @ C_9 ) )
% 3.36/1.73 => ( B_13 = C_9 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_795_add__imp__eq,axiom,
% 3.36/1.73 ! [A_12: nat,B_13: nat,C_9: nat] :
% 3.36/1.73 ( ( ( plus_plus_nat @ A_12 @ B_13 )
% 3.36/1.73 = ( plus_plus_nat @ A_12 @ C_9 ) )
% 3.36/1.73 => ( B_13 = C_9 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_796_add__right__imp__eq,axiom,
% 3.36/1.73 ! [B_12: int,A_11: int,C_8: int] :
% 3.36/1.73 ( ( ( plus_plus_int @ B_12 @ A_11 )
% 3.36/1.73 = ( plus_plus_int @ C_8 @ A_11 ) )
% 3.36/1.73 => ( B_12 = C_8 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_797_add__right__imp__eq,axiom,
% 3.36/1.73 ! [B_12: nat,A_11: nat,C_8: nat] :
% 3.36/1.73 ( ( ( plus_plus_nat @ B_12 @ A_11 )
% 3.36/1.73 = ( plus_plus_nat @ C_8 @ A_11 ) )
% 3.36/1.73 => ( B_12 = C_8 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_798_nat__add__commute,axiom,
% 3.36/1.73 ! [M: nat,N: nat] :
% 3.36/1.73 ( ( plus_plus_nat @ M @ N )
% 3.36/1.73 = ( plus_plus_nat @ N @ M ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_799_nat__add__left__commute,axiom,
% 3.36/1.73 ! [X: nat,Y: nat,Z: nat] :
% 3.36/1.73 ( ( plus_plus_nat @ X @ ( plus_plus_nat @ Y @ Z ) )
% 3.36/1.73 = ( plus_plus_nat @ Y @ ( plus_plus_nat @ X @ Z ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_800_nat__add__assoc,axiom,
% 3.36/1.73 ! [M: nat,N: nat,K: nat] :
% 3.36/1.73 ( ( plus_plus_nat @ ( plus_plus_nat @ M @ N ) @ K )
% 3.36/1.73 = ( plus_plus_nat @ M @ ( plus_plus_nat @ N @ K ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_801_nat__add__left__cancel,axiom,
% 3.36/1.73 ! [K: nat,M: nat,N: nat] :
% 3.36/1.73 ( ( ( plus_plus_nat @ K @ M )
% 3.36/1.73 = ( plus_plus_nat @ K @ N ) )
% 3.36/1.73 <=> ( M = N ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_802_nat__add__right__cancel,axiom,
% 3.36/1.73 ! [M: nat,K: nat,N: nat] :
% 3.36/1.73 ( ( ( plus_plus_nat @ M @ K )
% 3.36/1.73 = ( plus_plus_nat @ N @ K ) )
% 3.36/1.73 <=> ( M = N ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_803_add__le__cancel__right,axiom,
% 3.36/1.73 ! [A_10: int,C_7: int,B_11: int] :
% 3.36/1.73 ( ( ord_less_eq_int @ ( plus_plus_int @ A_10 @ C_7 ) @ ( plus_plus_int @ B_11 @ C_7 ) )
% 3.36/1.73 <=> ( ord_less_eq_int @ A_10 @ B_11 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_804_add__le__cancel__right,axiom,
% 3.36/1.73 ! [A_10: nat,C_7: nat,B_11: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A_10 @ C_7 ) @ ( plus_plus_nat @ B_11 @ C_7 ) )
% 3.36/1.73 <=> ( ord_less_eq_nat @ A_10 @ B_11 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_805_add__le__cancel__left,axiom,
% 3.36/1.73 ! [C_6: int,A_9: int,B_10: int] :
% 3.36/1.73 ( ( ord_less_eq_int @ ( plus_plus_int @ C_6 @ A_9 ) @ ( plus_plus_int @ C_6 @ B_10 ) )
% 3.36/1.73 <=> ( ord_less_eq_int @ A_9 @ B_10 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_806_add__le__cancel__left,axiom,
% 3.36/1.73 ! [C_6: nat,A_9: nat,B_10: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C_6 @ A_9 ) @ ( plus_plus_nat @ C_6 @ B_10 ) )
% 3.36/1.73 <=> ( ord_less_eq_nat @ A_9 @ B_10 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_807_add__right__mono,axiom,
% 3.36/1.73 ! [C_5: int,A_8: int,B_9: int] :
% 3.36/1.73 ( ( ord_less_eq_int @ A_8 @ B_9 )
% 3.36/1.73 => ( ord_less_eq_int @ ( plus_plus_int @ A_8 @ C_5 ) @ ( plus_plus_int @ B_9 @ C_5 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_808_add__right__mono,axiom,
% 3.36/1.73 ! [C_5: nat,A_8: nat,B_9: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ A_8 @ B_9 )
% 3.36/1.73 => ( ord_less_eq_nat @ ( plus_plus_nat @ A_8 @ C_5 ) @ ( plus_plus_nat @ B_9 @ C_5 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_809_add__left__mono,axiom,
% 3.36/1.73 ! [C_4: int,A_7: int,B_8: int] :
% 3.36/1.73 ( ( ord_less_eq_int @ A_7 @ B_8 )
% 3.36/1.73 => ( ord_less_eq_int @ ( plus_plus_int @ C_4 @ A_7 ) @ ( plus_plus_int @ C_4 @ B_8 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_810_add__left__mono,axiom,
% 3.36/1.73 ! [C_4: nat,A_7: nat,B_8: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ A_7 @ B_8 )
% 3.36/1.73 => ( ord_less_eq_nat @ ( plus_plus_nat @ C_4 @ A_7 ) @ ( plus_plus_nat @ C_4 @ B_8 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_811_add__mono,axiom,
% 3.36/1.73 ! [C_3: int,D_2: int,A_6: int,B_7: int] :
% 3.36/1.73 ( ( ord_less_eq_int @ A_6 @ B_7 )
% 3.36/1.73 => ( ( ord_less_eq_int @ C_3 @ D_2 )
% 3.36/1.73 => ( ord_less_eq_int @ ( plus_plus_int @ A_6 @ C_3 ) @ ( plus_plus_int @ B_7 @ D_2 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_812_add__mono,axiom,
% 3.36/1.73 ! [C_3: nat,D_2: nat,A_6: nat,B_7: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ A_6 @ B_7 )
% 3.36/1.73 => ( ( ord_less_eq_nat @ C_3 @ D_2 )
% 3.36/1.73 => ( ord_less_eq_nat @ ( plus_plus_nat @ A_6 @ C_3 ) @ ( plus_plus_nat @ B_7 @ D_2 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_813_add__le__imp__le__right,axiom,
% 3.36/1.73 ! [A_5: int,C_2: int,B_6: int] :
% 3.36/1.73 ( ( ord_less_eq_int @ ( plus_plus_int @ A_5 @ C_2 ) @ ( plus_plus_int @ B_6 @ C_2 ) )
% 3.36/1.73 => ( ord_less_eq_int @ A_5 @ B_6 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_814_add__le__imp__le__right,axiom,
% 3.36/1.73 ! [A_5: nat,C_2: nat,B_6: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A_5 @ C_2 ) @ ( plus_plus_nat @ B_6 @ C_2 ) )
% 3.36/1.73 => ( ord_less_eq_nat @ A_5 @ B_6 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_815_add__le__imp__le__left,axiom,
% 3.36/1.73 ! [C_1: int,A_4: int,B_5: int] :
% 3.36/1.73 ( ( ord_less_eq_int @ ( plus_plus_int @ C_1 @ A_4 ) @ ( plus_plus_int @ C_1 @ B_5 ) )
% 3.36/1.73 => ( ord_less_eq_int @ A_4 @ B_5 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_816_add__le__imp__le__left,axiom,
% 3.36/1.73 ! [C_1: nat,A_4: nat,B_5: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C_1 @ A_4 ) @ ( plus_plus_nat @ C_1 @ B_5 ) )
% 3.36/1.73 => ( ord_less_eq_nat @ A_4 @ B_5 ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_817_diff__add__cancel,axiom,
% 3.36/1.73 ! [A_3: int,B_4: int] :
% 3.36/1.73 ( ( plus_plus_int @ ( minus_minus_int @ A_3 @ B_4 ) @ B_4 )
% 3.36/1.73 = A_3 ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_818_add__diff__cancel,axiom,
% 3.36/1.73 ! [A_2: int,B_3: int] :
% 3.36/1.73 ( ( minus_minus_int @ ( plus_plus_int @ A_2 @ B_3 ) @ B_3 )
% 3.36/1.73 = A_2 ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_819_add__Suc__right,axiom,
% 3.36/1.73 ! [M: nat,N: nat] :
% 3.36/1.73 ( ( plus_plus_nat @ M @ ( suc @ N ) )
% 3.36/1.73 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_820_add__Suc,axiom,
% 3.36/1.73 ! [M: nat,N: nat] :
% 3.36/1.73 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 3.36/1.73 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_821_add__Suc__shift,axiom,
% 3.36/1.73 ! [M: nat,N: nat] :
% 3.36/1.73 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 3.36/1.73 = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_822_le__add2,axiom,
% 3.36/1.73 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_823_le__add1,axiom,
% 3.36/1.73 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_824_le__iff__add,axiom,
% 3.36/1.73 ! [M: nat,N: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ M @ N )
% 3.36/1.73 <=> ? [K_1: nat] :
% 3.36/1.73 ( N
% 3.36/1.73 = ( plus_plus_nat @ M @ K_1 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_825_nat__add__left__cancel__le,axiom,
% 3.36/1.73 ! [K: nat,M: nat,N: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 3.36/1.73 <=> ( ord_less_eq_nat @ M @ N ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_826_trans__le__add1,axiom,
% 3.36/1.73 ! [M: nat,I_1: nat,J: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ I_1 @ J )
% 3.36/1.73 => ( ord_less_eq_nat @ I_1 @ ( plus_plus_nat @ J @ M ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_827_trans__le__add2,axiom,
% 3.36/1.73 ! [M: nat,I_1: nat,J: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ I_1 @ J )
% 3.36/1.73 => ( ord_less_eq_nat @ I_1 @ ( plus_plus_nat @ M @ J ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_828_add__le__mono1,axiom,
% 3.36/1.73 ! [K: nat,I_1: nat,J: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ I_1 @ J )
% 3.36/1.73 => ( ord_less_eq_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_829_add__le__mono,axiom,
% 3.36/1.73 ! [K: nat,L: nat,I_1: nat,J: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ I_1 @ J )
% 3.36/1.73 => ( ( ord_less_eq_nat @ K @ L )
% 3.36/1.73 => ( ord_less_eq_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_830_add__leD2,axiom,
% 3.36/1.73 ! [M: nat,K: nat,N: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 3.36/1.73 => ( ord_less_eq_nat @ K @ N ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_831_add__leD1,axiom,
% 3.36/1.73 ! [M: nat,K: nat,N: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 3.36/1.73 => ( ord_less_eq_nat @ M @ N ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_832_add__leE,axiom,
% 3.36/1.73 ! [M: nat,K: nat,N: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 3.36/1.73 => ~ ( ( ord_less_eq_nat @ M @ N )
% 3.36/1.73 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_833_diff__add__inverse2,axiom,
% 3.36/1.73 ! [M: nat,N: nat] :
% 3.36/1.73 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 3.36/1.73 = M ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_834_diff__add__inverse,axiom,
% 3.36/1.73 ! [N: nat,M: nat] :
% 3.36/1.73 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 3.36/1.73 = M ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_835_diff__diff__left,axiom,
% 3.36/1.73 ! [I_1: nat,J: nat,K: nat] :
% 3.36/1.73 ( ( minus_minus_nat @ ( minus_minus_nat @ I_1 @ J ) @ K )
% 3.36/1.73 = ( minus_minus_nat @ I_1 @ ( plus_plus_nat @ J @ K ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_836_diff__cancel,axiom,
% 3.36/1.73 ! [K: nat,M: nat,N: nat] :
% 3.36/1.73 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 3.36/1.73 = ( minus_minus_nat @ M @ N ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_837_diff__cancel2,axiom,
% 3.36/1.73 ! [M: nat,K: nat,N: nat] :
% 3.36/1.73 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 3.36/1.73 = ( minus_minus_nat @ M @ N ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_838_diff__diff__right,axiom,
% 3.36/1.73 ! [I_1: nat,K: nat,J: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ K @ J )
% 3.36/1.73 => ( ( minus_minus_nat @ I_1 @ ( minus_minus_nat @ J @ K ) )
% 3.36/1.73 = ( minus_minus_nat @ ( plus_plus_nat @ I_1 @ K ) @ J ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_839_le__diff__conv,axiom,
% 3.36/1.73 ! [J: nat,K: nat,I_1: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I_1 )
% 3.36/1.73 <=> ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I_1 @ K ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_840_le__add__diff,axiom,
% 3.36/1.73 ! [M: nat,K: nat,N: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ K @ N )
% 3.36/1.73 => ( ord_less_eq_nat @ M @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_841_le__add__diff__inverse,axiom,
% 3.36/1.73 ! [N: nat,M: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ N @ M )
% 3.36/1.73 => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 3.36/1.73 = M ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_842_add__diff__assoc,axiom,
% 3.36/1.73 ! [I_1: nat,K: nat,J: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ K @ J )
% 3.36/1.73 => ( ( plus_plus_nat @ I_1 @ ( minus_minus_nat @ J @ K ) )
% 3.36/1.73 = ( minus_minus_nat @ ( plus_plus_nat @ I_1 @ J ) @ K ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_843_le__diff__conv2,axiom,
% 3.36/1.73 ! [I_1: nat,K: nat,J: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ K @ J )
% 3.36/1.73 => ( ( ord_less_eq_nat @ I_1 @ ( minus_minus_nat @ J @ K ) )
% 3.36/1.73 <=> ( ord_less_eq_nat @ ( plus_plus_nat @ I_1 @ K ) @ J ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_844_le__add__diff__inverse2,axiom,
% 3.36/1.73 ! [N: nat,M: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ N @ M )
% 3.36/1.73 => ( ( plus_plus_nat @ ( minus_minus_nat @ M @ N ) @ N )
% 3.36/1.73 = M ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_845_le__imp__diff__is__add,axiom,
% 3.36/1.73 ! [K: nat,I_1: nat,J: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ I_1 @ J )
% 3.36/1.73 => ( ( ( minus_minus_nat @ J @ I_1 )
% 3.36/1.73 = K )
% 3.36/1.73 <=> ( J
% 3.36/1.73 = ( plus_plus_nat @ K @ I_1 ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_846_diff__add__assoc,axiom,
% 3.36/1.73 ! [I_1: nat,K: nat,J: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ K @ J )
% 3.36/1.73 => ( ( minus_minus_nat @ ( plus_plus_nat @ I_1 @ J ) @ K )
% 3.36/1.73 = ( plus_plus_nat @ I_1 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_847_add__diff__assoc2,axiom,
% 3.36/1.73 ! [I_1: nat,K: nat,J: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ K @ J )
% 3.36/1.73 => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I_1 )
% 3.36/1.73 = ( minus_minus_nat @ ( plus_plus_nat @ J @ I_1 ) @ K ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_848_diff__add__assoc2,axiom,
% 3.36/1.73 ! [I_1: nat,K: nat,J: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ K @ J )
% 3.36/1.73 => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I_1 ) @ K )
% 3.36/1.73 = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I_1 ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_849_Suc__eq__plus1__left,axiom,
% 3.36/1.73 ! [N: nat] :
% 3.36/1.73 ( ( suc @ N )
% 3.36/1.73 = ( plus_plus_nat @ one_one_nat @ N ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_850_Suc__eq__plus1,axiom,
% 3.36/1.73 ! [N: nat] :
% 3.36/1.73 ( ( suc @ N )
% 3.36/1.73 = ( plus_plus_nat @ N @ one_one_nat ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_851_diff__Suc__diff__eq1,axiom,
% 3.36/1.73 ! [M: nat,K: nat,J: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ K @ J )
% 3.36/1.73 => ( ( minus_minus_nat @ M @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
% 3.36/1.73 = ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( suc @ J ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_852_diff__Suc__diff__eq2,axiom,
% 3.36/1.73 ! [M: nat,K: nat,J: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ K @ J )
% 3.36/1.73 => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ M )
% 3.36/1.73 = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ M ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_853_termination__basic__simps_I4_J,axiom,
% 3.36/1.73 ! [Y: nat,X: nat,Z: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ X @ Z )
% 3.36/1.73 => ( ord_less_eq_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_854_termination__basic__simps_I3_J,axiom,
% 3.36/1.73 ! [Z: nat,X: nat,Y: nat] :
% 3.36/1.73 ( ( ord_less_eq_nat @ X @ Y )
% 3.36/1.73 => ( ord_less_eq_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_855_lessI,axiom,
% 3.36/1.73 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_856_Suc__mono,axiom,
% 3.36/1.73 ! [M: nat,N: nat] :
% 3.36/1.73 ( ( ord_less_nat @ M @ N )
% 3.36/1.73 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_857_finite__Collect__less__nat,axiom,
% 3.36/1.73 ! [K: nat] :
% 3.36/1.73 ( finite_finite_nat
% 3.36/1.73 @ ( collect_nat
% 3.36/1.73 @ ^ [N_1: nat] : ( ord_less_nat @ N_1 @ K ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_858_less__not__refl,axiom,
% 3.36/1.73 ! [N: nat] :
% 3.36/1.73 ~ ( ord_less_nat @ N @ N ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_859_nat__neq__iff,axiom,
% 3.36/1.73 ! [M: nat,N: nat] :
% 3.36/1.73 ( ( M != N )
% 3.36/1.73 <=> ( ( ord_less_nat @ M @ N )
% 3.36/1.73 | ( ord_less_nat @ N @ M ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_860_linorder__neqE__nat,axiom,
% 3.36/1.73 ! [X: nat,Y: nat] :
% 3.36/1.73 ( ( X != Y )
% 3.36/1.73 => ( ~ ( ord_less_nat @ X @ Y )
% 3.36/1.73 => ( ord_less_nat @ Y @ X ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_861_less__irrefl__nat,axiom,
% 3.36/1.73 ! [N: nat] :
% 3.36/1.73 ~ ( ord_less_nat @ N @ N ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_862_less__not__refl2,axiom,
% 3.36/1.73 ! [N: nat,M: nat] :
% 3.36/1.73 ( ( ord_less_nat @ N @ M )
% 3.36/1.73 => ( M != N ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_863_less__not__refl3,axiom,
% 3.36/1.73 ! [S: nat,T: nat] :
% 3.36/1.73 ( ( ord_less_nat @ S @ T )
% 3.36/1.73 => ( S != T ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_864_nat__less__cases,axiom,
% 3.36/1.73 ! [P: nat > nat > $o,M: nat,N: nat] :
% 3.36/1.73 ( ( ( ord_less_nat @ M @ N )
% 3.36/1.73 => ( P @ N @ M ) )
% 3.36/1.73 => ( ( ( M = N )
% 3.36/1.73 => ( P @ N @ M ) )
% 3.36/1.73 => ( ( ( ord_less_nat @ N @ M )
% 3.36/1.73 => ( P @ N @ M ) )
% 3.36/1.73 => ( P @ N @ M ) ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_865_not__less__eq,axiom,
% 3.36/1.73 ! [M: nat,N: nat] :
% 3.36/1.73 ( ~ ( ord_less_nat @ M @ N )
% 3.36/1.73 <=> ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_866_less__Suc__eq,axiom,
% 3.36/1.73 ! [M: nat,N: nat] :
% 3.36/1.73 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 3.36/1.73 <=> ( ( ord_less_nat @ M @ N )
% 3.36/1.73 | ( M = N ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_867_Suc__less__eq,axiom,
% 3.36/1.73 ! [M: nat,N: nat] :
% 3.36/1.73 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 3.36/1.73 <=> ( ord_less_nat @ M @ N ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_868_not__less__less__Suc__eq,axiom,
% 3.36/1.73 ! [N: nat,M: nat] :
% 3.36/1.73 ( ~ ( ord_less_nat @ N @ M )
% 3.36/1.73 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 3.36/1.73 <=> ( N = M ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_869_less__antisym,axiom,
% 3.36/1.73 ! [N: nat,M: nat] :
% 3.36/1.73 ( ~ ( ord_less_nat @ N @ M )
% 3.36/1.73 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 3.36/1.73 => ( M = N ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_870_less__SucI,axiom,
% 3.36/1.73 ! [M: nat,N: nat] :
% 3.36/1.73 ( ( ord_less_nat @ M @ N )
% 3.36/1.73 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_871_Suc__lessI,axiom,
% 3.36/1.73 ! [M: nat,N: nat] :
% 3.36/1.73 ( ( ord_less_nat @ M @ N )
% 3.36/1.73 => ( ( ( suc @ M )
% 3.36/1.73 != N )
% 3.36/1.73 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_872_less__trans__Suc,axiom,
% 3.36/1.73 ! [K: nat,I_1: nat,J: nat] :
% 3.36/1.73 ( ( ord_less_nat @ I_1 @ J )
% 3.36/1.73 => ( ( ord_less_nat @ J @ K )
% 3.36/1.73 => ( ord_less_nat @ ( suc @ I_1 ) @ K ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_873_less__SucE,axiom,
% 3.36/1.73 ! [M: nat,N: nat] :
% 3.36/1.73 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 3.36/1.73 => ( ~ ( ord_less_nat @ M @ N )
% 3.36/1.73 => ( M = N ) ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_874_Suc__lessD,axiom,
% 3.36/1.73 ! [M: nat,N: nat] :
% 3.36/1.73 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 3.36/1.73 => ( ord_less_nat @ M @ N ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_875_Suc__less__SucD,axiom,
% 3.36/1.73 ! [M: nat,N: nat] :
% 3.36/1.73 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 3.36/1.73 => ( ord_less_nat @ M @ N ) ) ).
% 3.36/1.73
% 3.36/1.73 thf(fact_876_not__add__less1,axiom,
% 3.36/1.74 ! [I_1: nat,J: nat] :
% 3.36/1.74 ~ ( ord_less_nat @ ( plus_plus_nat @ I_1 @ J ) @ I_1 ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_877_not__add__less2,axiom,
% 3.36/1.74 ! [J: nat,I_1: nat] :
% 3.36/1.74 ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I_1 ) @ I_1 ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_878_nat__add__left__cancel__less,axiom,
% 3.36/1.74 ! [K: nat,M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 3.36/1.74 <=> ( ord_less_nat @ M @ N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_879_trans__less__add1,axiom,
% 3.36/1.74 ! [M: nat,I_1: nat,J: nat] :
% 3.36/1.74 ( ( ord_less_nat @ I_1 @ J )
% 3.36/1.74 => ( ord_less_nat @ I_1 @ ( plus_plus_nat @ J @ M ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_880_trans__less__add2,axiom,
% 3.36/1.74 ! [M: nat,I_1: nat,J: nat] :
% 3.36/1.74 ( ( ord_less_nat @ I_1 @ J )
% 3.36/1.74 => ( ord_less_nat @ I_1 @ ( plus_plus_nat @ M @ J ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_881_add__less__mono1,axiom,
% 3.36/1.74 ! [K: nat,I_1: nat,J: nat] :
% 3.36/1.74 ( ( ord_less_nat @ I_1 @ J )
% 3.36/1.74 => ( ord_less_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_882_add__less__mono,axiom,
% 3.36/1.74 ! [K: nat,L: nat,I_1: nat,J: nat] :
% 3.36/1.74 ( ( ord_less_nat @ I_1 @ J )
% 3.36/1.74 => ( ( ord_less_nat @ K @ L )
% 3.36/1.74 => ( ord_less_nat @ ( plus_plus_nat @ I_1 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_883_less__add__eq__less,axiom,
% 3.36/1.74 ! [M: nat,N: nat,K: nat,L: nat] :
% 3.36/1.74 ( ( ord_less_nat @ K @ L )
% 3.36/1.74 => ( ( ( plus_plus_nat @ M @ L )
% 3.36/1.74 = ( plus_plus_nat @ K @ N ) )
% 3.36/1.74 => ( ord_less_nat @ M @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_884_add__lessD1,axiom,
% 3.36/1.74 ! [I_1: nat,J: nat,K: nat] :
% 3.36/1.74 ( ( ord_less_nat @ ( plus_plus_nat @ I_1 @ J ) @ K )
% 3.36/1.74 => ( ord_less_nat @ I_1 @ K ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_885_termination__basic__simps_I1_J,axiom,
% 3.36/1.74 ! [Z: nat,X: nat,Y: nat] :
% 3.36/1.74 ( ( ord_less_nat @ X @ Y )
% 3.36/1.74 => ( ord_less_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_886_termination__basic__simps_I2_J,axiom,
% 3.36/1.74 ! [Y: nat,X: nat,Z: nat] :
% 3.36/1.74 ( ( ord_less_nat @ X @ Z )
% 3.36/1.74 => ( ord_less_nat @ X @ ( plus_plus_nat @ Y @ Z ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_887_nat__less__le,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ M @ N )
% 3.36/1.74 <=> ( ( ord_less_eq_nat @ M @ N )
% 3.36/1.74 & ( M != N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_888_le__eq__less__or__eq,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ M @ N )
% 3.36/1.74 <=> ( ( ord_less_nat @ M @ N )
% 3.36/1.74 | ( M = N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_889_less__imp__le__nat,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ M @ N )
% 3.36/1.74 => ( ord_less_eq_nat @ M @ N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_890_le__neq__implies__less,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ M @ N )
% 3.36/1.74 => ( ( M != N )
% 3.36/1.74 => ( ord_less_nat @ M @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_891_less__or__eq__imp__le,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ( ord_less_nat @ M @ N )
% 3.36/1.74 | ( M = N ) )
% 3.36/1.74 => ( ord_less_eq_nat @ M @ N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_892_termination__basic__simps_I5_J,axiom,
% 3.36/1.74 ! [X: nat,Y: nat] :
% 3.36/1.74 ( ( ord_less_nat @ X @ Y )
% 3.36/1.74 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_893_less__imp__diff__less,axiom,
% 3.36/1.74 ! [N: nat,J: nat,K: nat] :
% 3.36/1.74 ( ( ord_less_nat @ J @ K )
% 3.36/1.74 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_894_diff__less__mono2,axiom,
% 3.36/1.74 ! [L: nat,M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ M @ N )
% 3.36/1.74 => ( ( ord_less_nat @ M @ L )
% 3.36/1.74 => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_895_finite__nat__set__iff__bounded,axiom,
% 3.36/1.74 ! [N_2: nat > $o] :
% 3.36/1.74 ( ( finite_finite_nat @ N_2 )
% 3.36/1.74 <=> ? [M_1: nat] :
% 3.36/1.74 ! [X_1: nat] :
% 3.36/1.74 ( ( member_nat @ X_1 @ N_2 )
% 3.36/1.74 => ( ord_less_nat @ X_1 @ M_1 ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_896_card__Collect__less__nat,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ( ( finite_card_nat
% 3.36/1.74 @ ( collect_nat
% 3.36/1.74 @ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) )
% 3.36/1.74 = N ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_897_finite__M__bounded__by__nat,axiom,
% 3.36/1.74 ! [P: nat > $o,I_1: nat] :
% 3.36/1.74 ( finite_finite_nat
% 3.36/1.74 @ ( collect_nat
% 3.36/1.74 @ ^ [K_1: nat] : ( (&) @ ( P @ K_1 ) @ ( ord_less_nat @ K_1 @ I_1 ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_898_less__add__Suc1,axiom,
% 3.36/1.74 ! [I_1: nat,M: nat] : ( ord_less_nat @ I_1 @ ( suc @ ( plus_plus_nat @ I_1 @ M ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_899_less__add__Suc2,axiom,
% 3.36/1.74 ! [I_1: nat,M: nat] : ( ord_less_nat @ I_1 @ ( suc @ ( plus_plus_nat @ M @ I_1 ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_900_less__iff__Suc__add,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ M @ N )
% 3.36/1.74 <=> ? [K_1: nat] :
% 3.36/1.74 ( N
% 3.36/1.74 = ( suc @ ( plus_plus_nat @ M @ K_1 ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_901_less__eq__Suc__le,axiom,
% 3.36/1.74 ! [N: nat,M: nat] :
% 3.36/1.74 ( ( ord_less_nat @ N @ M )
% 3.36/1.74 <=> ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_902_less__Suc__eq__le,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 3.36/1.74 <=> ( ord_less_eq_nat @ M @ N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_903_Suc__le__eq,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 3.36/1.74 <=> ( ord_less_nat @ M @ N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_904_le__imp__less__Suc,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ M @ N )
% 3.36/1.74 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_905_Suc__leI,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ M @ N )
% 3.36/1.74 => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_906_le__less__Suc__eq,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ M @ N )
% 3.36/1.74 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 3.36/1.74 <=> ( N = M ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_907_Suc__le__lessD,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 3.36/1.74 => ( ord_less_nat @ M @ N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_908_diff__less__Suc,axiom,
% 3.36/1.74 ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_909_less__diff__conv,axiom,
% 3.36/1.74 ! [I_1: nat,J: nat,K: nat] :
% 3.36/1.74 ( ( ord_less_nat @ I_1 @ ( minus_minus_nat @ J @ K ) )
% 3.36/1.74 <=> ( ord_less_nat @ ( plus_plus_nat @ I_1 @ K ) @ J ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_910_add__diff__inverse,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ~ ( ord_less_nat @ M @ N )
% 3.36/1.74 => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 3.36/1.74 = M ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_911_diff__less__mono,axiom,
% 3.36/1.74 ! [C: nat,A_1: nat,B_2: nat] :
% 3.36/1.74 ( ( ord_less_nat @ A_1 @ B_2 )
% 3.36/1.74 => ( ( ord_less_eq_nat @ C @ A_1 )
% 3.36/1.74 => ( ord_less_nat @ ( minus_minus_nat @ A_1 @ C ) @ ( minus_minus_nat @ B_2 @ C ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_912_less__diff__iff,axiom,
% 3.36/1.74 ! [N: nat,K: nat,M: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ K @ M )
% 3.36/1.74 => ( ( ord_less_eq_nat @ K @ N )
% 3.36/1.74 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 3.36/1.74 <=> ( ord_less_nat @ M @ N ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_913_less__eq__Suc__le__raw,axiom,
% 3.36/1.74 ! [X_1: nat] :
% 3.36/1.74 ( ( ord_less_nat @ X_1 )
% 3.36/1.74 = ( ord_less_eq_nat @ ( suc @ X_1 ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_914_mono__nat__linear__lb,axiom,
% 3.36/1.74 ! [M: nat,K: nat,F: nat > nat] :
% 3.36/1.74 ( ! [M_1: nat,N_1: nat] :
% 3.36/1.74 ( ( ord_less_nat @ M_1 @ N_1 )
% 3.36/1.74 => ( ord_less_nat @ ( F @ M_1 ) @ ( F @ N_1 ) ) )
% 3.36/1.74 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_915_inc__induct,axiom,
% 3.36/1.74 ! [P: nat > $o,I_1: nat,J: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ I_1 @ J )
% 3.36/1.74 => ( ( P @ J )
% 3.36/1.74 => ( ! [I: nat] :
% 3.36/1.74 ( ( ord_less_nat @ I @ J )
% 3.36/1.74 => ( ( P @ ( suc @ I ) )
% 3.36/1.74 => ( P @ I ) ) )
% 3.36/1.74 => ( P @ I_1 ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_916_less__imp__Suc__add,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ M @ N )
% 3.36/1.74 => ? [K_1: nat] :
% 3.36/1.74 ( N
% 3.36/1.74 = ( suc @ ( plus_plus_nat @ M @ K_1 ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_917_bounded__nat__set__is__finite,axiom,
% 3.36/1.74 ! [N: nat,N_2: nat > $o] :
% 3.36/1.74 ( ! [X_1: nat] :
% 3.36/1.74 ( ( member_nat @ X_1 @ N_2 )
% 3.36/1.74 => ( ord_less_nat @ X_1 @ N ) )
% 3.36/1.74 => ( finite_finite_nat @ N_2 ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_918_less__mono__imp__le__mono,axiom,
% 3.36/1.74 ! [I_1: nat,J: nat,F: nat > nat] :
% 3.36/1.74 ( ! [I: nat,J_1: nat] :
% 3.36/1.74 ( ( ord_less_nat @ I @ J_1 )
% 3.36/1.74 => ( ord_less_nat @ ( F @ I ) @ ( F @ J_1 ) ) )
% 3.36/1.74 => ( ( ord_less_eq_nat @ I_1 @ J )
% 3.36/1.74 => ( ord_less_eq_nat @ ( F @ I_1 ) @ ( F @ J ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_919_lessE,axiom,
% 3.36/1.74 ! [I_1: nat,K: nat] :
% 3.36/1.74 ( ( ord_less_nat @ I_1 @ K )
% 3.36/1.74 => ( ( K
% 3.36/1.74 != ( suc @ I_1 ) )
% 3.36/1.74 => ~ ! [J_1: nat] :
% 3.36/1.74 ( ( ord_less_nat @ I_1 @ J_1 )
% 3.36/1.74 => ( K
% 3.36/1.74 != ( suc @ J_1 ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_920_Suc__lessE,axiom,
% 3.36/1.74 ! [I_1: nat,K: nat] :
% 3.36/1.74 ( ( ord_less_nat @ ( suc @ I_1 ) @ K )
% 3.36/1.74 => ~ ! [J_1: nat] :
% 3.36/1.74 ( ( ord_less_nat @ I_1 @ J_1 )
% 3.36/1.74 => ( K
% 3.36/1.74 != ( suc @ J_1 ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_921_less__zeroE,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_922_le0,axiom,
% 3.36/1.74 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_923_zero__less__Suc,axiom,
% 3.36/1.74 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_924_le__0__eq,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 3.36/1.74 <=> ( N = zero_zero_nat ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_925_less__eq__nat_Osimps_I1_J,axiom,
% 3.36/1.74 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_926_diffs0__imp__equal,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ( minus_minus_nat @ M @ N )
% 3.36/1.74 = zero_zero_nat )
% 3.36/1.74 => ( ( ( minus_minus_nat @ N @ M )
% 3.36/1.74 = zero_zero_nat )
% 3.36/1.74 => ( M = N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_927_diff__self__eq__0,axiom,
% 3.36/1.74 ! [M: nat] :
% 3.36/1.74 ( ( minus_minus_nat @ M @ M )
% 3.36/1.74 = zero_zero_nat ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_928_minus__nat_Odiff__0,axiom,
% 3.36/1.74 ! [M: nat] :
% 3.36/1.74 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 3.36/1.74 = M ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_929_diff__0__eq__0,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 3.36/1.74 = zero_zero_nat ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_930_bot__nat__def,axiom,
% 3.36/1.74 bot_bot_nat = zero_zero_nat ).
% 3.36/1.74
% 3.36/1.74 thf(fact_931_add__eq__self__zero,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ( plus_plus_nat @ M @ N )
% 3.36/1.74 = M )
% 3.36/1.74 => ( N = zero_zero_nat ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_932_add__is__0,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ( plus_plus_nat @ M @ N )
% 3.36/1.74 = zero_zero_nat )
% 3.36/1.74 <=> ( ( M = zero_zero_nat )
% 3.36/1.74 & ( N = zero_zero_nat ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_933_Nat_Oadd__0__right,axiom,
% 3.36/1.74 ! [M: nat] :
% 3.36/1.74 ( ( plus_plus_nat @ M @ zero_zero_nat )
% 3.36/1.74 = M ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_934_plus__nat_Oadd__0,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ( ( plus_plus_nat @ zero_zero_nat @ N )
% 3.36/1.74 = N ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_935_gr0I,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ( ( N != zero_zero_nat )
% 3.36/1.74 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_936_gr__implies__not0,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ M @ N )
% 3.36/1.74 => ( N != zero_zero_nat ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_937_less__nat__zero__code,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_938_neq0__conv,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ( ( N != zero_zero_nat )
% 3.36/1.74 <=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_939_not__less0,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_940_Suc__neq__Zero,axiom,
% 3.36/1.74 ! [M: nat] :
% 3.36/1.74 ( ( suc @ M )
% 3.36/1.74 != zero_zero_nat ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_941_Zero__neq__Suc,axiom,
% 3.36/1.74 ! [M: nat] :
% 3.36/1.74 ( zero_zero_nat
% 3.36/1.74 != ( suc @ M ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_942_nat_Osimps_I3_J,axiom,
% 3.36/1.74 ! [Nat_2: nat] :
% 3.36/1.74 ( ( suc @ Nat_2 )
% 3.36/1.74 != zero_zero_nat ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_943_Suc__not__Zero,axiom,
% 3.36/1.74 ! [M: nat] :
% 3.36/1.74 ( ( suc @ M )
% 3.36/1.74 != zero_zero_nat ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_944_nat_Osimps_I2_J,axiom,
% 3.36/1.74 ! [Nat_1: nat] :
% 3.36/1.74 ( zero_zero_nat
% 3.36/1.74 != ( suc @ Nat_1 ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_945_Zero__not__Suc,axiom,
% 3.36/1.74 ! [M: nat] :
% 3.36/1.74 ( zero_zero_nat
% 3.36/1.74 != ( suc @ M ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_946_gr0__conv__Suc,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.36/1.74 <=> ? [M_1: nat] :
% 3.36/1.74 ( N
% 3.36/1.74 = ( suc @ M_1 ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_947_less__Suc0,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 3.36/1.74 <=> ( N = zero_zero_nat ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_948_less__Suc__eq__0__disj,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 3.36/1.74 <=> ( ( M = zero_zero_nat )
% 3.36/1.74 | ? [J_1: nat] :
% 3.36/1.74 ( ( M
% 3.36/1.74 = ( suc @ J_1 ) )
% 3.36/1.74 & ( ord_less_nat @ J_1 @ N ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_949_one__is__add,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ( suc @ zero_zero_nat )
% 3.36/1.74 = ( plus_plus_nat @ M @ N ) )
% 3.36/1.74 <=> ( ( ( M
% 3.36/1.74 = ( suc @ zero_zero_nat ) )
% 3.36/1.74 & ( N = zero_zero_nat ) )
% 3.36/1.74 | ( ( M = zero_zero_nat )
% 3.36/1.74 & ( N
% 3.36/1.74 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_950_add__is__1,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ( plus_plus_nat @ M @ N )
% 3.36/1.74 = ( suc @ zero_zero_nat ) )
% 3.36/1.74 <=> ( ( ( M
% 3.36/1.74 = ( suc @ zero_zero_nat ) )
% 3.36/1.74 & ( N = zero_zero_nat ) )
% 3.36/1.74 | ( ( M = zero_zero_nat )
% 3.36/1.74 & ( N
% 3.36/1.74 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_951_add__gr__0,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 3.36/1.74 <=> ( ( ord_less_nat @ zero_zero_nat @ M )
% 3.36/1.74 | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_952_zero__less__diff,axiom,
% 3.36/1.74 ! [N: nat,M: nat] :
% 3.36/1.74 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 3.36/1.74 <=> ( ord_less_nat @ M @ N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_953_diff__less,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.36/1.74 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 3.36/1.74 => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_954_diff__add__0,axiom,
% 3.36/1.74 ! [N: nat,M: nat] :
% 3.36/1.74 ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 3.36/1.74 = zero_zero_nat ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_955_diff__is__0__eq,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ( minus_minus_nat @ M @ N )
% 3.36/1.74 = zero_zero_nat )
% 3.36/1.74 <=> ( ord_less_eq_nat @ M @ N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_956_diff__is__0__eq_H,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ M @ N )
% 3.36/1.74 => ( ( minus_minus_nat @ M @ N )
% 3.36/1.74 = zero_zero_nat ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_957_One__nat__def,axiom,
% 3.36/1.74 ( one_one_nat
% 3.36/1.74 = ( suc @ zero_zero_nat ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_958_diff__Suc,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 3.36/1.74 = ( nat_case_nat @ zero_zero_nat
% 3.36/1.74 @ ^ [K_1: nat] : K_1
% 3.36/1.74 @ ( minus_minus_nat @ M @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_959_Suc__pred,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.36/1.74 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 3.36/1.74 = N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_960_diff__Suc__less,axiom,
% 3.36/1.74 ! [I_1: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.36/1.74 => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I_1 ) ) @ N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_961_nat__diff__split,axiom,
% 3.36/1.74 ! [P: nat > $o,A_1: nat,B_2: nat] :
% 3.36/1.74 ( ( P @ ( minus_minus_nat @ A_1 @ B_2 ) )
% 3.36/1.74 <=> ( ( ( ord_less_nat @ A_1 @ B_2 )
% 3.36/1.74 => ( P @ zero_zero_nat ) )
% 3.36/1.74 & ! [D_1: nat] :
% 3.36/1.74 ( ( A_1
% 3.36/1.74 = ( plus_plus_nat @ B_2 @ D_1 ) )
% 3.36/1.74 => ( P @ D_1 ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_962_nat__diff__split__asm,axiom,
% 3.36/1.74 ! [P: nat > $o,A_1: nat,B_2: nat] :
% 3.36/1.74 ( ( P @ ( minus_minus_nat @ A_1 @ B_2 ) )
% 3.36/1.74 <=> ~ ( ( ( ord_less_nat @ A_1 @ B_2 )
% 3.36/1.74 & ~ ( P @ zero_zero_nat ) )
% 3.36/1.74 | ? [D_1: nat] :
% 3.36/1.74 ( ( A_1
% 3.36/1.74 = ( plus_plus_nat @ B_2 @ D_1 ) )
% 3.36/1.74 & ~ ( P @ D_1 ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_963_card__less__Suc,axiom,
% 3.36/1.74 ! [I_1: nat,M_2: nat > $o] :
% 3.36/1.74 ( ( member_nat @ zero_zero_nat @ M_2 )
% 3.36/1.74 => ( ( suc
% 3.36/1.74 @ ( finite_card_nat
% 3.36/1.74 @ ( collect_nat
% 3.36/1.74 @ ^ [K_1: nat] : ( (&) @ ( member_nat @ ( suc @ K_1 ) @ M_2 ) @ ( ord_less_nat @ K_1 @ I_1 ) ) ) ) )
% 3.36/1.74 = ( finite_card_nat
% 3.36/1.74 @ ( collect_nat
% 3.36/1.74 @ ^ [K_1: nat] : ( (&) @ ( member_nat @ K_1 @ M_2 ) @ ( ord_less_nat @ K_1 @ ( suc @ I_1 ) ) ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_964_card__less,axiom,
% 3.36/1.74 ! [I_1: nat,M_2: nat > $o] :
% 3.36/1.74 ( ( member_nat @ zero_zero_nat @ M_2 )
% 3.36/1.74 => ( ( finite_card_nat
% 3.36/1.74 @ ( collect_nat
% 3.36/1.74 @ ^ [K_1: nat] : ( (&) @ ( member_nat @ K_1 @ M_2 ) @ ( ord_less_nat @ K_1 @ ( suc @ I_1 ) ) ) ) )
% 3.36/1.74 != zero_zero_nat ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_965_card__less__Suc2,axiom,
% 3.36/1.74 ! [I_1: nat,M_2: nat > $o] :
% 3.36/1.74 ( ~ ( member_nat @ zero_zero_nat @ M_2 )
% 3.36/1.74 => ( ( finite_card_nat
% 3.36/1.74 @ ( collect_nat
% 3.36/1.74 @ ^ [K_1: nat] : ( (&) @ ( member_nat @ ( suc @ K_1 ) @ M_2 ) @ ( ord_less_nat @ K_1 @ I_1 ) ) ) )
% 3.36/1.74 = ( finite_card_nat
% 3.36/1.74 @ ( collect_nat
% 3.36/1.74 @ ^ [K_1: nat] : ( (&) @ ( member_nat @ K_1 @ M_2 ) @ ( ord_less_nat @ K_1 @ ( suc @ I_1 ) ) ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_966_Suc__diff__1,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.36/1.74 => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
% 3.36/1.74 = N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_967_Suc__pred_H,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.36/1.74 => ( N
% 3.36/1.74 = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_968_add__eq__if,axiom,
% 3.36/1.74 ! [N: nat,M: nat] :
% 3.36/1.74 ( ( ( M = zero_zero_nat )
% 3.36/1.74 => ( ( plus_plus_nat @ M @ N )
% 3.36/1.74 = N ) )
% 3.36/1.74 & ( ( M != zero_zero_nat )
% 3.36/1.74 => ( ( plus_plus_nat @ M @ N )
% 3.36/1.74 = ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_969_ex__least__nat__less,axiom,
% 3.36/1.74 ! [N: nat,P: nat > $o] :
% 3.36/1.74 ( ~ ( P @ zero_zero_nat )
% 3.36/1.74 => ( ( P @ N )
% 3.36/1.74 => ? [K_1: nat] :
% 3.36/1.74 ( ( ord_less_nat @ K_1 @ N )
% 3.36/1.74 & ! [I: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ I @ K_1 )
% 3.36/1.74 => ~ ( P @ I ) )
% 3.36/1.74 & ( P @ ( plus_plus_nat @ K_1 @ one_one_nat ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_970_ex__least__nat__le,axiom,
% 3.36/1.74 ! [N: nat,P: nat > $o] :
% 3.36/1.74 ( ~ ( P @ zero_zero_nat )
% 3.36/1.74 => ( ( P @ N )
% 3.36/1.74 => ? [K_1: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ K_1 @ N )
% 3.36/1.74 & ! [I: nat] :
% 3.36/1.74 ( ( ord_less_nat @ I @ K_1 )
% 3.36/1.74 => ~ ( P @ I ) )
% 3.36/1.74 & ( P @ K_1 ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_971_less__imp__add__positive,axiom,
% 3.36/1.74 ! [I_1: nat,J: nat] :
% 3.36/1.74 ( ( ord_less_nat @ I_1 @ J )
% 3.36/1.74 => ? [K_1: nat] :
% 3.36/1.74 ( ( ord_less_nat @ zero_zero_nat @ K_1 )
% 3.36/1.74 & ( ( plus_plus_nat @ I_1 @ K_1 )
% 3.36/1.74 = J ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_972_gr0__implies__Suc,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ zero_zero_nat @ N )
% 3.36/1.74 => ? [M_1: nat] :
% 3.36/1.74 ( N
% 3.36/1.74 = ( suc @ M_1 ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_973_nat_Oexhaust,axiom,
% 3.36/1.74 ! [Y: nat] :
% 3.36/1.74 ( ( Y != zero_zero_nat )
% 3.36/1.74 => ~ ! [Nat: nat] :
% 3.36/1.74 ( Y
% 3.36/1.74 != ( suc @ Nat ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_974_zero__induct,axiom,
% 3.36/1.74 ! [P: nat > $o,K: nat] :
% 3.36/1.74 ( ( P @ K )
% 3.36/1.74 => ( ! [N_1: nat] :
% 3.36/1.74 ( ( P @ ( suc @ N_1 ) )
% 3.36/1.74 => ( P @ N_1 ) )
% 3.36/1.74 => ( P @ zero_zero_nat ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_975_not0__implies__Suc,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ( ( N != zero_zero_nat )
% 3.36/1.74 => ? [M_1: nat] :
% 3.36/1.74 ( N
% 3.36/1.74 = ( suc @ M_1 ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_976_nat__induct,axiom,
% 3.36/1.74 ! [N: nat,P: nat > $o] :
% 3.36/1.74 ( ( P @ zero_zero_nat )
% 3.36/1.74 => ( ! [N_1: nat] :
% 3.36/1.74 ( ( P @ N_1 )
% 3.36/1.74 => ( P @ ( suc @ N_1 ) ) )
% 3.36/1.74 => ( P @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_977_expand__Suc,axiom,
% 3.36/1.74 ! [V: int] :
% 3.36/1.74 ( ( ord_less_nat @ zero_zero_nat @ ( number_number_of_nat @ V ) )
% 3.36/1.74 => ( ( number_number_of_nat @ V )
% 3.36/1.74 = ( suc @ ( minus_minus_nat @ ( number_number_of_nat @ V ) @ one_one_nat ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_978_mult__0,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ( ( times_times_nat @ zero_zero_nat @ N )
% 3.36/1.74 = zero_zero_nat ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_979_mult__0__right,axiom,
% 3.36/1.74 ! [M: nat] :
% 3.36/1.74 ( ( times_times_nat @ M @ zero_zero_nat )
% 3.36/1.74 = zero_zero_nat ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_980_mult__is__0,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ( times_times_nat @ M @ N )
% 3.36/1.74 = zero_zero_nat )
% 3.36/1.74 <=> ( ( M = zero_zero_nat )
% 3.36/1.74 | ( N = zero_zero_nat ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_981_mult__cancel1,axiom,
% 3.36/1.74 ! [K: nat,M: nat,N: nat] :
% 3.36/1.74 ( ( ( times_times_nat @ K @ M )
% 3.36/1.74 = ( times_times_nat @ K @ N ) )
% 3.36/1.74 <=> ( ( M = N )
% 3.36/1.74 | ( K = zero_zero_nat ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_982_mult__cancel2,axiom,
% 3.36/1.74 ! [M: nat,K: nat,N: nat] :
% 3.36/1.74 ( ( ( times_times_nat @ M @ K )
% 3.36/1.74 = ( times_times_nat @ N @ K ) )
% 3.36/1.74 <=> ( ( M = N )
% 3.36/1.74 | ( K = zero_zero_nat ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_983_Suc__mult__cancel1,axiom,
% 3.36/1.74 ! [K: nat,M: nat,N: nat] :
% 3.36/1.74 ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 3.36/1.74 = ( times_times_nat @ ( suc @ K ) @ N ) )
% 3.36/1.74 <=> ( M = N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_984_add__mult__distrib,axiom,
% 3.36/1.74 ! [M: nat,N: nat,K: nat] :
% 3.36/1.74 ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
% 3.36/1.74 = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_985_add__mult__distrib2,axiom,
% 3.36/1.74 ! [K: nat,M: nat,N: nat] :
% 3.36/1.74 ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
% 3.36/1.74 = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_986_mult__le__mono,axiom,
% 3.36/1.74 ! [K: nat,L: nat,I_1: nat,J: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ I_1 @ J )
% 3.36/1.74 => ( ( ord_less_eq_nat @ K @ L )
% 3.36/1.74 => ( ord_less_eq_nat @ ( times_times_nat @ I_1 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_987_mult__le__mono2,axiom,
% 3.36/1.74 ! [K: nat,I_1: nat,J: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ I_1 @ J )
% 3.36/1.74 => ( ord_less_eq_nat @ ( times_times_nat @ K @ I_1 ) @ ( times_times_nat @ K @ J ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_988_mult__le__mono1,axiom,
% 3.36/1.74 ! [K: nat,I_1: nat,J: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ I_1 @ J )
% 3.36/1.74 => ( ord_less_eq_nat @ ( times_times_nat @ I_1 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_989_le__cube,axiom,
% 3.36/1.74 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_990_le__square,axiom,
% 3.36/1.74 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_991_diff__mult__distrib,axiom,
% 3.36/1.74 ! [M: nat,N: nat,K: nat] :
% 3.36/1.74 ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
% 3.36/1.74 = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_992_diff__mult__distrib2,axiom,
% 3.36/1.74 ! [K: nat,M: nat,N: nat] :
% 3.36/1.74 ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 3.36/1.74 = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_993_nat__mult__eq__1__iff,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ( times_times_nat @ M @ N )
% 3.36/1.74 = one_one_nat )
% 3.36/1.74 <=> ( ( M = one_one_nat )
% 3.36/1.74 & ( N = one_one_nat ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_994_nat__mult__1__right,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ( ( times_times_nat @ N @ one_one_nat )
% 3.36/1.74 = N ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_995_nat__1__eq__mult__iff,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( one_one_nat
% 3.36/1.74 = ( times_times_nat @ M @ N ) )
% 3.36/1.74 <=> ( ( M = one_one_nat )
% 3.36/1.74 & ( N = one_one_nat ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_996_nat__mult__1,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ( ( times_times_nat @ one_one_nat @ N )
% 3.36/1.74 = N ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_997_mult__eq__1__iff,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ( times_times_nat @ M @ N )
% 3.36/1.74 = ( suc @ zero_zero_nat ) )
% 3.36/1.74 <=> ( ( M
% 3.36/1.74 = ( suc @ zero_zero_nat ) )
% 3.36/1.74 & ( N
% 3.36/1.74 = ( suc @ zero_zero_nat ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_998_mult__less__mono2,axiom,
% 3.36/1.74 ! [K: nat,I_1: nat,J: nat] :
% 3.36/1.74 ( ( ord_less_nat @ I_1 @ J )
% 3.36/1.74 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 3.36/1.74 => ( ord_less_nat @ ( times_times_nat @ K @ I_1 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_999_mult__less__mono1,axiom,
% 3.36/1.74 ! [K: nat,I_1: nat,J: nat] :
% 3.36/1.74 ( ( ord_less_nat @ I_1 @ J )
% 3.36/1.74 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 3.36/1.74 => ( ord_less_nat @ ( times_times_nat @ I_1 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1000_mult__less__cancel2,axiom,
% 3.36/1.74 ! [M: nat,K: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 3.36/1.74 <=> ( ( ord_less_nat @ zero_zero_nat @ K )
% 3.36/1.74 & ( ord_less_nat @ M @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1001_mult__less__cancel1,axiom,
% 3.36/1.74 ! [K: nat,M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 3.36/1.74 <=> ( ( ord_less_nat @ zero_zero_nat @ K )
% 3.36/1.74 & ( ord_less_nat @ M @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1002_nat__0__less__mult__iff,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 3.36/1.74 <=> ( ( ord_less_nat @ zero_zero_nat @ M )
% 3.36/1.74 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1003_Suc__mult__less__cancel1,axiom,
% 3.36/1.74 ! [K: nat,M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 3.36/1.74 <=> ( ord_less_nat @ M @ N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1004_mult__Suc__right,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( times_times_nat @ M @ ( suc @ N ) )
% 3.36/1.74 = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1005_mult__Suc,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( times_times_nat @ ( suc @ M ) @ N )
% 3.36/1.74 = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1006_Suc__mult__le__cancel1,axiom,
% 3.36/1.74 ! [K: nat,M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 3.36/1.74 <=> ( ord_less_eq_nat @ M @ N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1007_mult__eq__self__implies__10,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( M
% 3.36/1.74 = ( times_times_nat @ M @ N ) )
% 3.36/1.74 => ( ( N = one_one_nat )
% 3.36/1.74 | ( M = zero_zero_nat ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1008_n__less__m__mult__n,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 3.36/1.74 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 3.36/1.74 => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1009_n__less__n__mult__m,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 3.36/1.74 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 3.36/1.74 => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1010_one__less__mult,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 3.36/1.74 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 3.36/1.74 => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1011_one__le__mult__iff,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
% 3.36/1.74 <=> ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 3.36/1.74 & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1012_mult__le__cancel1,axiom,
% 3.36/1.74 ! [K: nat,M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 3.36/1.74 <=> ( ( ord_less_nat @ zero_zero_nat @ K )
% 3.36/1.74 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1013_mult__le__cancel2,axiom,
% 3.36/1.74 ! [M: nat,K: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 3.36/1.74 <=> ( ( ord_less_nat @ zero_zero_nat @ K )
% 3.36/1.74 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1014_mult__eq__if,axiom,
% 3.36/1.74 ! [N: nat,M: nat] :
% 3.36/1.74 ( ( ( M = zero_zero_nat )
% 3.36/1.74 => ( ( times_times_nat @ M @ N )
% 3.36/1.74 = zero_zero_nat ) )
% 3.36/1.74 & ( ( M != zero_zero_nat )
% 3.36/1.74 => ( ( times_times_nat @ M @ N )
% 3.36/1.74 = ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1015_nat__less__add__iff2,axiom,
% 3.36/1.74 ! [U: nat,M: nat,N: nat,I_1: nat,J: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ I_1 @ J )
% 3.36/1.74 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 3.36/1.74 <=> ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I_1 ) @ U ) @ N ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1016_nat__mult__commute,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( times_times_nat @ M @ N )
% 3.36/1.74 = ( times_times_nat @ N @ M ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1017_nat__mult__assoc,axiom,
% 3.36/1.74 ! [M: nat,N: nat,K: nat] :
% 3.36/1.74 ( ( times_times_nat @ ( times_times_nat @ M @ N ) @ K )
% 3.36/1.74 = ( times_times_nat @ M @ ( times_times_nat @ N @ K ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1018_less__number__of__int__code,axiom,
% 3.36/1.74 ! [K: int,L: int] :
% 3.36/1.74 ( ( ord_less_int @ ( number_number_of_int @ K ) @ ( number_number_of_int @ L ) )
% 3.36/1.74 <=> ( ord_less_int @ K @ L ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1019_zmult__1,axiom,
% 3.36/1.74 ! [Z: int] :
% 3.36/1.74 ( ( times_times_int @ one_one_int @ Z )
% 3.36/1.74 = Z ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1020_zmult__1__right,axiom,
% 3.36/1.74 ! [Z: int] :
% 3.36/1.74 ( ( times_times_int @ Z @ one_one_int )
% 3.36/1.74 = Z ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1021_zmult__commute,axiom,
% 3.36/1.74 ! [Z: int,W: int] :
% 3.36/1.74 ( ( times_times_int @ Z @ W )
% 3.36/1.74 = ( times_times_int @ W @ Z ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1022_times__numeral__code_I5_J,axiom,
% 3.36/1.74 ! [V: int,W: int] :
% 3.36/1.74 ( ( times_times_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ W ) )
% 3.36/1.74 = ( number_number_of_int @ ( times_times_int @ V @ W ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1023_zmult__assoc,axiom,
% 3.36/1.74 ! [Z1: int,Z2: int,Z3: int] :
% 3.36/1.74 ( ( times_times_int @ ( times_times_int @ Z1 @ Z2 ) @ Z3 )
% 3.36/1.74 = ( times_times_int @ Z1 @ ( times_times_int @ Z2 @ Z3 ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1024_less__eq__number__of__int__code,axiom,
% 3.36/1.74 ! [K: int,L: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ ( number_number_of_int @ K ) @ ( number_number_of_int @ L ) )
% 3.36/1.74 <=> ( ord_less_eq_int @ K @ L ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1025_zdiff__zmult__distrib,axiom,
% 3.36/1.74 ! [Z1: int,Z2: int,W: int] :
% 3.36/1.74 ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z2 ) @ W )
% 3.36/1.74 = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1026_zdiff__zmult__distrib2,axiom,
% 3.36/1.74 ! [W: int,Z1: int,Z2: int] :
% 3.36/1.74 ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z2 ) )
% 3.36/1.74 = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1027_zadd__zmult__distrib2,axiom,
% 3.36/1.74 ! [W: int,Z1: int,Z2: int] :
% 3.36/1.74 ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z2 ) )
% 3.36/1.74 = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1028_plus__numeral__code_I9_J,axiom,
% 3.36/1.74 ! [V: int,W: int] :
% 3.36/1.74 ( ( plus_plus_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ W ) )
% 3.36/1.74 = ( number_number_of_int @ ( plus_plus_int @ V @ W ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1029_zadd__zmult__distrib,axiom,
% 3.36/1.74 ! [Z1: int,Z2: int,W: int] :
% 3.36/1.74 ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z2 ) @ W )
% 3.36/1.74 = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1030_zmult__zless__mono2,axiom,
% 3.36/1.74 ! [K: int,I_1: int,J: int] :
% 3.36/1.74 ( ( ord_less_int @ I_1 @ J )
% 3.36/1.74 => ( ( ord_less_int @ zero_zero_int @ K )
% 3.36/1.74 => ( ord_less_int @ ( times_times_int @ K @ I_1 ) @ ( times_times_int @ K @ J ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1031_pos__zmult__eq__1__iff,axiom,
% 3.36/1.74 ! [N: int,M: int] :
% 3.36/1.74 ( ( ord_less_int @ zero_zero_int @ M )
% 3.36/1.74 => ( ( ( times_times_int @ M @ N )
% 3.36/1.74 = one_one_int )
% 3.36/1.74 <=> ( ( M = one_one_int )
% 3.36/1.74 & ( N = one_one_int ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1032_odd__nonzero,axiom,
% 3.36/1.74 ! [Z: int] :
% 3.36/1.74 ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
% 3.36/1.74 != zero_zero_int ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1033_zadd__0__right,axiom,
% 3.36/1.74 ! [Z: int] :
% 3.36/1.74 ( ( plus_plus_int @ Z @ zero_zero_int )
% 3.36/1.74 = Z ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1034_zadd__0,axiom,
% 3.36/1.74 ! [Z: int] :
% 3.36/1.74 ( ( plus_plus_int @ zero_zero_int @ Z )
% 3.36/1.74 = Z ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1035_int__0__less__1,axiom,
% 3.36/1.74 ord_less_int @ zero_zero_int @ one_one_int ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1036_int__one__le__iff__zero__less,axiom,
% 3.36/1.74 ! [Z: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ one_one_int @ Z )
% 3.36/1.74 <=> ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1037_less__bin__lemma,axiom,
% 3.36/1.74 ! [K: int,L: int] :
% 3.36/1.74 ( ( ord_less_int @ K @ L )
% 3.36/1.74 <=> ( ord_less_int @ ( minus_minus_int @ K @ L ) @ zero_zero_int ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1038_le__imp__0__less,axiom,
% 3.36/1.74 ! [Z: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 3.36/1.74 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1039_odd__less__0,axiom,
% 3.36/1.74 ! [Z: int] :
% 3.36/1.74 ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
% 3.36/1.74 <=> ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1040_zadd__left__mono,axiom,
% 3.36/1.74 ! [K: int,I_1: int,J: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ I_1 @ J )
% 3.36/1.74 => ( ord_less_eq_int @ ( plus_plus_int @ K @ I_1 ) @ ( plus_plus_int @ K @ J ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1041_zadd__assoc,axiom,
% 3.36/1.74 ! [Z1: int,Z2: int,Z3: int] :
% 3.36/1.74 ( ( plus_plus_int @ ( plus_plus_int @ Z1 @ Z2 ) @ Z3 )
% 3.36/1.74 = ( plus_plus_int @ Z1 @ ( plus_plus_int @ Z2 @ Z3 ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1042_zadd__left__commute,axiom,
% 3.36/1.74 ! [X: int,Y: int,Z: int] :
% 3.36/1.74 ( ( plus_plus_int @ X @ ( plus_plus_int @ Y @ Z ) )
% 3.36/1.74 = ( plus_plus_int @ Y @ ( plus_plus_int @ X @ Z ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1043_zadd__commute,axiom,
% 3.36/1.74 ! [Z: int,W: int] :
% 3.36/1.74 ( ( plus_plus_int @ Z @ W )
% 3.36/1.74 = ( plus_plus_int @ W @ Z ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1044_zle__refl,axiom,
% 3.36/1.74 ! [W: int] : ( ord_less_eq_int @ W @ W ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1045_zle__linear,axiom,
% 3.36/1.74 ! [Z: int,W: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ Z @ W )
% 3.36/1.74 | ( ord_less_eq_int @ W @ Z ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1046_zle__trans,axiom,
% 3.36/1.74 ! [K: int,I_1: int,J: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ I_1 @ J )
% 3.36/1.74 => ( ( ord_less_eq_int @ J @ K )
% 3.36/1.74 => ( ord_less_eq_int @ I_1 @ K ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1047_zle__antisym,axiom,
% 3.36/1.74 ! [Z: int,W: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ Z @ W )
% 3.36/1.74 => ( ( ord_less_eq_int @ W @ Z )
% 3.36/1.74 => ( Z = W ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1048_zle__diff1__eq,axiom,
% 3.36/1.74 ! [W: int,Z: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
% 3.36/1.74 <=> ( ord_less_int @ W @ Z ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1049_zless__linear,axiom,
% 3.36/1.74 ! [X: int,Y: int] :
% 3.36/1.74 ( ( ord_less_int @ X @ Y )
% 3.36/1.74 | ( X = Y )
% 3.36/1.74 | ( ord_less_int @ Y @ X ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1050_zless__le,axiom,
% 3.36/1.74 ! [Z: int,W: int] :
% 3.36/1.74 ( ( ord_less_int @ Z @ W )
% 3.36/1.74 <=> ( ( ord_less_eq_int @ Z @ W )
% 3.36/1.74 & ( Z != W ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1051_zadd__strict__right__mono,axiom,
% 3.36/1.74 ! [K: int,I_1: int,J: int] :
% 3.36/1.74 ( ( ord_less_int @ I_1 @ J )
% 3.36/1.74 => ( ord_less_int @ ( plus_plus_int @ I_1 @ K ) @ ( plus_plus_int @ J @ K ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1052_zadd__zless__mono,axiom,
% 3.36/1.74 ! [Z_3: int,Z: int,W_1: int,W: int] :
% 3.36/1.74 ( ( ord_less_int @ W_1 @ W )
% 3.36/1.74 => ( ( ord_less_eq_int @ Z_3 @ Z )
% 3.36/1.74 => ( ord_less_int @ ( plus_plus_int @ W_1 @ Z_3 ) @ ( plus_plus_int @ W @ Z ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1053_zle__add1__eq__le,axiom,
% 3.36/1.74 ! [W: int,Z: int] :
% 3.36/1.74 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 3.36/1.74 <=> ( ord_less_eq_int @ W @ Z ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1054_add1__zle__eq,axiom,
% 3.36/1.74 ! [W: int,Z: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
% 3.36/1.74 <=> ( ord_less_int @ W @ Z ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1055_zless__imp__add1__zle,axiom,
% 3.36/1.74 ! [W: int,Z: int] :
% 3.36/1.74 ( ( ord_less_int @ W @ Z )
% 3.36/1.74 => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1056_zless__add1__eq,axiom,
% 3.36/1.74 ! [W: int,Z: int] :
% 3.36/1.74 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 3.36/1.74 <=> ( ( ord_less_int @ W @ Z )
% 3.36/1.74 | ( W = Z ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1057_nat__mult__eq__cancel__disj,axiom,
% 3.36/1.74 ! [K: nat,M: nat,N: nat] :
% 3.36/1.74 ( ( ( times_times_nat @ K @ M )
% 3.36/1.74 = ( times_times_nat @ K @ N ) )
% 3.36/1.74 <=> ( ( K = zero_zero_nat )
% 3.36/1.74 | ( M = N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1058_left__add__mult__distrib,axiom,
% 3.36/1.74 ! [I_1: nat,U: nat,J: nat,K: nat] :
% 3.36/1.74 ( ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
% 3.36/1.74 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I_1 @ J ) @ U ) @ K ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1059_nat__mult__eq__cancel1,axiom,
% 3.36/1.74 ! [M: nat,N: nat,K: nat] :
% 3.36/1.74 ( ( ord_less_nat @ zero_zero_nat @ K )
% 3.36/1.74 => ( ( ( times_times_nat @ K @ M )
% 3.36/1.74 = ( times_times_nat @ K @ N ) )
% 3.36/1.74 <=> ( M = N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1060_nat__mult__less__cancel1,axiom,
% 3.36/1.74 ! [M: nat,N: nat,K: nat] :
% 3.36/1.74 ( ( ord_less_nat @ zero_zero_nat @ K )
% 3.36/1.74 => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 3.36/1.74 <=> ( ord_less_nat @ M @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1061_nat__mult__le__cancel1,axiom,
% 3.36/1.74 ! [M: nat,N: nat,K: nat] :
% 3.36/1.74 ( ( ord_less_nat @ zero_zero_nat @ K )
% 3.36/1.74 => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 3.36/1.74 <=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1062_nat__le__add__iff1,axiom,
% 3.36/1.74 ! [U: nat,M: nat,N: nat,J: nat,I_1: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ J @ I_1 )
% 3.36/1.74 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 3.36/1.74 <=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1063_nat__diff__add__eq1,axiom,
% 3.36/1.74 ! [U: nat,M: nat,N: nat,J: nat,I_1: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ J @ I_1 )
% 3.36/1.74 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 3.36/1.74 = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1064_nat__eq__add__iff1,axiom,
% 3.36/1.74 ! [U: nat,M: nat,N: nat,J: nat,I_1: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ J @ I_1 )
% 3.36/1.74 => ( ( ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M )
% 3.36/1.74 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 3.36/1.74 <=> ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J ) @ U ) @ M )
% 3.36/1.74 = N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1065_nat__le__add__iff2,axiom,
% 3.36/1.74 ! [U: nat,M: nat,N: nat,I_1: nat,J: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ I_1 @ J )
% 3.36/1.74 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 3.36/1.74 <=> ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I_1 ) @ U ) @ N ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1066_nat__diff__add__eq2,axiom,
% 3.36/1.74 ! [U: nat,M: nat,N: nat,I_1: nat,J: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ I_1 @ J )
% 3.36/1.74 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 3.36/1.74 = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I_1 ) @ U ) @ N ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1067_nat__eq__add__iff2,axiom,
% 3.36/1.74 ! [U: nat,M: nat,N: nat,I_1: nat,J: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ I_1 @ J )
% 3.36/1.74 => ( ( ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M )
% 3.36/1.74 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 3.36/1.74 <=> ( M
% 3.36/1.74 = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I_1 ) @ U ) @ N ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1068_nat__less__add__iff1,axiom,
% 3.36/1.74 ! [U: nat,M: nat,N: nat,J: nat,I_1: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ J @ I_1 )
% 3.36/1.74 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I_1 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 3.36/1.74 <=> ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I_1 @ J ) @ U ) @ M ) @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1069_zdiv__mono2__neg__lemma,axiom,
% 3.36/1.74 ! [B_2: int,Q_1: int,R_1: int,B_1: int,Q: int,R: int] :
% 3.36/1.74 ( ( ( plus_plus_int @ ( times_times_int @ B_2 @ Q_1 ) @ R_1 )
% 3.36/1.74 = ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R ) )
% 3.36/1.74 => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R ) @ zero_zero_int )
% 3.36/1.74 => ( ( ord_less_int @ R_1 @ B_2 )
% 3.36/1.74 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 3.36/1.74 => ( ( ord_less_int @ zero_zero_int @ B_1 )
% 3.36/1.74 => ( ( ord_less_eq_int @ B_1 @ B_2 )
% 3.36/1.74 => ( ord_less_eq_int @ Q @ Q_1 ) ) ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1070_unique__quotient__lemma__neg,axiom,
% 3.36/1.74 ! [B_2: int,Q: int,R: int,Q_1: int,R_1: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B_2 @ Q ) @ R ) @ ( plus_plus_int @ ( times_times_int @ B_2 @ Q_1 ) @ R_1 ) )
% 3.36/1.74 => ( ( ord_less_eq_int @ R_1 @ zero_zero_int )
% 3.36/1.74 => ( ( ord_less_int @ B_2 @ R_1 )
% 3.36/1.74 => ( ( ord_less_int @ B_2 @ R )
% 3.36/1.74 => ( ord_less_eq_int @ Q_1 @ Q ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1071_number__of__is__id,axiom,
% 3.36/1.74 ! [K: int] :
% 3.36/1.74 ( ( number_number_of_int @ K )
% 3.36/1.74 = K ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1072_int__0__neq__1,axiom,
% 3.36/1.74 zero_zero_int != one_one_int ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1073_self__quotient__aux1,axiom,
% 3.36/1.74 ! [R_1: int,Q_1: int,A_1: int] :
% 3.36/1.74 ( ( ord_less_int @ zero_zero_int @ A_1 )
% 3.36/1.74 => ( ( A_1
% 3.36/1.74 = ( plus_plus_int @ R_1 @ ( times_times_int @ A_1 @ Q_1 ) ) )
% 3.36/1.74 => ( ( ord_less_int @ R_1 @ A_1 )
% 3.36/1.74 => ( ord_less_eq_int @ one_one_int @ Q_1 ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1074_self__quotient__aux2,axiom,
% 3.36/1.74 ! [R_1: int,Q_1: int,A_1: int] :
% 3.36/1.74 ( ( ord_less_int @ zero_zero_int @ A_1 )
% 3.36/1.74 => ( ( A_1
% 3.36/1.74 = ( plus_plus_int @ R_1 @ ( times_times_int @ A_1 @ Q_1 ) ) )
% 3.36/1.74 => ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
% 3.36/1.74 => ( ord_less_eq_int @ Q_1 @ one_one_int ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1075_q__pos__lemma,axiom,
% 3.36/1.74 ! [B_1: int,Q: int,R: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R ) )
% 3.36/1.74 => ( ( ord_less_int @ R @ B_1 )
% 3.36/1.74 => ( ( ord_less_int @ zero_zero_int @ B_1 )
% 3.36/1.74 => ( ord_less_eq_int @ zero_zero_int @ Q ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1076_q__neg__lemma,axiom,
% 3.36/1.74 ! [B_1: int,Q: int,R: int] :
% 3.36/1.74 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R ) @ zero_zero_int )
% 3.36/1.74 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 3.36/1.74 => ( ( ord_less_int @ zero_zero_int @ B_1 )
% 3.36/1.74 => ( ord_less_eq_int @ Q @ zero_zero_int ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1077_unique__quotient__lemma,axiom,
% 3.36/1.74 ! [B_2: int,Q: int,R: int,Q_1: int,R_1: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B_2 @ Q ) @ R ) @ ( plus_plus_int @ ( times_times_int @ B_2 @ Q_1 ) @ R_1 ) )
% 3.36/1.74 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 3.36/1.74 => ( ( ord_less_int @ R @ B_2 )
% 3.36/1.74 => ( ( ord_less_int @ R_1 @ B_2 )
% 3.36/1.74 => ( ord_less_eq_int @ Q @ Q_1 ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1078_zdiv__mono2__lemma,axiom,
% 3.36/1.74 ! [B_2: int,Q_1: int,R_1: int,B_1: int,Q: int,R: int] :
% 3.36/1.74 ( ( ( plus_plus_int @ ( times_times_int @ B_2 @ Q_1 ) @ R_1 )
% 3.36/1.74 = ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R ) )
% 3.36/1.74 => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R ) )
% 3.36/1.74 => ( ( ord_less_int @ R @ B_1 )
% 3.36/1.74 => ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
% 3.36/1.74 => ( ( ord_less_int @ zero_zero_int @ B_1 )
% 3.36/1.74 => ( ( ord_less_eq_int @ B_1 @ B_2 )
% 3.36/1.74 => ( ord_less_eq_int @ Q_1 @ Q ) ) ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1079_int__less__induct,axiom,
% 3.36/1.74 ! [P: int > $o,I_1: int,K: int] :
% 3.36/1.74 ( ( ord_less_int @ I_1 @ K )
% 3.36/1.74 => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 3.36/1.74 => ( ! [I: int] :
% 3.36/1.74 ( ( ord_less_int @ I @ K )
% 3.36/1.74 => ( ( P @ I )
% 3.36/1.74 => ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
% 3.36/1.74 => ( P @ I_1 ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1080_int__le__induct,axiom,
% 3.36/1.74 ! [P: int > $o,I_1: int,K: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ I_1 @ K )
% 3.36/1.74 => ( ( P @ K )
% 3.36/1.74 => ( ! [I: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ I @ K )
% 3.36/1.74 => ( ( P @ I )
% 3.36/1.74 => ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
% 3.36/1.74 => ( P @ I_1 ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1081_int__gr__induct,axiom,
% 3.36/1.74 ! [P: int > $o,K: int,I_1: int] :
% 3.36/1.74 ( ( ord_less_int @ K @ I_1 )
% 3.36/1.74 => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
% 3.36/1.74 => ( ! [I: int] :
% 3.36/1.74 ( ( ord_less_int @ K @ I )
% 3.36/1.74 => ( ( P @ I )
% 3.36/1.74 => ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
% 3.36/1.74 => ( P @ I_1 ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1082_int__ge__induct,axiom,
% 3.36/1.74 ! [P: int > $o,K: int,I_1: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ K @ I_1 )
% 3.36/1.74 => ( ( P @ K )
% 3.36/1.74 => ( ! [I: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ K @ I )
% 3.36/1.74 => ( ( P @ I )
% 3.36/1.74 => ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
% 3.36/1.74 => ( P @ I_1 ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1083_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
% 3.36/1.74 ! [Y: int,X: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 3.36/1.74 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 3.36/1.74 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ X @ Y ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1084_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
% 3.36/1.74 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1085_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
% 3.36/1.74 ! [Y: int,X: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 3.36/1.74 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 3.36/1.74 => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ X @ Y ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1086_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
% 3.36/1.74 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1087_transfer__nat__int__set__cong,axiom,
% 3.36/1.74 ! [P_1: int > $o,P: int > $o] :
% 3.36/1.74 ( ! [X_1: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ zero_zero_int @ X_1 )
% 3.36/1.74 => ( ( P @ X_1 )
% 3.36/1.74 <=> ( P_1 @ X_1 ) ) )
% 3.36/1.74 => ( ( collect_int
% 3.36/1.74 @ ^ [X_1: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ X_1 ) @ ( P @ X_1 ) ) )
% 3.36/1.74 = ( collect_int
% 3.36/1.74 @ ^ [X_1: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ X_1 ) @ ( P_1 @ X_1 ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1088_decr__mult__lemma,axiom,
% 3.36/1.74 ! [K: int,P: int > $o,D: int] :
% 3.36/1.74 ( ( ord_less_int @ zero_zero_int @ D )
% 3.36/1.74 => ( ! [X_1: int] :
% 3.36/1.74 ( ( P @ X_1 )
% 3.36/1.74 => ( P @ ( minus_minus_int @ X_1 @ D ) ) )
% 3.36/1.74 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 3.36/1.74 => ! [X_1: int] :
% 3.36/1.74 ( ( P @ X_1 )
% 3.36/1.74 => ( P @ ( minus_minus_int @ X_1 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1089_conj__le__cong,axiom,
% 3.36/1.74 ! [P_1: $o,P: $o,X: int] :
% 3.36/1.74 ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 3.36/1.74 => ( P
% 3.36/1.74 <=> P_1 ) )
% 3.36/1.74 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 3.36/1.74 & P )
% 3.36/1.74 <=> ( ( ord_less_eq_int @ zero_zero_int @ X )
% 3.36/1.74 & P_1 ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1090_imp__le__cong,axiom,
% 3.36/1.74 ! [P_1: $o,P: $o,X: int] :
% 3.36/1.74 ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 3.36/1.74 => ( P
% 3.36/1.74 <=> P_1 ) )
% 3.36/1.74 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 3.36/1.74 => P )
% 3.36/1.74 <=> ( ( ord_less_eq_int @ zero_zero_int @ X )
% 3.36/1.74 => P_1 ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1091_incr__mult__lemma,axiom,
% 3.36/1.74 ! [K: int,P: int > $o,D: int] :
% 3.36/1.74 ( ( ord_less_int @ zero_zero_int @ D )
% 3.36/1.74 => ( ! [X_1: int] :
% 3.36/1.74 ( ( P @ X_1 )
% 3.36/1.74 => ( P @ ( plus_plus_int @ X_1 @ D ) ) )
% 3.36/1.74 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 3.36/1.74 => ! [X_1: int] :
% 3.36/1.74 ( ( P @ X_1 )
% 3.36/1.74 => ( P @ ( plus_plus_int @ X_1 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1092_int__induct,axiom,
% 3.36/1.74 ! [I_1: int,P: int > $o,K: int] :
% 3.36/1.74 ( ( P @ K )
% 3.36/1.74 => ( ! [I: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ K @ I )
% 3.36/1.74 => ( ( P @ I )
% 3.36/1.74 => ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
% 3.36/1.74 => ( ! [I: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ I @ K )
% 3.36/1.74 => ( ( P @ I )
% 3.36/1.74 => ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
% 3.36/1.74 => ( P @ I_1 ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1093_minusinfinity,axiom,
% 3.36/1.74 ! [P: int > $o,P1: int > $o,D: int] :
% 3.36/1.74 ( ( ord_less_int @ zero_zero_int @ D )
% 3.36/1.74 => ( ! [X_1: int,K_1: int] :
% 3.36/1.74 ( ( P1 @ X_1 )
% 3.36/1.74 <=> ( P1 @ ( minus_minus_int @ X_1 @ ( times_times_int @ K_1 @ D ) ) ) )
% 3.36/1.74 => ( ? [Z_2: int] :
% 3.36/1.74 ! [X_1: int] :
% 3.36/1.74 ( ( ord_less_int @ X_1 @ Z_2 )
% 3.36/1.74 => ( ( P @ X_1 )
% 3.36/1.74 <=> ( P1 @ X_1 ) ) )
% 3.36/1.74 => ( ( ex @ P1 )
% 3.36/1.74 => ( ex @ P ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1094_plusinfinity,axiom,
% 3.36/1.74 ! [P: int > $o,P_1: int > $o,D: int] :
% 3.36/1.74 ( ( ord_less_int @ zero_zero_int @ D )
% 3.36/1.74 => ( ! [X_1: int,K_1: int] :
% 3.36/1.74 ( ( P_1 @ X_1 )
% 3.36/1.74 <=> ( P_1 @ ( minus_minus_int @ X_1 @ ( times_times_int @ K_1 @ D ) ) ) )
% 3.36/1.74 => ( ? [Z_2: int] :
% 3.36/1.74 ! [X_1: int] :
% 3.36/1.74 ( ( ord_less_int @ Z_2 @ X_1 )
% 3.36/1.74 => ( ( P @ X_1 )
% 3.36/1.74 <=> ( P_1 @ X_1 ) ) )
% 3.36/1.74 => ( ( ex @ P_1 )
% 3.36/1.74 => ( ex @ P ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1095_tsub__def,axiom,
% 3.36/1.74 ! [Y: int,X: int] :
% 3.36/1.74 ( ( ( ord_less_eq_int @ Y @ X )
% 3.36/1.74 => ( ( nat_tsub @ X @ Y )
% 3.36/1.74 = ( minus_minus_int @ X @ Y ) ) )
% 3.36/1.74 & ( ~ ( ord_less_eq_int @ Y @ X )
% 3.36/1.74 => ( ( nat_tsub @ X @ Y )
% 3.36/1.74 = zero_zero_int ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1096_zero__zle__int,axiom,
% 3.36/1.74 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1621563631at_int @ N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1097_int__less__0__conv,axiom,
% 3.36/1.74 ! [K: nat] :
% 3.36/1.74 ~ ( ord_less_int @ ( semiri1621563631at_int @ K ) @ zero_zero_int ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1098_int__1,axiom,
% 3.36/1.74 ( ( semiri1621563631at_int @ one_one_nat )
% 3.36/1.74 = one_one_int ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1099_int__0,axiom,
% 3.36/1.74 ( ( semiri1621563631at_int @ zero_zero_nat )
% 3.36/1.74 = zero_zero_int ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1100_int__eq__0__conv,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ( ( ( semiri1621563631at_int @ N )
% 3.36/1.74 = zero_zero_int )
% 3.36/1.74 <=> ( N = zero_zero_nat ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1101_zless__int,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
% 3.36/1.74 <=> ( ord_less_nat @ M @ N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1102_zle__int,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ord_less_eq_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
% 3.36/1.74 <=> ( ord_less_eq_nat @ M @ N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1103_zadd__int,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( plus_plus_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
% 3.36/1.74 = ( semiri1621563631at_int @ ( plus_plus_nat @ M @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1104_zadd__int__left,axiom,
% 3.36/1.74 ! [M: nat,N: nat,Z: int] :
% 3.36/1.74 ( ( plus_plus_int @ ( semiri1621563631at_int @ M ) @ ( plus_plus_int @ ( semiri1621563631at_int @ N ) @ Z ) )
% 3.36/1.74 = ( plus_plus_int @ ( semiri1621563631at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1105_zle__iff__zadd,axiom,
% 3.36/1.74 ! [W: int,Z: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ W @ Z )
% 3.36/1.74 <=> ? [N_1: nat] :
% 3.36/1.74 ( Z
% 3.36/1.74 = ( plus_plus_int @ W @ ( semiri1621563631at_int @ N_1 ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1106_zmult__int,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( times_times_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
% 3.36/1.74 = ( semiri1621563631at_int @ ( times_times_nat @ M @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1107_int__mult,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( semiri1621563631at_int @ ( times_times_nat @ M @ N ) )
% 3.36/1.74 = ( times_times_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1108_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,
% 3.36/1.74 ! [X: nat,Y: nat] :
% 3.36/1.74 ( ( times_times_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
% 3.36/1.74 = ( semiri1621563631at_int @ ( times_times_nat @ X @ Y ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1109_transfer__int__nat__relations_I2_J,axiom,
% 3.36/1.74 ! [X: nat,Y: nat] :
% 3.36/1.74 ( ( ord_less_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
% 3.36/1.74 <=> ( ord_less_nat @ X @ Y ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1110_transfer__int__nat__relations_I3_J,axiom,
% 3.36/1.74 ! [X: nat,Y: nat] :
% 3.36/1.74 ( ( ord_less_eq_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
% 3.36/1.74 <=> ( ord_less_eq_nat @ X @ Y ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1111_Nat__Transfer_Otransfer__int__nat__functions_I3_J,axiom,
% 3.36/1.74 ! [X: nat,Y: nat] :
% 3.36/1.74 ( ( nat_tsub @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
% 3.36/1.74 = ( semiri1621563631at_int @ ( minus_minus_nat @ X @ Y ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1112_Nat__Transfer_Otransfer__int__nat__functions_I1_J,axiom,
% 3.36/1.74 ! [X: nat,Y: nat] :
% 3.36/1.74 ( ( plus_plus_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
% 3.36/1.74 = ( semiri1621563631at_int @ ( plus_plus_nat @ X @ Y ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1113_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J,axiom,
% 3.36/1.74 ! [A: nat > $o] :
% 3.36/1.74 ( ( finite_card_nat @ A )
% 3.36/1.74 = ( finite_card_int @ ( image_nat_int @ semiri1621563631at_int @ A ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1114_transfer__nat__int__set__relations_I1_J,axiom,
% 3.36/1.74 ! [A: nat > $o] :
% 3.36/1.74 ( ( finite_finite_nat @ A )
% 3.36/1.74 <=> ( finite_finite_int @ ( image_nat_int @ semiri1621563631at_int @ A ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1115_transfer__int__nat__numerals_I2_J,axiom,
% 3.36/1.74 ( one_one_int
% 3.36/1.74 = ( semiri1621563631at_int @ one_one_nat ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1116_transfer__int__nat__numerals_I1_J,axiom,
% 3.36/1.74 ( zero_zero_int
% 3.36/1.74 = ( semiri1621563631at_int @ zero_zero_nat ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1117_Nat__Transfer_Otransfer__int__nat__set__functions_I5_J,axiom,
% 3.36/1.74 ! [P: int > $o] :
% 3.36/1.74 ( ( collect_int
% 3.36/1.74 @ ^ [X_1: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ X_1 ) @ ( P @ X_1 ) ) )
% 3.36/1.74 = ( image_nat_int @ semiri1621563631at_int
% 3.36/1.74 @ ( collect_nat
% 3.36/1.74 @ ^ [X_1: nat] : ( P @ ( semiri1621563631at_int @ X_1 ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1118_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
% 3.36/1.74 ! [Z: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1621563631at_int @ Z ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1119_transfer__int__nat__quantifiers_I2_J,axiom,
% 3.36/1.74 ! [P: int > $o] :
% 3.36/1.74 ( ? [X_1: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ zero_zero_int @ X_1 )
% 3.36/1.74 & ( P @ X_1 ) )
% 3.36/1.74 <=> ? [X_1: nat] : ( P @ ( semiri1621563631at_int @ X_1 ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1120_transfer__int__nat__quantifiers_I1_J,axiom,
% 3.36/1.74 ! [P: int > $o] :
% 3.36/1.74 ( ! [X_1: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ zero_zero_int @ X_1 )
% 3.36/1.74 => ( P @ X_1 ) )
% 3.36/1.74 <=> ! [X_1: nat] : ( P @ ( semiri1621563631at_int @ X_1 ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1121_int__le__0__conv,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ( ( ord_less_eq_int @ ( semiri1621563631at_int @ N ) @ zero_zero_int )
% 3.36/1.74 <=> ( N = zero_zero_nat ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1122_int__Suc0__eq__1,axiom,
% 3.36/1.74 ( ( semiri1621563631at_int @ ( suc @ zero_zero_nat ) )
% 3.36/1.74 = one_one_int ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1123_zless__iff__Suc__zadd,axiom,
% 3.36/1.74 ! [W: int,Z: int] :
% 3.36/1.74 ( ( ord_less_int @ W @ Z )
% 3.36/1.74 <=> ? [N_1: nat] :
% 3.36/1.74 ( Z
% 3.36/1.74 = ( plus_plus_int @ W @ ( semiri1621563631at_int @ ( suc @ N_1 ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1124_int__Suc,axiom,
% 3.36/1.74 ! [M: nat] :
% 3.36/1.74 ( ( semiri1621563631at_int @ ( suc @ M ) )
% 3.36/1.74 = ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ M ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1125_zdiff__int,axiom,
% 3.36/1.74 ! [N: nat,M: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ N @ M )
% 3.36/1.74 => ( ( minus_minus_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
% 3.36/1.74 = ( semiri1621563631at_int @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1126_zero__less__int__conv,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ( ( ord_less_int @ zero_zero_int @ ( semiri1621563631at_int @ N ) )
% 3.36/1.74 <=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1127_zmult__zless__mono2__lemma,axiom,
% 3.36/1.74 ! [K: nat,I_1: int,J: int] :
% 3.36/1.74 ( ( ord_less_int @ I_1 @ J )
% 3.36/1.74 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 3.36/1.74 => ( ord_less_int @ ( times_times_int @ ( semiri1621563631at_int @ K ) @ I_1 ) @ ( times_times_int @ ( semiri1621563631at_int @ K ) @ J ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1128_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J,axiom,
% 3.36/1.74 ! [Y: int,X: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 3.36/1.74 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 3.36/1.74 => ( ord_less_eq_int @ zero_zero_int @ ( nat_tsub @ X @ Y ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1129_zdiff__int__split,axiom,
% 3.36/1.74 ! [P: int > $o,X: nat,Y: nat] :
% 3.36/1.74 ( ( P @ ( semiri1621563631at_int @ ( minus_minus_nat @ X @ Y ) ) )
% 3.36/1.74 <=> ( ( ( ord_less_eq_nat @ Y @ X )
% 3.36/1.74 => ( P @ ( minus_minus_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) ) ) )
% 3.36/1.74 & ( ( ord_less_nat @ X @ Y )
% 3.36/1.74 => ( P @ zero_zero_int ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1130_tsub__eq,axiom,
% 3.36/1.74 ! [Y: int,X: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ Y @ X )
% 3.36/1.74 => ( ( nat_tsub @ X @ Y )
% 3.36/1.74 = ( minus_minus_int @ X @ Y ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1131_zero__less__imp__eq__int,axiom,
% 3.36/1.74 ! [K: int] :
% 3.36/1.74 ( ( ord_less_int @ zero_zero_int @ K )
% 3.36/1.74 => ? [N_1: nat] :
% 3.36/1.74 ( ( ord_less_nat @ zero_zero_nat @ N_1 )
% 3.36/1.74 & ( K
% 3.36/1.74 = ( semiri1621563631at_int @ N_1 ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1132_int__int__eq,axiom,
% 3.36/1.74 ! [M: nat,N: nat] :
% 3.36/1.74 ( ( ( semiri1621563631at_int @ M )
% 3.36/1.74 = ( semiri1621563631at_int @ N ) )
% 3.36/1.74 <=> ( M = N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1133_transfer__nat__int__set__relations_I4_J,axiom,
% 3.36/1.74 ! [A: nat > $o,B: nat > $o] :
% 3.36/1.74 ( ( ord_less_nat_o @ A @ B )
% 3.36/1.74 <=> ( ord_less_int_o @ ( image_nat_int @ semiri1621563631at_int @ A ) @ ( image_nat_int @ semiri1621563631at_int @ B ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1134_transfer__nat__int__set__relations_I5_J,axiom,
% 3.36/1.74 ! [A: nat > $o,B: nat > $o] :
% 3.36/1.74 ( ( ord_less_eq_nat_o @ A @ B )
% 3.36/1.74 <=> ( ord_less_eq_int_o @ ( image_nat_int @ semiri1621563631at_int @ A ) @ ( image_nat_int @ semiri1621563631at_int @ B ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1135_Nat__Transfer_Otransfer__int__nat__set__functions_I2_J,axiom,
% 3.36/1.74 ( bot_bot_int_o
% 3.36/1.74 = ( image_nat_int @ semiri1621563631at_int @ bot_bot_nat_o ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1136_transfer__nat__int__set__relations_I3_J,axiom,
% 3.36/1.74 ! [A: nat > $o,B: nat > $o] :
% 3.36/1.74 ( ( A = B )
% 3.36/1.74 <=> ( ( image_nat_int @ semiri1621563631at_int @ A )
% 3.36/1.74 = ( image_nat_int @ semiri1621563631at_int @ B ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1137_transfer__nat__int__set__relations_I2_J,axiom,
% 3.36/1.74 ! [X: nat,A: nat > $o] :
% 3.36/1.74 ( ( member_nat @ X @ A )
% 3.36/1.74 <=> ( member_int @ ( semiri1621563631at_int @ X ) @ ( image_nat_int @ semiri1621563631at_int @ A ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1138_int__if__cong,axiom,
% 3.36/1.74 ! [X: nat,Y: nat,P: $o] :
% 3.36/1.74 ( ( P
% 3.36/1.74 => ( ( semiri1621563631at_int @ X )
% 3.36/1.74 = ( semiri1621563631at_int @ ( if_nat @ P @ X @ Y ) ) ) )
% 3.36/1.74 & ( ~ P
% 3.36/1.74 => ( ( semiri1621563631at_int @ Y )
% 3.36/1.74 = ( semiri1621563631at_int @ ( if_nat @ P @ X @ Y ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1139_transfer__int__nat__relations_I1_J,axiom,
% 3.36/1.74 ! [X: nat,Y: nat] :
% 3.36/1.74 ( ( ( semiri1621563631at_int @ X )
% 3.36/1.74 = ( semiri1621563631at_int @ Y ) )
% 3.36/1.74 <=> ( X = Y ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1140_nonneg__int__cases,axiom,
% 3.36/1.74 ! [K: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 3.36/1.74 => ~ ! [N_1: nat] :
% 3.36/1.74 ( K
% 3.36/1.74 != ( semiri1621563631at_int @ N_1 ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1141_nonneg__eq__int,axiom,
% 3.36/1.74 ! [Z: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 3.36/1.74 => ~ ! [M_1: nat] :
% 3.36/1.74 ( Z
% 3.36/1.74 != ( semiri1621563631at_int @ M_1 ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1142_zero__le__imp__eq__int,axiom,
% 3.36/1.74 ! [K: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 3.36/1.74 => ? [N_1: nat] :
% 3.36/1.74 ( K
% 3.36/1.74 = ( semiri1621563631at_int @ N_1 ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1143_int__diff__cases,axiom,
% 3.36/1.74 ! [Z: int] :
% 3.36/1.74 ~ ! [M_1: nat,N_1: nat] :
% 3.36/1.74 ( Z
% 3.36/1.74 != ( minus_minus_int @ ( semiri1621563631at_int @ M_1 ) @ ( semiri1621563631at_int @ N_1 ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1144_decr__lemma,axiom,
% 3.36/1.74 ! [X: int,Z: int,D: int] :
% 3.36/1.74 ( ( ord_less_int @ zero_zero_int @ D )
% 3.36/1.74 => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1145_zabs__less__one__iff,axiom,
% 3.36/1.74 ! [Z: int] :
% 3.36/1.74 ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
% 3.36/1.74 <=> ( Z = zero_zero_int ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1146_neg__def,axiom,
% 3.36/1.74 ! [Z_1: int] :
% 3.36/1.74 ( ( nat_neg @ Z_1 )
% 3.36/1.74 <=> ( ord_less_int @ Z_1 @ zero_zero_int ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1147_not__neg__eq__ge__0,axiom,
% 3.36/1.74 ! [X: int] :
% 3.36/1.74 ( ~ ( nat_neg @ X )
% 3.36/1.74 <=> ( ord_less_eq_int @ zero_zero_int @ X ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1148_not__neg__1,axiom,
% 3.36/1.74 ~ ( nat_neg @ one_one_int ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1149_not__neg__0,axiom,
% 3.36/1.74 ~ ( nat_neg @ zero_zero_int ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1150_abs__zmult__eq__1,axiom,
% 3.36/1.74 ! [M: int,N: int] :
% 3.36/1.74 ( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
% 3.36/1.74 = one_one_int )
% 3.36/1.74 => ( ( abs_abs_int @ M )
% 3.36/1.74 = one_one_int ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1151_not__neg__int,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ~ ( nat_neg @ ( semiri1621563631at_int @ N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1152_abs__int__eq,axiom,
% 3.36/1.74 ! [M: nat] :
% 3.36/1.74 ( ( abs_abs_int @ ( semiri1621563631at_int @ M ) )
% 3.36/1.74 = ( semiri1621563631at_int @ M ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1153_neg__imp__number__of__eq__0,axiom,
% 3.36/1.74 ! [V: int] :
% 3.36/1.74 ( ( nat_neg @ ( number_number_of_int @ V ) )
% 3.36/1.74 => ( ( number_number_of_nat @ V )
% 3.36/1.74 = zero_zero_nat ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1154_eq__nat__number__of,axiom,
% 3.36/1.74 ! [V: int,V_1: int] :
% 3.36/1.74 ( ( ( number_number_of_nat @ V )
% 3.36/1.74 = ( number_number_of_nat @ V_1 ) )
% 3.36/1.74 <=> ( ( ( nat_neg @ ( number_number_of_int @ V ) )
% 3.36/1.74 => ( ord_less_eq_int @ ( number_number_of_int @ V_1 ) @ zero_zero_int ) )
% 3.36/1.74 & ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
% 3.36/1.74 => ( ( ( nat_neg @ ( number_number_of_int @ V_1 ) )
% 3.36/1.74 => ( ( number_number_of_int @ V )
% 3.36/1.74 = zero_zero_int ) )
% 3.36/1.74 & ( ~ ( nat_neg @ ( number_number_of_int @ V_1 ) )
% 3.36/1.74 => ( V = V_1 ) ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1155_nat__number__of__add__left,axiom,
% 3.36/1.74 ! [V_1: int,K: nat,V: int] :
% 3.36/1.74 ( ( ( nat_neg @ ( number_number_of_int @ V ) )
% 3.36/1.74 => ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K ) )
% 3.36/1.74 = ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K ) ) )
% 3.36/1.74 & ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
% 3.36/1.74 => ( ( ( nat_neg @ ( number_number_of_int @ V_1 ) )
% 3.36/1.74 => ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K ) )
% 3.36/1.74 = ( plus_plus_nat @ ( number_number_of_nat @ V ) @ K ) ) )
% 3.36/1.74 & ( ~ ( nat_neg @ ( number_number_of_int @ V_1 ) )
% 3.36/1.74 => ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K ) )
% 3.36/1.74 = ( plus_plus_nat @ ( number_number_of_nat @ ( plus_plus_int @ V @ V_1 ) ) @ K ) ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1156_int__nat__number__of,axiom,
% 3.36/1.74 ! [V: int] :
% 3.36/1.74 ( ( ( nat_neg @ ( number_number_of_int @ V ) )
% 3.36/1.74 => ( ( semiri1621563631at_int @ ( number_number_of_nat @ V ) )
% 3.36/1.74 = zero_zero_int ) )
% 3.36/1.74 & ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
% 3.36/1.74 => ( ( semiri1621563631at_int @ ( number_number_of_nat @ V ) )
% 3.36/1.74 = ( number_number_of_int @ V ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1157_incr__lemma,axiom,
% 3.36/1.74 ! [Z: int,X: int,D: int] :
% 3.36/1.74 ( ( ord_less_int @ zero_zero_int @ D )
% 3.36/1.74 => ( 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 ) @ D ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1158_int__val__lemma,axiom,
% 3.36/1.74 ! [K: int,F: nat > int,N: nat] :
% 3.36/1.74 ( ! [I: nat] :
% 3.36/1.74 ( ( ord_less_nat @ I @ N )
% 3.36/1.74 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( F @ I ) ) ) @ one_one_int ) )
% 3.36/1.74 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 3.36/1.74 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 3.36/1.74 => ? [I: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ I @ N )
% 3.36/1.74 & ( ( F @ I )
% 3.36/1.74 = K ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1159_nat0__intermed__int__val,axiom,
% 3.36/1.74 ! [K: int,F: nat > int,N: nat] :
% 3.36/1.74 ( ! [I: nat] :
% 3.36/1.74 ( ( ord_less_nat @ I @ N )
% 3.36/1.74 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( F @ I ) ) ) @ one_one_int ) )
% 3.36/1.74 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 3.36/1.74 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 3.36/1.74 => ? [I: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ I @ N )
% 3.36/1.74 & ( ( F @ I )
% 3.36/1.74 = K ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1160_nat__intermed__int__val,axiom,
% 3.36/1.74 ! [K: int,F: nat > int,N: nat,M: nat] :
% 3.36/1.74 ( ! [I: nat] :
% 3.36/1.74 ( ( ( ord_less_eq_nat @ M @ I )
% 3.36/1.74 & ( ord_less_nat @ I @ N ) )
% 3.36/1.74 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( F @ I ) ) ) @ one_one_int ) )
% 3.36/1.74 => ( ( ord_less_nat @ M @ N )
% 3.36/1.74 => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 3.36/1.74 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 3.36/1.74 => ? [I: nat] :
% 3.36/1.74 ( ( ord_less_eq_nat @ M @ I )
% 3.36/1.74 & ( ord_less_eq_nat @ I @ N )
% 3.36/1.74 & ( ( F @ I )
% 3.36/1.74 = K ) ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1161_Suc__nat__number__of__add,axiom,
% 3.36/1.74 ! [N: nat,V: int] :
% 3.36/1.74 ( ( ( nat_neg @ ( number_number_of_int @ V ) )
% 3.36/1.74 => ( ( suc @ ( plus_plus_nat @ ( number_number_of_nat @ V ) @ N ) )
% 3.36/1.74 = ( plus_plus_nat @ one_one_nat @ N ) ) )
% 3.36/1.74 & ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
% 3.36/1.74 => ( ( suc @ ( plus_plus_nat @ ( number_number_of_nat @ V ) @ N ) )
% 3.36/1.74 = ( plus_plus_nat @ ( number_number_of_nat @ ( succ @ V ) ) @ N ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1162_succ__def,axiom,
% 3.36/1.74 ! [K: int] :
% 3.36/1.74 ( ( succ @ K )
% 3.36/1.74 = ( plus_plus_int @ K @ one_one_int ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1163_Suc__nat__number__of,axiom,
% 3.36/1.74 ! [V: int] :
% 3.36/1.74 ( ( ( nat_neg @ ( number_number_of_int @ V ) )
% 3.36/1.74 => ( ( suc @ ( number_number_of_nat @ V ) )
% 3.36/1.74 = one_one_nat ) )
% 3.36/1.74 & ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
% 3.36/1.74 => ( ( suc @ ( number_number_of_nat @ V ) )
% 3.36/1.74 = ( number_number_of_nat @ ( succ @ V ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1164_nat__number__of__Bit1,axiom,
% 3.36/1.74 ! [W: int] :
% 3.36/1.74 ( ( ( nat_neg @ ( number_number_of_int @ W ) )
% 3.36/1.74 => ( ( number_number_of_nat @ ( bit1 @ W ) )
% 3.36/1.74 = zero_zero_nat ) )
% 3.36/1.74 & ( ~ ( nat_neg @ ( number_number_of_int @ W ) )
% 3.36/1.74 => ( ( number_number_of_nat @ ( bit1 @ W ) )
% 3.36/1.74 = ( suc @ ( plus_plus_nat @ ( number_number_of_nat @ W ) @ ( number_number_of_nat @ W ) ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1165_nat__1__add__number__of,axiom,
% 3.36/1.74 ! [V: int] :
% 3.36/1.74 ( ( ( ord_less_int @ V @ pls )
% 3.36/1.74 => ( ( plus_plus_nat @ one_one_nat @ ( number_number_of_nat @ V ) )
% 3.36/1.74 = one_one_nat ) )
% 3.36/1.74 & ( ~ ( ord_less_int @ V @ pls )
% 3.36/1.74 => ( ( plus_plus_nat @ one_one_nat @ ( number_number_of_nat @ V ) )
% 3.36/1.74 = ( number_number_of_nat @ ( succ @ V ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1166_succ__Pls,axiom,
% 3.36/1.74 ( ( succ @ pls )
% 3.36/1.74 = ( bit1 @ pls ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1167_neg__number__of__Bit1,axiom,
% 3.36/1.74 ! [W: int] :
% 3.36/1.74 ( ( nat_neg @ ( number_number_of_int @ ( bit1 @ W ) ) )
% 3.36/1.74 <=> ( nat_neg @ ( number_number_of_int @ W ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1168_not__neg__number__of__Pls,axiom,
% 3.36/1.74 ~ ( nat_neg @ ( number_number_of_int @ pls ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1169_transfer__int__nat__numerals_I4_J,axiom,
% 3.36/1.74 ( ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) )
% 3.36/1.74 = ( semiri1621563631at_int @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1170_numeral__1__eq__Suc__0,axiom,
% 3.36/1.74 ( ( number_number_of_nat @ ( bit1 @ pls ) )
% 3.36/1.74 = ( suc @ zero_zero_nat ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1171_numeral__3__eq__3,axiom,
% 3.36/1.74 ( ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) )
% 3.36/1.74 = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1172_Suc3__eq__add__3,axiom,
% 3.36/1.74 ! [N: nat] :
% 3.36/1.74 ( ( suc @ ( suc @ ( suc @ N ) ) )
% 3.36/1.74 = ( plus_plus_nat @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) @ N ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1173_Numeral1__eq1__nat,axiom,
% 3.36/1.74 ( one_one_nat
% 3.36/1.74 = ( number_number_of_nat @ ( bit1 @ pls ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1174_nat__numeral__1__eq__1,axiom,
% 3.36/1.74 ( ( number_number_of_nat @ ( bit1 @ pls ) )
% 3.36/1.74 = one_one_nat ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1175_one__is__num__one,axiom,
% 3.36/1.74 ( one_one_int
% 3.36/1.74 = ( number_number_of_int @ ( bit1 @ pls ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1176_zero__is__num__zero,axiom,
% 3.36/1.74 ( zero_zero_int
% 3.36/1.74 = ( number_number_of_int @ pls ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1177_Bit1__def,axiom,
% 3.36/1.74 ! [K: int] :
% 3.36/1.74 ( ( bit1 @ K )
% 3.36/1.74 = ( plus_plus_int @ ( plus_plus_int @ one_one_int @ K ) @ K ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1178_nat__number__of__Pls,axiom,
% 3.36/1.74 ( ( number_number_of_nat @ pls )
% 3.36/1.74 = zero_zero_nat ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1179_semiring__norm_I113_J,axiom,
% 3.36/1.74 ( zero_zero_nat
% 3.36/1.74 = ( number_number_of_nat @ pls ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1180_Pls__def,axiom,
% 3.36/1.74 pls = zero_zero_int ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1181_rel__simps_I51_J,axiom,
% 3.36/1.74 ! [K: int,L: int] :
% 3.36/1.74 ( ( ( bit1 @ K )
% 3.36/1.74 = ( bit1 @ L ) )
% 3.36/1.74 <=> ( K = L ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1182_rel__simps_I46_J,axiom,
% 3.36/1.74 ! [K: int] :
% 3.36/1.74 ( ( bit1 @ K )
% 3.36/1.74 != pls ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1183_rel__simps_I39_J,axiom,
% 3.36/1.74 ! [L: int] :
% 3.36/1.74 ( pls
% 3.36/1.74 != ( bit1 @ L ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1184_diff__bin__simps_I1_J,axiom,
% 3.36/1.74 ! [K: int] :
% 3.36/1.74 ( ( minus_minus_int @ K @ pls )
% 3.36/1.74 = K ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1185_add__Pls,axiom,
% 3.36/1.74 ! [K: int] :
% 3.36/1.74 ( ( plus_plus_int @ pls @ K )
% 3.36/1.74 = K ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1186_add__Pls__right,axiom,
% 3.36/1.74 ! [K: int] :
% 3.36/1.74 ( ( plus_plus_int @ K @ pls )
% 3.36/1.74 = K ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1187_mult__Pls,axiom,
% 3.36/1.74 ! [W: int] :
% 3.36/1.74 ( ( times_times_int @ pls @ W )
% 3.36/1.74 = pls ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1188_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J,axiom,
% 3.36/1.74 ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1189_rel__simps_I19_J,axiom,
% 3.36/1.74 ord_less_eq_int @ pls @ pls ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1190_rel__simps_I22_J,axiom,
% 3.36/1.74 ! [K: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ pls @ ( bit1 @ K ) )
% 3.36/1.74 <=> ( ord_less_eq_int @ pls @ K ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1191_rel__simps_I34_J,axiom,
% 3.36/1.74 ! [K: int,L: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ ( bit1 @ K ) @ ( bit1 @ L ) )
% 3.36/1.74 <=> ( ord_less_eq_int @ K @ L ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1192_less__eq__int__code_I16_J,axiom,
% 3.36/1.74 ! [K1: int,K2: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ ( bit1 @ K1 ) @ ( bit1 @ K2 ) )
% 3.36/1.74 <=> ( ord_less_eq_int @ K1 @ K2 ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1193_rel__simps_I2_J,axiom,
% 3.36/1.74 ~ ( ord_less_int @ pls @ pls ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1194_rel__simps_I12_J,axiom,
% 3.36/1.74 ! [K: int] :
% 3.36/1.74 ( ( ord_less_int @ ( bit1 @ K ) @ pls )
% 3.36/1.74 <=> ( ord_less_int @ K @ pls ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1195_rel__simps_I17_J,axiom,
% 3.36/1.74 ! [K: int,L: int] :
% 3.36/1.74 ( ( ord_less_int @ ( bit1 @ K ) @ ( bit1 @ L ) )
% 3.36/1.74 <=> ( ord_less_int @ K @ L ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1196_less__int__code_I16_J,axiom,
% 3.36/1.74 ! [K1: int,K2: int] :
% 3.36/1.74 ( ( ord_less_int @ ( bit1 @ K1 ) @ ( bit1 @ K2 ) )
% 3.36/1.74 <=> ( ord_less_int @ K1 @ K2 ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1197_rel__simps_I29_J,axiom,
% 3.36/1.74 ! [K: int] :
% 3.36/1.74 ( ( ord_less_eq_int @ ( bit1 @ K ) @ pls )
% 3.36/1.74 <=> ( ord_less_int @ K @ pls ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(fact_1198_rel__simps_I5_J,axiom,
% 3.36/1.74 ! [K: int] :
% 3.36/1.74 ( ( ord_less_int @ pls @ ( bit1 @ K ) )
% 3.36/1.74 <=> ( ord_less_eq_int @ pls @ K ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(help_fequal_1_1_fequal_000t__a_T,axiom,
% 3.36/1.74 ! [X: x_a,Y: x_a] :
% 3.36/1.74 ( ~ ( fequal_a @ X @ Y )
% 3.36/1.74 | ( X = Y ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(help_fequal_2_1_fequal_000t__a_T,axiom,
% 3.36/1.74 ! [X: x_a,Y: x_a] :
% 3.36/1.74 ( ( X != Y )
% 3.36/1.74 | ( fequal_a @ X @ Y ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(help_If_1_1_If_000tc__Nat__Onat_T,axiom,
% 3.36/1.74 ! [X: nat,Y: nat] :
% 3.36/1.74 ( ( if_nat @ $true @ X @ Y )
% 3.36/1.74 = X ) ).
% 3.36/1.74
% 3.36/1.74 thf(help_If_2_1_If_000tc__Nat__Onat_T,axiom,
% 3.36/1.74 ! [X: nat,Y: nat] :
% 3.36/1.74 ( ( if_nat @ $false @ X @ Y )
% 3.36/1.74 = Y ) ).
% 3.36/1.74
% 3.36/1.74 thf(help_If_3_1_If_000tc__Nat__Onat_T,axiom,
% 3.36/1.74 ! [P: $o] :
% 3.36/1.74 ( ( P = $true )
% 3.36/1.74 | ( P = $false ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(help_fequal_1_1_fequal_000tc__Int__Oint_T,axiom,
% 3.36/1.74 ! [X: int,Y: int] :
% 3.36/1.74 ( ~ ( fequal_int @ X @ Y )
% 3.36/1.74 | ( X = Y ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(help_fequal_2_1_fequal_000tc__Int__Oint_T,axiom,
% 3.36/1.74 ! [X: int,Y: int] :
% 3.36/1.74 ( ( X != Y )
% 3.36/1.74 | ( fequal_int @ X @ Y ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(help_fequal_1_1_fequal_000tc__Nat__Onat_T,axiom,
% 3.36/1.74 ! [X: nat,Y: nat] :
% 3.36/1.74 ( ~ ( fequal_nat @ X @ Y )
% 3.36/1.74 | ( X = Y ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(help_fequal_2_1_fequal_000tc__Nat__Onat_T,axiom,
% 3.36/1.74 ! [X: nat,Y: nat] :
% 3.36/1.74 ( ( X != Y )
% 3.36/1.74 | ( fequal_nat @ X @ Y ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(conj_0,hypothesis,
% 3.36/1.74 finite_finite_pname @ u ).
% 3.36/1.74
% 3.36/1.74 thf(conj_1,hypothesis,
% 3.36/1.74 ord_less_eq_a_o @ g @ ( image_pname_a @ mgt_call @ u ) ).
% 3.36/1.74
% 3.36/1.74 thf(conj_2,hypothesis,
% 3.36/1.74 ord_less_eq_nat @ ( suc @ na ) @ ( finite_card_a @ ( image_pname_a @ mgt_call @ u ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(conj_3,hypothesis,
% 3.36/1.74 ( ( finite_card_a @ g )
% 3.36/1.74 = ( minus_minus_nat @ ( finite_card_a @ ( image_pname_a @ mgt_call @ u ) ) @ ( suc @ na ) ) ) ).
% 3.36/1.74
% 3.36/1.74 thf(conj_4,hypothesis,
% 3.36/1.74 member_pname @ pn @ u ).
% 3.36/1.74
% 3.36/1.74 thf(conj_5,hypothesis,
% 3.36/1.74 ~ ( member_a @ ( mgt_call @ pn ) @ g ) ).
% 3.36/1.74
% 3.36/1.74 thf(conj_6,conjecture,
% 3.36/1.74 ord_less_eq_a_o @ ( insert_a @ ( mgt_call @ pn ) @ g ) @ ( image_pname_a @ mgt_call @ u ) ).
% 3.36/1.74 ------------------------
% 3.43/1.81 % (2491)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_2194 for (2999ds/183Mi)
% 3.43/1.81 % (2492)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_2194 for (2999ds/4Mi)
% 3.43/1.81 % (2493)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_2194 for (2999ds/27Mi)
% 3.43/1.81 % (2494)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_2194 for (2999ds/2Mi)
% 3.43/1.81 % (2495)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_2194 for (2999ds/2Mi)
% 3.43/1.81 % (2496)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_2194 for (2999ds/275Mi)
% 3.43/1.81 % (2497)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_2194 for (2999ds/18Mi)
% 3.43/1.82 % (2494)Instruction limit reached!
% 3.43/1.82 % (2494)------------------------------
% 3.43/1.82 % (2494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.82 % (2494)Termination reason: Unknown
% 3.43/1.82 % (2494)Termination phase: shuffling
% 3.43/1.82
% 3.43/1.82 % (2494)Memory used [KB]: 2686
% 3.43/1.82 % (2494)Time elapsed: 0.004 s
% 3.43/1.82 % (2494)Instructions burned: 2 (million)
% 3.43/1.82 % (2494)------------------------------
% 3.43/1.82 % (2494)------------------------------
% 3.43/1.82 % (2495)Instruction limit reached!
% 3.43/1.82 % (2495)------------------------------
% 3.43/1.82 % (2495)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.82 % (2495)Termination reason: Unknown
% 3.43/1.82 % (2495)Termination phase: shuffling
% 3.43/1.82
% 3.43/1.82 % (2495)Memory used [KB]: 2686
% 3.43/1.82 % (2495)Time elapsed: 0.004 s
% 3.43/1.82 % (2495)Instructions burned: 2 (million)
% 3.43/1.82 % (2495)------------------------------
% 3.43/1.82 % (2495)------------------------------
% 3.43/1.82 % (2492)Instruction limit reached!
% 3.43/1.82 % (2492)------------------------------
% 3.43/1.82 % (2492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.82 % (2492)Termination reason: Unknown
% 3.43/1.82 % (2492)Termination phase: shuffling
% 3.43/1.82
% 3.43/1.82 % (2492)Memory used [KB]: 2814
% 3.43/1.82 % (2492)Time elapsed: 0.006 s
% 3.43/1.82 % (2492)Instructions burned: 4 (million)
% 3.43/1.82 % (2492)------------------------------
% 3.43/1.82 % (2492)------------------------------
% 3.43/1.83 % (2497)Instruction limit reached!
% 3.43/1.83 % (2497)------------------------------
% 3.43/1.83 % (2497)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.83 % (2497)Termination reason: Unknown
% 3.43/1.83 % (2497)Termination phase: shuffling
% 3.43/1.83
% 3.43/1.83 % (2497)Memory used [KB]: 3198
% 3.43/1.83 % (2497)Time elapsed: 0.015 s
% 3.43/1.83 % (2497)Instructions burned: 18 (million)
% 3.43/1.83 % (2497)------------------------------
% 3.43/1.83 % (2497)------------------------------
% 3.43/1.83 % (2493)Instruction limit reached!
% 3.43/1.83 % (2493)------------------------------
% 3.43/1.83 % (2493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.83 % (2493)Termination reason: Unknown
% 3.43/1.83 % (2493)Termination phase: shuffling
% 3.43/1.83
% 3.43/1.83 % (2493)Memory used [KB]: 3454
% 3.43/1.83 % (2493)Time elapsed: 0.021 s
% 3.43/1.83 % (2493)Instructions burned: 28 (million)
% 3.43/1.83 % (2493)------------------------------
% 3.43/1.83 % (2493)------------------------------
% 3.43/1.83 % (2498)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_2194 for (2999ds/3Mi)
% 3.43/1.83 % (2499)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_2194 for (2999ds/37Mi)
% 3.43/1.84 % (2500)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_2194 for (2999ds/15Mi)
% 3.43/1.84 % (2498)Instruction limit reached!
% 3.43/1.84 % (2498)------------------------------
% 3.43/1.84 % (2498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.84 % (2498)Termination reason: Unknown
% 3.43/1.84 % (2498)Termination phase: shuffling
% 3.43/1.84
% 3.43/1.84 % (2498)Memory used [KB]: 2814
% 3.43/1.84 % (2498)Time elapsed: 0.005 s
% 3.43/1.84 % (2498)Instructions burned: 3 (million)
% 3.43/1.84 % (2498)------------------------------
% 3.43/1.84 % (2498)------------------------------
% 3.43/1.84 % (2501)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_2194 for (2999ds/3Mi)
% 3.43/1.85 % (2500)Instruction limit reached!
% 3.43/1.85 % (2500)------------------------------
% 3.43/1.85 % (2500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.85 % (2500)Termination reason: Unknown
% 3.43/1.85 % (2500)Termination phase: shuffling
% 3.43/1.85
% 3.43/1.85 % (2500)Memory used [KB]: 3198
% 3.43/1.85 % (2500)Time elapsed: 0.013 s
% 3.43/1.85 % (2500)Instructions burned: 16 (million)
% 3.43/1.85 % (2500)------------------------------
% 3.43/1.85 % (2500)------------------------------
% 3.43/1.85 % (2501)Instruction limit reached!
% 3.43/1.85 % (2501)------------------------------
% 3.43/1.85 % (2501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.85 % (2501)Termination reason: Unknown
% 3.43/1.85 % (2501)Termination phase: shuffling
% 3.43/1.85
% 3.43/1.85 % (2501)Memory used [KB]: 2814
% 3.43/1.85 % (2501)Time elapsed: 0.005 s
% 3.43/1.85 % (2501)Instructions burned: 4 (million)
% 3.43/1.85 % (2501)------------------------------
% 3.43/1.85 % (2501)------------------------------
% 3.43/1.85 % (2502)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_2194 for (2999ds/1041Mi)
% 3.43/1.85 % (2503)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_2194 for (2999ds/7Mi)
% 3.43/1.85 % (2499)Instruction limit reached!
% 3.43/1.85 % (2499)------------------------------
% 3.43/1.85 % (2499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.85 % (2499)Termination reason: Unknown
% 3.43/1.85 % (2499)Termination phase: shuffling
% 3.43/1.85
% 3.43/1.85 % (2499)Memory used [KB]: 3837
% 3.43/1.85 % (2499)Time elapsed: 0.023 s
% 3.43/1.85 % (2499)Instructions burned: 38 (million)
% 3.43/1.85 % (2499)------------------------------
% 3.43/1.85 % (2499)------------------------------
% 3.43/1.86 % (2505)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_2194 for (2998ds/3Mi)
% 3.43/1.86 % (2503)Instruction limit reached!
% 3.43/1.86 % (2503)------------------------------
% 3.43/1.86 % (2503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.86 % (2505)Instruction limit reached!
% 3.43/1.86 % (2505)------------------------------
% 3.43/1.86 % (2505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.86 % (2505)Termination reason: Unknown
% 3.43/1.86 % (2503)Termination reason: Unknown
% 3.43/1.86 % (2505)Termination phase: shuffling
% 3.43/1.86
% 3.43/1.86 % (2503)Termination phase: shuffling
% 3.43/1.86
% 3.43/1.86 % (2505)Memory used [KB]: 2814
% 3.43/1.86 % (2505)Time elapsed: 0.003 s
% 3.43/1.86 % (2503)Memory used [KB]: 2942
% 3.43/1.86 % (2505)Instructions burned: 4 (million)
% 3.43/1.86 % (2503)Time elapsed: 0.008 s
% 3.43/1.86 % (2505)------------------------------
% 3.43/1.86 % (2505)------------------------------
% 3.43/1.86 % (2503)Instructions burned: 8 (million)
% 3.43/1.86 % (2503)------------------------------
% 3.43/1.86 % (2503)------------------------------
% 3.43/1.86 % (2504)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_2194 for (2998ds/16Mi)
% 3.43/1.86 % (2491)Instruction limit reached!
% 3.43/1.86 % (2491)------------------------------
% 3.43/1.86 % (2491)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.86 % (2491)Termination reason: Unknown
% 3.43/1.86 % (2491)Termination phase: Preprocessing 3
% 3.43/1.86
% 3.43/1.86 % (2491)Memory used [KB]: 6268
% 3.43/1.86 % (2491)Time elapsed: 0.060 s
% 3.43/1.86 % (2491)Instructions burned: 183 (million)
% 3.43/1.86 % (2491)------------------------------
% 3.43/1.86 % (2491)------------------------------
% 3.43/1.87 % (2507)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_2194 for (2998ds/7Mi)
% 3.43/1.87 % (2507)Instruction limit reached!
% 3.43/1.87 % (2507)------------------------------
% 3.43/1.87 % (2507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.87 % (2507)Termination reason: Unknown
% 3.43/1.87 % (2507)Termination phase: shuffling
% 3.43/1.87
% 3.43/1.87 % (2507)Memory used [KB]: 2942
% 3.43/1.87 % (2507)Time elapsed: 0.004 s
% 3.43/1.87 % (2507)Instructions burned: 7 (million)
% 3.43/1.87 % (2507)------------------------------
% 3.43/1.87 % (2507)------------------------------
% 3.43/1.87 % (2506)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_2194 for (2998ds/3Mi)
% 3.43/1.87 % (2509)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_2194 for (2998ds/4Mi)
% 3.43/1.87 % (2504)Instruction limit reached!
% 3.43/1.87 % (2504)------------------------------
% 3.43/1.87 % (2504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.87 % (2504)Termination reason: Unknown
% 3.43/1.87 % (2504)Termination phase: shuffling
% 3.43/1.87
% 3.43/1.87 % (2504)Memory used [KB]: 3198
% 3.43/1.87 % (2504)Time elapsed: 0.013 s
% 3.43/1.87 % (2504)Instructions burned: 16 (million)
% 3.43/1.87 % (2504)------------------------------
% 3.43/1.87 % (2504)------------------------------
% 3.43/1.88 % (2506)Instruction limit reached!
% 3.43/1.88 % (2506)------------------------------
% 3.43/1.88 % (2506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.88 % (2506)Termination reason: Unknown
% 3.43/1.88 % (2506)Termination phase: shuffling
% 3.43/1.88
% 3.43/1.88 % (2506)Memory used [KB]: 2814
% 3.43/1.88 % (2506)Time elapsed: 0.005 s
% 3.43/1.88 % (2506)Instructions burned: 4 (million)
% 3.43/1.88 % (2506)------------------------------
% 3.43/1.88 % (2506)------------------------------
% 3.43/1.88 % (2509)Instruction limit reached!
% 3.43/1.88 % (2509)------------------------------
% 3.43/1.88 % (2509)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.88 % (2509)Termination reason: Unknown
% 3.43/1.88 % (2509)Termination phase: shuffling
% 3.43/1.88
% 3.43/1.88 % (2509)Memory used [KB]: 2814
% 3.43/1.88 % (2509)Time elapsed: 0.003 s
% 3.43/1.88 % (2509)Instructions burned: 4 (million)
% 3.43/1.88 % (2509)------------------------------
% 3.43/1.88 % (2509)------------------------------
% 3.43/1.88 % (2508)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_2194 for (2998ds/3Mi)
% 3.43/1.88 % (2508)Instruction limit reached!
% 3.43/1.88 % (2508)------------------------------
% 3.43/1.88 % (2508)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.88 % (2508)Termination reason: Unknown
% 3.43/1.88 % (2508)Termination phase: shuffling
% 3.43/1.88
% 3.43/1.88 % (2508)Memory used [KB]: 2814
% 3.43/1.88 % (2508)Time elapsed: 0.005 s
% 3.43/1.88 % (2508)Instructions burned: 3 (million)
% 3.43/1.88 % (2508)------------------------------
% 3.43/1.88 % (2508)------------------------------
% 3.43/1.88 % (2510)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_2194 for (2998ds/18Mi)
% 3.43/1.89 % (2513)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_2194 for (2998ds/902Mi)
% 3.43/1.89 % (2510)Instruction limit reached!
% 3.43/1.89 % (2510)------------------------------
% 3.43/1.89 % (2510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.89 % (2510)Termination reason: Unknown
% 3.43/1.89 % (2510)Termination phase: shuffling
% 3.43/1.89
% 3.43/1.89 % (2510)Memory used [KB]: 3326
% 3.43/1.89 % (2510)Time elapsed: 0.008 s
% 3.43/1.89 % (2510)Instructions burned: 19 (million)
% 3.43/1.89 % (2510)------------------------------
% 3.43/1.89 % (2510)------------------------------
% 3.43/1.89 % (2512)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_2194 for (2998ds/6Mi)
% 3.43/1.89 % (2512)Instruction limit reached!
% 3.43/1.89 % (2512)------------------------------
% 3.43/1.89 % (2512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.89 % (2512)Termination reason: Unknown
% 3.43/1.89 % (2512)Termination phase: shuffling
% 3.43/1.89
% 3.43/1.89 % (2512)Memory used [KB]: 2814
% 3.43/1.89 % (2512)Time elapsed: 0.005 s
% 3.43/1.89 % (2512)Instructions burned: 7 (million)
% 3.43/1.89 % (2512)------------------------------
% 3.43/1.89 % (2512)------------------------------
% 3.43/1.89 % (2511)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_2194 for (2998ds/710Mi)
% 3.43/1.90 % (2515)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_2194 for (2998ds/5Mi)
% 3.43/1.90 % (2514)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_2194 for (2998ds/21Mi)
% 3.43/1.90 % (2515)Instruction limit reached!
% 3.43/1.90 % (2515)------------------------------
% 3.43/1.90 % (2515)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.90 % (2515)Termination reason: Unknown
% 3.43/1.90 % (2515)Termination phase: shuffling
% 3.43/1.90
% 3.43/1.90 % (2515)Memory used [KB]: 2942
% 3.43/1.90 % (2515)Time elapsed: 0.004 s
% 3.43/1.90 % (2515)Instructions burned: 7 (million)
% 3.43/1.90 % (2515)------------------------------
% 3.43/1.90 % (2515)------------------------------
% 3.43/1.90 % (2516)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_2194 for (2998ds/6Mi)
% 3.43/1.91 % (2516)Instruction limit reached!
% 3.43/1.91 % (2516)------------------------------
% 3.43/1.91 % (2516)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.91 % (2516)Termination reason: Unknown
% 3.43/1.91 % (2516)Termination phase: shuffling
% 3.43/1.91
% 3.43/1.91 % (2516)Memory used [KB]: 2942
% 3.43/1.91 % (2516)Time elapsed: 0.004 s
% 3.43/1.91 % (2516)Instructions burned: 7 (million)
% 3.43/1.91 % (2516)------------------------------
% 3.43/1.91 % (2516)------------------------------
% 3.43/1.91 % (2517)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_2194 for (2998ds/377Mi)
% 3.43/1.91 % (2514)Instruction limit reached!
% 3.43/1.91 % (2514)------------------------------
% 3.43/1.91 % (2514)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.91 % (2514)Termination reason: Unknown
% 3.43/1.91 % (2514)Termination phase: shuffling
% 3.43/1.91
% 3.43/1.91 % (2514)Memory used [KB]: 3326
% 3.43/1.91 % (2514)Time elapsed: 0.017 s
% 3.43/1.91 % (2514)Instructions burned: 22 (million)
% 3.43/1.91 % (2514)------------------------------
% 3.43/1.91 % (2514)------------------------------
% 3.43/1.92 % (2518)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_2194 for (2998ds/779Mi)
% 3.43/1.93 % (2519)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_2194 for (2998ds/19Mi)
% 3.43/1.94 % (2519)Instruction limit reached!
% 3.43/1.94 % (2519)------------------------------
% 3.43/1.94 % (2519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.94 % (2519)Termination reason: Unknown
% 3.43/1.94 % (2519)Termination phase: shuffling
% 3.43/1.94
% 3.43/1.94 % (2519)Memory used [KB]: 3326
% 3.43/1.94 % (2519)Time elapsed: 0.010 s
% 3.43/1.94 % (2519)Instructions burned: 19 (million)
% 3.43/1.94 % (2519)------------------------------
% 3.43/1.94 % (2519)------------------------------
% 3.43/1.95 % (2496)Instruction limit reached!
% 3.43/1.95 % (2496)------------------------------
% 3.43/1.95 % (2496)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.95 % (2496)Termination reason: Unknown
% 3.43/1.95 % (2496)Termination phase: Saturation
% 3.43/1.95
% 3.43/1.95 % (2496)Memory used [KB]: 10746
% 3.43/1.95 % (2496)Time elapsed: 0.136 s
% 3.43/1.95 % (2496)Instructions burned: 276 (million)
% 3.43/1.95 % (2496)------------------------------
% 3.43/1.95 % (2496)------------------------------
% 3.43/1.95 % (2520)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_2194 for (2998ds/879Mi)
% 3.43/1.96 % (2521)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_2194 for (2998ds/17Mi)
% 3.43/1.97 % (2521)Instruction limit reached!
% 3.43/1.97 % (2521)------------------------------
% 3.43/1.97 % (2521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.97 % (2521)Termination reason: Unknown
% 3.43/1.97 % (2521)Termination phase: shuffling
% 3.43/1.97
% 3.43/1.97 % (2521)Memory used [KB]: 3198
% 3.43/1.97 % (2521)Time elapsed: 0.008 s
% 3.43/1.97 % (2521)Instructions burned: 17 (million)
% 3.43/1.97 % (2521)------------------------------
% 3.43/1.97 % (2521)------------------------------
% 3.43/1.98 % (2522)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_2194 for (2997ds/3Mi)
% 3.43/1.98 % (2522)Instruction limit reached!
% 3.43/1.98 % (2522)------------------------------
% 3.43/1.98 % (2522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.98 % (2522)Termination reason: Unknown
% 3.43/1.98 % (2522)Termination phase: shuffling
% 3.43/1.98
% 3.43/1.98 % (2522)Memory used [KB]: 2814
% 3.43/1.98 % (2522)Time elapsed: 0.003 s
% 3.43/1.98 % (2522)Instructions burned: 3 (million)
% 3.43/1.98 % (2522)------------------------------
% 3.43/1.98 % (2522)------------------------------
% 3.43/1.99 % (2523)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_2194 for (2997ds/30Mi)
% 3.43/2.00 % (2523)Instruction limit reached!
% 3.43/2.00 % (2523)------------------------------
% 3.43/2.00 % (2523)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/2.00 % (2523)Termination reason: Unknown
% 3.43/2.00 % (2523)Termination phase: shuffling
% 3.43/2.00
% 3.43/2.00 % (2523)Memory used [KB]: 3837
% 3.43/2.00 % (2523)Time elapsed: 0.014 s
% 3.43/2.00 % (2523)Instructions burned: 31 (million)
% 3.43/2.00 % (2523)------------------------------
% 3.43/2.00 % (2523)------------------------------
% 3.43/2.01 % (2524)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_2194 for (2997ds/127Mi)
% 4.90/2.02 % (2517)Instruction limit reached!
% 4.90/2.02 % (2517)------------------------------
% 4.90/2.02 % (2517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.90/2.02 % (2517)Termination reason: Unknown
% 4.90/2.02 % (2517)Termination phase: Saturation
% 4.90/2.02
% 4.90/2.02 % (2517)Memory used [KB]: 11641
% 4.90/2.02 % (2517)Time elapsed: 0.111 s
% 4.90/2.02 % (2517)Instructions burned: 379 (million)
% 4.90/2.02 % (2517)------------------------------
% 4.90/2.02 % (2517)------------------------------
% 4.90/2.03 % (2525)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_2194 for (2997ds/100Mi)
% 4.90/2.06 % (2524)Instruction limit reached!
% 4.90/2.06 % (2524)------------------------------
% 4.90/2.06 % (2524)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.90/2.06 % (2524)Termination reason: Unknown
% 4.90/2.06 % (2524)Termination phase: Property scanning
% 4.90/2.06
% 4.90/2.06 % (2524)Memory used [KB]: 5245
% 4.90/2.06 % (2524)Time elapsed: 0.047 s
% 4.90/2.06 % (2524)Instructions burned: 129 (million)
% 4.90/2.06 % (2524)------------------------------
% 4.90/2.06 % (2524)------------------------------
% 4.90/2.06 % (2525)Instruction limit reached!
% 4.90/2.06 % (2525)------------------------------
% 4.90/2.06 % (2525)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.90/2.06 % (2525)Termination reason: Unknown
% 4.90/2.06 % (2525)Termination phase: shuffling
% 4.90/2.06
% 4.90/2.06 % (2525)Memory used [KB]: 5756
% 4.90/2.06 % (2525)Time elapsed: 0.030 s
% 4.90/2.06 % (2525)Instructions burned: 102 (million)
% 4.90/2.06 % (2525)------------------------------
% 4.90/2.06 % (2525)------------------------------
% 4.90/2.07 % (2526)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_2194 for (2996ds/3Mi)
% 4.90/2.07 % (2527)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_2194 for (2996ds/20Mi)
% 4.90/2.07 % (2526)Instruction limit reached!
% 4.90/2.07 % (2526)------------------------------
% 4.90/2.07 % (2526)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.90/2.07 % (2526)Termination reason: Unknown
% 4.90/2.07 % (2526)Termination phase: shuffling
% 4.90/2.07
% 4.90/2.07 % (2526)Memory used [KB]: 2814
% 4.90/2.07 % (2526)Time elapsed: 0.003 s
% 4.90/2.07 % (2526)Instructions burned: 3 (million)
% 4.90/2.07 % (2526)------------------------------
% 4.90/2.07 % (2526)------------------------------
% 4.90/2.08 % (2527)Instruction limit reached!
% 4.90/2.08 % (2527)------------------------------
% 4.90/2.08 % (2527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.90/2.08 % (2527)Termination reason: Unknown
% 4.90/2.08 % (2527)Termination phase: shuffling
% 4.90/2.08
% 4.90/2.08 % (2527)Memory used [KB]: 3326
% 4.90/2.08 % (2527)Time elapsed: 0.009 s
% 4.90/2.08 % (2527)Instructions burned: 22 (million)
% 4.90/2.08 % (2527)------------------------------
% 4.90/2.08 % (2527)------------------------------
% 4.90/2.08 % (2513)First to succeed.
% 4.90/2.08 % (2528)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_2194 for (2996ds/86Mi)
% 4.90/2.08 % (2513)Refutation found. Thanks to Tanya!
% 4.90/2.08 % SZS status Theorem for DTF2THF_2194
% 4.90/2.08 % SZS output start Proof for DTF2THF_2194
% 4.90/2.08 thf(type_def_5, type, pname: $tType).
% 4.90/2.08 thf(type_def_6, type, option_com: $tType).
% 4.90/2.08 thf(type_def_8, type, int: $tType).
% 4.90/2.08 thf(type_def_9, type, x_a: $tType).
% 4.90/2.08 thf(type_def_10, type, nat: $tType).
% 4.90/2.08 thf(type_def_11, type, com: $tType).
% 4.90/2.08 thf(func_def_0, type, x_a: $tType).
% 4.90/2.08 thf(func_def_1, type, com: $tType).
% 4.90/2.08 thf(func_def_2, type, pname: $tType).
% 4.90/2.08 thf(func_def_3, type, int: $tType).
% 4.90/2.08 thf(func_def_4, type, nat: $tType).
% 4.90/2.08 thf(func_def_5, type, option_com: $tType).
% 4.90/2.08 thf(func_def_6, type, body: pname > option_com).
% 4.90/2.08 thf(func_def_7, type, ex: (int > $o) > $o).
% 4.90/2.08 thf(func_def_8, type, finite_card_a_o: ((x_a > $o) > $o) > nat).
% 4.90/2.08 thf(func_def_9, type, finite_card_pname_o: ((pname > $o) > $o) > nat).
% 4.90/2.08 thf(func_def_10, type, finite_card_int_o: ((int > $o) > $o) > nat).
% 4.90/2.08 thf(func_def_11, type, finite_card_nat_o: ((nat > $o) > $o) > nat).
% 4.90/2.08 thf(func_def_12, type, finite_card_a: (x_a > $o) > nat).
% 4.90/2.08 thf(func_def_13, type, finite_card_pname: (pname > $o) > nat).
% 4.90/2.08 thf(func_def_14, type, finite_card_int: (int > $o) > nat).
% 4.90/2.08 thf(func_def_15, type, finite_card_nat: (nat > $o) > nat).
% 4.90/2.08 thf(func_def_16, type, finite_finite_a_o_o: (((x_a > $o) > $o) > $o) > $o).
% 4.90/2.08 thf(func_def_17, type, finite1066544169me_o_o: (((pname > $o) > $o) > $o) > $o).
% 4.90/2.08 thf(func_def_18, type, finite229719499nt_o_o: (((int > $o) > $o) > $o) > $o).
% 4.90/2.08 thf(func_def_19, type, finite1676163439at_o_o: (((nat > $o) > $o) > $o) > $o).
% 4.90/2.08 thf(func_def_20, type, finite_finite_a_o: ((x_a > $o) > $o) > $o).
% 4.90/2.08 thf(func_def_21, type, finite297249702name_o: ((pname > $o) > $o) > $o).
% 4.90/2.08 thf(func_def_22, type, finite_finite_int_o: ((int > $o) > $o) > $o).
% 4.90/2.08 thf(func_def_23, type, finite_finite_nat_o: ((nat > $o) > $o) > $o).
% 4.90/2.08 thf(func_def_24, type, finite_finite_a: (x_a > $o) > $o).
% 4.90/2.08 thf(func_def_25, type, finite_finite_pname: (pname > $o) > $o).
% 4.90/2.08 thf(func_def_26, type, finite_finite_int: (int > $o) > $o).
% 4.90/2.08 thf(func_def_27, type, finite_finite_nat: (nat > $o) > $o).
% 4.90/2.08 thf(func_def_28, type, finite_folding_one_a: (x_a > x_a > x_a) > ((x_a > $o) > x_a) > $o).
% 4.90/2.08 thf(func_def_29, type, finite1282449217_pname: (pname > pname > pname) > ((pname > $o) > pname) > $o).
% 4.90/2.08 thf(func_def_30, type, finite1626084323ne_int: (int > int > int) > ((int > $o) > int) > $o).
% 4.90/2.08 thf(func_def_31, type, finite988810631ne_nat: (nat > nat > nat) > ((nat > $o) > nat) > $o).
% 4.90/2.08 thf(func_def_32, type, finite1819937229idem_a: (x_a > x_a > x_a) > ((x_a > $o) > x_a) > $o).
% 4.90/2.08 thf(func_def_33, type, finite89670078_pname: (pname > pname > pname) > ((pname > $o) > pname) > $o).
% 4.90/2.08 thf(func_def_34, type, finite1432773856em_int: (int > int > int) > ((int > $o) > int) > $o).
% 4.90/2.08 thf(func_def_35, type, finite795500164em_nat: (nat > nat > nat) > ((nat > $o) > nat) > $o).
% 4.90/2.08 thf(func_def_36, type, abs_abs_int: int > int).
% 4.90/2.08 thf(func_def_37, type, minus_minus_a_o: (x_a > $o) > (x_a > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_38, type, minus_minus_pname_o: (pname > $o) > (pname > $o) > pname > $o).
% 4.90/2.08 thf(func_def_39, type, minus_minus_int_o: (int > $o) > (int > $o) > int > $o).
% 4.90/2.08 thf(func_def_40, type, minus_minus_nat_o: (nat > $o) > (nat > $o) > nat > $o).
% 4.90/2.08 thf(func_def_41, type, minus_minus_int: int > int > int).
% 4.90/2.08 thf(func_def_42, type, minus_minus_nat: nat > nat > nat).
% 4.90/2.08 thf(func_def_43, type, one_one_int: int).
% 4.90/2.08 thf(func_def_44, type, one_one_nat: nat).
% 4.90/2.08 thf(func_def_45, type, plus_plus_int: int > int > int).
% 4.90/2.08 thf(func_def_46, type, plus_plus_nat: nat > nat > nat).
% 4.90/2.08 thf(func_def_47, type, times_times_int: int > int > int).
% 4.90/2.08 thf(func_def_48, type, times_times_nat: nat > nat > nat).
% 4.90/2.08 thf(func_def_49, type, zero_zero_int: int).
% 4.90/2.08 thf(func_def_50, type, zero_zero_nat: nat).
% 4.90/2.08 thf(func_def_51, type, the_a: (x_a > $o) > x_a).
% 4.90/2.08 thf(func_def_52, type, the_int: (int > $o) > int).
% 4.90/2.08 thf(func_def_53, type, the_nat: (nat > $o) > nat).
% 4.90/2.08 thf(func_def_54, type, if_nat: $o > nat > nat > nat).
% 4.90/2.08 thf(func_def_55, type, bit1: int > int).
% 4.90/2.08 thf(func_def_56, type, pls: int).
% 4.90/2.08 thf(func_def_57, type, number_number_of_int: int > int).
% 4.90/2.08 thf(func_def_58, type, number_number_of_nat: int > nat).
% 4.90/2.08 thf(func_def_59, type, succ: int > int).
% 4.90/2.08 thf(func_def_60, type, suc: nat > nat).
% 4.90/2.08 thf(func_def_61, type, nat_case_o: $o > (nat > $o) > nat > $o).
% 4.90/2.08 thf(func_def_62, type, nat_case_nat: nat > (nat > nat) > nat > nat).
% 4.90/2.08 thf(func_def_63, type, semiri1621563631at_int: nat > int).
% 4.90/2.08 thf(func_def_64, type, nat_neg: int > $o).
% 4.90/2.08 thf(func_def_65, type, nat_tsub: int > int > int).
% 4.90/2.08 thf(func_def_66, type, the_com: option_com > com).
% 4.90/2.08 thf(func_def_67, type, bot_bot_a_o: x_a > $o).
% 4.90/2.08 thf(func_def_68, type, bot_bot_pname_o: pname > $o).
% 4.90/2.08 thf(func_def_69, type, bot_bot_int_o: int > $o).
% 4.90/2.08 thf(func_def_70, type, bot_bot_nat_o: nat > $o).
% 4.90/2.08 thf(func_def_72, type, bot_bot_nat: nat).
% 4.90/2.08 thf(func_def_73, type, ord_less_int_o: (int > $o) > (int > $o) > $o).
% 4.90/2.08 thf(func_def_74, type, ord_less_nat_o: (nat > $o) > (nat > $o) > $o).
% 4.90/2.08 thf(func_def_75, type, ord_less_int: int > int > $o).
% 4.90/2.08 thf(func_def_76, type, ord_less_nat: nat > nat > $o).
% 4.90/2.08 thf(func_def_77, type, ord_less_eq_a_o_o: ((x_a > $o) > $o) > ((x_a > $o) > $o) > $o).
% 4.90/2.08 thf(func_def_78, type, ord_le1205211808me_o_o: ((pname > $o) > $o) > ((pname > $o) > $o) > $o).
% 4.90/2.08 thf(func_def_79, type, ord_less_eq_int_o_o: ((int > $o) > $o) > ((int > $o) > $o) > $o).
% 4.90/2.08 thf(func_def_80, type, ord_less_eq_nat_o_o: ((nat > $o) > $o) > ((nat > $o) > $o) > $o).
% 4.90/2.08 thf(func_def_81, type, ord_less_eq_a_o: (x_a > $o) > (x_a > $o) > $o).
% 4.90/2.08 thf(func_def_82, type, ord_less_eq_pname_o: (pname > $o) > (pname > $o) > $o).
% 4.90/2.08 thf(func_def_83, type, ord_less_eq_int_o: (int > $o) > (int > $o) > $o).
% 4.90/2.08 thf(func_def_84, type, ord_less_eq_nat_o: (nat > $o) > (nat > $o) > $o).
% 4.90/2.08 thf(func_def_85, type, ord_less_eq_o: $o > $o > $o).
% 4.90/2.08 thf(func_def_86, type, ord_less_eq_int: int > int > $o).
% 4.90/2.08 thf(func_def_87, type, ord_less_eq_nat: nat > nat > $o).
% 4.90/2.08 thf(func_def_88, type, collect_a_o_o: (((x_a > $o) > $o) > $o) > ((x_a > $o) > $o) > $o).
% 4.90/2.08 thf(func_def_89, type, collect_pname_o_o: (((pname > $o) > $o) > $o) > ((pname > $o) > $o) > $o).
% 4.90/2.08 thf(func_def_90, type, collect_int_o_o: (((int > $o) > $o) > $o) > ((int > $o) > $o) > $o).
% 4.90/2.08 thf(func_def_91, type, collect_nat_o_o: (((nat > $o) > $o) > $o) > ((nat > $o) > $o) > $o).
% 4.90/2.08 thf(func_def_92, type, collect_a_o: ((x_a > $o) > $o) > (x_a > $o) > $o).
% 4.90/2.08 thf(func_def_93, type, collect_pname_o: ((pname > $o) > $o) > (pname > $o) > $o).
% 4.90/2.08 thf(func_def_94, type, collect_int_o: ((int > $o) > $o) > (int > $o) > $o).
% 4.90/2.08 thf(func_def_95, type, collect_nat_o: ((nat > $o) > $o) > (nat > $o) > $o).
% 4.90/2.08 thf(func_def_96, type, collect_a: (x_a > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_97, type, collect_pname: (pname > $o) > pname > $o).
% 4.90/2.08 thf(func_def_98, type, collect_int: (int > $o) > int > $o).
% 4.90/2.08 thf(func_def_99, type, collect_nat: (nat > $o) > nat > $o).
% 4.90/2.08 thf(func_def_100, type, image_a_o_a: ((x_a > $o) > x_a) > ((x_a > $o) > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_101, type, image_a_o_pname: ((x_a > $o) > pname) > ((x_a > $o) > $o) > pname > $o).
% 4.90/2.08 thf(func_def_102, type, image_a_o_int: ((x_a > $o) > int) > ((x_a > $o) > $o) > int > $o).
% 4.90/2.08 thf(func_def_103, type, image_a_o_nat: ((x_a > $o) > nat) > ((x_a > $o) > $o) > nat > $o).
% 4.90/2.08 thf(func_def_104, type, image_pname_o_a: ((pname > $o) > x_a) > ((pname > $o) > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_105, type, image_pname_o_pname: ((pname > $o) > pname) > ((pname > $o) > $o) > pname > $o).
% 4.90/2.08 thf(func_def_106, type, image_pname_o_int: ((pname > $o) > int) > ((pname > $o) > $o) > int > $o).
% 4.90/2.08 thf(func_def_107, type, image_pname_o_nat: ((pname > $o) > nat) > ((pname > $o) > $o) > nat > $o).
% 4.90/2.08 thf(func_def_108, type, image_int_o_a: ((int > $o) > x_a) > ((int > $o) > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_109, type, image_int_o_pname: ((int > $o) > pname) > ((int > $o) > $o) > pname > $o).
% 4.90/2.08 thf(func_def_110, type, image_int_o_int: ((int > $o) > int) > ((int > $o) > $o) > int > $o).
% 4.90/2.08 thf(func_def_111, type, image_int_o_nat: ((int > $o) > nat) > ((int > $o) > $o) > nat > $o).
% 4.90/2.08 thf(func_def_112, type, image_nat_o_a: ((nat > $o) > x_a) > ((nat > $o) > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_113, type, image_nat_o_pname: ((nat > $o) > pname) > ((nat > $o) > $o) > pname > $o).
% 4.90/2.08 thf(func_def_114, type, image_nat_o_int: ((nat > $o) > int) > ((nat > $o) > $o) > int > $o).
% 4.90/2.08 thf(func_def_115, type, image_nat_o_nat: ((nat > $o) > nat) > ((nat > $o) > $o) > nat > $o).
% 4.90/2.08 thf(func_def_116, type, image_a_a_o: (x_a > x_a > $o) > (x_a > $o) > (x_a > $o) > $o).
% 4.90/2.08 thf(func_def_117, type, image_a_pname_o: (x_a > pname > $o) > (x_a > $o) > (pname > $o) > $o).
% 4.90/2.08 thf(func_def_118, type, image_a_int_o: (x_a > int > $o) > (x_a > $o) > (int > $o) > $o).
% 4.90/2.08 thf(func_def_119, type, image_a_nat_o: (x_a > nat > $o) > (x_a > $o) > (nat > $o) > $o).
% 4.90/2.08 thf(func_def_120, type, image_a_a: (x_a > x_a) > (x_a > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_121, type, image_a_pname: (x_a > pname) > (x_a > $o) > pname > $o).
% 4.90/2.08 thf(func_def_122, type, image_a_int: (x_a > int) > (x_a > $o) > int > $o).
% 4.90/2.08 thf(func_def_123, type, image_a_nat: (x_a > nat) > (x_a > $o) > nat > $o).
% 4.90/2.08 thf(func_def_124, type, image_pname_a_o: (pname > x_a > $o) > (pname > $o) > (x_a > $o) > $o).
% 4.90/2.08 thf(func_def_125, type, image_pname_pname_o: (pname > pname > $o) > (pname > $o) > (pname > $o) > $o).
% 4.90/2.08 thf(func_def_126, type, image_pname_int_o: (pname > int > $o) > (pname > $o) > (int > $o) > $o).
% 4.90/2.08 thf(func_def_127, type, image_pname_nat_o: (pname > nat > $o) > (pname > $o) > (nat > $o) > $o).
% 4.90/2.08 thf(func_def_128, type, image_pname_a: (pname > x_a) > (pname > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_129, type, image_pname_pname: (pname > pname) > (pname > $o) > pname > $o).
% 4.90/2.08 thf(func_def_130, type, image_pname_int: (pname > int) > (pname > $o) > int > $o).
% 4.90/2.08 thf(func_def_131, type, image_pname_nat: (pname > nat) > (pname > $o) > nat > $o).
% 4.90/2.08 thf(func_def_132, type, image_int_a_o: (int > x_a > $o) > (int > $o) > (x_a > $o) > $o).
% 4.90/2.08 thf(func_def_133, type, image_int_pname_o: (int > pname > $o) > (int > $o) > (pname > $o) > $o).
% 4.90/2.08 thf(func_def_134, type, image_int_int_o: (int > int > $o) > (int > $o) > (int > $o) > $o).
% 4.90/2.08 thf(func_def_135, type, image_int_nat_o: (int > nat > $o) > (int > $o) > (nat > $o) > $o).
% 4.90/2.08 thf(func_def_136, type, image_int_a: (int > x_a) > (int > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_137, type, image_int_pname: (int > pname) > (int > $o) > pname > $o).
% 4.90/2.08 thf(func_def_138, type, image_nat_a_o: (nat > x_a > $o) > (nat > $o) > (x_a > $o) > $o).
% 4.90/2.08 thf(func_def_139, type, image_nat_pname_o: (nat > pname > $o) > (nat > $o) > (pname > $o) > $o).
% 4.90/2.08 thf(func_def_140, type, image_nat_int_o: (nat > int > $o) > (nat > $o) > (int > $o) > $o).
% 4.90/2.08 thf(func_def_141, type, image_nat_nat_o: (nat > nat > $o) > (nat > $o) > (nat > $o) > $o).
% 4.90/2.08 thf(func_def_142, type, image_nat_a: (nat > x_a) > (nat > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_143, type, image_nat_pname: (nat > pname) > (nat > $o) > pname > $o).
% 4.90/2.08 thf(func_def_144, type, image_nat_int: (nat > int) > (nat > $o) > int > $o).
% 4.90/2.08 thf(func_def_145, type, insert_a_o: (x_a > $o) > ((x_a > $o) > $o) > (x_a > $o) > $o).
% 4.90/2.08 thf(func_def_146, type, insert_pname_o: (pname > $o) > ((pname > $o) > $o) > (pname > $o) > $o).
% 4.90/2.08 thf(func_def_147, type, insert_int_o: (int > $o) > ((int > $o) > $o) > (int > $o) > $o).
% 4.90/2.08 thf(func_def_148, type, insert_nat_o: (nat > $o) > ((nat > $o) > $o) > (nat > $o) > $o).
% 4.90/2.08 thf(func_def_149, type, insert_a: x_a > (x_a > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_150, type, insert_pname: pname > (pname > $o) > pname > $o).
% 4.90/2.08 thf(func_def_151, type, insert_int: int > (int > $o) > int > $o).
% 4.90/2.08 thf(func_def_152, type, insert_nat: nat > (nat > $o) > nat > $o).
% 4.90/2.08 thf(func_def_153, type, the_elem_a: (x_a > $o) > x_a).
% 4.90/2.08 thf(func_def_154, type, the_elem_int: (int > $o) > int).
% 4.90/2.08 thf(func_def_155, type, the_elem_nat: (nat > $o) > nat).
% 4.90/2.08 thf(func_def_156, type, fequal_a: x_a > x_a > $o).
% 4.90/2.08 thf(func_def_157, type, fequal_int: int > int > $o).
% 4.90/2.08 thf(func_def_158, type, fequal_nat: nat > nat > $o).
% 4.90/2.08 thf(func_def_159, type, member_a_o: (x_a > $o) > ((x_a > $o) > $o) > $o).
% 4.90/2.08 thf(func_def_160, type, member_pname_o: (pname > $o) > ((pname > $o) > $o) > $o).
% 4.90/2.08 thf(func_def_161, type, member_int_o: (int > $o) > ((int > $o) > $o) > $o).
% 4.90/2.08 thf(func_def_162, type, member_nat_o: (nat > $o) > ((nat > $o) > $o) > $o).
% 4.90/2.08 thf(func_def_163, type, member_a: x_a > (x_a > $o) > $o).
% 4.90/2.08 thf(func_def_164, type, member_pname: pname > (pname > $o) > $o).
% 4.90/2.08 thf(func_def_165, type, member_int: int > (int > $o) > $o).
% 4.90/2.08 thf(func_def_166, type, member_nat: nat > (nat > $o) > $o).
% 4.90/2.08 thf(func_def_167, type, g: x_a > $o).
% 4.90/2.08 thf(func_def_168, type, p: (x_a > $o) > (x_a > $o) > $o).
% 4.90/2.08 thf(func_def_169, type, u: pname > $o).
% 4.90/2.08 thf(func_def_170, type, mgt: com > x_a).
% 4.90/2.08 thf(func_def_171, type, mgt_call: pname > x_a).
% 4.90/2.08 thf(func_def_172, type, na: nat).
% 4.90/2.08 thf(func_def_173, type, pn: pname).
% 4.90/2.08 thf(func_def_174, type, wt: com > $o).
% 4.90/2.08 thf(func_def_185, type, sP0: int > int > $o).
% 4.90/2.08 thf(func_def_186, type, sP1: int > (int > $o) > $o).
% 4.90/2.08 thf(func_def_187, type, sP2: nat > nat > (nat > int) > int > $o).
% 4.90/2.08 thf(func_def_188, type, sK3: (nat > $o) > (nat > pname) > (pname > $o) > nat > $o).
% 4.90/2.08 thf(func_def_189, type, sK4: (int > $o) > ((x_a > $o) > $o) > ((x_a > $o) > int) > (x_a > $o) > $o).
% 4.90/2.08 thf(func_def_190, type, sK5: (int > $o) > (int > int > $o) > int).
% 4.90/2.08 thf(func_def_191, type, sK6: (nat > $o) > nat).
% 4.90/2.08 thf(func_def_192, type, sK7: (int > $o) > ((int > $o) > $o) > ((int > $o) > int) > (int > $o) > $o).
% 4.90/2.08 thf(func_def_193, type, sK8: (pname > pname > pname) > pname).
% 4.90/2.08 thf(func_def_194, type, sK9: (pname > pname > pname) > pname).
% 4.90/2.08 thf(func_def_195, type, sK10: (nat > $o) > (nat > nat > $o) > nat).
% 4.90/2.08 thf(func_def_196, type, sK11: ((nat > $o) > x_a) > (x_a > $o) > ((nat > $o) > $o) > (nat > $o) > $o).
% 4.90/2.08 thf(func_def_197, type, sK12: (x_a > $o) > (x_a > x_a) > (x_a > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_198, type, sK13: (pname > x_a) > (x_a > $o) > (pname > $o) > pname > $o).
% 4.90/2.08 thf(func_def_199, type, sK14: ((int > $o) > $o) > int).
% 4.90/2.08 thf(func_def_200, type, sK15: ((int > $o) > $o) > int > $o).
% 4.90/2.08 thf(func_def_201, type, sK16: (x_a > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_202, type, sK17: (x_a > $o) > x_a).
% 4.90/2.08 thf(func_def_203, type, sK18: nat > nat > (nat > $o) > nat).
% 4.90/2.08 thf(func_def_204, type, sK19: ((x_a > $o) > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_205, type, sK20: ((x_a > $o) > $o) > x_a).
% 4.90/2.08 thf(func_def_206, type, sK21: (nat > $o) > nat).
% 4.90/2.08 thf(func_def_207, type, sK22: (int > $o) > (int > pname > $o) > int).
% 4.90/2.08 thf(func_def_208, type, sK23: (nat > $o) > nat).
% 4.90/2.08 thf(func_def_209, type, sK24: (nat > $o) > nat > $o).
% 4.90/2.08 thf(func_def_210, type, sK25: (int > $o) > (x_a > int) > (x_a > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_211, type, sK26: (nat > $o) > nat).
% 4.90/2.08 thf(func_def_212, type, sK27: int > (nat > $o) > (nat > int) > nat).
% 4.90/2.08 thf(func_def_213, type, sK28: (pname > $o) > ((pname > $o) > pname) > ((pname > $o) > $o) > (pname > $o) > $o).
% 4.90/2.08 thf(func_def_214, type, sK29: int > (int > $o) > int).
% 4.90/2.08 thf(func_def_215, type, sK30: (int > $o) > int).
% 4.90/2.08 thf(func_def_216, type, sK31: (pname > $o) > ((pname > $o) > $o) > (pname > pname > $o) > pname > $o).
% 4.90/2.08 thf(func_def_217, type, sK32: ((int > $o) > $o) > (nat > $o) > (nat > int > $o) > nat > $o).
% 4.90/2.08 thf(func_def_218, type, sK33: nat > nat > nat).
% 4.90/2.08 thf(func_def_219, type, sK34: (int > $o) > int).
% 4.90/2.08 thf(func_def_220, type, sK35: ((x_a > $o) > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_221, type, sK36: ((x_a > $o) > $o) > x_a).
% 4.90/2.08 thf(func_def_222, type, sK37: (pname > x_a) > (pname > $o) > (x_a > $o) > pname).
% 4.90/2.08 thf(func_def_223, type, sK38: (pname > $o) > ((x_a > $o) > $o) > ((x_a > $o) > pname) > (x_a > $o) > $o).
% 4.90/2.08 thf(func_def_224, type, sK39: (int > $o) > int).
% 4.90/2.08 thf(func_def_225, type, sK40: ((int > $o) > $o) > int > $o).
% 4.90/2.08 thf(func_def_226, type, sK41: ((int > $o) > $o) > int).
% 4.90/2.08 thf(func_def_227, type, sK42: ((int > $o) > $o) > int).
% 4.90/2.08 thf(func_def_228, type, sK43: int > (int > $o) > int).
% 4.90/2.08 thf(func_def_229, type, sK44: (nat > int) > nat > nat).
% 4.90/2.08 thf(func_def_230, type, sK45: int > (nat > int) > nat > nat).
% 4.90/2.08 thf(func_def_231, type, sK46: (pname > $o) > ((pname > $o) > $o) > pname).
% 4.90/2.08 thf(func_def_232, type, sK47: (pname > $o) > ((pname > $o) > $o) > pname > $o).
% 4.90/2.08 thf(func_def_233, type, sK48: int > (nat > int) > (nat > $o) > nat).
% 4.90/2.08 thf(func_def_234, type, sK49: (int > $o) > (int > $o) > int).
% 4.90/2.08 thf(func_def_235, type, sK50: (nat > x_a > $o) > (nat > $o) > nat).
% 4.90/2.08 thf(func_def_236, type, sK51: (int > $o) > (int > $o) > int).
% 4.90/2.08 thf(func_def_237, type, sK52: ((pname > $o) > $o) > pname > $o).
% 4.90/2.08 thf(func_def_238, type, sK53: ((pname > $o) > $o) > pname).
% 4.90/2.08 thf(func_def_239, type, sK54: x_a > (pname > x_a) > (pname > $o) > pname).
% 4.90/2.08 thf(func_def_240, type, sK55: ((pname > $o) > nat) > ((pname > $o) > $o) > (nat > $o) > (pname > $o) > $o).
% 4.90/2.08 thf(func_def_241, type, sK56: nat > nat).
% 4.90/2.08 thf(func_def_242, type, sK57: nat > nat).
% 4.90/2.08 thf(func_def_243, type, sK58: (nat > x_a > $o) > nat).
% 4.90/2.08 thf(func_def_244, type, sK59: (nat > nat) > nat).
% 4.90/2.08 thf(func_def_245, type, sK60: nat > nat).
% 4.90/2.08 thf(func_def_246, type, sK61: (nat > int) > nat > nat).
% 4.90/2.08 thf(func_def_247, type, sK62: (nat > int) > int > nat > nat).
% 4.90/2.08 thf(func_def_248, type, sK63: (int > $o) > (int > $o) > int).
% 4.90/2.08 thf(func_def_249, type, sK64: (nat > $o) > nat).
% 4.90/2.08 thf(func_def_250, type, sK65: nat > nat > nat).
% 4.90/2.08 thf(func_def_251, type, sK66: (int > nat > $o) > (int > $o) > int).
% 4.90/2.08 thf(func_def_252, type, sK67: ((nat > $o) > int) > (int > $o) > ((nat > $o) > $o) > (nat > $o) > $o).
% 4.90/2.08 thf(func_def_253, type, sK68: (nat > pname > $o) > (nat > $o) > nat).
% 4.90/2.08 thf(func_def_254, type, sK69: (nat > $o) > (nat > int) > nat).
% 4.90/2.08 thf(func_def_255, type, sK70: (pname > $o) > (pname > pname) > (pname > $o) > pname > $o).
% 4.90/2.08 thf(func_def_256, type, sK71: int > (int > $o) > int).
% 4.90/2.08 thf(func_def_257, type, sK72: int > (int > $o) > int).
% 4.90/2.08 thf(func_def_258, type, sK73: int > (int > $o) > (int > $o) > int).
% 4.90/2.08 thf(func_def_259, type, sK74: (x_a > $o) > x_a).
% 4.90/2.08 thf(func_def_260, type, sK75: (x_a > $o) > x_a).
% 4.90/2.08 thf(func_def_261, type, sK76: (x_a > $o) > (x_a > pname > $o) > ((pname > $o) > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_262, type, sK77: ((nat > $o) > $o) > (pname > $o) > (pname > nat > $o) > pname > $o).
% 4.90/2.08 thf(func_def_263, type, sK78: (int > $o) > (int > pname > $o) > ((pname > $o) > $o) > int > $o).
% 4.90/2.08 thf(func_def_264, type, sK79: (x_a > x_a > x_a) > x_a).
% 4.90/2.08 thf(func_def_265, type, sK80: (x_a > x_a > x_a) > x_a).
% 4.90/2.08 thf(func_def_266, type, sK81: nat > (nat > $o) > nat).
% 4.90/2.08 thf(func_def_267, type, sK82: (nat > nat > nat) > nat).
% 4.90/2.08 thf(func_def_268, type, sK83: (nat > nat > nat) > nat).
% 4.90/2.08 thf(func_def_269, type, sK84: (nat > nat) > nat).
% 4.90/2.08 thf(func_def_270, type, sK85: (int > $o) > (int > x_a > $o) > int).
% 4.90/2.08 thf(func_def_271, type, sK86: (pname > $o) > pname).
% 4.90/2.08 thf(func_def_272, type, sK87: (pname > $o) > pname > $o).
% 4.90/2.08 thf(func_def_273, type, sK88: (int > $o) > (int > $o) > int).
% 4.90/2.08 thf(func_def_274, type, sK89: (int > $o) > (pname > int) > (pname > $o) > pname > $o).
% 4.90/2.08 thf(func_def_275, type, sK90: (x_a > $o) > ((nat > $o) > $o) > (x_a > nat > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_276, type, sK91: int > nat).
% 4.90/2.08 thf(func_def_277, type, sK92: (nat > $o) > (nat > $o) > nat).
% 4.90/2.08 thf(func_def_278, type, sK93: (x_a > $o) > (nat > $o) > (nat > x_a) > nat > $o).
% 4.90/2.08 thf(func_def_279, type, sK94: ((int > $o) > $o) > (x_a > $o) > (x_a > int > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_280, type, sK95: (nat > int) > (nat > $o) > (int > $o) > nat).
% 4.90/2.08 thf(func_def_281, type, sK96: int > nat).
% 4.90/2.08 thf(func_def_282, type, sK97: ((pname > $o) > $o) > pname).
% 4.90/2.08 thf(func_def_283, type, sK98: ((pname > $o) > $o) > pname > $o).
% 4.90/2.08 thf(func_def_284, type, sK99: ((pname > $o) > $o) > pname).
% 4.90/2.08 thf(func_def_285, type, sK100: (pname > x_a) > (pname > $o) > x_a > pname).
% 4.90/2.08 thf(func_def_286, type, sK101: int > int > nat).
% 4.90/2.08 thf(func_def_287, type, sK102: (int > int > int) > int).
% 4.90/2.08 thf(func_def_288, type, sK103: (int > int > int) > int).
% 4.90/2.08 thf(func_def_289, type, sK104: (x_a > $o) > x_a).
% 4.90/2.08 thf(func_def_290, type, sK105: nat > nat).
% 4.90/2.08 thf(func_def_291, type, sK106: (int > $o) > int > int).
% 4.90/2.08 thf(func_def_292, type, sK107: nat > nat).
% 4.90/2.08 thf(func_def_293, type, sK108: (pname > $o) > pname).
% 4.90/2.08 thf(func_def_294, type, sK109: (pname > $o) > (pname > $o) > pname).
% 4.90/2.08 thf(func_def_295, type, sK110: int > (int > $o) > int).
% 4.90/2.08 thf(func_def_296, type, sK111: (int > $o) > int > int).
% 4.90/2.08 thf(func_def_297, type, sK112: int > nat).
% 4.90/2.08 thf(func_def_298, type, sK113: (pname > $o) > (pname > x_a > $o) > ((x_a > $o) > $o) > pname > $o).
% 4.90/2.08 thf(func_def_299, type, sK114: (nat > pname > $o) > nat).
% 4.90/2.08 thf(func_def_300, type, sK115: nat > nat > nat).
% 4.90/2.08 thf(func_def_301, type, sK116: ((nat > $o) > $o) > ((nat > $o) > pname) > (pname > $o) > (nat > $o) > $o).
% 4.90/2.08 thf(func_def_302, type, sK117: (pname > $o) > pname).
% 4.90/2.08 thf(func_def_303, type, sK118: (pname > $o) > pname > $o).
% 4.90/2.08 thf(func_def_304, type, sK119: (x_a > $o) > x_a).
% 4.90/2.08 thf(func_def_305, type, sK120: (x_a > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_306, type, sK121: (int > $o) > (nat > int) > (nat > $o) > nat > $o).
% 4.90/2.08 thf(func_def_307, type, sK122: (nat > $o) > nat).
% 4.90/2.08 thf(func_def_308, type, sK123: ((pname > $o) > $o) > pname > $o).
% 4.90/2.08 thf(func_def_309, type, sK124: ((pname > $o) > $o) > pname).
% 4.90/2.08 thf(func_def_310, type, sK125: nat > nat > (nat > int) > int > nat).
% 4.90/2.08 thf(func_def_311, type, sK126: (nat > int) > nat > nat > nat).
% 4.90/2.08 thf(func_def_312, type, sK127: nat > nat > nat).
% 4.90/2.08 thf(func_def_313, type, sK128: nat > (nat > $o) > nat).
% 4.90/2.08 thf(func_def_314, type, sK129: (int > $o) > (nat > $o) > (nat > int) > nat > $o).
% 4.90/2.08 thf(func_def_315, type, sK130: (pname > $o) > pname).
% 4.90/2.08 thf(func_def_316, type, sK131: (nat > nat) > nat).
% 4.90/2.08 thf(func_def_317, type, sK132: (nat > nat) > nat).
% 4.90/2.08 thf(func_def_318, type, sK133: (int > $o) > int > int).
% 4.90/2.08 thf(func_def_319, type, sK134: (pname > x_a) > (pname > $o) > (x_a > $o) > pname > $o).
% 4.90/2.08 thf(func_def_320, type, sK135: ((pname > $o) > $o) > (x_a > $o) > ((pname > $o) > x_a) > (pname > $o) > $o).
% 4.90/2.08 thf(func_def_321, type, sK136: (nat > int > $o) > nat).
% 4.90/2.08 thf(func_def_322, type, sK137: (nat > pname > $o) > ((pname > $o) > $o) > (nat > $o) > nat > $o).
% 4.90/2.08 thf(func_def_323, type, sK138: ((x_a > $o) > $o) > (x_a > $o) > ((x_a > $o) > x_a) > (x_a > $o) > $o).
% 4.90/2.08 thf(func_def_324, type, sK139: (int > $o) > int).
% 4.90/2.08 thf(func_def_325, type, sK140: (int > $o) > int > $o).
% 4.90/2.08 thf(func_def_326, type, sK141: (nat > $o) > nat > nat).
% 4.90/2.08 thf(func_def_327, type, sK142: (nat > $o) > nat).
% 4.90/2.08 thf(func_def_328, type, sK143: nat > (nat > $o) > nat).
% 4.90/2.08 thf(func_def_329, type, sK144: (nat > $o) > nat).
% 4.90/2.08 thf(func_def_330, type, sK145: ((nat > $o) > $o) > nat > $o).
% 4.90/2.08 thf(func_def_331, type, sK146: ((nat > $o) > $o) > nat).
% 4.90/2.08 thf(func_def_332, type, sK147: (nat > $o) > ((nat > $o) > $o) > nat).
% 4.90/2.08 thf(func_def_333, type, sK148: (nat > $o) > ((nat > $o) > $o) > nat > $o).
% 4.90/2.08 thf(func_def_334, type, sK149: nat > (nat > $o) > nat).
% 4.90/2.08 thf(func_def_335, type, sK150: (nat > $o) > nat).
% 4.90/2.08 thf(func_def_336, type, sK151: ((int > $o) > $o) > int > $o).
% 4.90/2.08 thf(func_def_337, type, sK152: ((int > $o) > $o) > int).
% 4.90/2.08 thf(func_def_338, type, sK153: (int > $o) > int).
% 4.90/2.08 thf(func_def_339, type, sK154: (int > $o) > nat).
% 4.90/2.08 thf(func_def_340, type, sK155: (int > $o) > int).
% 4.90/2.08 thf(func_def_341, type, sK156: (pname > x_a) > (pname > $o) > pname).
% 4.90/2.08 thf(func_def_342, type, sK157: (int > $o) > ((int > $o) > $o) > int).
% 4.90/2.08 thf(func_def_343, type, sK158: (int > $o) > ((int > $o) > $o) > int > $o).
% 4.90/2.08 thf(func_def_344, type, sK159: (x_a > $o) > pname).
% 4.90/2.08 thf(func_def_345, type, sK160: int > (int > $o) > (int > $o) > int).
% 4.90/2.08 thf(func_def_346, type, sK161: int > (int > $o) > int).
% 4.90/2.08 thf(func_def_347, type, sK162: int > (int > $o) > int).
% 4.90/2.08 thf(func_def_348, type, sK163: (nat > $o) > (nat > $o) > nat).
% 4.90/2.08 thf(func_def_349, type, sK164: (x_a > $o) > (int > x_a) > (int > $o) > int > $o).
% 4.90/2.08 thf(func_def_350, type, sK165: (x_a > nat) > (nat > $o) > (x_a > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_351, type, sK166: (nat > nat > $o) > nat).
% 4.90/2.08 thf(func_def_352, type, sK167: ((int > $o) > $o) > (int > int > $o) > (int > $o) > int > $o).
% 4.90/2.08 thf(func_def_353, type, sK168: (nat > $o) > nat > nat > nat).
% 4.90/2.08 thf(func_def_354, type, sK169: (nat > x_a > $o) > (nat > $o) > ((x_a > $o) > $o) > nat > $o).
% 4.90/2.08 thf(func_def_355, type, sK170: (x_a > $o) > x_a).
% 4.90/2.08 thf(func_def_356, type, sK171: (x_a > pname) > (x_a > $o) > (pname > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_357, type, sK172: ((int > $o) > nat) > (nat > $o) > ((int > $o) > $o) > (int > $o) > $o).
% 4.90/2.08 thf(func_def_358, type, sK173: (x_a > x_a > $o) > (x_a > $o) > ((x_a > $o) > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_359, type, sK174: (nat > $o) > (nat > nat > $o) > ((nat > $o) > $o) > nat > $o).
% 4.90/2.08 thf(func_def_360, type, sK175: ((pname > $o) > $o) > (int > $o) > ((pname > $o) > int) > (pname > $o) > $o).
% 4.90/2.08 thf(func_def_361, type, sK176: (nat > $o) > nat).
% 4.90/2.08 thf(func_def_362, type, sK177: (nat > $o) > nat).
% 4.90/2.08 thf(func_def_363, type, sK178: (int > $o) > int > int).
% 4.90/2.08 thf(func_def_364, type, sK179: (nat > int) > nat).
% 4.90/2.08 thf(func_def_365, type, sK180: (pname > $o) > (pname > nat) > (nat > $o) > pname > $o).
% 4.90/2.08 thf(func_def_366, type, sK181: ((nat > $o) > $o) > (nat > $o) > ((nat > $o) > nat) > (nat > $o) > $o).
% 4.90/2.08 thf(func_def_367, type, sK182: (int > $o) > int > int).
% 4.90/2.08 thf(func_def_368, type, sK183: (x_a > $o) > (x_a > $o) > x_a).
% 4.90/2.08 thf(func_def_369, type, sK184: ((nat > $o) > $o) > nat).
% 4.90/2.08 thf(func_def_370, type, sK185: ((nat > $o) > $o) > nat > $o).
% 4.90/2.08 thf(func_def_371, type, sK186: ((x_a > $o) > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_372, type, sK187: ((x_a > $o) > $o) > x_a).
% 4.90/2.08 thf(func_def_373, type, sK188: ((x_a > $o) > $o) > x_a).
% 4.90/2.08 thf(func_def_374, type, sK189: (nat > $o) > nat > nat).
% 4.90/2.08 thf(func_def_375, type, sK190: nat > nat > nat).
% 4.90/2.08 thf(func_def_376, type, sK191: int > nat).
% 4.90/2.08 thf(func_def_377, type, sK192: (int > nat > $o) > (int > $o) > ((nat > $o) > $o) > int > $o).
% 4.90/2.08 thf(func_def_378, type, sK193: ((int > $o) > $o) > (pname > $o) > ((int > $o) > pname) > (int > $o) > $o).
% 4.90/2.08 thf(func_def_379, type, sK194: (nat > $o) > nat).
% 4.90/2.08 thf(func_def_380, type, sK195: (nat > $o) > nat > $o).
% 4.90/2.08 thf(func_def_381, type, sK196: int > nat).
% 4.90/2.08 thf(func_def_382, type, sK197: int > nat).
% 4.90/2.08 thf(func_def_383, type, sK198: nat > nat > nat).
% 4.90/2.08 thf(func_def_384, type, sK199: (x_a > $o) > (x_a > $o) > x_a).
% 4.90/2.08 thf(func_def_385, type, sK200: (int > $o) > nat).
% 4.90/2.08 thf(func_def_386, type, sK201: (int > $o) > int).
% 4.90/2.08 thf(func_def_387, type, sK202: (int > $o) > (int > x_a) > int).
% 4.90/2.08 thf(func_def_388, type, sK203: (int > pname) > (pname > $o) > (int > $o) > int > $o).
% 4.90/2.08 thf(func_def_389, type, sK204: (nat > nat) > nat).
% 4.90/2.08 thf(func_def_390, type, sK205: (nat > nat) > nat).
% 4.90/2.08 thf(func_def_391, type, sK206: nat > nat > nat).
% 4.90/2.08 thf(func_def_392, type, sK207: int > int > nat).
% 4.90/2.08 thf(func_def_393, type, sK208: (int > $o) > int > $o).
% 4.90/2.08 thf(func_def_394, type, sK209: (int > $o) > int).
% 4.90/2.08 thf(func_def_395, type, sK210: ((nat > $o) > $o) > nat > $o).
% 4.90/2.08 thf(func_def_396, type, sK211: ((nat > $o) > $o) > nat).
% 4.90/2.08 thf(func_def_397, type, sK212: ((nat > $o) > $o) > nat).
% 4.90/2.08 thf(func_def_398, type, sK213: ((x_a > $o) > $o) > (int > x_a > $o) > (int > $o) > int > $o).
% 4.90/2.08 thf(func_def_399, type, sK214: ((x_a > $o) > $o) > (x_a > $o) > x_a).
% 4.90/2.08 thf(func_def_400, type, sK215: ((x_a > $o) > $o) > (x_a > $o) > x_a > $o).
% 4.90/2.08 thf(func_def_401, type, sK216: ((int > $o) > x_a) > ((int > $o) > $o) > (x_a > $o) > (int > $o) > $o).
% 4.90/2.08 thf(func_def_402, type, sK217: (nat > $o) > (nat > $o) > nat).
% 4.90/2.08 thf(func_def_403, type, sK218: (nat > $o) > ((x_a > $o) > $o) > ((x_a > $o) > nat) > (x_a > $o) > $o).
% 4.90/2.08 thf(func_def_404, type, sK219: (pname > $o) > (pname > int > $o) > ((int > $o) > $o) > pname > $o).
% 4.90/2.08 thf(func_def_405, type, sK220: (nat > $o) > (nat > int > $o) > nat).
% 4.90/2.08 thf(func_def_406, type, sK221: (x_a > $o) > (x_a > $o) > x_a).
% 4.90/2.08 thf(func_def_408, type, ph223: !>[X0: $tType]:(X0)).
% 4.90/2.08 thf(f8243,plain,(
% 4.90/2.08 $false),
% 4.90/2.08 inference(trivial_inequality_removal,[status(thm)],[f8241])).
% 4.90/2.08 thf(f8241,plain,(
% 4.90/2.08 ($true != $true)),
% 4.90/2.08 inference(superposition,[status(thm)],[f8138,f5785])).
% 4.90/2.08 thf(f5785,plain,(
% 4.90/2.08 ((member_pname @ pn @ u) = $true)),
% 4.90/2.08 inference(cnf_transformation,[status(thm)],[f2417])).
% 4.90/2.08 thf(f2417,plain,(
% 4.90/2.08 ((member_pname @ pn @ u) = $true)),
% 4.90/2.08 inference(fool_elimination,[status(thm)],[f2416])).
% 4.90/2.08 thf(f2416,plain,(
% 4.90/2.08 (member_pname @ pn @ u)),
% 4.90/2.08 inference(rectify,[status(thm)],[f1213])).
% 4.90/2.08 thf(f1213,axiom,(
% 4.90/2.08 (member_pname @ pn @ u)),
% 4.90/2.08 file('/export/starexec/sandbox/tmp/tmp.KmoHMpFwsY/DTF2THF_2194.p',conj_4)).
% 4.90/2.08 thf(f8138,plain,(
% 4.90/2.08 ((member_pname @ pn @ u) != $true)),
% 4.90/2.08 inference(trivial_inequality_removal,[status(thm)],[f8136])).
% 4.90/2.08 thf(f8136,plain,(
% 4.90/2.08 ((member_pname @ pn @ u) != $true) | ($true != $true)),
% 4.90/2.08 inference(superposition,[status(thm)],[f7814,f6903])).
% 4.90/2.08 thf(f6903,plain,(
% 4.90/2.08 ( ! [X2 : pname,X0 : pname > x_a,X1 : pname > $o] : (((member_a @ (X0 @ X2) @ (image_pname_a @ X0 @ X1)) = $true) | ((member_pname @ X2 @ X1) != $true)) )),
% 4.90/2.08 inference(cnf_transformation,[status(thm)],[f3997])).
% 4.90/2.08 thf(f3997,plain,(
% 4.90/2.08 ! [X0 : pname > x_a,X1 : pname > $o,X2 : pname] : (((member_a @ (X0 @ X2) @ (image_pname_a @ X0 @ X1)) = $true) | ((member_pname @ X2 @ X1) != $true))),
% 4.90/2.08 inference(ennf_transformation,[status(thm)],[f1289])).
% 4.90/2.08 thf(f1289,plain,(
% 4.90/2.08 ! [X0 : pname > x_a,X1 : pname > $o,X2 : pname] : (((member_pname @ X2 @ X1) = $true) => ((member_a @ (X0 @ X2) @ (image_pname_a @ X0 @ X1)) = $true))),
% 4.90/2.08 inference(fool_elimination,[status(thm)],[f1288])).
% 4.90/2.08 thf(f1288,plain,(
% 4.90/2.08 ! [X0 : pname > x_a,X1 : pname > $o,X2 : pname] : ((member_pname @ X2 @ X1) => (member_a @ (X0 @ X2) @ (image_pname_a @ X0 @ X1)))),
% 4.90/2.08 inference(rectify,[status(thm)],[f435])).
% 4.90/2.08 thf(f435,axiom,(
% 4.90/2.08 ! [X160 : pname > x_a,X162 : pname > $o,X161 : pname] : ((member_pname @ X161 @ X162) => (member_a @ (X160 @ X161) @ (image_pname_a @ X160 @ X162)))),
% 4.90/2.08 file('/export/starexec/sandbox/tmp/tmp.KmoHMpFwsY/DTF2THF_2194.p',fact_434_imageI)).
% 4.90/2.08 thf(f7814,plain,(
% 4.90/2.08 ($true != (member_a @ (mgt_call @ pn) @ (image_pname_a @ mgt_call @ u)))),
% 4.90/2.08 inference(trivial_inequality_removal,[status(thm)],[f7813])).
% 4.90/2.08 thf(f7813,plain,(
% 4.90/2.08 ($true != (member_a @ (mgt_call @ pn) @ (image_pname_a @ mgt_call @ u))) | ($true != $true)),
% 4.90/2.08 inference(forward_demodulation,[status(thm)],[f7801,f6937])).
% 4.90/2.08 thf(f6937,plain,(
% 4.90/2.08 ((ord_less_eq_a_o @ g @ (image_pname_a @ mgt_call @ u)) = $true)),
% 4.90/2.08 inference(cnf_transformation,[status(thm)],[f2737])).
% 4.90/2.08 thf(f2737,plain,(
% 4.90/2.08 ((ord_less_eq_a_o @ g @ (image_pname_a @ mgt_call @ u)) = $true)),
% 4.90/2.08 inference(fool_elimination,[status(thm)],[f2736])).
% 4.90/2.08 thf(f2736,plain,(
% 4.90/2.08 (ord_less_eq_a_o @ g @ (image_pname_a @ mgt_call @ u))),
% 4.90/2.08 inference(rectify,[status(thm)],[f1210])).
% 4.90/2.08 thf(f1210,axiom,(
% 4.90/2.08 (ord_less_eq_a_o @ g @ (image_pname_a @ mgt_call @ u))),
% 4.90/2.08 file('/export/starexec/sandbox/tmp/tmp.KmoHMpFwsY/DTF2THF_2194.p',conj_1)).
% 4.90/2.08 thf(f7801,plain,(
% 4.90/2.08 ((ord_less_eq_a_o @ g @ (image_pname_a @ mgt_call @ u)) != $true) | ($true != (member_a @ (mgt_call @ pn) @ (image_pname_a @ mgt_call @ u)))),
% 4.90/2.08 inference(trivial_inequality_removal,[status(thm)],[f7798])).
% 4.90/2.08 thf(f7798,plain,(
% 4.90/2.08 ($true != (member_a @ (mgt_call @ pn) @ (image_pname_a @ mgt_call @ u))) | ((ord_less_eq_a_o @ g @ (image_pname_a @ mgt_call @ u)) != $true) | ($true != $true)),
% 4.90/2.08 inference(superposition,[status(thm)],[f5434,f6060])).
% 4.90/2.08 thf(f6060,plain,(
% 4.90/2.08 ( ! [X2 : x_a,X0 : x_a > $o,X1 : x_a > $o] : (((ord_less_eq_a_o @ (insert_a @ X2 @ X1) @ X0) = $true) | ((ord_less_eq_a_o @ X1 @ X0) != $true) | ((member_a @ X2 @ X0) != $true)) )),
% 4.90/2.08 inference(cnf_transformation,[status(thm)],[f4958])).
% 4.90/2.08 thf(f4958,plain,(
% 4.90/2.08 ! [X0 : x_a > $o,X1 : x_a > $o,X2 : x_a] : (((((member_a @ X2 @ X0) = $true) & ((ord_less_eq_a_o @ X1 @ X0) = $true)) | ((ord_less_eq_a_o @ (insert_a @ X2 @ X1) @ X0) != $true)) & (((ord_less_eq_a_o @ (insert_a @ X2 @ X1) @ X0) = $true) | ((member_a @ X2 @ X0) != $true) | ((ord_less_eq_a_o @ X1 @ X0) != $true)))),
% 4.90/2.08 inference(flattening,[status(thm)],[f4957])).
% 4.90/2.08 thf(f4957,plain,(
% 4.90/2.08 ! [X0 : x_a > $o,X1 : x_a > $o,X2 : x_a] : (((((member_a @ X2 @ X0) = $true) & ((ord_less_eq_a_o @ X1 @ X0) = $true)) | ((ord_less_eq_a_o @ (insert_a @ X2 @ X1) @ X0) != $true)) & (((ord_less_eq_a_o @ (insert_a @ X2 @ X1) @ X0) = $true) | (((member_a @ X2 @ X0) != $true) | ((ord_less_eq_a_o @ X1 @ X0) != $true))))),
% 4.90/2.08 inference(nnf_transformation,[status(thm)],[f1857])).
% 4.90/2.08 thf(f1857,plain,(
% 4.90/2.08 ! [X0 : x_a > $o,X1 : x_a > $o,X2 : x_a] : ((((member_a @ X2 @ X0) = $true) & ((ord_less_eq_a_o @ X1 @ X0) = $true)) <=> ((ord_less_eq_a_o @ (insert_a @ X2 @ X1) @ X0) = $true))),
% 4.90/2.08 inference(fool_elimination,[status(thm)],[f1856])).
% 4.90/2.08 thf(f1856,plain,(
% 4.90/2.08 ! [X0 : x_a > $o,X1 : x_a > $o,X2 : x_a] : ((ord_less_eq_a_o @ (insert_a @ X2 @ X1) @ X0) <=> ((member_a @ X2 @ X0) & (ord_less_eq_a_o @ X1 @ X0)))),
% 4.90/2.08 inference(rectify,[status(thm)],[f454])).
% 4.90/2.08 thf(f454,axiom,(
% 4.90/2.08 ! [X181 : x_a > $o,X180 : x_a > $o,X179 : x_a] : ((ord_less_eq_a_o @ (insert_a @ X179 @ X180) @ X181) <=> ((member_a @ X179 @ X181) & (ord_less_eq_a_o @ X180 @ X181)))),
% 4.90/2.08 file('/export/starexec/sandbox/tmp/tmp.KmoHMpFwsY/DTF2THF_2194.p',fact_453_insert__subset)).
% 4.90/2.08 thf(f5434,plain,(
% 4.90/2.08 ((ord_less_eq_a_o @ (insert_a @ (mgt_call @ pn) @ g) @ (image_pname_a @ mgt_call @ u)) != $true)),
% 4.90/2.08 inference(cnf_transformation,[status(thm)],[f3346])).
% 4.90/2.08 thf(f3346,plain,(
% 4.90/2.08 ((ord_less_eq_a_o @ (insert_a @ (mgt_call @ pn) @ g) @ (image_pname_a @ mgt_call @ u)) != $true)),
% 4.90/2.08 inference(flattening,[status(thm)],[f1807])).
% 4.90/2.08 thf(f1807,plain,(
% 4.90/2.08 ~((ord_less_eq_a_o @ (insert_a @ (mgt_call @ pn) @ g) @ (image_pname_a @ mgt_call @ u)) = $true)),
% 4.90/2.08 inference(fool_elimination,[status(thm)],[f1806])).
% 4.90/2.08 thf(f1806,plain,(
% 4.90/2.08 ~(ord_less_eq_a_o @ (insert_a @ (mgt_call @ pn) @ g) @ (image_pname_a @ mgt_call @ u))),
% 4.90/2.08 inference(rectify,[status(thm)],[f1216])).
% 4.90/2.08 thf(f1216,negated_conjecture,(
% 4.90/2.08 ~(ord_less_eq_a_o @ (insert_a @ (mgt_call @ pn) @ g) @ (image_pname_a @ mgt_call @ u))),
% 4.90/2.08 inference(negated_conjecture,[status(cth)],[f1215])).
% 4.90/2.08 thf(f1215,conjecture,(
% 4.90/2.08 (ord_less_eq_a_o @ (insert_a @ (mgt_call @ pn) @ g) @ (image_pname_a @ mgt_call @ u))),
% 4.90/2.08 file('/export/starexec/sandbox/tmp/tmp.KmoHMpFwsY/DTF2THF_2194.p',conj_6)).
% 4.90/2.08 % SZS output end Proof for DTF2THF_2194
% 4.90/2.08 % (2513)------------------------------
% 4.90/2.08 % (2513)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.90/2.08 % (2513)Termination reason: Refutation
% 4.90/2.08
% 4.90/2.08 % (2513)Memory used [KB]: 13432
% 4.90/2.08 % (2513)Time elapsed: 0.197 s
% 4.90/2.08 % (2513)Instructions burned: 643 (million)
% 4.90/2.08 % (2513)------------------------------
% 4.90/2.08 % (2513)------------------------------
% 4.90/2.08 % (2490)Success in time 0.344 s
%------------------------------------------------------------------------------