%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP027^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n031.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 08:54:57 AM UTC 2026
% Result : Theorem 3.23s 1.95s
% Output : Refutation 3.23s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 3
% Syntax : Number of formulae : 27 ( 7 unt; 0 typ; 0 def)
% Number of atoms : 235 ( 99 equ; 0 cnn)
% Maximal formula atoms : 14 ( 8 avg)
% Number of connectives : 912 ( 36 ~; 27 |; 40 &; 780 @)
% ( 0 <=>; 29 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Number of types : 55 ( 54 usr)
% Number of type conns : 72 ( 72 >; 0 *; 0 +; 0 <<)
% Number of symbols : 239 ( 236 usr; 5 con; 0-7 aty)
% Number of variables : 72 ( 18 ^; 52 !; 2 ?; 72 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
carrie1954260683t_unit: $tType ).
thf(type_def_6,type,
set_c_d: $tType ).
thf(type_def_8,type,
carrie1105631105xt_a_b: $tType ).
thf(type_def_9,type,
set_a: $tType ).
thf(type_def_10,type,
carrie1948989069_d_a_e: $tType ).
thf(type_def_11,type,
set_d: $tType ).
thf(type_def_12,type,
a: $tType ).
thf(type_def_13,type,
c: $tType ).
thf(type_def_14,type,
d: $tType ).
thf(type_def_15,type,
set_a_a: $tType ).
thf(type_def_16,type,
set_c: $tType ).
thf(type_def_17,type,
set_c_d_c_d: $tType ).
thf(type_def_18,type,
set_c_d_c_d_a2: $tType ).
thf(type_def_19,type,
set_c_d_c_d_d: $tType ).
thf(type_def_20,type,
set_a_a_a_a_a: $tType ).
thf(type_def_21,type,
set_a_a_a_a_a_d: $tType ).
thf(type_def_22,type,
set_a_a_a_a: $tType ).
thf(type_def_23,type,
set_a_a_a_a_a3: $tType ).
thf(type_def_24,type,
set_a_a_a_a_d: $tType ).
thf(type_def_25,type,
set_a_a_c_d: $tType ).
thf(type_def_26,type,
set_a_a_c_d_a: $tType ).
thf(type_def_27,type,
set_a_a_c_d_d: $tType ).
thf(type_def_28,type,
set_a_a_a: $tType ).
thf(type_def_29,type,
set_a_a_a_a4: $tType ).
thf(type_def_30,type,
set_a_a_a_d: $tType ).
thf(type_def_31,type,
set_a_a_a2: $tType ).
thf(type_def_32,type,
set_a_a_d: $tType ).
thf(type_def_33,type,
set_c_d_a: $tType ).
thf(type_def_34,type,
set_c_d_d: $tType ).
thf(type_def_35,type,
set_a_c_d: $tType ).
thf(type_def_36,type,
set_a_d: $tType ).
thf(type_def_37,type,
set_c_c_d: $tType ).
thf(type_def_38,type,
set_c_a: $tType ).
thf(type_def_39,type,
set_d_c_d: $tType ).
thf(type_def_40,type,
set_d_a: $tType ).
thf(type_def_41,type,
set_d_d: $tType ).
thf(type_def_42,type,
nat: $tType ).
thf(type_def_43,type,
set_a_a_a_a_a2: $tType ).
thf(type_def_44,type,
set_a_a_a_c_d: $tType ).
thf(type_def_45,type,
set_a_a_a_a3: $tType ).
thf(type_def_46,type,
set_a_a_c_d2: $tType ).
thf(type_def_47,type,
set_c_d_a_a_a2: $tType ).
thf(type_def_48,type,
set_c_d_c_d_c_d: $tType ).
thf(type_def_49,type,
set_c_d_c_d_a: $tType ).
thf(type_def_50,type,
set_c_d_a_a_a: $tType ).
thf(type_def_51,type,
set_c_d_a_a: $tType ).
thf(type_def_52,type,
set_a_a_a_a2: $tType ).
thf(type_def_53,type,
set_a_c_d_a: $tType ).
thf(type_def_54,type,
set_a_c_d_c_d: $tType ).
thf(type_def_55,type,
set_a_a_c_d_c_d: $tType ).
thf(type_def_56,type,
set_a_c: $tType ).
thf(type_def_57,type,
set_c_c: $tType ).
thf(type_def_58,type,
set_d_c: $tType ).
thf(type_def_59,type,
set_nat: $tType ).
thf(func_def_0,type,
carrie1954260683t_unit: $tType ).
thf(func_def_1,type,
carrie1948989069_d_a_e: $tType ).
thf(func_def_2,type,
carrie1105631105xt_a_b: $tType ).
thf(func_def_3,type,
set_a_a_c_d_c_d: $tType ).
thf(func_def_4,type,
set_c_d_c_d_c_d: $tType ).
thf(func_def_5,type,
set_a_a_a_a_a_d: $tType ).
thf(func_def_6,type,
set_a_a_a_a_a: $tType ).
thf(func_def_7,type,
set_a_c_d_c_d: $tType ).
thf(func_def_8,type,
set_c_d_a_a_a: $tType ).
thf(func_def_9,type,
set_c_d_c_d_a: $tType ).
thf(func_def_10,type,
set_c_d_a_a_a2: $tType ).
thf(func_def_11,type,
set_a_a_a_c_d: $tType ).
thf(func_def_12,type,
set_a_a_a_a_a2: $tType ).
thf(func_def_13,type,
set_a_a_c_d_d: $tType ).
thf(func_def_14,type,
set_a_a_c_d_a: $tType ).
thf(func_def_15,type,
set_a_a_a_a_d: $tType ).
thf(func_def_16,type,
set_a_a_a_a_a3: $tType ).
thf(func_def_17,type,
set_c_d_c_d_d: $tType ).
thf(func_def_18,type,
set_c_d_c_d_a2: $tType ).
thf(func_def_19,type,
set_a_a_c_d: $tType ).
thf(func_def_20,type,
set_a_a_a_a: $tType ).
thf(func_def_21,type,
set_a_c_d_a: $tType ).
thf(func_def_22,type,
set_a_a_a_a2: $tType ).
thf(func_def_23,type,
set_c_d_c_d: $tType ).
thf(func_def_24,type,
set_c_d_a_a: $tType ).
thf(func_def_25,type,
set_a_a_c_d2: $tType ).
thf(func_def_26,type,
set_a_a_a_a3: $tType ).
thf(func_def_27,type,
set_a_a_a_d: $tType ).
thf(func_def_28,type,
set_a_a_a_a4: $tType ).
thf(func_def_29,type,
set_d_c_d: $tType ).
thf(func_def_30,type,
set_c_c_d: $tType ).
thf(func_def_31,type,
set_a_c_d: $tType ).
thf(func_def_32,type,
set_a_a_a: $tType ).
thf(func_def_33,type,
set_c_d_d: $tType ).
thf(func_def_34,type,
set_c_d_a: $tType ).
thf(func_def_35,type,
set_a_a_d: $tType ).
thf(func_def_36,type,
set_a_a_a2: $tType ).
thf(func_def_37,type,
set_d_d: $tType ).
thf(func_def_38,type,
set_d_c: $tType ).
thf(func_def_39,type,
set_d_a: $tType ).
thf(func_def_40,type,
set_c_d: $tType ).
thf(func_def_41,type,
set_c_c: $tType ).
thf(func_def_42,type,
set_c_a: $tType ).
thf(func_def_43,type,
set_a_d: $tType ).
thf(func_def_44,type,
set_a_c: $tType ).
thf(func_def_45,type,
set_a_a: $tType ).
thf(func_def_46,type,
set_nat: $tType ).
thf(func_def_47,type,
set_d: $tType ).
thf(func_def_48,type,
set_c: $tType ).
thf(func_def_49,type,
set_a: $tType ).
thf(func_def_50,type,
nat: $tType ).
thf(func_def_51,type,
d: $tType ).
thf(func_def_52,type,
c: $tType ).
thf(func_def_53,type,
a: $tType ).
thf(func_def_54,type,
carrie20288686t_unit: carrie1954260683t_unit > set_c_d ).
thf(func_def_55,type,
carrie867757212xt_a_b: carrie1105631105xt_a_b > set_a ).
thf(func_def_56,type,
carrie11369756_d_a_e: carrie1948989069_d_a_e > set_d ).
thf(func_def_57,type,
j_rad_a_b: carrie1105631105xt_a_b > set_a ).
thf(func_def_58,type,
ring_a_b: carrie1105631105xt_a_b > $o ).
thf(func_def_59,type,
tp_a_b: carrie1105631105xt_a_b > a > a > a ).
thf(func_def_60,type,
un_a_b: carrie1105631105xt_a_b > a ).
thf(func_def_61,type,
sr_a_b: carrie1105631105xt_a_b > set_a > carrie1105631105xt_a_b ).
thf(func_def_62,type,
subring_a_b_b: carrie1105631105xt_a_b > carrie1105631105xt_a_b > $o ).
thf(func_def_63,type,
unit_a_b: carrie1105631105xt_a_b > a > $o ).
thf(func_def_64,type,
aGroup2131905830t_unit: carrie1954260683t_unit > $o ).
thf(func_def_65,type,
aGroup2097840802xt_a_b: carrie1105631105xt_a_b > $o ).
thf(func_def_66,type,
aGroup1940521469_d_a_e: carrie1948989069_d_a_e > $o ).
thf(func_def_67,type,
mop_a_Ring_ext_a_b: carrie1105631105xt_a_b > a > a ).
thf(func_def_68,type,
pop_c_449750139t_unit: carrie1954260683t_unit > ( c > d ) > ( c > d ) > c > d ).
thf(func_def_69,type,
pop_a_Ring_ext_a_b: carrie1105631105xt_a_b > a > a > a ).
thf(func_def_70,type,
pop_d_261713874_d_a_e: carrie1948989069_d_a_e > d > d > d ).
thf(func_def_71,type,
aset_s1743553894xt_a_b: carrie1105631105xt_a_b > set_a > set_a > set_a ).
thf(func_def_72,type,
carr_d267359390xt_a_b: set_a > ( a > carrie1105631105xt_a_b ) > set_a_a ).
thf(func_def_73,type,
carr_d56066307_d_a_e: set_c > ( c > carrie1948989069_d_a_e ) > set_c_d ).
thf(func_def_74,type,
carr_p1617094073xt_a_b: set_c_d_c_d > ( ( ( c > d ) > c > d ) > carrie1105631105xt_a_b ) > set_c_d_c_d_a2 ).
thf(func_def_75,type,
carr_p428609062_d_a_e: set_c_d_c_d > ( ( ( c > d ) > c > d ) > carrie1948989069_d_a_e ) > set_c_d_c_d_d ).
thf(func_def_76,type,
carr_p1980447933_d_a_e: set_a_a_a_a_a > ( ( a > a > a > a > a ) > carrie1948989069_d_a_e ) > set_a_a_a_a_a_d ).
thf(func_def_77,type,
carr_p1850824937xt_a_b: set_a_a_a_a > ( ( a > a > a > a ) > carrie1105631105xt_a_b ) > set_a_a_a_a_a3 ).
thf(func_def_78,type,
carr_p305487606_d_a_e: set_a_a_a_a > ( ( a > a > a > a ) > carrie1948989069_d_a_e ) > set_a_a_a_a_d ).
thf(func_def_79,type,
carr_p109409832xt_a_b: set_a_a_c_d > ( ( a > a > c > d ) > carrie1105631105xt_a_b ) > set_a_a_c_d_a ).
thf(func_def_80,type,
carr_p577380087_d_a_e: set_a_a_c_d > ( ( a > a > c > d ) > carrie1948989069_d_a_e ) > set_a_a_c_d_d ).
thf(func_def_81,type,
carr_p1481099064xt_a_b: set_a_a_a > ( ( a > a > a ) > carrie1105631105xt_a_b ) > set_a_a_a_a4 ).
thf(func_def_82,type,
carr_p1364179431_d_a_e: set_a_a_a > ( ( a > a > a ) > carrie1948989069_d_a_e ) > set_a_a_a_d ).
thf(func_def_83,type,
carr_p740328767xt_a_b: set_a_a > ( ( a > a ) > carrie1105631105xt_a_b ) > set_a_a_a2 ).
thf(func_def_84,type,
carr_p1646427936_d_a_e: set_a_a > ( ( a > a ) > carrie1948989069_d_a_e ) > set_a_a_d ).
thf(func_def_85,type,
carr_p996491646xt_a_b: set_c_d > ( ( c > d ) > carrie1105631105xt_a_b ) > set_c_d_a ).
thf(func_def_86,type,
carr_p124510241_d_a_e: set_c_d > ( ( c > d ) > carrie1948989069_d_a_e ) > set_c_d_d ).
thf(func_def_87,type,
carr_p923087418t_unit: set_a > ( a > carrie1954260683t_unit ) > set_a_c_d ).
thf(func_def_88,type,
carr_p290614414xt_a_b: set_a > ( a > carrie1105631105xt_a_b ) > set_a_a ).
thf(func_def_89,type,
carr_p1084104977_d_a_e: set_a > ( a > carrie1948989069_d_a_e ) > set_a_d ).
thf(func_def_90,type,
carr_p1080792380t_unit: set_c > ( c > carrie1954260683t_unit ) > set_c_c_d ).
thf(func_def_91,type,
carr_p295160588xt_a_b: set_c > ( c > carrie1105631105xt_a_b ) > set_c_a ).
thf(func_def_92,type,
carr_p1636662547_d_a_e: set_c > ( c > carrie1948989069_d_a_e ) > set_c_d ).
thf(func_def_93,type,
carr_p1159644861t_unit: set_d > ( d > carrie1954260683t_unit ) > set_d_c_d ).
thf(func_def_94,type,
carr_p297433675xt_a_b: set_d > ( d > carrie1105631105xt_a_b ) > set_d_a ).
thf(func_def_95,type,
carr_p1912941332_d_a_e: set_d > ( d > carrie1948989069_d_a_e ) > set_d_d ).
thf(func_def_96,type,
local_ring_a_b: carrie1105631105xt_a_b > $o ).
thf(func_def_97,type,
maximal_ideal_a_b: carrie1105631105xt_a_b > set_a > $o ).
thf(func_def_98,type,
mul_closed_set_a_b: carrie1105631105xt_a_b > set_a > $o ).
thf(func_def_99,type,
nscal_a_b: carrie1105631105xt_a_b > a > nat > a ).
thf(func_def_100,type,
primary_ideal_a_b: carrie1105631105xt_a_b > set_a > $o ).
thf(func_def_101,type,
prod_m2022989242xt_a_b: set_a > ( a > carrie1105631105xt_a_b ) > ( a > a ) > a > a ).
thf(func_def_102,type,
prod_m854557927_d_a_e: set_c > ( c > carrie1948989069_d_a_e ) > ( c > d ) > c > d ).
thf(func_def_103,type,
prod_p1959959370xt_a_b: set_c_d_c_d > ( ( ( c > d ) > c > d ) > carrie1105631105xt_a_b ) > ( ( ( c > d ) > c > d ) > a ) > ( ( ( c > d ) > c > d ) > a ) > ( ( c > d ) > c > d ) > a ).
thf(func_def_104,type,
prod_p46206586xt_a_b: set_a_a_a_a > ( ( a > a > a > a ) > carrie1105631105xt_a_b ) > ( ( a > a > a > a ) > a ) > ( ( a > a > a > a ) > a ) > ( a > a > a > a ) > a ).
thf(func_def_105,type,
prod_p452275129xt_a_b: set_a_a_c_d > ( ( a > a > c > d ) > carrie1105631105xt_a_b ) > ( ( a > a > c > d ) > a ) > ( ( a > a > c > d ) > a ) > ( a > a > c > d ) > a ).
thf(func_def_106,type,
prod_p1460711911xt_a_b: set_a_a_a > ( ( a > a > a ) > carrie1105631105xt_a_b ) > ( ( a > a > a ) > a ) > ( ( a > a > a ) > a ) > ( a > a > a ) > a ).
thf(func_def_107,type,
prod_p1210452560xt_a_b: set_a_a > ( ( a > a ) > carrie1105631105xt_a_b ) > ( ( a > a ) > a ) > ( ( a > a ) > a ) > ( a > a ) > a ).
thf(func_def_108,type,
prod_p1466615439xt_a_b: set_c_d > ( ( c > d ) > carrie1105631105xt_a_b ) > ( ( c > d ) > a ) > ( ( c > d ) > a ) > ( c > d ) > a ).
thf(func_def_109,type,
prod_p704341949xt_a_b: set_a > ( a > carrie1105631105xt_a_b ) > ( a > a ) > ( a > a ) > a > a ).
thf(func_def_110,type,
prod_p708888123xt_a_b: set_c > ( c > carrie1105631105xt_a_b ) > ( c > a ) > ( c > a ) > c > a ).
thf(func_def_111,type,
prod_p131511076_d_a_e: set_c > ( c > carrie1948989069_d_a_e ) > ( c > d ) > ( c > d ) > c > d ).
thf(func_def_112,type,
prod_p711161210xt_a_b: set_d > ( d > carrie1105631105xt_a_b ) > ( d > a ) > ( d > a ) > d > a ).
thf(func_def_113,type,
prod_z851058290xt_a_b: set_a > ( a > carrie1105631105xt_a_b ) > a > a ).
thf(func_def_114,type,
prod_z1214987823_d_a_e: set_c > ( c > carrie1948989069_d_a_e ) > c > d ).
thf(func_def_115,type,
sr_a_b2: carrie1105631105xt_a_b > set_a > $o ).
thf(func_def_116,type,
zeroring_a_b: carrie1105631105xt_a_b > $o ).
thf(func_def_117,type,
sum_mult_a_b: carrie1105631105xt_a_b > set_a > set_a > set_a ).
thf(func_def_118,type,
module1440397025unit_b: carrie1954260683t_unit > carrie1105631105xt_a_b > $o ).
thf(func_def_119,type,
module_d_a_e_b: carrie1948989069_d_a_e > carrie1105631105xt_a_b > $o ).
thf(func_def_120,type,
sprod_1929599593t_unit: carrie1954260683t_unit > a > ( c > d ) > c > d ).
thf(func_def_121,type,
sprod_d_a_e: carrie1948989069_d_a_e > a > d > d ).
thf(func_def_122,type,
mHom_a1526663829t_unit: carrie1105631105xt_a_b > carrie1954260683t_unit > carrie1954260683t_unit > set_c_d_c_d ).
thf(func_def_123,type,
mHom_a764575953it_d_e: carrie1105631105xt_a_b > carrie1954260683t_unit > carrie1948989069_d_a_e > set_c_d_d ).
thf(func_def_124,type,
mHom_a_b_d_e_d_e: carrie1105631105xt_a_b > carrie1948989069_d_a_e > carrie1948989069_d_a_e > set_d_d ).
thf(func_def_125,type,
algebr1968153176_c_d_e: carrie1105631105xt_a_b > set_c > ( c > carrie1948989069_d_a_e ) > carrie1954260683t_unit ).
thf(func_def_126,type,
algebr828196337_d_d_e: carrie1105631105xt_a_b > set_c_d_c_d > ( ( ( c > d ) > c > d ) > carrie1948989069_d_a_e ) > a > ( ( ( c > d ) > c > d ) > d ) > ( ( c > d ) > c > d ) > d ).
thf(func_def_127,type,
algebr2122715238_a_d_e: carrie1105631105xt_a_b > set_a_a_a_a_a > ( ( a > a > a > a > a ) > carrie1948989069_d_a_e ) > a > ( ( a > a > a > a > a ) > d ) > ( a > a > a > a > a ) > d ).
thf(func_def_128,type,
algebr303206081_a_d_e: carrie1105631105xt_a_b > set_a_a_a_a > ( ( a > a > a > a ) > carrie1948989069_d_a_e ) > a > ( ( a > a > a > a ) > d ) > ( a > a > a > a ) > d ).
thf(func_def_129,type,
algebr1620716610_d_d_e: carrie1105631105xt_a_b > set_a_a_c_d > ( ( a > a > c > d ) > carrie1948989069_d_a_e ) > a > ( ( a > a > c > d ) > d ) > ( a > a > c > d ) > d ).
thf(func_def_130,type,
algebr1655014672_a_d_e: carrie1105631105xt_a_b > set_a_a_a > ( ( a > a > a ) > carrie1948989069_d_a_e ) > a > ( ( a > a > a ) > d ) > ( a > a > a ) > d ).
thf(func_def_131,type,
algebr137561451_a_d_e: carrie1105631105xt_a_b > set_a_a > ( ( a > a ) > carrie1948989069_d_a_e ) > a > ( ( a > a ) > d ) > ( a > a ) > d ).
thf(func_def_132,type,
algebr924632044_d_d_e: carrie1105631105xt_a_b > set_c_d > ( ( c > d ) > carrie1948989069_d_a_e ) > a > ( ( c > d ) > d ) > ( c > d ) > d ).
thf(func_def_133,type,
algebr2109842362_a_d_e: carrie1105631105xt_a_b > set_a > ( a > carrie1948989069_d_a_e ) > a > ( a > d ) > a > d ).
thf(func_def_134,type,
algebr1697692348_c_d_e: carrie1105631105xt_a_b > set_c > ( c > carrie1948989069_d_a_e ) > a > ( c > d ) > c > d ).
thf(func_def_135,type,
algebr417875517_d_d_e: carrie1105631105xt_a_b > set_d > ( d > carrie1948989069_d_a_e ) > a > ( d > d ) > d > d ).
thf(func_def_136,type,
pi_a_a_a_a_a: set_a_a_a > ( ( a > a > a ) > set_a_a ) > set_a_a_a_a_a2 ).
thf(func_def_137,type,
pi_a_a_a_c_d: set_a_a_a > ( ( a > a > a ) > set_c_d ) > set_a_a_a_c_d ).
thf(func_def_138,type,
pi_a_a_a_a: set_a_a_a > ( ( a > a > a ) > set_a ) > set_a_a_a_a4 ).
thf(func_def_139,type,
pi_a_a_a_a2: set_a_a > ( ( a > a ) > set_a_a ) > set_a_a_a_a3 ).
thf(func_def_140,type,
pi_a_a_c_d: set_a_a > ( ( a > a ) > set_c_d ) > set_a_a_c_d2 ).
thf(func_def_141,type,
pi_a_a_a: set_a_a > ( ( a > a ) > set_a ) > set_a_a_a2 ).
thf(func_def_142,type,
pi_c_d_a_a_a: set_c_d > ( ( c > d ) > set_a_a_a2 ) > set_c_d_a_a_a2 ).
thf(func_def_143,type,
pi_c_d_c_d_c_d: set_c_d > ( ( c > d ) > set_c_d_c_d ) > set_c_d_c_d_c_d ).
thf(func_def_144,type,
pi_c_d_c_d_a: set_c_d > ( ( c > d ) > set_c_d_a ) > set_c_d_c_d_a ).
thf(func_def_145,type,
pi_c_d_a_a_a2: set_c_d > ( ( c > d ) > set_a_a_a ) > set_c_d_a_a_a ).
thf(func_def_146,type,
pi_c_d_a_a: set_c_d > ( ( c > d ) > set_a_a ) > set_c_d_a_a ).
thf(func_def_147,type,
pi_c_d_c_d: set_c_d > ( ( c > d ) > set_c_d ) > set_c_d_c_d ).
thf(func_def_148,type,
pi_c_d_a: set_c_d > ( ( c > d ) > set_a ) > set_c_d_a ).
thf(func_def_149,type,
pi_a_a_a_a3: set_a > ( a > set_a_a_a2 ) > set_a_a_a_a2 ).
thf(func_def_150,type,
pi_a_c_d_a: set_a > ( a > set_c_d_a ) > set_a_c_d_a ).
thf(func_def_151,type,
pi_a_a_c_d_c_d: set_a > ( a > set_a_c_d_c_d ) > set_a_a_c_d_c_d ).
thf(func_def_152,type,
pi_a_a_a_a_a2: set_a > ( a > set_a_a_a_a ) > set_a_a_a_a_a ).
thf(func_def_153,type,
pi_a_a_a_a4: set_a > ( a > set_a_a_a ) > set_a_a_a_a ).
thf(func_def_154,type,
pi_a_a_c_d2: set_a > ( a > set_a_c_d ) > set_a_a_c_d ).
thf(func_def_155,type,
pi_a_a_a2: set_a > ( a > set_a_a ) > set_a_a_a ).
thf(func_def_156,type,
pi_a_c_d: set_a > ( a > set_c_d ) > set_a_c_d ).
thf(func_def_157,type,
pi_a_a: set_a > ( a > set_a ) > set_a_a ).
thf(func_def_158,type,
pi_a_c: set_a > ( a > set_c ) > set_a_c ).
thf(func_def_159,type,
pi_a_d: set_a > ( a > set_d ) > set_a_d ).
thf(func_def_160,type,
pi_c_a: set_c > ( c > set_a ) > set_c_a ).
thf(func_def_161,type,
pi_c_c: set_c > ( c > set_c ) > set_c_c ).
thf(func_def_162,type,
pi_c_d: set_c > ( c > set_d ) > set_c_d ).
thf(func_def_163,type,
pi_d_a: set_d > ( d > set_a ) > set_d_a ).
thf(func_def_164,type,
pi_d_c: set_d > ( d > set_c ) > set_d_c ).
thf(func_def_165,type,
pi_d_d: set_d > ( d > set_d ) > set_d_d ).
thf(func_def_166,type,
plus_plus_nat: nat > nat > nat ).
thf(func_def_167,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_168,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(func_def_169,type,
collect_c_d_c_d: ( ( ( c > d ) > c > d ) > $o ) > set_c_d_c_d ).
thf(func_def_170,type,
collect_a_a_a_a: ( ( a > a > a > a ) > $o ) > set_a_a_a_a ).
thf(func_def_171,type,
collect_a_a_c_d: ( ( a > a > c > d ) > $o ) > set_a_a_c_d ).
thf(func_def_172,type,
collect_a_a_a: ( ( a > a > a ) > $o ) > set_a_a_a ).
thf(func_def_173,type,
collect_a_a: ( ( a > a ) > $o ) > set_a_a ).
thf(func_def_174,type,
collect_c_d: ( ( c > d ) > $o ) > set_c_d ).
thf(func_def_175,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(func_def_176,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_177,type,
collect_c: ( c > $o ) > set_c ).
thf(func_def_178,type,
collect_d: ( d > $o ) > set_d ).
thf(func_def_179,type,
member_c_d_c_d_a: ( ( ( c > d ) > c > d ) > a ) > set_c_d_c_d_a2 > $o ).
thf(func_def_180,type,
member_c_d_c_d_d: ( ( ( c > d ) > c > d ) > d ) > set_c_d_c_d_d > $o ).
thf(func_def_181,type,
member_a_a_a_a_a_d: ( ( a > a > a > a > a ) > d ) > set_a_a_a_a_a_d > $o ).
thf(func_def_182,type,
member_a_a_a_a_a: ( ( a > a > a > a ) > a ) > set_a_a_a_a_a3 > $o ).
thf(func_def_183,type,
member_a_a_a_a_d: ( ( a > a > a > a ) > d ) > set_a_a_a_a_d > $o ).
thf(func_def_184,type,
member_a_a_c_d_a: ( ( a > a > c > d ) > a ) > set_a_a_c_d_a > $o ).
thf(func_def_185,type,
member_a_a_c_d_d: ( ( a > a > c > d ) > d ) > set_a_a_c_d_d > $o ).
thf(func_def_186,type,
member_a_a_a_a_a2: ( ( a > a > a ) > a > a ) > set_a_a_a_a_a2 > $o ).
thf(func_def_187,type,
member_a_a_a_c_d: ( ( a > a > a ) > c > d ) > set_a_a_a_c_d > $o ).
thf(func_def_188,type,
member_a_a_a_a: ( ( a > a > a ) > a ) > set_a_a_a_a4 > $o ).
thf(func_def_189,type,
member_a_a_a_d: ( ( a > a > a ) > d ) > set_a_a_a_d > $o ).
thf(func_def_190,type,
member_a_a_a_a2: ( ( a > a ) > a > a ) > set_a_a_a_a3 > $o ).
thf(func_def_191,type,
member_a_a_c_d: ( ( a > a ) > c > d ) > set_a_a_c_d2 > $o ).
thf(func_def_192,type,
member_a_a_a: ( ( a > a ) > a ) > set_a_a_a2 > $o ).
thf(func_def_193,type,
member_a_a_d: ( ( a > a ) > d ) > set_a_a_d > $o ).
thf(func_def_194,type,
member_c_d_a_a_a: ( ( c > d ) > ( a > a ) > a ) > set_c_d_a_a_a2 > $o ).
thf(func_def_195,type,
member_c_d_c_d_c_d: ( ( c > d ) > ( c > d ) > c > d ) > set_c_d_c_d_c_d > $o ).
thf(func_def_196,type,
member_c_d_c_d_a2: ( ( c > d ) > ( c > d ) > a ) > set_c_d_c_d_a > $o ).
thf(func_def_197,type,
member_c_d_a_a_a2: ( ( c > d ) > a > a > a ) > set_c_d_a_a_a > $o ).
thf(func_def_198,type,
member_c_d_a_a: ( ( c > d ) > a > a ) > set_c_d_a_a > $o ).
thf(func_def_199,type,
member_c_d_c_d: ( ( c > d ) > c > d ) > set_c_d_c_d > $o ).
thf(func_def_200,type,
member_c_d_a: ( ( c > d ) > a ) > set_c_d_a > $o ).
thf(func_def_201,type,
member_c_d_d: ( ( c > d ) > d ) > set_c_d_d > $o ).
thf(func_def_202,type,
member_a_a_a_a3: ( a > ( a > a ) > a ) > set_a_a_a_a2 > $o ).
thf(func_def_203,type,
member_a_c_d_c_d: ( a > ( c > d ) > c > d ) > set_a_c_d_c_d > $o ).
thf(func_def_204,type,
member_a_c_d_a: ( a > ( c > d ) > a ) > set_a_c_d_a > $o ).
thf(func_def_205,type,
member_a_a_c_d_c_d: ( a > a > ( c > d ) > c > d ) > set_a_a_c_d_c_d > $o ).
thf(func_def_206,type,
member_a_a_a_a_a3: ( a > a > a > a > a ) > set_a_a_a_a_a > $o ).
thf(func_def_207,type,
member_a_a_a_a4: ( a > a > a > a ) > set_a_a_a_a > $o ).
thf(func_def_208,type,
member_a_a_c_d2: ( a > a > c > d ) > set_a_a_c_d > $o ).
thf(func_def_209,type,
member_a_a_a2: ( a > a > a ) > set_a_a_a > $o ).
thf(func_def_210,type,
member_a_c_d: ( a > c > d ) > set_a_c_d > $o ).
thf(func_def_211,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(func_def_212,type,
member_a_c: ( a > c ) > set_a_c > $o ).
thf(func_def_213,type,
member_a_d: ( a > d ) > set_a_d > $o ).
thf(func_def_214,type,
member_c_c_d: ( c > c > d ) > set_c_c_d > $o ).
thf(func_def_215,type,
member_c_a: ( c > a ) > set_c_a > $o ).
thf(func_def_216,type,
member_c_c: ( c > c ) > set_c_c > $o ).
thf(func_def_217,type,
member_c_d: ( c > d ) > set_c_d > $o ).
thf(func_def_218,type,
member_d_c_d: ( d > c > d ) > set_d_c_d > $o ).
thf(func_def_219,type,
member_d_a: ( d > a ) > set_d_a > $o ).
thf(func_def_220,type,
member_d_c: ( d > c ) > set_d_c > $o ).
thf(func_def_221,type,
member_d_d: ( d > d ) > set_d_d > $o ).
thf(func_def_222,type,
member_nat: nat > set_nat > $o ).
thf(func_def_223,type,
member_a: a > set_a > $o ).
thf(func_def_224,type,
member_c: c > set_c > $o ).
thf(func_def_225,type,
member_d: d > set_d > $o ).
thf(func_def_226,type,
i: set_c ).
thf(func_def_227,type,
m: c > carrie1948989069_d_a_e ).
thf(func_def_228,type,
r: carrie1105631105xt_a_b ).
thf(func_def_236,type,
sK0: set_c > ( c > carrie1948989069_d_a_e ) > c ).
thf(func_def_237,type,
sK1: ( c > d ) > set_c > ( c > d ) > c ).
thf(func_def_238,type,
sK2: set_c > ( c > carrie1948989069_d_a_e ) > c ).
thf(func_def_239,type,
sK3: ( c > carrie1948989069_d_a_e ) > set_c > carrie1105631105xt_a_b > c ).
thf(func_def_240,type,
sK4: ( ( c > d ) > ( c > d ) > c > d ) > ( ( c > d ) > ( c > d ) > c > d ) > set_c_d > c > d ).
thf(func_def_241,type,
sK5: c > d ).
thf(func_def_242,type,
sK6: ( c > carrie1948989069_d_a_e ) > carrie1105631105xt_a_b > set_c > c ).
thf(func_def_243,type,
sK7: ( c > carrie1948989069_d_a_e ) > set_c > c ).
thf(func_def_244,type,
sK8: ( c > carrie1948989069_d_a_e ) > set_c > c ).
thf(func_def_245,type,
sK9: set_a_a > ( ( a > a ) > carrie1948989069_d_a_e ) > a > a ).
thf(func_def_246,type,
sK10: set_a_a > ( ( a > a ) > carrie1948989069_d_a_e ) > carrie1105631105xt_a_b > a > a ).
thf(func_def_247,type,
sK11: ( d > carrie1948989069_d_a_e ) > carrie1105631105xt_a_b > set_d > d ).
thf(func_def_248,type,
sK12: ( ( a > a > a > a > a ) > carrie1948989069_d_a_e ) > set_a_a_a_a_a > a > a > a > a > a ).
thf(func_def_249,type,
sK13: ( c > carrie1948989069_d_a_e ) > set_c > c ).
thf(func_def_250,type,
sK14: ( ( c > d ) > carrie1948989069_d_a_e ) > set_c_d > c > d ).
thf(func_def_251,type,
sK15: ( ( ( c > d ) > c > d ) > carrie1948989069_d_a_e ) > set_c_d_c_d > ( c > d ) > c > d ).
thf(func_def_252,type,
sK16: set_a > ( a > carrie1948989069_d_a_e ) > a ).
thf(func_def_253,type,
sK17: set_c > ( c > d ) > ( c > d ) > c ).
thf(func_def_254,type,
sK18: set_c > ( c > carrie1948989069_d_a_e ) > c ).
thf(func_def_255,type,
sK19: set_a_a_a_a > ( ( a > a > a > a ) > carrie1948989069_d_a_e ) > a > a > a > a ).
thf(func_def_256,type,
sK20: ( ( a > a > a > a ) > carrie1948989069_d_a_e ) > set_a_a_a_a > carrie1105631105xt_a_b > a > a > a > a ).
thf(func_def_257,type,
sK21: ( ( c > d ) > c > d ) > set_c_d > set_c_d > c > d ).
thf(func_def_258,type,
sK22: set_c > ( c > carrie1948989069_d_a_e ) > c ).
thf(func_def_259,type,
sK23: set_c > ( c > carrie1948989069_d_a_e ) > ( c > carrie1948989069_d_a_e ) > c ).
thf(func_def_260,type,
sK24: ( c > carrie1948989069_d_a_e ) > set_c > carrie1105631105xt_a_b > c ).
thf(func_def_261,type,
sK25: set_c > ( c > carrie1948989069_d_a_e ) > c ).
thf(func_def_262,type,
sK26: ( ( ( c > d ) > c > d ) > carrie1105631105xt_a_b ) > set_c_d_c_d > ( c > d ) > c > d ).
thf(func_def_263,type,
sK27: set_a > ( a > carrie1948989069_d_a_e ) > carrie1105631105xt_a_b > a ).
thf(func_def_264,type,
sK28: set_c > ( c > carrie1948989069_d_a_e ) > ( c > carrie1948989069_d_a_e ) > c ).
thf(func_def_265,type,
sK29: set_c > ( c > carrie1948989069_d_a_e ) > c ).
thf(func_def_266,type,
sK30: ( c > carrie1948989069_d_a_e ) > set_c > c ).
thf(func_def_267,type,
sK31: set_c > ( c > carrie1948989069_d_a_e ) > c ).
thf(func_def_268,type,
sK32: set_c > ( c > carrie1948989069_d_a_e ) > c ).
thf(func_def_269,type,
sK33: ( ( c > d ) > set_c_d ) > set_c_d > ( ( c > d ) > c > d ) > c > d ).
thf(func_def_270,type,
sK34: ( d > carrie1948989069_d_a_e ) > set_d > d ).
thf(func_def_271,type,
sK35: set_c_d > ( ( c > d ) > c > d ) > ( ( c > d ) > c > d ) > c > d ).
thf(func_def_272,type,
sK36: set_a_a_a > ( ( a > a > a ) > carrie1948989069_d_a_e ) > carrie1105631105xt_a_b > a > a > a ).
thf(func_def_273,type,
sK37: ( c > carrie1948989069_d_a_e ) > set_c > c ).
thf(func_def_274,type,
sK38: set_a_a_a > ( ( a > a > a ) > carrie1948989069_d_a_e ) > a > a > a ).
thf(func_def_275,type,
sK39: ( c > carrie1948989069_d_a_e ) > set_c > c ).
thf(func_def_276,type,
sK40: set_c > ( c > carrie1948989069_d_a_e ) > c ).
thf(func_def_277,type,
sK41: ( c > carrie1948989069_d_a_e ) > set_c > c ).
thf(func_def_278,type,
sK42: set_c > ( c > carrie1948989069_d_a_e ) > c ).
thf(func_def_279,type,
sK43: set_c > ( c > carrie1948989069_d_a_e ) > ( c > carrie1948989069_d_a_e ) > c ).
thf(func_def_280,type,
sK44: set_c > ( c > carrie1948989069_d_a_e ) > c ).
thf(func_def_281,type,
sK45: ( c > carrie1948989069_d_a_e ) > set_c > c ).
thf(func_def_282,type,
sK46: ( c > carrie1948989069_d_a_e ) > set_c > carrie1105631105xt_a_b > c ).
thf(func_def_283,type,
sK47: ( c > d ) > set_c > ( c > d ) > c ).
thf(func_def_284,type,
sK48: set_a_a_c_d > ( ( a > a > c > d ) > carrie1948989069_d_a_e ) > carrie1105631105xt_a_b > a > a > c > d ).
thf(func_def_285,type,
sK49: ( ( a > a > a > a > a ) > carrie1948989069_d_a_e ) > set_a_a_a_a_a > carrie1105631105xt_a_b > a > a > a > a > a ).
thf(func_def_286,type,
sK50: ( c > carrie1948989069_d_a_e ) > set_c > c ).
thf(func_def_287,type,
sK51: carrie1105631105xt_a_b > ( ( c > d ) > carrie1948989069_d_a_e ) > set_c_d > c > d ).
thf(func_def_288,type,
sK52: ( c > carrie1948989069_d_a_e ) > set_c > ( c > carrie1948989069_d_a_e ) > c ).
thf(func_def_289,type,
sK53: ( c > carrie1948989069_d_a_e ) > set_c > c ).
thf(func_def_290,type,
sK54: set_c_d_c_d > ( ( ( c > d ) > c > d ) > carrie1948989069_d_a_e ) > carrie1105631105xt_a_b > ( c > d ) > c > d ).
thf(func_def_291,type,
sK55: set_a_a_c_d > ( ( a > a > c > d ) > carrie1948989069_d_a_e ) > a > a > c > d ).
thf(func_def_292,type,
sK56: set_c > ( c > carrie1948989069_d_a_e ) > c ).
thf(func_def_294,type,
sK58: ( ( c > d ) > set_c_d_c_d ) > ( ( c > d ) > ( c > d ) > c > d ) > set_c_d > c > d ).
thf(func_def_295,type,
sK59: set_c_d > ( ( c > d ) > c > d ) > ( ( c > d ) > set_c_d ) > c > d ).
thf(func_def_296,type,
ph60:
!>[X0: $tType] : X0 ).
thf(func_def_298,type,
sK61: c ).
thf(f1791,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1790,f1299]) ).
thf(f1299,plain,
( sK5
!= ( prod_p131511076_d_a_e @ i @ m @ sK5 @ ( prod_z1214987823_d_a_e @ i @ m ) ) ),
inference(cnf_transformation,[status(thm)],[f1185]) ).
thf(f1185,plain,
( ( sK5
!= ( prod_p131511076_d_a_e @ i @ m @ sK5 @ ( prod_z1214987823_d_a_e @ i @ m ) ) )
& ( ( member_c_d @ sK5 @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK5])],[f1085,f1184]) ).
thf(f1184,plain,
( ? [X0: c > d] :
( ( ( prod_p131511076_d_a_e @ i @ m @ X0 @ ( prod_z1214987823_d_a_e @ i @ m ) )
!= X0 )
& ( ( member_c_d @ X0 @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true ) )
=> ( ( sK5
!= ( prod_p131511076_d_a_e @ i @ m @ sK5 @ ( prod_z1214987823_d_a_e @ i @ m ) ) )
& ( ( member_c_d @ sK5 @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1085,plain,
? [X0: c > d] :
( ( ( prod_p131511076_d_a_e @ i @ m @ X0 @ ( prod_z1214987823_d_a_e @ i @ m ) )
!= X0 )
& ( ( member_c_d @ X0 @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f1054]) ).
thf(f1054,plain,
~ ! [X0: c > d] :
( ( ( prod_p131511076_d_a_e @ i @ m @ X0 @ ( prod_z1214987823_d_a_e @ i @ m ) )
= X0 )
| ( ( member_c_d @ X0 @ ( carr_p1636662547_d_a_e @ i @ m ) )
!= $true ) ),
inference(flattening,[status(thm)],[f914]) ).
thf(f914,plain,
~ ! [X0: c > d] :
( ( ( member_c_d @ X0 @ ( carr_p1636662547_d_a_e @ i @ m ) )
!= $true )
| ( ( prod_p131511076_d_a_e @ i @ m @ X0 @ ( prod_z1214987823_d_a_e @ i @ m ) )
= X0 ) ),
inference(fool_elimination,[status(thm)],[f913]) ).
thf(f913,plain,
~ ! [X0: c > d] :
( ~ ( member_c_d @ X0 @ ( carr_p1636662547_d_a_e @ i @ m ) )
| ( ( prod_p131511076_d_a_e @ i @ m @ X0 @ ( prod_z1214987823_d_a_e @ i @ m ) )
= X0 ) ),
inference(rectify,[status(thm)],[f359]) ).
thf(f359,negated_conjecture,
~ ! [X53: c > d] :
( ~ ( member_c_d @ X53 @ ( carr_p1636662547_d_a_e @ i @ m ) )
| ( ( prod_p131511076_d_a_e @ i @ m @ X53 @ ( prod_z1214987823_d_a_e @ i @ m ) )
= X53 ) ),
inference(negated_conjecture,[status(cth)],[f358]) ).
thf(f358,conjecture,
! [X53: c > d] :
( ~ ( member_c_d @ X53 @ ( carr_p1636662547_d_a_e @ i @ m ) )
| ( ( prod_p131511076_d_a_e @ i @ m @ X53 @ ( prod_z1214987823_d_a_e @ i @ m ) )
= X53 ) ),
file('/export/starexec/sandbox/tmp/tmp.2el5MKjZyq/DTF2THF_12139.p',conj_2) ).
thf(f1790,plain,
( sK5
= ( prod_p131511076_d_a_e @ i @ m @ sK5 @ ( prod_z1214987823_d_a_e @ i @ m ) ) ),
inference(forward_demodulation,[status(thm)],[f1781,f1492]) ).
thf(f1492,plain,
( sK5
= ( prod_p131511076_d_a_e @ i @ m @ ( prod_z1214987823_d_a_e @ i @ m ) @ sK5 ) ),
inference(trivial_inequality_removal,[status(thm)],[f1491]) ).
thf(f1491,plain,
( ( $true != $true )
| ( sK5
= ( prod_p131511076_d_a_e @ i @ m @ ( prod_z1214987823_d_a_e @ i @ m ) @ sK5 ) ) ),
inference(superposition,[status(thm)],[f1328,f1298]) ).
thf(f1298,plain,
( ( member_c_d @ sK5 @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f1185]) ).
thf(f1328,plain,
! [X3: c > d] :
( ( ( member_c_d @ X3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
!= $true )
| ( ( prod_p131511076_d_a_e @ i @ m @ ( prod_z1214987823_d_a_e @ i @ m ) @ X3 )
= X3 ) ),
inference(cnf_transformation,[status(thm)],[f1204]) ).
thf(f1204,plain,
( ( ( member_c_d @ ( prod_z1214987823_d_a_e @ i @ m ) @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true )
& ! [X0: c > d] :
( ! [X1: c > d] :
( ( ( member_c_d @ X1 @ ( carr_p1636662547_d_a_e @ i @ m ) )
!= $true )
| ! [X2: c > d] :
( ( ( member_c_d @ X2 @ ( carr_p1636662547_d_a_e @ i @ m ) )
!= $true )
| ( ( prod_p131511076_d_a_e @ i @ m @ ( prod_p131511076_d_a_e @ i @ m @ X0 @ X1 ) @ X2 )
= ( prod_p131511076_d_a_e @ i @ m @ X0 @ ( prod_p131511076_d_a_e @ i @ m @ X1 @ X2 ) ) ) ) )
| ( ( member_c_d @ X0 @ ( carr_p1636662547_d_a_e @ i @ m ) )
!= $true ) )
& ! [X3: c > d] :
( ( ( member_c_d @ X3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
!= $true )
| ( ( prod_p131511076_d_a_e @ i @ m @ ( prod_z1214987823_d_a_e @ i @ m ) @ X3 )
= X3 ) )
& ( ( member_c_d_c_d_c_d @ ( prod_p131511076_d_a_e @ i @ m )
@ ( pi_c_d_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
@ ^ [Y0: c > d] :
( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
@ ^ [Y1: c > d] : ( carr_p1636662547_d_a_e @ i @ m ) ) ) )
= $true )
& ! [X4: c > d] :
( ( ( prod_z1214987823_d_a_e @ i @ m )
= ( prod_p131511076_d_a_e @ i @ m @ ( prod_m854557927_d_a_e @ i @ m @ X4 ) @ X4 ) )
| ( ( member_c_d @ X4 @ ( carr_p1636662547_d_a_e @ i @ m ) )
!= $true ) )
& ! [X5: c > d] :
( ( ( member_c_d @ X5 @ ( carr_p1636662547_d_a_e @ i @ m ) )
!= $true )
| ! [X6: c > d] :
( ( ( prod_p131511076_d_a_e @ i @ m @ X6 @ X5 )
= ( prod_p131511076_d_a_e @ i @ m @ X5 @ X6 ) )
| ( ( member_c_d @ X6 @ ( carr_p1636662547_d_a_e @ i @ m ) )
!= $true ) ) )
& ( ( member_c_d_c_d @ ( prod_m854557927_d_a_e @ i @ m )
@ ( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
@ ^ [Y0: c > d] : ( carr_p1636662547_d_a_e @ i @ m ) ) )
= $true ) ),
inference(rectify,[status(thm)],[f1080]) ).
thf(f1080,plain,
( ( ( member_c_d @ ( prod_z1214987823_d_a_e @ i @ m ) @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true )
& ! [X0: c > d] :
( ! [X1: c > d] :
( ( ( member_c_d @ X1 @ ( carr_p1636662547_d_a_e @ i @ m ) )
!= $true )
| ! [X2: c > d] :
( ( ( member_c_d @ X2 @ ( carr_p1636662547_d_a_e @ i @ m ) )
!= $true )
| ( ( prod_p131511076_d_a_e @ i @ m @ ( prod_p131511076_d_a_e @ i @ m @ X0 @ X1 ) @ X2 )
= ( prod_p131511076_d_a_e @ i @ m @ X0 @ ( prod_p131511076_d_a_e @ i @ m @ X1 @ X2 ) ) ) ) )
| ( ( member_c_d @ X0 @ ( carr_p1636662547_d_a_e @ i @ m ) )
!= $true ) )
& ! [X4: c > d] :
( ( ( member_c_d @ X4 @ ( carr_p1636662547_d_a_e @ i @ m ) )
!= $true )
| ( ( prod_p131511076_d_a_e @ i @ m @ ( prod_z1214987823_d_a_e @ i @ m ) @ X4 )
= X4 ) )
& ( ( member_c_d_c_d_c_d @ ( prod_p131511076_d_a_e @ i @ m )
@ ( pi_c_d_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
@ ^ [Y0: c > d] :
( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
@ ^ [Y1: c > d] : ( carr_p1636662547_d_a_e @ i @ m ) ) ) )
= $true )
& ! [X3: c > d] :
( ( ( prod_z1214987823_d_a_e @ i @ m )
= ( prod_p131511076_d_a_e @ i @ m @ ( prod_m854557927_d_a_e @ i @ m @ X3 ) @ X3 ) )
| ( ( member_c_d @ X3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
!= $true ) )
& ! [X5: c > d] :
( ( ( member_c_d @ X5 @ ( carr_p1636662547_d_a_e @ i @ m ) )
!= $true )
| ! [X6: c > d] :
( ( ( prod_p131511076_d_a_e @ i @ m @ X6 @ X5 )
= ( prod_p131511076_d_a_e @ i @ m @ X5 @ X6 ) )
| ( ( member_c_d @ X6 @ ( carr_p1636662547_d_a_e @ i @ m ) )
!= $true ) ) )
& ( ( member_c_d_c_d @ ( prod_m854557927_d_a_e @ i @ m )
@ ( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
@ ^ [Y0: c > d] : ( carr_p1636662547_d_a_e @ i @ m ) ) )
= $true ) ),
inference(ennf_transformation,[status(thm)],[f1053]) ).
thf(f1053,plain,
( ! [X5: c > d] :
( ( ( member_c_d @ X5 @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true )
=> ! [X6: c > d] :
( ( ( member_c_d @ X6 @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true )
=> ( ( prod_p131511076_d_a_e @ i @ m @ X6 @ X5 )
= ( prod_p131511076_d_a_e @ i @ m @ X5 @ X6 ) ) ) )
& ! [X0: c > d] :
( ( ( member_c_d @ X0 @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true )
=> ! [X1: c > d] :
( ( ( member_c_d @ X1 @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true )
=> ! [X2: c > d] :
( ( ( member_c_d @ X2 @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true )
=> ( ( prod_p131511076_d_a_e @ i @ m @ ( prod_p131511076_d_a_e @ i @ m @ X0 @ X1 ) @ X2 )
= ( prod_p131511076_d_a_e @ i @ m @ X0 @ ( prod_p131511076_d_a_e @ i @ m @ X1 @ X2 ) ) ) ) ) )
& ! [X4: c > d] :
( ( ( member_c_d @ X4 @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true )
=> ( ( prod_p131511076_d_a_e @ i @ m @ ( prod_z1214987823_d_a_e @ i @ m ) @ X4 )
= X4 ) )
& ( ( member_c_d_c_d @ ( prod_m854557927_d_a_e @ i @ m )
@ ( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
@ ^ [Y0: c > d] : ( carr_p1636662547_d_a_e @ i @ m ) ) )
= $true )
& ! [X3: c > d] :
( ( ( member_c_d @ X3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true )
=> ( ( prod_z1214987823_d_a_e @ i @ m )
= ( prod_p131511076_d_a_e @ i @ m @ ( prod_m854557927_d_a_e @ i @ m @ X3 ) @ X3 ) ) )
& ( ( member_c_d @ ( prod_z1214987823_d_a_e @ i @ m ) @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true )
& ( ( member_c_d_c_d_c_d @ ( prod_p131511076_d_a_e @ i @ m )
@ ( pi_c_d_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
@ ^ [Y0: c > d] :
( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
@ ^ [Y1: c > d] : ( carr_p1636662547_d_a_e @ i @ m ) ) ) )
= $true ) ),
inference(rectify,[status(thm)],[f388]) ).
thf(f388,plain,
( ! [X0: c > d] :
( ( ( member_c_d @ X0 @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true )
=> ! [X1: c > d] :
( ( ( member_c_d @ X1 @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true )
=> ! [X2: c > d] :
( ( ( member_c_d @ X2 @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true )
=> ( ( prod_p131511076_d_a_e @ i @ m @ ( prod_p131511076_d_a_e @ i @ m @ X0 @ X1 ) @ X2 )
= ( prod_p131511076_d_a_e @ i @ m @ X0 @ ( prod_p131511076_d_a_e @ i @ m @ X1 @ X2 ) ) ) ) ) )
& ( ( member_c_d_c_d_c_d @ ( prod_p131511076_d_a_e @ i @ m )
@ ( pi_c_d_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
@ ^ [Y0: c > d] :
( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
@ ^ [Y1: c > d] : ( carr_p1636662547_d_a_e @ i @ m ) ) ) )
= $true )
& ( ( member_c_d @ ( prod_z1214987823_d_a_e @ i @ m ) @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true )
& ! [X5: c > d] :
( ( ( member_c_d @ X5 @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true )
=> ( ( prod_z1214987823_d_a_e @ i @ m )
= ( prod_p131511076_d_a_e @ i @ m @ ( prod_m854557927_d_a_e @ i @ m @ X5 ) @ X5 ) ) )
& ( ( member_c_d_c_d @ ( prod_m854557927_d_a_e @ i @ m )
@ ( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
@ ^ [Y0: c > d] : ( carr_p1636662547_d_a_e @ i @ m ) ) )
= $true )
& ! [X7: c > d] :
( ( ( member_c_d @ X7 @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true )
=> ( ( prod_p131511076_d_a_e @ i @ m @ ( prod_z1214987823_d_a_e @ i @ m ) @ X7 )
= X7 ) )
& ! [X8: c > d] :
( ( ( member_c_d @ X8 @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true )
=> ! [X9: c > d] :
( ( ( member_c_d @ X9 @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true )
=> ( ( prod_p131511076_d_a_e @ i @ m @ X8 @ X9 )
= ( prod_p131511076_d_a_e @ i @ m @ X9 @ X8 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f387]) ).
thf(f387,plain,
( ! [X0: c > d] :
( ( member_c_d @ X0 @ ( carr_p1636662547_d_a_e @ i @ m ) )
=> ! [X1: c > d] :
( ( member_c_d @ X1 @ ( carr_p1636662547_d_a_e @ i @ m ) )
=> ! [X2: c > d] :
( ( member_c_d @ X2 @ ( carr_p1636662547_d_a_e @ i @ m ) )
=> ( ( prod_p131511076_d_a_e @ i @ m @ ( prod_p131511076_d_a_e @ i @ m @ X0 @ X1 ) @ X2 )
= ( prod_p131511076_d_a_e @ i @ m @ X0 @ ( prod_p131511076_d_a_e @ i @ m @ X1 @ X2 ) ) ) ) ) )
& ( member_c_d_c_d_c_d @ ( prod_p131511076_d_a_e @ i @ m )
@ ( pi_c_d_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
@ ^ [X3: c > d] :
( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
@ ^ [X4: c > d] : ( carr_p1636662547_d_a_e @ i @ m ) ) ) )
& ( member_c_d @ ( prod_z1214987823_d_a_e @ i @ m ) @ ( carr_p1636662547_d_a_e @ i @ m ) )
& ! [X5: c > d] :
( ( member_c_d @ X5 @ ( carr_p1636662547_d_a_e @ i @ m ) )
=> ( ( prod_z1214987823_d_a_e @ i @ m )
= ( prod_p131511076_d_a_e @ i @ m @ ( prod_m854557927_d_a_e @ i @ m @ X5 ) @ X5 ) ) )
& ( member_c_d_c_d @ ( prod_m854557927_d_a_e @ i @ m )
@ ( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
@ ^ [X6: c > d] : ( carr_p1636662547_d_a_e @ i @ m ) ) )
& ! [X7: c > d] :
( ( member_c_d @ X7 @ ( carr_p1636662547_d_a_e @ i @ m ) )
=> ( ( prod_p131511076_d_a_e @ i @ m @ ( prod_z1214987823_d_a_e @ i @ m ) @ X7 )
= X7 ) )
& ! [X8: c > d] :
( ( member_c_d @ X8 @ ( carr_p1636662547_d_a_e @ i @ m ) )
=> ! [X9: c > d] :
( ( member_c_d @ X9 @ ( carr_p1636662547_d_a_e @ i @ m ) )
=> ( ( prod_p131511076_d_a_e @ i @ m @ X8 @ X9 )
= ( prod_p131511076_d_a_e @ i @ m @ X9 @ X8 ) ) ) ) ),
inference(rectify,[status(thm)],[f357]) ).
thf(f357,axiom,
( ! [X50: c > d] :
( ( member_c_d @ X50 @ ( carr_p1636662547_d_a_e @ i @ m ) )
=> ! [X51: c > d] :
( ( member_c_d @ X51 @ ( carr_p1636662547_d_a_e @ i @ m ) )
=> ! [X52: c > d] :
( ( member_c_d @ X52 @ ( carr_p1636662547_d_a_e @ i @ m ) )
=> ( ( prod_p131511076_d_a_e @ i @ m @ ( prod_p131511076_d_a_e @ i @ m @ X50 @ X51 ) @ X52 )
= ( prod_p131511076_d_a_e @ i @ m @ X50 @ ( prod_p131511076_d_a_e @ i @ m @ X51 @ X52 ) ) ) ) ) )
& ( member_c_d_c_d_c_d @ ( prod_p131511076_d_a_e @ i @ m )
@ ( pi_c_d_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
@ ^ [X5: c > d] :
( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
@ ^ [X6: c > d] : ( carr_p1636662547_d_a_e @ i @ m ) ) ) )
& ( member_c_d @ ( prod_z1214987823_d_a_e @ i @ m ) @ ( carr_p1636662547_d_a_e @ i @ m ) )
& ! [X50: c > d] :
( ( member_c_d @ X50 @ ( carr_p1636662547_d_a_e @ i @ m ) )
=> ( ( prod_p131511076_d_a_e @ i @ m @ ( prod_m854557927_d_a_e @ i @ m @ X50 ) @ X50 )
= ( prod_z1214987823_d_a_e @ i @ m ) ) )
& ( member_c_d_c_d @ ( prod_m854557927_d_a_e @ i @ m )
@ ( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
@ ^ [X5: c > d] : ( carr_p1636662547_d_a_e @ i @ m ) ) )
& ! [X50: c > d] :
( ( member_c_d @ X50 @ ( carr_p1636662547_d_a_e @ i @ m ) )
=> ( ( prod_p131511076_d_a_e @ i @ m @ ( prod_z1214987823_d_a_e @ i @ m ) @ X50 )
= X50 ) )
& ! [X50: c > d] :
( ( member_c_d @ X50 @ ( carr_p1636662547_d_a_e @ i @ m ) )
=> ! [X51: c > d] :
( ( member_c_d @ X51 @ ( carr_p1636662547_d_a_e @ i @ m ) )
=> ( ( prod_p131511076_d_a_e @ i @ m @ X50 @ X51 )
= ( prod_p131511076_d_a_e @ i @ m @ X51 @ X50 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.2el5MKjZyq/DTF2THF_12139.p',conj_1) ).
thf(f1781,plain,
( ( prod_p131511076_d_a_e @ i @ m @ sK5 @ ( prod_z1214987823_d_a_e @ i @ m ) )
= ( prod_p131511076_d_a_e @ i @ m @ ( prod_z1214987823_d_a_e @ i @ m ) @ sK5 ) ),
inference(trivial_inequality_removal,[status(thm)],[f1775]) ).
thf(f1775,plain,
( ( $true != $true )
| ( ( prod_p131511076_d_a_e @ i @ m @ sK5 @ ( prod_z1214987823_d_a_e @ i @ m ) )
= ( prod_p131511076_d_a_e @ i @ m @ ( prod_z1214987823_d_a_e @ i @ m ) @ sK5 ) ) ),
inference(superposition,[status(thm)],[f1509,f1330]) ).
thf(f1330,plain,
( ( member_c_d @ ( prod_z1214987823_d_a_e @ i @ m ) @ ( carr_p1636662547_d_a_e @ i @ m ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f1204]) ).
thf(f1509,plain,
! [X0: c > d] :
( ( ( member_c_d @ X0 @ ( carr_p1636662547_d_a_e @ i @ m ) )
!= $true )
| ( ( prod_p131511076_d_a_e @ i @ m @ sK5 @ X0 )
= ( prod_p131511076_d_a_e @ i @ m @ X0 @ sK5 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f1507]) ).
thf(f1507,plain,
! [X0: c > d] :
( ( ( prod_p131511076_d_a_e @ i @ m @ sK5 @ X0 )
= ( prod_p131511076_d_a_e @ i @ m @ X0 @ sK5 ) )
| ( $true != $true )
| ( ( member_c_d @ X0 @ ( carr_p1636662547_d_a_e @ i @ m ) )
!= $true ) ),
inference(superposition,[status(thm)],[f1325,f1298]) ).
thf(f1325,plain,
! [X6: c > d,X5: c > d] :
( ( ( member_c_d @ X6 @ ( carr_p1636662547_d_a_e @ i @ m ) )
!= $true )
| ( ( prod_p131511076_d_a_e @ i @ m @ X6 @ X5 )
= ( prod_p131511076_d_a_e @ i @ m @ X5 @ X6 ) )
| ( ( member_c_d @ X5 @ ( carr_p1636662547_d_a_e @ i @ m ) )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1204]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : ITP027^1 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.14 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.35 % Computer : n031.cluster.edu
% 0.16/0.35 % Model : x86_64 x86_64
% 0.16/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.35 % Memory : 8042.1875MB
% 0.16/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.35 % CPULimit : 300
% 0.16/0.35 % WCLimit : 300
% 0.16/0.35 % DateTime : Tue Feb 24 08:24:48 EST 2026
% 0.16/0.35 % CPUTime :
% 0.16/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.27/0.48 ---- Original DTF file ---
% 0.27/0.48 thf(spec,logic,$$dhol).
% 0.27/0.48 %------------------------------------------------------------------------------
% 0.27/0.48 % File : ITP027^1 : TPTP v9.2.1. Released v7.5.0.
% 0.27/0.48 % Domain : Interactive Theorem Proving
% 0.27/0.48 % Problem : Sledgehammer Algebra8 problem prob_2341__6471840_1
% 0.27/0.48 % Version : Especial.
% 0.27/0.48 % English :
% 0.27/0.48
% 0.27/0.48 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.27/0.48 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.27/0.48 % Source : [Des21]
% 0.27/0.48 % Names : Algebra8/prob_2341__6471840_1 [Des21]
% 0.27/0.48
% 0.27/0.48 % Status : Theorem
% 0.27/0.48 % Rating : 0.22 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% 0.27/0.48 % Syntax : Number of formulae : 587 ( 39 unt; 229 typ; 0 def)
% 0.27/0.48 % Number of atoms : 1402 ( 259 equ; 0 cnn)
% 0.27/0.48 % Maximal formula atoms : 14 ( 3 avg)
% 0.27/0.48 % Number of connectives : 6019 ( 29 ~; 1 |; 8 &;5008 @)
% 0.27/0.48 % ( 0 <=>; 973 =>; 0 <=; 0 <~>)
% 0.27/0.48 % Maximal formula depth : 17 ( 10 avg)
% 0.27/0.48 % Number of types : 55 ( 54 usr)
% 0.27/0.48 % Number of type conns : 1891 (1891 >; 0 *; 0 +; 0 <<)
% 0.27/0.48 % Number of symbols : 176 ( 175 usr; 2 con; 0-6 aty)
% 0.27/0.48 % Number of variables : 1543 ( 170 ^;1369 !; 4 ?;1543 :)
% 0.27/0.48 % SPC : TH0_THM_EQU_NAR
% 0.27/0.48
% 0.27/0.48 % Comments : This file was generated by Sledgehammer 2021-02-23 15:48:27.663
% 0.27/0.48 %------------------------------------------------------------------------------
% 0.27/0.48 % Could-be-implicit typings (54)
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% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_Mtf__a_J_J,type,
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% 0.27/0.48
% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_I_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_Mtf__a_J_J,type,
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% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J,type,
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% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_J,type,
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% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_I_062_Itf__c_Mtf__d_J_Mtf__a_J_J_J,type,
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% 0.27/0.48
% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_J_J,type,
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% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_J,type,
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% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__a_Mtf__a_J_J_J,type,
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% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_Mtf__a_J_M_062_Itf__c_Mtf__d_J_J_J,type,
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% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_Mtf__d_J_J,type,
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% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_Mtf__a_J_J,type,
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% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_Itf__c_M_062_Itf__c_Mtf__d_J_J_J,type,
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% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_Itf__c_Mtf__d_J_J_J,type,
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% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J,type,
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% 0.27/0.48
% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mtf__d_J_J,type,
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% 0.27/0.48
% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mtf__a_J_J,type,
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% 0.27/0.48
% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_Mtf__a_J_Mtf__d_J_J,type,
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% 0.27/0.48
% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
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% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_Itf__d_Mtf__a_J_J,type,
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% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__d_J_J,type,
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% 0.27/0.48
% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__c_J_J,type,
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% 0.27/0.48
% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__a_J_J,type,
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% 0.27/0.48
% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__d_J_J,type,
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% 0.27/0.48
% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__c_J_J,type,
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% 0.27/0.48
% 0.27/0.48 thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
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% 0.27/0.48
% 0.27/0.48 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 0.27/0.48 set_nat: $tType ).
% 0.27/0.48
% 0.27/0.48 thf(ty_n_t__Set__Oset_Itf__d_J,type,
% 0.27/0.48 set_d: $tType ).
% 0.27/0.48
% 0.27/0.48 thf(ty_n_t__Set__Oset_Itf__c_J,type,
% 0.27/0.48 set_c: $tType ).
% 0.27/0.48
% 0.27/0.48 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.27/0.48 set_a: $tType ).
% 0.27/0.48
% 0.27/0.48 thf(ty_n_t__Nat__Onat,type,
% 0.27/0.48 nat: $tType ).
% 0.27/0.48
% 0.27/0.48 thf(ty_n_tf__d,type,
% 0.27/0.48 d: $tType ).
% 0.27/0.48
% 0.27/0.48 thf(ty_n_tf__c,type,
% 0.27/0.48 c: $tType ).
% 0.27/0.48
% 0.27/0.48 thf(ty_n_tf__a,type,
% 0.27/0.48 a: $tType ).
% 0.27/0.48
% 0.27/0.48 % Explicit typings (175)
% 0.27/0.48 thf(sy_c_Algebra1_Ocarrier_Ocarrier_001_062_Itf__c_Mtf__d_J_001t__Algebra4__OaGroup__OaGroup____ext_I_062_Itf__c_Mtf__d_J_Mt__Algebra7__OModule__OModule____ext_I_062_Itf__c_Mtf__d_J_Mtf__a_Mt__Product____Type__Ounit_J_J,type,
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% 0.27/0.48 thf(sy_c_Algebra1_Ocarrier_Ocarrier_001tf__a_001t__Algebra4__OaGroup__OaGroup____ext_Itf__a_Mt__Algebra4__ORing__ORing____ext_Itf__a_Mtf__b_J_J,type,
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% 0.27/0.48 thf(sy_c_Algebra1_Ocarrier_Ocarrier_001tf__d_001t__Algebra4__OaGroup__OaGroup____ext_Itf__d_Mt__Algebra7__OModule__OModule____ext_Itf__d_Mtf__a_Mtf__e_J_J,type,
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% 0.27/0.48 thf(sy_c_Algebra4_OJ__rad_001tf__a_001tf__b,type,
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% 0.27/0.48
% 0.27/0.48 thf(sy_c_Algebra4_ORing_001tf__a_001tf__b,type,
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% 0.27/0.48
% 0.27/0.48 thf(sy_c_Algebra4_ORing_Otp_001tf__a_001tf__b,type,
% 0.27/0.48 tp_a_b: carrie1105631105xt_a_b > a > a > a ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Algebra4_ORing_Oun_001tf__a_001tf__b,type,
% 0.27/0.48 un_a_b: carrie1105631105xt_a_b > a ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Algebra4_OSr_001tf__a_001tf__b,type,
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% 0.27/0.48
% 0.27/0.48 thf(sy_c_Algebra4_OSubring_001tf__a_001tf__b_001tf__b,type,
% 0.27/0.48 subring_a_b_b: carrie1105631105xt_a_b > carrie1105631105xt_a_b > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Algebra4_OUnit_001tf__a_001tf__b,type,
% 0.27/0.48 unit_a_b: carrie1105631105xt_a_b > a > $o ).
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% 0.27/0.48 thf(sy_c_Algebra4_OaGroup_001_062_Itf__c_Mtf__d_J_001t__Algebra7__OModule__OModule____ext_I_062_Itf__c_Mtf__d_J_Mtf__a_Mt__Product____Type__Ounit_J,type,
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% 0.27/0.48 thf(sy_c_Algebra4_OaGroup_001tf__a_001t__Algebra4__ORing__ORing____ext_Itf__a_Mtf__b_J,type,
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% 0.27/0.48 mHom_a764575953it_d_e: carrie1105631105xt_a_b > carrie1954260683t_unit > carrie1948989069_d_a_e > set_c_d_d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Algebra7_OmHom_001tf__a_001tf__b_001tf__d_001tf__e_001tf__d_001tf__e,type,
% 0.27/0.48 mHom_a_b_d_e_d_e: carrie1105631105xt_a_b > carrie1948989069_d_a_e > carrie1948989069_d_a_e > set_d_d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Algebra8__Mirabelle__tltoludslr_OprodM_001tf__a_001tf__b_001tf__c_001tf__d_001tf__e,type,
% 0.27/0.48 algebr1968153176_c_d_e: carrie1105631105xt_a_b > set_c > ( c > carrie1948989069_d_a_e ) > carrie1954260683t_unit ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Algebra8__Mirabelle__tltoludslr_OprodM__sprod_001tf__a_001tf__b_001_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_001tf__d_001tf__e,type,
% 0.27/0.48 algebr828196337_d_d_e: carrie1105631105xt_a_b > set_c_d_c_d > ( ( ( c > d ) > c > d ) > carrie1948989069_d_a_e ) > a > ( ( ( c > d ) > c > d ) > d ) > ( ( c > d ) > c > d ) > d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Algebra8__Mirabelle__tltoludslr_OprodM__sprod_001tf__a_001tf__b_001_062_Itf__a_M_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_J_001tf__d_001tf__e,type,
% 0.27/0.48 algebr2122715238_a_d_e: carrie1105631105xt_a_b > set_a_a_a_a_a > ( ( a > a > a > a > a ) > carrie1948989069_d_a_e ) > a > ( ( a > a > a > a > a ) > d ) > ( a > a > a > a > a ) > d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Algebra8__Mirabelle__tltoludslr_OprodM__sprod_001tf__a_001tf__b_001_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_001tf__d_001tf__e,type,
% 0.27/0.48 algebr303206081_a_d_e: carrie1105631105xt_a_b > set_a_a_a_a > ( ( a > a > a > a ) > carrie1948989069_d_a_e ) > a > ( ( a > a > a > a ) > d ) > ( a > a > a > a ) > d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Algebra8__Mirabelle__tltoludslr_OprodM__sprod_001tf__a_001tf__b_001_062_Itf__a_M_062_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_001tf__d_001tf__e,type,
% 0.27/0.48 algebr1620716610_d_d_e: carrie1105631105xt_a_b > set_a_a_c_d > ( ( a > a > c > d ) > carrie1948989069_d_a_e ) > a > ( ( a > a > c > d ) > d ) > ( a > a > c > d ) > d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Algebra8__Mirabelle__tltoludslr_OprodM__sprod_001tf__a_001tf__b_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_001tf__d_001tf__e,type,
% 0.27/0.48 algebr1655014672_a_d_e: carrie1105631105xt_a_b > set_a_a_a > ( ( a > a > a ) > carrie1948989069_d_a_e ) > a > ( ( a > a > a ) > d ) > ( a > a > a ) > d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Algebra8__Mirabelle__tltoludslr_OprodM__sprod_001tf__a_001tf__b_001_062_Itf__a_Mtf__a_J_001tf__d_001tf__e,type,
% 0.27/0.48 algebr137561451_a_d_e: carrie1105631105xt_a_b > set_a_a > ( ( a > a ) > carrie1948989069_d_a_e ) > a > ( ( a > a ) > d ) > ( a > a ) > d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Algebra8__Mirabelle__tltoludslr_OprodM__sprod_001tf__a_001tf__b_001_062_Itf__c_Mtf__d_J_001tf__d_001tf__e,type,
% 0.27/0.48 algebr924632044_d_d_e: carrie1105631105xt_a_b > set_c_d > ( ( c > d ) > carrie1948989069_d_a_e ) > a > ( ( c > d ) > d ) > ( c > d ) > d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Algebra8__Mirabelle__tltoludslr_OprodM__sprod_001tf__a_001tf__b_001tf__a_001tf__d_001tf__e,type,
% 0.27/0.48 algebr2109842362_a_d_e: carrie1105631105xt_a_b > set_a > ( a > carrie1948989069_d_a_e ) > a > ( a > d ) > a > d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Algebra8__Mirabelle__tltoludslr_OprodM__sprod_001tf__a_001tf__b_001tf__c_001tf__d_001tf__e,type,
% 0.27/0.48 algebr1697692348_c_d_e: carrie1105631105xt_a_b > set_c > ( c > carrie1948989069_d_a_e ) > a > ( c > d ) > c > d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Algebra8__Mirabelle__tltoludslr_OprodM__sprod_001tf__a_001tf__b_001tf__d_001tf__d_001tf__e,type,
% 0.27/0.48 algebr417875517_d_d_e: carrie1105631105xt_a_b > set_d > ( d > carrie1948989069_d_a_e ) > a > ( d > d ) > d > d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_001_062_Itf__a_Mtf__a_J,type,
% 0.27/0.48 pi_a_a_a_a_a: set_a_a_a > ( ( a > a > a ) > set_a_a ) > set_a_a_a_a_a2 ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_001_062_Itf__c_Mtf__d_J,type,
% 0.27/0.48 pi_a_a_a_c_d: set_a_a_a > ( ( a > a > a ) > set_c_d ) > set_a_a_a_c_d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_001tf__a,type,
% 0.27/0.48 pi_a_a_a_a: set_a_a_a > ( ( a > a > a ) > set_a ) > set_a_a_a_a4 ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001_062_Itf__a_Mtf__a_J_001_062_Itf__a_Mtf__a_J,type,
% 0.27/0.48 pi_a_a_a_a2: set_a_a > ( ( a > a ) > set_a_a ) > set_a_a_a_a3 ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001_062_Itf__a_Mtf__a_J_001_062_Itf__c_Mtf__d_J,type,
% 0.27/0.48 pi_a_a_c_d: set_a_a > ( ( a > a ) > set_c_d ) > set_a_a_c_d2 ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001_062_Itf__a_Mtf__a_J_001tf__a,type,
% 0.27/0.48 pi_a_a_a: set_a_a > ( ( a > a ) > set_a ) > set_a_a_a2 ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001_062_Itf__c_Mtf__d_J_001_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J,type,
% 0.27/0.48 pi_c_d_a_a_a: set_c_d > ( ( c > d ) > set_a_a_a2 ) > set_c_d_a_a_a2 ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001_062_Itf__c_Mtf__d_J_001_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J,type,
% 0.27/0.48 pi_c_d_c_d_c_d: set_c_d > ( ( c > d ) > set_c_d_c_d ) > set_c_d_c_d_c_d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001_062_Itf__c_Mtf__d_J_001_062_I_062_Itf__c_Mtf__d_J_Mtf__a_J,type,
% 0.27/0.48 pi_c_d_c_d_a: set_c_d > ( ( c > d ) > set_c_d_a ) > set_c_d_c_d_a ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001_062_Itf__c_Mtf__d_J_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J,type,
% 0.27/0.48 pi_c_d_a_a_a2: set_c_d > ( ( c > d ) > set_a_a_a ) > set_c_d_a_a_a ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001_062_Itf__c_Mtf__d_J_001_062_Itf__a_Mtf__a_J,type,
% 0.27/0.48 pi_c_d_a_a: set_c_d > ( ( c > d ) > set_a_a ) > set_c_d_a_a ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001_062_Itf__c_Mtf__d_J_001_062_Itf__c_Mtf__d_J,type,
% 0.27/0.48 pi_c_d_c_d: set_c_d > ( ( c > d ) > set_c_d ) > set_c_d_c_d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001_062_Itf__c_Mtf__d_J_001tf__a,type,
% 0.27/0.48 pi_c_d_a: set_c_d > ( ( c > d ) > set_a ) > set_c_d_a ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001tf__a_001_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J,type,
% 0.27/0.48 pi_a_a_a_a3: set_a > ( a > set_a_a_a2 ) > set_a_a_a_a2 ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001tf__a_001_062_I_062_Itf__c_Mtf__d_J_Mtf__a_J,type,
% 0.27/0.48 pi_a_c_d_a: set_a > ( a > set_c_d_a ) > set_a_c_d_a ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001tf__a_001_062_Itf__a_M_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_J,type,
% 0.27/0.48 pi_a_a_c_d_c_d: set_a > ( a > set_a_c_d_c_d ) > set_a_a_c_d_c_d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001tf__a_001_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J,type,
% 0.27/0.48 pi_a_a_a_a_a2: set_a > ( a > set_a_a_a_a ) > set_a_a_a_a_a ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001tf__a_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J,type,
% 0.27/0.48 pi_a_a_a_a4: set_a > ( a > set_a_a_a ) > set_a_a_a_a ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001tf__a_001_062_Itf__a_M_062_Itf__c_Mtf__d_J_J,type,
% 0.27/0.48 pi_a_a_c_d2: set_a > ( a > set_a_c_d ) > set_a_a_c_d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001tf__a_001_062_Itf__a_Mtf__a_J,type,
% 0.27/0.48 pi_a_a_a2: set_a > ( a > set_a_a ) > set_a_a_a ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001tf__a_001_062_Itf__c_Mtf__d_J,type,
% 0.27/0.48 pi_a_c_d: set_a > ( a > set_c_d ) > set_a_c_d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001tf__a_001tf__a,type,
% 0.27/0.48 pi_a_a: set_a > ( a > set_a ) > set_a_a ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001tf__a_001tf__c,type,
% 0.27/0.48 pi_a_c: set_a > ( a > set_c ) > set_a_c ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001tf__a_001tf__d,type,
% 0.27/0.48 pi_a_d: set_a > ( a > set_d ) > set_a_d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001tf__c_001tf__a,type,
% 0.27/0.48 pi_c_a: set_c > ( c > set_a ) > set_c_a ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001tf__c_001tf__c,type,
% 0.27/0.48 pi_c_c: set_c > ( c > set_c ) > set_c_c ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001tf__c_001tf__d,type,
% 0.27/0.48 pi_c_d: set_c > ( c > set_d ) > set_c_d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001tf__d_001tf__a,type,
% 0.27/0.48 pi_d_a: set_d > ( d > set_a ) > set_d_a ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001tf__d_001tf__c,type,
% 0.27/0.48 pi_d_c: set_d > ( d > set_c ) > set_d_c ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_FuncSet_OPi_001tf__d_001tf__d,type,
% 0.27/0.48 pi_d_d: set_d > ( d > set_d ) > set_d_d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 0.27/0.48 plus_plus_nat: nat > nat > nat ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.27/0.48 ord_less_eq_nat: nat > nat > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 0.27/0.48 ord_less_eq_set_a: set_a > set_a > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Set_OCollect_001_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J,type,
% 0.27/0.48 collect_c_d_c_d: ( ( ( c > d ) > c > d ) > $o ) > set_c_d_c_d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Set_OCollect_001_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J,type,
% 0.27/0.48 collect_a_a_a_a: ( ( a > a > a > a ) > $o ) > set_a_a_a_a ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Set_OCollect_001_062_Itf__a_M_062_Itf__a_M_062_Itf__c_Mtf__d_J_J_J,type,
% 0.27/0.48 collect_a_a_c_d: ( ( a > a > c > d ) > $o ) > set_a_a_c_d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Set_OCollect_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J,type,
% 0.27/0.48 collect_a_a_a: ( ( a > a > a ) > $o ) > set_a_a_a ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Set_OCollect_001_062_Itf__a_Mtf__a_J,type,
% 0.27/0.48 collect_a_a: ( ( a > a ) > $o ) > set_a_a ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Set_OCollect_001_062_Itf__c_Mtf__d_J,type,
% 0.27/0.48 collect_c_d: ( ( c > d ) > $o ) > set_c_d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 0.27/0.48 collect_nat: ( nat > $o ) > set_nat ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.27/0.48 collect_a: ( a > $o ) > set_a ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Set_OCollect_001tf__c,type,
% 0.27/0.48 collect_c: ( c > $o ) > set_c ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_Set_OCollect_001tf__d,type,
% 0.27/0.48 collect_d: ( d > $o ) > set_d ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_I_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_Mtf__a_J,type,
% 0.27/0.48 member_c_d_c_d_a: ( ( ( c > d ) > c > d ) > a ) > set_c_d_c_d_a2 > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_I_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_Mtf__d_J,type,
% 0.27/0.48 member_c_d_c_d_d: ( ( ( c > d ) > c > d ) > d ) > set_c_d_c_d_d > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_J_Mtf__d_J,type,
% 0.27/0.48 member_a_a_a_a_a_d: ( ( a > a > a > a > a ) > d ) > set_a_a_a_a_a_d > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_Mtf__a_J,type,
% 0.27/0.48 member_a_a_a_a_a: ( ( a > a > a > a ) > a ) > set_a_a_a_a_a3 > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_Mtf__d_J,type,
% 0.27/0.48 member_a_a_a_a_d: ( ( a > a > a > a ) > d ) > set_a_a_a_a_d > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_Mtf__a_J,type,
% 0.27/0.48 member_a_a_c_d_a: ( ( a > a > c > d ) > a ) > set_a_a_c_d_a > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_Mtf__d_J,type,
% 0.27/0.48 member_a_a_c_d_d: ( ( a > a > c > d ) > d ) > set_a_a_c_d_d > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_M_062_Itf__a_Mtf__a_J_J,type,
% 0.27/0.48 member_a_a_a_a_a2: ( ( a > a > a ) > a > a ) > set_a_a_a_a_a2 > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_M_062_Itf__c_Mtf__d_J_J,type,
% 0.27/0.48 member_a_a_a_c_d: ( ( a > a > a ) > c > d ) > set_a_a_a_c_d > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_Mtf__a_J,type,
% 0.27/0.48 member_a_a_a_a: ( ( a > a > a ) > a ) > set_a_a_a_a4 > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_Mtf__d_J,type,
% 0.27/0.48 member_a_a_a_d: ( ( a > a > a ) > d ) > set_a_a_a_d > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_I_062_Itf__a_Mtf__a_J_M_062_Itf__a_Mtf__a_J_J,type,
% 0.27/0.48 member_a_a_a_a2: ( ( a > a ) > a > a ) > set_a_a_a_a3 > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_I_062_Itf__a_Mtf__a_J_M_062_Itf__c_Mtf__d_J_J,type,
% 0.27/0.48 member_a_a_c_d: ( ( a > a ) > c > d ) > set_a_a_c_d2 > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J,type,
% 0.27/0.48 member_a_a_a: ( ( a > a ) > a ) > set_a_a_a2 > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_I_062_Itf__a_Mtf__a_J_Mtf__d_J,type,
% 0.27/0.48 member_a_a_d: ( ( a > a ) > d ) > set_a_a_d > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_M_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
% 0.27/0.48 member_c_d_a_a_a: ( ( c > d ) > ( a > a ) > a ) > set_c_d_a_a_a2 > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_M_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_J,type,
% 0.27/0.48 member_c_d_c_d_c_d: ( ( c > d ) > ( c > d ) > c > d ) > set_c_d_c_d_c_d > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_M_062_I_062_Itf__c_Mtf__d_J_Mtf__a_J_J,type,
% 0.27/0.48 member_c_d_c_d_a2: ( ( c > d ) > ( c > d ) > a ) > set_c_d_c_d_a > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J,type,
% 0.27/0.48 member_c_d_a_a_a2: ( ( c > d ) > a > a > a ) > set_c_d_a_a_a > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__a_Mtf__a_J_J,type,
% 0.27/0.48 member_c_d_a_a: ( ( c > d ) > a > a ) > set_c_d_a_a > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J,type,
% 0.27/0.48 member_c_d_c_d: ( ( c > d ) > c > d ) > set_c_d_c_d > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_Mtf__a_J,type,
% 0.27/0.48 member_c_d_a: ( ( c > d ) > a ) > set_c_d_a > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_Mtf__d_J,type,
% 0.27/0.48 member_c_d_d: ( ( c > d ) > d ) > set_c_d_d > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_Itf__a_M_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
% 0.27/0.48 member_a_a_a_a3: ( a > ( a > a ) > a ) > set_a_a_a_a2 > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_Itf__a_M_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_J,type,
% 0.27/0.48 member_a_c_d_c_d: ( a > ( c > d ) > c > d ) > set_a_c_d_c_d > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_Itf__a_M_062_I_062_Itf__c_Mtf__d_J_Mtf__a_J_J,type,
% 0.27/0.48 member_a_c_d_a: ( a > ( c > d ) > a ) > set_a_c_d_a > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_Itf__a_M_062_Itf__a_M_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_J_J,type,
% 0.27/0.48 member_a_a_c_d_c_d: ( a > a > ( c > d ) > c > d ) > set_a_a_c_d_c_d > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_Itf__a_M_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_J,type,
% 0.27/0.48 member_a_a_a_a_a3: ( a > a > a > a > a ) > set_a_a_a_a_a > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J,type,
% 0.27/0.48 member_a_a_a_a4: ( a > a > a > a ) > set_a_a_a_a > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_Itf__a_M_062_Itf__a_M_062_Itf__c_Mtf__d_J_J_J,type,
% 0.27/0.48 member_a_a_c_d2: ( a > a > c > d ) > set_a_a_c_d > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J,type,
% 0.27/0.48 member_a_a_a2: ( a > a > a ) > set_a_a_a > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_Itf__a_M_062_Itf__c_Mtf__d_J_J,type,
% 0.27/0.48 member_a_c_d: ( a > c > d ) > set_a_c_d > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
% 0.27/0.48 member_a_a: ( a > a ) > set_a_a > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_Itf__a_Mtf__c_J,type,
% 0.27/0.48 member_a_c: ( a > c ) > set_a_c > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_Itf__a_Mtf__d_J,type,
% 0.27/0.48 member_a_d: ( a > d ) > set_a_d > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_Itf__c_M_062_Itf__c_Mtf__d_J_J,type,
% 0.27/0.48 member_c_c_d: ( c > c > d ) > set_c_c_d > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_Itf__c_Mtf__a_J,type,
% 0.27/0.48 member_c_a: ( c > a ) > set_c_a > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_Itf__c_Mtf__c_J,type,
% 0.27/0.48 member_c_c: ( c > c ) > set_c_c > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_Itf__c_Mtf__d_J,type,
% 0.27/0.48 member_c_d: ( c > d ) > set_c_d > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_Itf__d_M_062_Itf__c_Mtf__d_J_J,type,
% 0.27/0.48 member_d_c_d: ( d > c > d ) > set_d_c_d > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_Itf__d_Mtf__a_J,type,
% 0.27/0.48 member_d_a: ( d > a ) > set_d_a > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_Itf__d_Mtf__c_J,type,
% 0.27/0.48 member_d_c: ( d > c ) > set_d_c > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001_062_Itf__d_Mtf__d_J,type,
% 0.27/0.48 member_d_d: ( d > d ) > set_d_d > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001t__Nat__Onat,type,
% 0.27/0.48 member_nat: nat > set_nat > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001tf__a,type,
% 0.27/0.48 member_a: a > set_a > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001tf__c,type,
% 0.27/0.48 member_c: c > set_c > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_c_member_001tf__d,type,
% 0.27/0.48 member_d: d > set_d > $o ).
% 0.27/0.48
% 0.27/0.48 thf(sy_v_I,type,
% 0.27/0.48 i: set_c ).
% 0.27/0.48
% 0.27/0.48 thf(sy_v_M,type,
% 0.27/0.48 m: c > carrie1948989069_d_a_e ).
% 0.27/0.48
% 0.27/0.48 thf(sy_v_R,type,
% 0.27/0.48 r: carrie1105631105xt_a_b ).
% 0.27/0.48
% 0.27/0.48 % Relevant facts (355)
% 0.27/0.48 thf(fact_0_Ring__axioms,axiom,
% 0.27/0.48 ring_a_b @ r ).
% 0.27/0.48
% 0.27/0.48 % Ring_axioms
% 0.27/0.48 thf(fact_1_bivar__fun,axiom,
% 0.27/0.48 ! [F: a > c > d,A: set_a,B: set_c,C: set_d,A2: a] :
% 0.27/0.48 ( ( member_a_c_d @ F
% 0.27/0.48 @ ( pi_a_c_d @ A
% 0.27/0.48 @ ^ [Uu: a] :
% 0.27/0.48 ( pi_c_d @ B
% 0.27/0.48 @ ^ [Uv: c] : C ) ) )
% 0.27/0.48 => ( ( member_a @ A2 @ A )
% 0.27/0.48 => ( member_c_d @ ( F @ A2 )
% 0.27/0.48 @ ( pi_c_d @ B
% 0.27/0.48 @ ^ [Uu: c] : C ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % bivar_fun
% 0.27/0.48 thf(fact_2_bivar__fun,axiom,
% 0.27/0.48 ! [F: a > a > a,A: set_a,B: set_a,C: set_a,A2: a] :
% 0.27/0.48 ( ( member_a_a_a2 @ F
% 0.27/0.48 @ ( pi_a_a_a2 @ A
% 0.27/0.48 @ ^ [Uu: a] :
% 0.27/0.48 ( pi_a_a @ B
% 0.27/0.48 @ ^ [Uv: a] : C ) ) )
% 0.27/0.48 => ( ( member_a @ A2 @ A )
% 0.27/0.48 => ( member_a_a @ ( F @ A2 )
% 0.27/0.48 @ ( pi_a_a @ B
% 0.27/0.48 @ ^ [Uu: a] : C ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % bivar_fun
% 0.27/0.48 thf(fact_3_bivar__fun,axiom,
% 0.27/0.48 ! [F: ( c > d ) > a > a,A: set_c_d,B: set_a,C: set_a,A2: c > d] :
% 0.27/0.48 ( ( member_c_d_a_a @ F
% 0.27/0.48 @ ( pi_c_d_a_a @ A
% 0.27/0.48 @ ^ [Uu: c > d] :
% 0.27/0.48 ( pi_a_a @ B
% 0.27/0.48 @ ^ [Uv: a] : C ) ) )
% 0.27/0.48 => ( ( member_c_d @ A2 @ A )
% 0.27/0.48 => ( member_a_a @ ( F @ A2 )
% 0.27/0.48 @ ( pi_a_a @ B
% 0.27/0.48 @ ^ [Uu: a] : C ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % bivar_fun
% 0.27/0.48 thf(fact_4_bivar__fun,axiom,
% 0.27/0.48 ! [F: ( a > a ) > c > d,A: set_a_a,B: set_c,C: set_d,A2: a > a] :
% 0.27/0.48 ( ( member_a_a_c_d @ F
% 0.27/0.48 @ ( pi_a_a_c_d @ A
% 0.27/0.48 @ ^ [Uu: a > a] :
% 0.27/0.48 ( pi_c_d @ B
% 0.27/0.48 @ ^ [Uv: c] : C ) ) )
% 0.27/0.48 => ( ( member_a_a @ A2 @ A )
% 0.27/0.48 => ( member_c_d @ ( F @ A2 )
% 0.27/0.48 @ ( pi_c_d @ B
% 0.27/0.48 @ ^ [Uu: c] : C ) ) ) ) ).
% 0.27/0.48
% 0.27/0.48 % bivar_fun
% 0.27/0.48 thf(fact_5_bivar__fun,axiom,
% 0.27/0.48 ! [F: ( a > a ) > a > a,A: set_a_a,B: set_a,C: set_a,A2: a > a] :
% 0.27/0.48 ( ( member_a_a_a_a2 @ F
% 0.27/0.48 @ ( pi_a_a_a_a2 @ A
% 0.27/0.48 @ ^ [Uu: a > a] :
% 0.27/0.48 ( pi_a_a @ B
% 0.27/0.49 @ ^ [Uv: a] : C ) ) )
% 0.27/0.49 => ( ( member_a_a @ A2 @ A )
% 0.27/0.49 => ( member_a_a @ ( F @ A2 )
% 0.27/0.49 @ ( pi_a_a @ B
% 0.27/0.49 @ ^ [Uu: a] : C ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % bivar_fun
% 0.27/0.49 thf(fact_6_bivar__fun,axiom,
% 0.27/0.49 ! [F: a > a > a > a,A: set_a,B: set_a,C: set_a_a,A2: a] :
% 0.27/0.49 ( ( member_a_a_a_a4 @ F
% 0.27/0.49 @ ( pi_a_a_a_a4 @ A
% 0.27/0.49 @ ^ [Uu: a] :
% 0.27/0.49 ( pi_a_a_a2 @ B
% 0.27/0.49 @ ^ [Uv: a] : C ) ) )
% 0.27/0.49 => ( ( member_a @ A2 @ A )
% 0.27/0.49 => ( member_a_a_a2 @ ( F @ A2 )
% 0.27/0.49 @ ( pi_a_a_a2 @ B
% 0.27/0.49 @ ^ [Uu: a] : C ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % bivar_fun
% 0.27/0.49 thf(fact_7_bivar__fun,axiom,
% 0.27/0.49 ! [F: ( c > d ) > c > d,A: set_c_d,B: set_c,C: set_d,A2: c > d] :
% 0.27/0.49 ( ( member_c_d_c_d @ F
% 0.27/0.49 @ ( pi_c_d_c_d @ A
% 0.27/0.49 @ ^ [Uu: c > d] :
% 0.27/0.49 ( pi_c_d @ B
% 0.27/0.49 @ ^ [Uv: c] : C ) ) )
% 0.27/0.49 => ( ( member_c_d @ A2 @ A )
% 0.27/0.49 => ( member_c_d @ ( F @ A2 )
% 0.27/0.49 @ ( pi_c_d @ B
% 0.27/0.49 @ ^ [Uu: c] : C ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % bivar_fun
% 0.27/0.49 thf(fact_8_bivar__fun,axiom,
% 0.27/0.49 ! [F: ( c > d ) > a > a > a,A: set_c_d,B: set_a,C: set_a_a,A2: c > d] :
% 0.27/0.49 ( ( member_c_d_a_a_a2 @ F
% 0.27/0.49 @ ( pi_c_d_a_a_a2 @ A
% 0.27/0.49 @ ^ [Uu: c > d] :
% 0.27/0.49 ( pi_a_a_a2 @ B
% 0.27/0.49 @ ^ [Uv: a] : C ) ) )
% 0.27/0.49 => ( ( member_c_d @ A2 @ A )
% 0.27/0.49 => ( member_a_a_a2 @ ( F @ A2 )
% 0.27/0.49 @ ( pi_a_a_a2 @ B
% 0.27/0.49 @ ^ [Uu: a] : C ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % bivar_fun
% 0.27/0.49 thf(fact_9_bivar__fun,axiom,
% 0.27/0.49 ! [F: ( a > a > a ) > c > d,A: set_a_a_a,B: set_c,C: set_d,A2: a > a > a] :
% 0.27/0.49 ( ( member_a_a_a_c_d @ F
% 0.27/0.49 @ ( pi_a_a_a_c_d @ A
% 0.27/0.49 @ ^ [Uu: a > a > a] :
% 0.27/0.49 ( pi_c_d @ B
% 0.27/0.49 @ ^ [Uv: c] : C ) ) )
% 0.27/0.49 => ( ( member_a_a_a2 @ A2 @ A )
% 0.27/0.49 => ( member_c_d @ ( F @ A2 )
% 0.27/0.49 @ ( pi_c_d @ B
% 0.27/0.49 @ ^ [Uu: c] : C ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % bivar_fun
% 0.27/0.49 thf(fact_10_bivar__fun,axiom,
% 0.27/0.49 ! [F: ( a > a > a ) > a > a,A: set_a_a_a,B: set_a,C: set_a,A2: a > a > a] :
% 0.27/0.49 ( ( member_a_a_a_a_a2 @ F
% 0.27/0.49 @ ( pi_a_a_a_a_a @ A
% 0.27/0.49 @ ^ [Uu: a > a > a] :
% 0.27/0.49 ( pi_a_a @ B
% 0.27/0.49 @ ^ [Uv: a] : C ) ) )
% 0.27/0.49 => ( ( member_a_a_a2 @ A2 @ A )
% 0.27/0.49 => ( member_a_a @ ( F @ A2 )
% 0.27/0.49 @ ( pi_a_a @ B
% 0.27/0.49 @ ^ [Uu: a] : C ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % bivar_fun
% 0.27/0.49 thf(fact_11_bivar__fun__mem,axiom,
% 0.27/0.49 ! [F: a > a > a,A: set_a,B: set_a,C: set_a,A2: a,B2: a] :
% 0.27/0.49 ( ( member_a_a_a2 @ F
% 0.27/0.49 @ ( pi_a_a_a2 @ A
% 0.27/0.49 @ ^ [Uu: a] :
% 0.27/0.49 ( pi_a_a @ B
% 0.27/0.49 @ ^ [Uv: a] : C ) ) )
% 0.27/0.49 => ( ( member_a @ A2 @ A )
% 0.27/0.49 => ( ( member_a @ B2 @ B )
% 0.27/0.49 => ( member_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % bivar_fun_mem
% 0.27/0.49 thf(fact_12_bivar__fun__mem,axiom,
% 0.27/0.49 ! [F: ( c > d ) > a > a,A: set_c_d,B: set_a,C: set_a,A2: c > d,B2: a] :
% 0.27/0.49 ( ( member_c_d_a_a @ F
% 0.27/0.49 @ ( pi_c_d_a_a @ A
% 0.27/0.49 @ ^ [Uu: c > d] :
% 0.27/0.49 ( pi_a_a @ B
% 0.27/0.49 @ ^ [Uv: a] : C ) ) )
% 0.27/0.49 => ( ( member_c_d @ A2 @ A )
% 0.27/0.49 => ( ( member_a @ B2 @ B )
% 0.27/0.49 => ( member_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % bivar_fun_mem
% 0.27/0.49 thf(fact_13_bivar__fun__mem,axiom,
% 0.27/0.49 ! [F: ( a > a ) > a > a,A: set_a_a,B: set_a,C: set_a,A2: a > a,B2: a] :
% 0.27/0.49 ( ( member_a_a_a_a2 @ F
% 0.27/0.49 @ ( pi_a_a_a_a2 @ A
% 0.27/0.49 @ ^ [Uu: a > a] :
% 0.27/0.49 ( pi_a_a @ B
% 0.27/0.49 @ ^ [Uv: a] : C ) ) )
% 0.27/0.49 => ( ( member_a_a @ A2 @ A )
% 0.27/0.49 => ( ( member_a @ B2 @ B )
% 0.27/0.49 => ( member_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % bivar_fun_mem
% 0.27/0.49 thf(fact_14_bivar__fun__mem,axiom,
% 0.27/0.49 ! [F: a > ( c > d ) > a,A: set_a,B: set_c_d,C: set_a,A2: a,B2: c > d] :
% 0.27/0.49 ( ( member_a_c_d_a @ F
% 0.27/0.49 @ ( pi_a_c_d_a @ A
% 0.27/0.49 @ ^ [Uu: a] :
% 0.27/0.49 ( pi_c_d_a @ B
% 0.27/0.49 @ ^ [Uv: c > d] : C ) ) )
% 0.27/0.49 => ( ( member_a @ A2 @ A )
% 0.27/0.49 => ( ( member_c_d @ B2 @ B )
% 0.27/0.49 => ( member_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % bivar_fun_mem
% 0.27/0.49 thf(fact_15_bivar__fun__mem,axiom,
% 0.27/0.49 ! [F: a > ( a > a ) > a,A: set_a,B: set_a_a,C: set_a,A2: a,B2: a > a] :
% 0.27/0.49 ( ( member_a_a_a_a3 @ F
% 0.27/0.49 @ ( pi_a_a_a_a3 @ A
% 0.27/0.49 @ ^ [Uu: a] :
% 0.27/0.49 ( pi_a_a_a @ B
% 0.27/0.49 @ ^ [Uv: a > a] : C ) ) )
% 0.27/0.49 => ( ( member_a @ A2 @ A )
% 0.27/0.49 => ( ( member_a_a @ B2 @ B )
% 0.27/0.49 => ( member_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % bivar_fun_mem
% 0.27/0.49 thf(fact_16_bivar__fun__mem,axiom,
% 0.27/0.49 ! [F: a > a > c > d,A: set_a,B: set_a,C: set_c_d,A2: a,B2: a] :
% 0.27/0.49 ( ( member_a_a_c_d2 @ F
% 0.27/0.49 @ ( pi_a_a_c_d2 @ A
% 0.27/0.49 @ ^ [Uu: a] :
% 0.27/0.49 ( pi_a_c_d @ B
% 0.27/0.49 @ ^ [Uv: a] : C ) ) )
% 0.27/0.49 => ( ( member_a @ A2 @ A )
% 0.27/0.49 => ( ( member_a @ B2 @ B )
% 0.27/0.49 => ( member_c_d @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % bivar_fun_mem
% 0.27/0.49 thf(fact_17_bivar__fun__mem,axiom,
% 0.27/0.49 ! [F: a > a > a > a,A: set_a,B: set_a,C: set_a_a,A2: a,B2: a] :
% 0.27/0.49 ( ( member_a_a_a_a4 @ F
% 0.27/0.49 @ ( pi_a_a_a_a4 @ A
% 0.27/0.49 @ ^ [Uu: a] :
% 0.27/0.49 ( pi_a_a_a2 @ B
% 0.27/0.49 @ ^ [Uv: a] : C ) ) )
% 0.27/0.49 => ( ( member_a @ A2 @ A )
% 0.27/0.49 => ( ( member_a @ B2 @ B )
% 0.27/0.49 => ( member_a_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % bivar_fun_mem
% 0.27/0.49 thf(fact_18_bivar__fun__mem,axiom,
% 0.27/0.49 ! [F: ( c > d ) > c > d,A: set_c_d,B: set_c,C: set_d,A2: c > d,B2: c] :
% 0.27/0.49 ( ( member_c_d_c_d @ F
% 0.27/0.49 @ ( pi_c_d_c_d @ A
% 0.27/0.49 @ ^ [Uu: c > d] :
% 0.27/0.49 ( pi_c_d @ B
% 0.27/0.49 @ ^ [Uv: c] : C ) ) )
% 0.27/0.49 => ( ( member_c_d @ A2 @ A )
% 0.27/0.49 => ( ( member_c @ B2 @ B )
% 0.27/0.49 => ( member_d @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % bivar_fun_mem
% 0.27/0.49 thf(fact_19_bivar__fun__mem,axiom,
% 0.27/0.49 ! [F: ( c > d ) > ( c > d ) > a,A: set_c_d,B: set_c_d,C: set_a,A2: c > d,B2: c > d] :
% 0.27/0.49 ( ( member_c_d_c_d_a2 @ F
% 0.27/0.49 @ ( pi_c_d_c_d_a @ A
% 0.27/0.49 @ ^ [Uu: c > d] :
% 0.27/0.49 ( pi_c_d_a @ B
% 0.27/0.49 @ ^ [Uv: c > d] : C ) ) )
% 0.27/0.49 => ( ( member_c_d @ A2 @ A )
% 0.27/0.49 => ( ( member_c_d @ B2 @ B )
% 0.27/0.49 => ( member_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % bivar_fun_mem
% 0.27/0.49 thf(fact_20_bivar__fun__mem,axiom,
% 0.27/0.49 ! [F: ( c > d ) > ( a > a ) > a,A: set_c_d,B: set_a_a,C: set_a,A2: c > d,B2: a > a] :
% 0.27/0.49 ( ( member_c_d_a_a_a @ F
% 0.27/0.49 @ ( pi_c_d_a_a_a @ A
% 0.27/0.49 @ ^ [Uu: c > d] :
% 0.27/0.49 ( pi_a_a_a @ B
% 0.27/0.49 @ ^ [Uv: a > a] : C ) ) )
% 0.27/0.49 => ( ( member_c_d @ A2 @ A )
% 0.27/0.49 => ( ( member_a_a @ B2 @ B )
% 0.27/0.49 => ( member_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % bivar_fun_mem
% 0.27/0.49 thf(fact_21_Pi__I,axiom,
% 0.27/0.49 ! [A: set_c,F: c > d,B: c > set_d] :
% 0.27/0.49 ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ A )
% 0.27/0.49 => ( member_d @ ( F @ X ) @ ( B @ X ) ) )
% 0.27/0.49 => ( member_c_d @ F @ ( pi_c_d @ A @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_I
% 0.27/0.49 thf(fact_22_Pi__I,axiom,
% 0.27/0.49 ! [A: set_a,F: a > a,B: a > set_a] :
% 0.27/0.49 ( ! [X: a] :
% 0.27/0.49 ( ( member_a @ X @ A )
% 0.27/0.49 => ( member_a @ ( F @ X ) @ ( B @ X ) ) )
% 0.27/0.49 => ( member_a_a @ F @ ( pi_a_a @ A @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_I
% 0.27/0.49 thf(fact_23_Pi__I,axiom,
% 0.27/0.49 ! [A: set_c_d,F: ( c > d ) > a,B: ( c > d ) > set_a] :
% 0.27/0.49 ( ! [X: c > d] :
% 0.27/0.49 ( ( member_c_d @ X @ A )
% 0.27/0.49 => ( member_a @ ( F @ X ) @ ( B @ X ) ) )
% 0.27/0.49 => ( member_c_d_a @ F @ ( pi_c_d_a @ A @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_I
% 0.27/0.49 thf(fact_24_Pi__I,axiom,
% 0.27/0.49 ! [A: set_a_a,F: ( a > a ) > a,B: ( a > a ) > set_a] :
% 0.27/0.49 ( ! [X: a > a] :
% 0.27/0.49 ( ( member_a_a @ X @ A )
% 0.27/0.49 => ( member_a @ ( F @ X ) @ ( B @ X ) ) )
% 0.27/0.49 => ( member_a_a_a @ F @ ( pi_a_a_a @ A @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_I
% 0.27/0.49 thf(fact_25_Pi__I,axiom,
% 0.27/0.49 ! [A: set_a,F: a > c > d,B: a > set_c_d] :
% 0.27/0.49 ( ! [X: a] :
% 0.27/0.49 ( ( member_a @ X @ A )
% 0.27/0.49 => ( member_c_d @ ( F @ X ) @ ( B @ X ) ) )
% 0.27/0.49 => ( member_a_c_d @ F @ ( pi_a_c_d @ A @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_I
% 0.27/0.49 thf(fact_26_Pi__I,axiom,
% 0.27/0.49 ! [A: set_a,F: a > a > a,B: a > set_a_a] :
% 0.27/0.49 ( ! [X: a] :
% 0.27/0.49 ( ( member_a @ X @ A )
% 0.27/0.49 => ( member_a_a @ ( F @ X ) @ ( B @ X ) ) )
% 0.27/0.49 => ( member_a_a_a2 @ F @ ( pi_a_a_a2 @ A @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_I
% 0.27/0.49 thf(fact_27_Pi__I,axiom,
% 0.27/0.49 ! [A: set_c_d,F: ( c > d ) > c > d,B: ( c > d ) > set_c_d] :
% 0.27/0.49 ( ! [X: c > d] :
% 0.27/0.49 ( ( member_c_d @ X @ A )
% 0.27/0.49 => ( member_c_d @ ( F @ X ) @ ( B @ X ) ) )
% 0.27/0.49 => ( member_c_d_c_d @ F @ ( pi_c_d_c_d @ A @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_I
% 0.27/0.49 thf(fact_28_Pi__I,axiom,
% 0.27/0.49 ! [A: set_c_d,F: ( c > d ) > a > a,B: ( c > d ) > set_a_a] :
% 0.27/0.49 ( ! [X: c > d] :
% 0.27/0.49 ( ( member_c_d @ X @ A )
% 0.27/0.49 => ( member_a_a @ ( F @ X ) @ ( B @ X ) ) )
% 0.27/0.49 => ( member_c_d_a_a @ F @ ( pi_c_d_a_a @ A @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_I
% 0.27/0.49 thf(fact_29_Pi__I,axiom,
% 0.27/0.49 ! [A: set_a_a_a,F: ( a > a > a ) > a,B: ( a > a > a ) > set_a] :
% 0.27/0.49 ( ! [X: a > a > a] :
% 0.27/0.49 ( ( member_a_a_a2 @ X @ A )
% 0.27/0.49 => ( member_a @ ( F @ X ) @ ( B @ X ) ) )
% 0.27/0.49 => ( member_a_a_a_a @ F @ ( pi_a_a_a_a @ A @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_I
% 0.27/0.49 thf(fact_30_Pi__I,axiom,
% 0.27/0.49 ! [A: set_a_a,F: ( a > a ) > c > d,B: ( a > a ) > set_c_d] :
% 0.27/0.49 ( ! [X: a > a] :
% 0.27/0.49 ( ( member_a_a @ X @ A )
% 0.27/0.49 => ( member_c_d @ ( F @ X ) @ ( B @ X ) ) )
% 0.27/0.49 => ( member_a_a_c_d @ F @ ( pi_a_a_c_d @ A @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_I
% 0.27/0.49 thf(fact_31_prodM__sprod__mem,axiom,
% 0.27/0.49 ! [I: set_c,M: c > carrie1948989069_d_a_e,A2: a,M2: c > d] :
% 0.27/0.49 ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 0.27/0.49 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.49 => ( ( member_c_d @ M2 @ ( carr_p1636662547_d_a_e @ I @ M ) )
% 0.27/0.49 => ( member_c_d @ ( algebr1697692348_c_d_e @ r @ I @ M @ A2 @ M2 ) @ ( carr_p1636662547_d_a_e @ I @ M ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodM_sprod_mem
% 0.27/0.49 thf(fact_32_prodM__carr,axiom,
% 0.27/0.49 ! [I: set_c,M: c > carrie1948989069_d_a_e] :
% 0.27/0.49 ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 0.27/0.49 => ( ( carrie20288686t_unit @ ( algebr1968153176_c_d_e @ r @ I @ M ) )
% 0.27/0.49 = ( carr_p1636662547_d_a_e @ I @ M ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodM_carr
% 0.27/0.49 thf(fact_33_eq__fun,axiom,
% 0.27/0.49 ! [F: a > c > d,A: set_a,B: set_c_d,G: a > c > d] :
% 0.27/0.49 ( ( member_a_c_d @ F
% 0.27/0.49 @ ( pi_a_c_d @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( member_a_c_d @ G
% 0.27/0.49 @ ( pi_a_c_d @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % eq_fun
% 0.27/0.49 thf(fact_34_eq__fun,axiom,
% 0.27/0.49 ! [F: a > a > ( c > d ) > c > d,A: set_a,B: set_a_c_d_c_d,G: a > a > ( c > d ) > c > d] :
% 0.27/0.49 ( ( member_a_a_c_d_c_d @ F
% 0.27/0.49 @ ( pi_a_a_c_d_c_d @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( member_a_a_c_d_c_d @ G
% 0.27/0.49 @ ( pi_a_a_c_d_c_d @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % eq_fun
% 0.27/0.49 thf(fact_35_eq__fun,axiom,
% 0.27/0.49 ! [F: a > a > c > d,A: set_a,B: set_a_c_d,G: a > a > c > d] :
% 0.27/0.49 ( ( member_a_a_c_d2 @ F
% 0.27/0.49 @ ( pi_a_a_c_d2 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( member_a_a_c_d2 @ G
% 0.27/0.49 @ ( pi_a_a_c_d2 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % eq_fun
% 0.27/0.49 thf(fact_36_eq__fun,axiom,
% 0.27/0.49 ! [F: a > a > a > a > a,A: set_a,B: set_a_a_a_a,G: a > a > a > a > a] :
% 0.27/0.49 ( ( member_a_a_a_a_a3 @ F
% 0.27/0.49 @ ( pi_a_a_a_a_a2 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( member_a_a_a_a_a3 @ G
% 0.27/0.49 @ ( pi_a_a_a_a_a2 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % eq_fun
% 0.27/0.49 thf(fact_37_eq__fun,axiom,
% 0.27/0.49 ! [F: a > a > a > a,A: set_a,B: set_a_a_a,G: a > a > a > a] :
% 0.27/0.49 ( ( member_a_a_a_a4 @ F
% 0.27/0.49 @ ( pi_a_a_a_a4 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( member_a_a_a_a4 @ G
% 0.27/0.49 @ ( pi_a_a_a_a4 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % eq_fun
% 0.27/0.49 thf(fact_38_eq__fun,axiom,
% 0.27/0.49 ! [F: ( c > d ) > ( c > d ) > c > d,A: set_c_d,B: set_c_d_c_d,G: ( c > d ) > ( c > d ) > c > d] :
% 0.27/0.49 ( ( member_c_d_c_d_c_d @ F
% 0.27/0.49 @ ( pi_c_d_c_d_c_d @ A
% 0.27/0.49 @ ^ [Uu: c > d] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( member_c_d_c_d_c_d @ G
% 0.27/0.49 @ ( pi_c_d_c_d_c_d @ A
% 0.27/0.49 @ ^ [Uu: c > d] : B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % eq_fun
% 0.27/0.49 thf(fact_39_eq__fun,axiom,
% 0.27/0.49 ! [F: ( c > d ) > c > d,A: set_c_d,B: set_c_d,G: ( c > d ) > c > d] :
% 0.27/0.49 ( ( member_c_d_c_d @ F
% 0.27/0.49 @ ( pi_c_d_c_d @ A
% 0.27/0.49 @ ^ [Uu: c > d] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( member_c_d_c_d @ G
% 0.27/0.49 @ ( pi_c_d_c_d @ A
% 0.27/0.49 @ ^ [Uu: c > d] : B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % eq_fun
% 0.27/0.49 thf(fact_40_eq__fun,axiom,
% 0.27/0.49 ! [F: c > d,A: set_c,B: set_d,G: c > d] :
% 0.27/0.49 ( ( member_c_d @ F
% 0.27/0.49 @ ( pi_c_d @ A
% 0.27/0.49 @ ^ [Uu: c] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( member_c_d @ G
% 0.27/0.49 @ ( pi_c_d @ A
% 0.27/0.49 @ ^ [Uu: c] : B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % eq_fun
% 0.27/0.49 thf(fact_41_eq__fun,axiom,
% 0.27/0.49 ! [F: a > a > a,A: set_a,B: set_a_a,G: a > a > a] :
% 0.27/0.49 ( ( member_a_a_a2 @ F
% 0.27/0.49 @ ( pi_a_a_a2 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( member_a_a_a2 @ G
% 0.27/0.49 @ ( pi_a_a_a2 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % eq_fun
% 0.27/0.49 thf(fact_42_eq__fun,axiom,
% 0.27/0.49 ! [F: a > a,A: set_a,B: set_a,G: a > a] :
% 0.27/0.49 ( ( member_a_a @ F
% 0.27/0.49 @ ( pi_a_a @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( member_a_a @ G
% 0.27/0.49 @ ( pi_a_a @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % eq_fun
% 0.27/0.49 thf(fact_43_funcTr,axiom,
% 0.27/0.49 ! [F: a > c > d,A: set_a,B: set_c_d,G: a > c > d,A2: a] :
% 0.27/0.49 ( ( member_a_c_d @ F
% 0.27/0.49 @ ( pi_a_c_d @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( member_a_c_d @ G
% 0.27/0.49 @ ( pi_a_c_d @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( ( member_a @ A2 @ A )
% 0.27/0.49 => ( ( F @ A2 )
% 0.27/0.49 = ( G @ A2 ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % funcTr
% 0.27/0.49 thf(fact_44_funcTr,axiom,
% 0.27/0.49 ! [F: a > a > ( c > d ) > c > d,A: set_a,B: set_a_c_d_c_d,G: a > a > ( c > d ) > c > d,A2: a] :
% 0.27/0.49 ( ( member_a_a_c_d_c_d @ F
% 0.27/0.49 @ ( pi_a_a_c_d_c_d @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( member_a_a_c_d_c_d @ G
% 0.27/0.49 @ ( pi_a_a_c_d_c_d @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( ( member_a @ A2 @ A )
% 0.27/0.49 => ( ( F @ A2 )
% 0.27/0.49 = ( G @ A2 ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % funcTr
% 0.27/0.49 thf(fact_45_funcTr,axiom,
% 0.27/0.49 ! [F: a > a > c > d,A: set_a,B: set_a_c_d,G: a > a > c > d,A2: a] :
% 0.27/0.49 ( ( member_a_a_c_d2 @ F
% 0.27/0.49 @ ( pi_a_a_c_d2 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( member_a_a_c_d2 @ G
% 0.27/0.49 @ ( pi_a_a_c_d2 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( ( member_a @ A2 @ A )
% 0.27/0.49 => ( ( F @ A2 )
% 0.27/0.49 = ( G @ A2 ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % funcTr
% 0.27/0.49 thf(fact_46_funcTr,axiom,
% 0.27/0.49 ! [F: a > a > a > a > a,A: set_a,B: set_a_a_a_a,G: a > a > a > a > a,A2: a] :
% 0.27/0.49 ( ( member_a_a_a_a_a3 @ F
% 0.27/0.49 @ ( pi_a_a_a_a_a2 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( member_a_a_a_a_a3 @ G
% 0.27/0.49 @ ( pi_a_a_a_a_a2 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( ( member_a @ A2 @ A )
% 0.27/0.49 => ( ( F @ A2 )
% 0.27/0.49 = ( G @ A2 ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % funcTr
% 0.27/0.49 thf(fact_47_funcTr,axiom,
% 0.27/0.49 ! [F: a > a > a > a,A: set_a,B: set_a_a_a,G: a > a > a > a,A2: a] :
% 0.27/0.49 ( ( member_a_a_a_a4 @ F
% 0.27/0.49 @ ( pi_a_a_a_a4 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( member_a_a_a_a4 @ G
% 0.27/0.49 @ ( pi_a_a_a_a4 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( ( member_a @ A2 @ A )
% 0.27/0.49 => ( ( F @ A2 )
% 0.27/0.49 = ( G @ A2 ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % funcTr
% 0.27/0.49 thf(fact_48_funcTr,axiom,
% 0.27/0.49 ! [F: ( c > d ) > ( c > d ) > c > d,A: set_c_d,B: set_c_d_c_d,G: ( c > d ) > ( c > d ) > c > d,A2: c > d] :
% 0.27/0.49 ( ( member_c_d_c_d_c_d @ F
% 0.27/0.49 @ ( pi_c_d_c_d_c_d @ A
% 0.27/0.49 @ ^ [Uu: c > d] : B ) )
% 0.27/0.49 => ( ( member_c_d_c_d_c_d @ G
% 0.27/0.49 @ ( pi_c_d_c_d_c_d @ A
% 0.27/0.49 @ ^ [Uu: c > d] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( ( member_c_d @ A2 @ A )
% 0.27/0.49 => ( ( F @ A2 )
% 0.27/0.49 = ( G @ A2 ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % funcTr
% 0.27/0.49 thf(fact_49_funcTr,axiom,
% 0.27/0.49 ! [F: ( c > d ) > c > d,A: set_c_d,B: set_c_d,G: ( c > d ) > c > d,A2: c > d] :
% 0.27/0.49 ( ( member_c_d_c_d @ F
% 0.27/0.49 @ ( pi_c_d_c_d @ A
% 0.27/0.49 @ ^ [Uu: c > d] : B ) )
% 0.27/0.49 => ( ( member_c_d_c_d @ G
% 0.27/0.49 @ ( pi_c_d_c_d @ A
% 0.27/0.49 @ ^ [Uu: c > d] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( ( member_c_d @ A2 @ A )
% 0.27/0.49 => ( ( F @ A2 )
% 0.27/0.49 = ( G @ A2 ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % funcTr
% 0.27/0.49 thf(fact_50_funcTr,axiom,
% 0.27/0.49 ! [F: c > d,A: set_c,B: set_d,G: c > d,A2: c] :
% 0.27/0.49 ( ( member_c_d @ F
% 0.27/0.49 @ ( pi_c_d @ A
% 0.27/0.49 @ ^ [Uu: c] : B ) )
% 0.27/0.49 => ( ( member_c_d @ G
% 0.27/0.49 @ ( pi_c_d @ A
% 0.27/0.49 @ ^ [Uu: c] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( ( member_c @ A2 @ A )
% 0.27/0.49 => ( ( F @ A2 )
% 0.27/0.49 = ( G @ A2 ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % funcTr
% 0.27/0.49 thf(fact_51_funcTr,axiom,
% 0.27/0.49 ! [F: a > a > a,A: set_a,B: set_a_a,G: a > a > a,A2: a] :
% 0.27/0.49 ( ( member_a_a_a2 @ F
% 0.27/0.49 @ ( pi_a_a_a2 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( member_a_a_a2 @ G
% 0.27/0.49 @ ( pi_a_a_a2 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( ( member_a @ A2 @ A )
% 0.27/0.49 => ( ( F @ A2 )
% 0.27/0.49 = ( G @ A2 ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % funcTr
% 0.27/0.49 thf(fact_52_funcTr,axiom,
% 0.27/0.49 ! [F: a > a,A: set_a,B: set_a,G: a > a,A2: a] :
% 0.27/0.49 ( ( member_a_a @ F
% 0.27/0.49 @ ( pi_a_a @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( member_a_a @ G
% 0.27/0.49 @ ( pi_a_a @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( ( member_a @ A2 @ A )
% 0.27/0.49 => ( ( F @ A2 )
% 0.27/0.49 = ( G @ A2 ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % funcTr
% 0.27/0.49 thf(fact_53_funcsetI,axiom,
% 0.27/0.49 ! [A: set_c,F: c > d,B: set_d] :
% 0.27/0.49 ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ A )
% 0.27/0.49 => ( member_d @ ( F @ X ) @ B ) )
% 0.27/0.49 => ( member_c_d @ F
% 0.27/0.49 @ ( pi_c_d @ A
% 0.27/0.49 @ ^ [Uu: c] : B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % funcsetI
% 0.27/0.49 thf(fact_54_funcsetI,axiom,
% 0.27/0.49 ! [A: set_a,F: a > a,B: set_a] :
% 0.27/0.49 ( ! [X: a] :
% 0.27/0.49 ( ( member_a @ X @ A )
% 0.27/0.49 => ( member_a @ ( F @ X ) @ B ) )
% 0.27/0.49 => ( member_a_a @ F
% 0.27/0.49 @ ( pi_a_a @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % funcsetI
% 0.27/0.49 thf(fact_55_funcsetI,axiom,
% 0.27/0.49 ! [A: set_c_d,F: ( c > d ) > a,B: set_a] :
% 0.27/0.49 ( ! [X: c > d] :
% 0.27/0.49 ( ( member_c_d @ X @ A )
% 0.27/0.49 => ( member_a @ ( F @ X ) @ B ) )
% 0.27/0.49 => ( member_c_d_a @ F
% 0.27/0.49 @ ( pi_c_d_a @ A
% 0.27/0.49 @ ^ [Uu: c > d] : B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % funcsetI
% 0.27/0.49 thf(fact_56_funcsetI,axiom,
% 0.27/0.49 ! [A: set_a_a,F: ( a > a ) > a,B: set_a] :
% 0.27/0.49 ( ! [X: a > a] :
% 0.27/0.49 ( ( member_a_a @ X @ A )
% 0.27/0.49 => ( member_a @ ( F @ X ) @ B ) )
% 0.27/0.49 => ( member_a_a_a @ F
% 0.27/0.49 @ ( pi_a_a_a @ A
% 0.27/0.49 @ ^ [Uu: a > a] : B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % funcsetI
% 0.27/0.49 thf(fact_57_funcsetI,axiom,
% 0.27/0.49 ! [A: set_a,F: a > c > d,B: set_c_d] :
% 0.27/0.49 ( ! [X: a] :
% 0.27/0.49 ( ( member_a @ X @ A )
% 0.27/0.49 => ( member_c_d @ ( F @ X ) @ B ) )
% 0.27/0.49 => ( member_a_c_d @ F
% 0.27/0.49 @ ( pi_a_c_d @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % funcsetI
% 0.27/0.49 thf(fact_58_funcsetI,axiom,
% 0.27/0.49 ! [A: set_a,F: a > a > a,B: set_a_a] :
% 0.27/0.49 ( ! [X: a] :
% 0.27/0.49 ( ( member_a @ X @ A )
% 0.27/0.49 => ( member_a_a @ ( F @ X ) @ B ) )
% 0.27/0.49 => ( member_a_a_a2 @ F
% 0.27/0.49 @ ( pi_a_a_a2 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % funcsetI
% 0.27/0.49 thf(fact_59_funcsetI,axiom,
% 0.27/0.49 ! [A: set_c_d,F: ( c > d ) > c > d,B: set_c_d] :
% 0.27/0.49 ( ! [X: c > d] :
% 0.27/0.49 ( ( member_c_d @ X @ A )
% 0.27/0.49 => ( member_c_d @ ( F @ X ) @ B ) )
% 0.27/0.49 => ( member_c_d_c_d @ F
% 0.27/0.49 @ ( pi_c_d_c_d @ A
% 0.27/0.49 @ ^ [Uu: c > d] : B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % funcsetI
% 0.27/0.49 thf(fact_60_funcsetI,axiom,
% 0.27/0.49 ! [A: set_c_d,F: ( c > d ) > a > a,B: set_a_a] :
% 0.27/0.49 ( ! [X: c > d] :
% 0.27/0.49 ( ( member_c_d @ X @ A )
% 0.27/0.49 => ( member_a_a @ ( F @ X ) @ B ) )
% 0.27/0.49 => ( member_c_d_a_a @ F
% 0.27/0.49 @ ( pi_c_d_a_a @ A
% 0.27/0.49 @ ^ [Uu: c > d] : B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % funcsetI
% 0.27/0.49 thf(fact_61_funcsetI,axiom,
% 0.27/0.49 ! [A: set_a_a_a,F: ( a > a > a ) > a,B: set_a] :
% 0.27/0.49 ( ! [X: a > a > a] :
% 0.27/0.49 ( ( member_a_a_a2 @ X @ A )
% 0.27/0.49 => ( member_a @ ( F @ X ) @ B ) )
% 0.27/0.49 => ( member_a_a_a_a @ F
% 0.27/0.49 @ ( pi_a_a_a_a @ A
% 0.27/0.49 @ ^ [Uu: a > a > a] : B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % funcsetI
% 0.27/0.49 thf(fact_62_funcsetI,axiom,
% 0.27/0.49 ! [A: set_a_a,F: ( a > a ) > c > d,B: set_c_d] :
% 0.27/0.49 ( ! [X: a > a] :
% 0.27/0.49 ( ( member_a_a @ X @ A )
% 0.27/0.49 => ( member_c_d @ ( F @ X ) @ B ) )
% 0.27/0.49 => ( member_a_a_c_d @ F
% 0.27/0.49 @ ( pi_a_a_c_d @ A
% 0.27/0.49 @ ^ [Uu: a > a] : B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % funcsetI
% 0.27/0.49 thf(fact_63_eq__funcs,axiom,
% 0.27/0.49 ! [F: a > c > d,A: set_a,B: set_c_d,G: a > c > d,X2: a] :
% 0.27/0.49 ( ( member_a_c_d @ F
% 0.27/0.49 @ ( pi_a_c_d @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( member_a_c_d @ G
% 0.27/0.49 @ ( pi_a_c_d @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( ( member_a @ X2 @ A )
% 0.27/0.49 => ( ( F @ X2 )
% 0.27/0.49 = ( G @ X2 ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % eq_funcs
% 0.27/0.49 thf(fact_64_eq__funcs,axiom,
% 0.27/0.49 ! [F: a > a > ( c > d ) > c > d,A: set_a,B: set_a_c_d_c_d,G: a > a > ( c > d ) > c > d,X2: a] :
% 0.27/0.49 ( ( member_a_a_c_d_c_d @ F
% 0.27/0.49 @ ( pi_a_a_c_d_c_d @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( member_a_a_c_d_c_d @ G
% 0.27/0.49 @ ( pi_a_a_c_d_c_d @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( ( member_a @ X2 @ A )
% 0.27/0.49 => ( ( F @ X2 )
% 0.27/0.49 = ( G @ X2 ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % eq_funcs
% 0.27/0.49 thf(fact_65_eq__funcs,axiom,
% 0.27/0.49 ! [F: a > a > c > d,A: set_a,B: set_a_c_d,G: a > a > c > d,X2: a] :
% 0.27/0.49 ( ( member_a_a_c_d2 @ F
% 0.27/0.49 @ ( pi_a_a_c_d2 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( member_a_a_c_d2 @ G
% 0.27/0.49 @ ( pi_a_a_c_d2 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( ( member_a @ X2 @ A )
% 0.27/0.49 => ( ( F @ X2 )
% 0.27/0.49 = ( G @ X2 ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % eq_funcs
% 0.27/0.49 thf(fact_66_eq__funcs,axiom,
% 0.27/0.49 ! [F: a > a > a > a > a,A: set_a,B: set_a_a_a_a,G: a > a > a > a > a,X2: a] :
% 0.27/0.49 ( ( member_a_a_a_a_a3 @ F
% 0.27/0.49 @ ( pi_a_a_a_a_a2 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( member_a_a_a_a_a3 @ G
% 0.27/0.49 @ ( pi_a_a_a_a_a2 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( ( member_a @ X2 @ A )
% 0.27/0.49 => ( ( F @ X2 )
% 0.27/0.49 = ( G @ X2 ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % eq_funcs
% 0.27/0.49 thf(fact_67_eq__funcs,axiom,
% 0.27/0.49 ! [F: a > a > a > a,A: set_a,B: set_a_a_a,G: a > a > a > a,X2: a] :
% 0.27/0.49 ( ( member_a_a_a_a4 @ F
% 0.27/0.49 @ ( pi_a_a_a_a4 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( member_a_a_a_a4 @ G
% 0.27/0.49 @ ( pi_a_a_a_a4 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( ( member_a @ X2 @ A )
% 0.27/0.49 => ( ( F @ X2 )
% 0.27/0.49 = ( G @ X2 ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % eq_funcs
% 0.27/0.49 thf(fact_68_eq__funcs,axiom,
% 0.27/0.49 ! [F: ( c > d ) > ( c > d ) > c > d,A: set_c_d,B: set_c_d_c_d,G: ( c > d ) > ( c > d ) > c > d,X2: c > d] :
% 0.27/0.49 ( ( member_c_d_c_d_c_d @ F
% 0.27/0.49 @ ( pi_c_d_c_d_c_d @ A
% 0.27/0.49 @ ^ [Uu: c > d] : B ) )
% 0.27/0.49 => ( ( member_c_d_c_d_c_d @ G
% 0.27/0.49 @ ( pi_c_d_c_d_c_d @ A
% 0.27/0.49 @ ^ [Uu: c > d] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( ( member_c_d @ X2 @ A )
% 0.27/0.49 => ( ( F @ X2 )
% 0.27/0.49 = ( G @ X2 ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % eq_funcs
% 0.27/0.49 thf(fact_69_eq__funcs,axiom,
% 0.27/0.49 ! [F: ( c > d ) > c > d,A: set_c_d,B: set_c_d,G: ( c > d ) > c > d,X2: c > d] :
% 0.27/0.49 ( ( member_c_d_c_d @ F
% 0.27/0.49 @ ( pi_c_d_c_d @ A
% 0.27/0.49 @ ^ [Uu: c > d] : B ) )
% 0.27/0.49 => ( ( member_c_d_c_d @ G
% 0.27/0.49 @ ( pi_c_d_c_d @ A
% 0.27/0.49 @ ^ [Uu: c > d] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( ( member_c_d @ X2 @ A )
% 0.27/0.49 => ( ( F @ X2 )
% 0.27/0.49 = ( G @ X2 ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % eq_funcs
% 0.27/0.49 thf(fact_70_eq__funcs,axiom,
% 0.27/0.49 ! [F: c > d,A: set_c,B: set_d,G: c > d,X2: c] :
% 0.27/0.49 ( ( member_c_d @ F
% 0.27/0.49 @ ( pi_c_d @ A
% 0.27/0.49 @ ^ [Uu: c] : B ) )
% 0.27/0.49 => ( ( member_c_d @ G
% 0.27/0.49 @ ( pi_c_d @ A
% 0.27/0.49 @ ^ [Uu: c] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( ( member_c @ X2 @ A )
% 0.27/0.49 => ( ( F @ X2 )
% 0.27/0.49 = ( G @ X2 ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % eq_funcs
% 0.27/0.49 thf(fact_71_eq__funcs,axiom,
% 0.27/0.49 ! [F: a > a > a,A: set_a,B: set_a_a,G: a > a > a,X2: a] :
% 0.27/0.49 ( ( member_a_a_a2 @ F
% 0.27/0.49 @ ( pi_a_a_a2 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( member_a_a_a2 @ G
% 0.27/0.49 @ ( pi_a_a_a2 @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( ( member_a @ X2 @ A )
% 0.27/0.49 => ( ( F @ X2 )
% 0.27/0.49 = ( G @ X2 ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % eq_funcs
% 0.27/0.49 thf(fact_72_eq__funcs,axiom,
% 0.27/0.49 ! [F: a > a,A: set_a,B: set_a,G: a > a,X2: a] :
% 0.27/0.49 ( ( member_a_a @ F
% 0.27/0.49 @ ( pi_a_a @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( member_a_a @ G
% 0.27/0.49 @ ( pi_a_a @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( F = G )
% 0.27/0.49 => ( ( member_a @ X2 @ A )
% 0.27/0.49 => ( ( F @ X2 )
% 0.27/0.49 = ( G @ X2 ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % eq_funcs
% 0.27/0.49 thf(fact_73_funcset__id,axiom,
% 0.27/0.49 ! [A: set_c_d] :
% 0.27/0.49 ( member_c_d_c_d
% 0.27/0.49 @ ^ [X3: c > d] : X3
% 0.27/0.49 @ ( pi_c_d_c_d @ A
% 0.27/0.49 @ ^ [Uu: c > d] : A ) ) ).
% 0.27/0.49
% 0.27/0.49 % funcset_id
% 0.27/0.49 thf(fact_74_funcset__id,axiom,
% 0.27/0.49 ! [A: set_a] :
% 0.27/0.49 ( member_a_a
% 0.27/0.49 @ ^ [X3: a] : X3
% 0.27/0.49 @ ( pi_a_a @ A
% 0.27/0.49 @ ^ [Uu: a] : A ) ) ).
% 0.27/0.49
% 0.27/0.49 % funcset_id
% 0.27/0.49 thf(fact_75_Algebra1_Ofuncset__mem,axiom,
% 0.27/0.49 ! [F: a > d,A: set_a,B: set_d,X2: a] :
% 0.27/0.49 ( ( member_a_d @ F
% 0.27/0.49 @ ( pi_a_d @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( member_a @ X2 @ A )
% 0.27/0.49 => ( member_d @ ( F @ X2 ) @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Algebra1.funcset_mem
% 0.27/0.49 thf(fact_76_Algebra1_Ofuncset__mem,axiom,
% 0.27/0.49 ! [F: a > c,A: set_a,B: set_c,X2: a] :
% 0.27/0.49 ( ( member_a_c @ F
% 0.27/0.49 @ ( pi_a_c @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( member_a @ X2 @ A )
% 0.27/0.49 => ( member_c @ ( F @ X2 ) @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Algebra1.funcset_mem
% 0.27/0.49 thf(fact_77_Algebra1_Ofuncset__mem,axiom,
% 0.27/0.49 ! [F: d > a,A: set_d,B: set_a,X2: d] :
% 0.27/0.49 ( ( member_d_a @ F
% 0.27/0.49 @ ( pi_d_a @ A
% 0.27/0.49 @ ^ [Uu: d] : B ) )
% 0.27/0.49 => ( ( member_d @ X2 @ A )
% 0.27/0.49 => ( member_a @ ( F @ X2 ) @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Algebra1.funcset_mem
% 0.27/0.49 thf(fact_78_Algebra1_Ofuncset__mem,axiom,
% 0.27/0.49 ! [F: d > d,A: set_d,B: set_d,X2: d] :
% 0.27/0.49 ( ( member_d_d @ F
% 0.27/0.49 @ ( pi_d_d @ A
% 0.27/0.49 @ ^ [Uu: d] : B ) )
% 0.27/0.49 => ( ( member_d @ X2 @ A )
% 0.27/0.49 => ( member_d @ ( F @ X2 ) @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Algebra1.funcset_mem
% 0.27/0.49 thf(fact_79_Algebra1_Ofuncset__mem,axiom,
% 0.27/0.49 ! [F: d > c,A: set_d,B: set_c,X2: d] :
% 0.27/0.49 ( ( member_d_c @ F
% 0.27/0.49 @ ( pi_d_c @ A
% 0.27/0.49 @ ^ [Uu: d] : B ) )
% 0.27/0.49 => ( ( member_d @ X2 @ A )
% 0.27/0.49 => ( member_c @ ( F @ X2 ) @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Algebra1.funcset_mem
% 0.27/0.49 thf(fact_80_Algebra1_Ofuncset__mem,axiom,
% 0.27/0.49 ! [F: c > a,A: set_c,B: set_a,X2: c] :
% 0.27/0.49 ( ( member_c_a @ F
% 0.27/0.49 @ ( pi_c_a @ A
% 0.27/0.49 @ ^ [Uu: c] : B ) )
% 0.27/0.49 => ( ( member_c @ X2 @ A )
% 0.27/0.49 => ( member_a @ ( F @ X2 ) @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Algebra1.funcset_mem
% 0.27/0.49 thf(fact_81_Algebra1_Ofuncset__mem,axiom,
% 0.27/0.49 ! [F: c > c,A: set_c,B: set_c,X2: c] :
% 0.27/0.49 ( ( member_c_c @ F
% 0.27/0.49 @ ( pi_c_c @ A
% 0.27/0.49 @ ^ [Uu: c] : B ) )
% 0.27/0.49 => ( ( member_c @ X2 @ A )
% 0.27/0.49 => ( member_c @ ( F @ X2 ) @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Algebra1.funcset_mem
% 0.27/0.49 thf(fact_82_Algebra1_Ofuncset__mem,axiom,
% 0.27/0.49 ! [F: c > d,A: set_c,B: set_d,X2: c] :
% 0.27/0.49 ( ( member_c_d @ F
% 0.27/0.49 @ ( pi_c_d @ A
% 0.27/0.49 @ ^ [Uu: c] : B ) )
% 0.27/0.49 => ( ( member_c @ X2 @ A )
% 0.27/0.49 => ( member_d @ ( F @ X2 ) @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Algebra1.funcset_mem
% 0.27/0.49 thf(fact_83_Algebra1_Ofuncset__mem,axiom,
% 0.27/0.49 ! [F: a > a,A: set_a,B: set_a,X2: a] :
% 0.27/0.49 ( ( member_a_a @ F
% 0.27/0.49 @ ( pi_a_a @ A
% 0.27/0.49 @ ^ [Uu: a] : B ) )
% 0.27/0.49 => ( ( member_a @ X2 @ A )
% 0.27/0.49 => ( member_a @ ( F @ X2 ) @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Algebra1.funcset_mem
% 0.27/0.49 thf(fact_84_Algebra1_Ofuncset__mem,axiom,
% 0.27/0.49 ! [F: ( c > d ) > a,A: set_c_d,B: set_a,X2: c > d] :
% 0.27/0.49 ( ( member_c_d_a @ F
% 0.27/0.49 @ ( pi_c_d_a @ A
% 0.27/0.49 @ ^ [Uu: c > d] : B ) )
% 0.27/0.49 => ( ( member_c_d @ X2 @ A )
% 0.27/0.49 => ( member_a @ ( F @ X2 ) @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Algebra1.funcset_mem
% 0.27/0.49 thf(fact_85_prod__pOp__func,axiom,
% 0.27/0.49 ! [I: set_c,A: c > carrie1948989069_d_a_e] :
% 0.27/0.49 ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 0.27/0.49 => ( member_c_d_c_d_c_d @ ( prod_p131511076_d_a_e @ I @ A )
% 0.27/0.49 @ ( pi_c_d_c_d_c_d @ ( carr_p1636662547_d_a_e @ I @ A )
% 0.27/0.49 @ ^ [Uu: c > d] :
% 0.27/0.49 ( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ I @ A )
% 0.27/0.49 @ ^ [Uv: c > d] : ( carr_p1636662547_d_a_e @ I @ A ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prod_pOp_func
% 0.27/0.49 thf(fact_86_ring__is__ag,axiom,
% 0.27/0.49 aGroup2097840802xt_a_b @ r ).
% 0.27/0.49
% 0.27/0.49 % ring_is_ag
% 0.27/0.49 thf(fact_87_prodM__mem__eq,axiom,
% 0.27/0.49 ! [I: set_c,M: c > carrie1948989069_d_a_e,M1: c > d,M22: c > d] :
% 0.27/0.49 ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 0.27/0.49 => ( ( member_c_d @ M1 @ ( carrie20288686t_unit @ ( algebr1968153176_c_d_e @ r @ I @ M ) ) )
% 0.27/0.49 => ( ( member_c_d @ M22 @ ( carrie20288686t_unit @ ( algebr1968153176_c_d_e @ r @ I @ M ) ) )
% 0.27/0.49 => ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( ( M1 @ X )
% 0.27/0.49 = ( M22 @ X ) ) )
% 0.27/0.49 => ( M1 = M22 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodM_mem_eq
% 0.27/0.49 thf(fact_88_J__rad__mem,axiom,
% 0.27/0.49 ! [X2: a] :
% 0.27/0.49 ( ( member_a @ X2 @ ( j_rad_a_b @ r ) )
% 0.27/0.49 => ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % J_rad_mem
% 0.27/0.49 thf(fact_89_Ring_ORing,axiom,
% 0.27/0.49 ! [R: carrie1105631105xt_a_b] :
% 0.27/0.49 ( ( ring_a_b @ R )
% 0.27/0.49 => ( ring_a_b @ R ) ) ).
% 0.27/0.49
% 0.27/0.49 % Ring.Ring
% 0.27/0.49 thf(fact_90_Ring_Oring__is__ag,axiom,
% 0.27/0.49 ! [R: carrie1105631105xt_a_b] :
% 0.27/0.49 ( ( ring_a_b @ R )
% 0.27/0.49 => ( aGroup2097840802xt_a_b @ R ) ) ).
% 0.27/0.49
% 0.27/0.49 % Ring.ring_is_ag
% 0.27/0.49 thf(fact_91_Ring_OprodM__mem__eq,axiom,
% 0.27/0.49 ! [R: carrie1105631105xt_a_b,I: set_c,M: c > carrie1948989069_d_a_e,M1: c > d,M22: c > d] :
% 0.27/0.49 ( ( ring_a_b @ R )
% 0.27/0.49 => ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 0.27/0.49 => ( ( member_c_d @ M1 @ ( carrie20288686t_unit @ ( algebr1968153176_c_d_e @ R @ I @ M ) ) )
% 0.27/0.49 => ( ( member_c_d @ M22 @ ( carrie20288686t_unit @ ( algebr1968153176_c_d_e @ R @ I @ M ) ) )
% 0.27/0.49 => ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( ( M1 @ X )
% 0.27/0.49 = ( M22 @ X ) ) )
% 0.27/0.49 => ( M1 = M22 ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Ring.prodM_mem_eq
% 0.27/0.49 thf(fact_92_carr__prodag__mem__eq,axiom,
% 0.27/0.49 ! [I: set_a,A: a > carrie1105631105xt_a_b,X4: a > a,Y: a > a] :
% 0.27/0.49 ( ! [X: a] :
% 0.27/0.49 ( ( member_a @ X @ I )
% 0.27/0.49 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 0.27/0.49 => ( ( member_a_a @ X4 @ ( carr_p290614414xt_a_b @ I @ A ) )
% 0.27/0.49 => ( ( member_a_a @ Y @ ( carr_p290614414xt_a_b @ I @ A ) )
% 0.27/0.49 => ( ! [X: a] :
% 0.27/0.49 ( ( member_a @ X @ I )
% 0.27/0.49 => ( ( X4 @ X )
% 0.27/0.49 = ( Y @ X ) ) )
% 0.27/0.49 => ( X4 = Y ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % carr_prodag_mem_eq
% 0.27/0.49 thf(fact_93_carr__prodag__mem__eq,axiom,
% 0.27/0.49 ! [I: set_c,A: c > carrie1948989069_d_a_e,X4: c > d,Y: c > d] :
% 0.27/0.49 ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 0.27/0.49 => ( ( member_c_d @ X4 @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 0.27/0.49 => ( ( member_c_d @ Y @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 0.27/0.49 => ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( ( X4 @ X )
% 0.27/0.49 = ( Y @ X ) ) )
% 0.27/0.49 => ( X4 = Y ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % carr_prodag_mem_eq
% 0.27/0.49 thf(fact_94_prodag__sameTr1,axiom,
% 0.27/0.49 ! [I: set_c,A: c > carrie1948989069_d_a_e,B: c > carrie1948989069_d_a_e] :
% 0.27/0.49 ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 0.27/0.49 => ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( ( A @ X )
% 0.27/0.49 = ( B @ X ) ) )
% 0.27/0.49 => ( ( carr_p1636662547_d_a_e @ I @ A )
% 0.27/0.49 = ( carr_p1636662547_d_a_e @ I @ B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodag_sameTr1
% 0.27/0.49 thf(fact_95_prodag__comp__i,axiom,
% 0.27/0.49 ! [A2: d > a,I: set_d,A: d > carrie1105631105xt_a_b,I2: d] :
% 0.27/0.49 ( ( member_d_a @ A2 @ ( carr_p297433675xt_a_b @ I @ A ) )
% 0.27/0.49 => ( ( member_d @ I2 @ I )
% 0.27/0.49 => ( member_a @ ( A2 @ I2 ) @ ( carrie867757212xt_a_b @ ( A @ I2 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodag_comp_i
% 0.27/0.49 thf(fact_96_prodag__comp__i,axiom,
% 0.27/0.49 ! [A2: c > a,I: set_c,A: c > carrie1105631105xt_a_b,I2: c] :
% 0.27/0.49 ( ( member_c_a @ A2 @ ( carr_p295160588xt_a_b @ I @ A ) )
% 0.27/0.49 => ( ( member_c @ I2 @ I )
% 0.27/0.49 => ( member_a @ ( A2 @ I2 ) @ ( carrie867757212xt_a_b @ ( A @ I2 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodag_comp_i
% 0.27/0.49 thf(fact_97_prodag__comp__i,axiom,
% 0.27/0.49 ! [A2: a > a,I: set_a,A: a > carrie1105631105xt_a_b,I2: a] :
% 0.27/0.49 ( ( member_a_a @ A2 @ ( carr_p290614414xt_a_b @ I @ A ) )
% 0.27/0.49 => ( ( member_a @ I2 @ I )
% 0.27/0.49 => ( member_a @ ( A2 @ I2 ) @ ( carrie867757212xt_a_b @ ( A @ I2 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodag_comp_i
% 0.27/0.49 thf(fact_98_prodag__comp__i,axiom,
% 0.27/0.49 ! [A2: c > d,I: set_c,A: c > carrie1948989069_d_a_e,I2: c] :
% 0.27/0.49 ( ( member_c_d @ A2 @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 0.27/0.49 => ( ( member_c @ I2 @ I )
% 0.27/0.49 => ( member_d @ ( A2 @ I2 ) @ ( carrie11369756_d_a_e @ ( A @ I2 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodag_comp_i
% 0.27/0.49 thf(fact_99_prodag__comp__i,axiom,
% 0.27/0.49 ! [A2: ( c > d ) > a,I: set_c_d,A: ( c > d ) > carrie1105631105xt_a_b,I2: c > d] :
% 0.27/0.49 ( ( member_c_d_a @ A2 @ ( carr_p996491646xt_a_b @ I @ A ) )
% 0.27/0.49 => ( ( member_c_d @ I2 @ I )
% 0.27/0.49 => ( member_a @ ( A2 @ I2 ) @ ( carrie867757212xt_a_b @ ( A @ I2 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodag_comp_i
% 0.27/0.49 thf(fact_100_prodag__comp__i,axiom,
% 0.27/0.49 ! [A2: ( a > a ) > a,I: set_a_a,A: ( a > a ) > carrie1105631105xt_a_b,I2: a > a] :
% 0.27/0.49 ( ( member_a_a_a @ A2 @ ( carr_p740328767xt_a_b @ I @ A ) )
% 0.27/0.49 => ( ( member_a_a @ I2 @ I )
% 0.27/0.49 => ( member_a @ ( A2 @ I2 ) @ ( carrie867757212xt_a_b @ ( A @ I2 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodag_comp_i
% 0.27/0.49 thf(fact_101_prodag__comp__i,axiom,
% 0.27/0.49 ! [A2: ( a > a > a ) > a,I: set_a_a_a,A: ( a > a > a ) > carrie1105631105xt_a_b,I2: a > a > a] :
% 0.27/0.49 ( ( member_a_a_a_a @ A2 @ ( carr_p1481099064xt_a_b @ I @ A ) )
% 0.27/0.49 => ( ( member_a_a_a2 @ I2 @ I )
% 0.27/0.49 => ( member_a @ ( A2 @ I2 ) @ ( carrie867757212xt_a_b @ ( A @ I2 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodag_comp_i
% 0.27/0.49 thf(fact_102_prodag__comp__i,axiom,
% 0.27/0.49 ! [A2: a > c > d,I: set_a,A: a > carrie1954260683t_unit,I2: a] :
% 0.27/0.49 ( ( member_a_c_d @ A2 @ ( carr_p923087418t_unit @ I @ A ) )
% 0.27/0.49 => ( ( member_a @ I2 @ I )
% 0.27/0.49 => ( member_c_d @ ( A2 @ I2 ) @ ( carrie20288686t_unit @ ( A @ I2 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodag_comp_i
% 0.27/0.49 thf(fact_103_prodag__comp__i,axiom,
% 0.27/0.49 ! [A2: d > c > d,I: set_d,A: d > carrie1954260683t_unit,I2: d] :
% 0.27/0.49 ( ( member_d_c_d @ A2 @ ( carr_p1159644861t_unit @ I @ A ) )
% 0.27/0.49 => ( ( member_d @ I2 @ I )
% 0.27/0.49 => ( member_c_d @ ( A2 @ I2 ) @ ( carrie20288686t_unit @ ( A @ I2 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodag_comp_i
% 0.27/0.49 thf(fact_104_prodag__comp__i,axiom,
% 0.27/0.49 ! [A2: c > c > d,I: set_c,A: c > carrie1954260683t_unit,I2: c] :
% 0.27/0.49 ( ( member_c_c_d @ A2 @ ( carr_p1080792380t_unit @ I @ A ) )
% 0.27/0.49 => ( ( member_c @ I2 @ I )
% 0.27/0.49 => ( member_c_d @ ( A2 @ I2 ) @ ( carrie20288686t_unit @ ( A @ I2 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodag_comp_i
% 0.27/0.49 thf(fact_105_prodag__sameTr2,axiom,
% 0.27/0.49 ! [I: set_c,A: c > carrie1948989069_d_a_e,B: c > carrie1948989069_d_a_e] :
% 0.27/0.49 ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 0.27/0.49 => ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( ( A @ X )
% 0.27/0.49 = ( B @ X ) ) )
% 0.27/0.49 => ( ( prod_p131511076_d_a_e @ I @ A )
% 0.27/0.49 = ( prod_p131511076_d_a_e @ I @ B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodag_sameTr2
% 0.27/0.49 thf(fact_106_prodag__sameTr3,axiom,
% 0.27/0.49 ! [I: set_c,A: c > carrie1948989069_d_a_e,B: c > carrie1948989069_d_a_e] :
% 0.27/0.49 ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 0.27/0.49 => ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( ( A @ X )
% 0.27/0.49 = ( B @ X ) ) )
% 0.27/0.49 => ( ( prod_m854557927_d_a_e @ I @ A )
% 0.27/0.49 = ( prod_m854557927_d_a_e @ I @ B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodag_sameTr3
% 0.27/0.49 thf(fact_107_prodag__sameTr4,axiom,
% 0.27/0.49 ! [I: set_c,A: c > carrie1948989069_d_a_e,B: c > carrie1948989069_d_a_e] :
% 0.27/0.49 ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 0.27/0.49 => ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( ( A @ X )
% 0.27/0.49 = ( B @ X ) ) )
% 0.27/0.49 => ( ( prod_z1214987823_d_a_e @ I @ A )
% 0.27/0.49 = ( prod_z1214987823_d_a_e @ I @ B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodag_sameTr4
% 0.27/0.49 thf(fact_108_Ring_OprodM__carr,axiom,
% 0.27/0.49 ! [R: carrie1105631105xt_a_b,I: set_c,M: c > carrie1948989069_d_a_e] :
% 0.27/0.49 ( ( ring_a_b @ R )
% 0.27/0.49 => ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 0.27/0.49 => ( ( carrie20288686t_unit @ ( algebr1968153176_c_d_e @ R @ I @ M ) )
% 0.27/0.49 = ( carr_p1636662547_d_a_e @ I @ M ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Ring.prodM_carr
% 0.27/0.49 thf(fact_109_Ring_OprodM__sprod__mem,axiom,
% 0.27/0.49 ! [R: carrie1105631105xt_a_b,I: set_c,M: c > carrie1948989069_d_a_e,A2: a,M2: c > d] :
% 0.27/0.49 ( ( ring_a_b @ R )
% 0.27/0.49 => ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 0.27/0.49 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.49 => ( ( member_c_d @ M2 @ ( carr_p1636662547_d_a_e @ I @ M ) )
% 0.27/0.49 => ( member_c_d @ ( algebr1697692348_c_d_e @ R @ I @ M @ A2 @ M2 ) @ ( carr_p1636662547_d_a_e @ I @ M ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Ring.prodM_sprod_mem
% 0.27/0.49 thf(fact_110_prod__pOp__commute,axiom,
% 0.27/0.49 ! [I: set_a,A: a > carrie1105631105xt_a_b,A2: a > a,B2: a > a] :
% 0.27/0.49 ( ! [X: a] :
% 0.27/0.49 ( ( member_a @ X @ I )
% 0.27/0.49 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 0.27/0.49 => ( ( member_a_a @ A2 @ ( carr_p290614414xt_a_b @ I @ A ) )
% 0.27/0.49 => ( ( member_a_a @ B2 @ ( carr_p290614414xt_a_b @ I @ A ) )
% 0.27/0.49 => ( ( prod_p704341949xt_a_b @ I @ A @ A2 @ B2 )
% 0.27/0.49 = ( prod_p704341949xt_a_b @ I @ A @ B2 @ A2 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prod_pOp_commute
% 0.27/0.49 thf(fact_111_prod__pOp__commute,axiom,
% 0.27/0.49 ! [I: set_c,A: c > carrie1948989069_d_a_e,A2: c > d,B2: c > d] :
% 0.27/0.49 ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 0.27/0.49 => ( ( member_c_d @ A2 @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 0.27/0.49 => ( ( member_c_d @ B2 @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 0.27/0.49 => ( ( prod_p131511076_d_a_e @ I @ A @ A2 @ B2 )
% 0.27/0.49 = ( prod_p131511076_d_a_e @ I @ A @ B2 @ A2 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prod_pOp_commute
% 0.27/0.49 thf(fact_112_prod__pOp__assoc,axiom,
% 0.27/0.49 ! [I: set_a,A: a > carrie1105631105xt_a_b,A2: a > a,B2: a > a,C2: a > a] :
% 0.27/0.49 ( ! [X: a] :
% 0.27/0.49 ( ( member_a @ X @ I )
% 0.27/0.49 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 0.27/0.49 => ( ( member_a_a @ A2 @ ( carr_p290614414xt_a_b @ I @ A ) )
% 0.27/0.49 => ( ( member_a_a @ B2 @ ( carr_p290614414xt_a_b @ I @ A ) )
% 0.27/0.49 => ( ( member_a_a @ C2 @ ( carr_p290614414xt_a_b @ I @ A ) )
% 0.27/0.49 => ( ( prod_p704341949xt_a_b @ I @ A @ ( prod_p704341949xt_a_b @ I @ A @ A2 @ B2 ) @ C2 )
% 0.27/0.49 = ( prod_p704341949xt_a_b @ I @ A @ A2 @ ( prod_p704341949xt_a_b @ I @ A @ B2 @ C2 ) ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prod_pOp_assoc
% 0.27/0.49 thf(fact_113_prod__pOp__assoc,axiom,
% 0.27/0.49 ! [I: set_c,A: c > carrie1948989069_d_a_e,A2: c > d,B2: c > d,C2: c > d] :
% 0.27/0.49 ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 0.27/0.49 => ( ( member_c_d @ A2 @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 0.27/0.49 => ( ( member_c_d @ B2 @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 0.27/0.49 => ( ( member_c_d @ C2 @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 0.27/0.49 => ( ( prod_p131511076_d_a_e @ I @ A @ ( prod_p131511076_d_a_e @ I @ A @ A2 @ B2 ) @ C2 )
% 0.27/0.49 = ( prod_p131511076_d_a_e @ I @ A @ A2 @ ( prod_p131511076_d_a_e @ I @ A @ B2 @ C2 ) ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prod_pOp_assoc
% 0.27/0.49 thf(fact_114_prod__pOp__mem,axiom,
% 0.27/0.49 ! [I: set_a,A: a > carrie1105631105xt_a_b,X4: a > a,Y: a > a] :
% 0.27/0.49 ( ! [X: a] :
% 0.27/0.49 ( ( member_a @ X @ I )
% 0.27/0.49 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 0.27/0.49 => ( ( member_a_a @ X4 @ ( carr_p290614414xt_a_b @ I @ A ) )
% 0.27/0.49 => ( ( member_a_a @ Y @ ( carr_p290614414xt_a_b @ I @ A ) )
% 0.27/0.49 => ( member_a_a @ ( prod_p704341949xt_a_b @ I @ A @ X4 @ Y ) @ ( carr_p290614414xt_a_b @ I @ A ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prod_pOp_mem
% 0.27/0.49 thf(fact_115_prod__pOp__mem,axiom,
% 0.27/0.49 ! [I: set_c,A: c > carrie1948989069_d_a_e,X4: c > d,Y: c > d] :
% 0.27/0.49 ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 0.27/0.49 => ( ( member_c_d @ X4 @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 0.27/0.49 => ( ( member_c_d @ Y @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 0.27/0.49 => ( member_c_d @ ( prod_p131511076_d_a_e @ I @ A @ X4 @ Y ) @ ( carr_p1636662547_d_a_e @ I @ A ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prod_pOp_mem
% 0.27/0.49 thf(fact_116_prod__mOp__mem,axiom,
% 0.27/0.49 ! [I: set_a,A: a > carrie1105631105xt_a_b,X4: a > a] :
% 0.27/0.49 ( ! [X: a] :
% 0.27/0.49 ( ( member_a @ X @ I )
% 0.27/0.49 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 0.27/0.49 => ( ( member_a_a @ X4 @ ( carr_p290614414xt_a_b @ I @ A ) )
% 0.27/0.49 => ( member_a_a @ ( prod_m2022989242xt_a_b @ I @ A @ X4 ) @ ( carr_p290614414xt_a_b @ I @ A ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prod_mOp_mem
% 0.27/0.49 thf(fact_117_prod__mOp__mem,axiom,
% 0.27/0.49 ! [I: set_c,A: c > carrie1948989069_d_a_e,X4: c > d] :
% 0.27/0.49 ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 0.27/0.49 => ( ( member_c_d @ X4 @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 0.27/0.49 => ( member_c_d @ ( prod_m854557927_d_a_e @ I @ A @ X4 ) @ ( carr_p1636662547_d_a_e @ I @ A ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prod_mOp_mem
% 0.27/0.49 thf(fact_118_prod__zero__func,axiom,
% 0.27/0.49 ! [I: set_a,A: a > carrie1105631105xt_a_b] :
% 0.27/0.49 ( ! [X: a] :
% 0.27/0.49 ( ( member_a @ X @ I )
% 0.27/0.49 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 0.27/0.49 => ( member_a_a @ ( prod_z851058290xt_a_b @ I @ A ) @ ( carr_p290614414xt_a_b @ I @ A ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prod_zero_func
% 0.27/0.49 thf(fact_119_prod__zero__func,axiom,
% 0.27/0.49 ! [I: set_c,A: c > carrie1948989069_d_a_e] :
% 0.27/0.49 ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 0.27/0.49 => ( member_c_d @ ( prod_z1214987823_d_a_e @ I @ A ) @ ( carr_p1636662547_d_a_e @ I @ A ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prod_zero_func
% 0.27/0.49 thf(fact_120_prod__mOp__func,axiom,
% 0.27/0.49 ! [I: set_c,A: c > carrie1948989069_d_a_e] :
% 0.27/0.49 ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 0.27/0.49 => ( member_c_d_c_d @ ( prod_m854557927_d_a_e @ I @ A )
% 0.27/0.49 @ ( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ I @ A )
% 0.27/0.49 @ ^ [Uu: c > d] : ( carr_p1636662547_d_a_e @ I @ A ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prod_mOp_func
% 0.27/0.49 thf(fact_121_Pi__cong,axiom,
% 0.27/0.49 ! [A: set_c_d,F: ( c > d ) > ( c > d ) > c > d,G: ( c > d ) > ( c > d ) > c > d,B: ( c > d ) > set_c_d_c_d] :
% 0.27/0.49 ( ! [W: c > d] :
% 0.27/0.49 ( ( member_c_d @ W @ A )
% 0.27/0.49 => ( ( F @ W )
% 0.27/0.49 = ( G @ W ) ) )
% 0.27/0.49 => ( ( member_c_d_c_d_c_d @ F @ ( pi_c_d_c_d_c_d @ A @ B ) )
% 0.27/0.49 = ( member_c_d_c_d_c_d @ G @ ( pi_c_d_c_d_c_d @ A @ B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_cong
% 0.27/0.49 thf(fact_122_Pi__cong,axiom,
% 0.27/0.49 ! [A: set_c_d,F: ( c > d ) > c > d,G: ( c > d ) > c > d,B: ( c > d ) > set_c_d] :
% 0.27/0.49 ( ! [W: c > d] :
% 0.27/0.49 ( ( member_c_d @ W @ A )
% 0.27/0.49 => ( ( F @ W )
% 0.27/0.49 = ( G @ W ) ) )
% 0.27/0.49 => ( ( member_c_d_c_d @ F @ ( pi_c_d_c_d @ A @ B ) )
% 0.27/0.49 = ( member_c_d_c_d @ G @ ( pi_c_d_c_d @ A @ B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_cong
% 0.27/0.49 thf(fact_123_Pi__cong,axiom,
% 0.27/0.49 ! [A: set_a,F: a > c > d,G: a > c > d,B: a > set_c_d] :
% 0.27/0.49 ( ! [W: a] :
% 0.27/0.49 ( ( member_a @ W @ A )
% 0.27/0.49 => ( ( F @ W )
% 0.27/0.49 = ( G @ W ) ) )
% 0.27/0.49 => ( ( member_a_c_d @ F @ ( pi_a_c_d @ A @ B ) )
% 0.27/0.49 = ( member_a_c_d @ G @ ( pi_a_c_d @ A @ B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_cong
% 0.27/0.49 thf(fact_124_Pi__cong,axiom,
% 0.27/0.49 ! [A: set_a,F: a > a > ( c > d ) > c > d,G: a > a > ( c > d ) > c > d,B: a > set_a_c_d_c_d] :
% 0.27/0.49 ( ! [W: a] :
% 0.27/0.49 ( ( member_a @ W @ A )
% 0.27/0.49 => ( ( F @ W )
% 0.27/0.49 = ( G @ W ) ) )
% 0.27/0.49 => ( ( member_a_a_c_d_c_d @ F @ ( pi_a_a_c_d_c_d @ A @ B ) )
% 0.27/0.49 = ( member_a_a_c_d_c_d @ G @ ( pi_a_a_c_d_c_d @ A @ B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_cong
% 0.27/0.49 thf(fact_125_Pi__cong,axiom,
% 0.27/0.49 ! [A: set_a,F: a > a > a,G: a > a > a,B: a > set_a_a] :
% 0.27/0.49 ( ! [W: a] :
% 0.27/0.49 ( ( member_a @ W @ A )
% 0.27/0.49 => ( ( F @ W )
% 0.27/0.49 = ( G @ W ) ) )
% 0.27/0.49 => ( ( member_a_a_a2 @ F @ ( pi_a_a_a2 @ A @ B ) )
% 0.27/0.49 = ( member_a_a_a2 @ G @ ( pi_a_a_a2 @ A @ B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_cong
% 0.27/0.49 thf(fact_126_Pi__cong,axiom,
% 0.27/0.49 ! [A: set_a,F: a > a,G: a > a,B: a > set_a] :
% 0.27/0.49 ( ! [W: a] :
% 0.27/0.49 ( ( member_a @ W @ A )
% 0.27/0.49 => ( ( F @ W )
% 0.27/0.49 = ( G @ W ) ) )
% 0.27/0.49 => ( ( member_a_a @ F @ ( pi_a_a @ A @ B ) )
% 0.27/0.49 = ( member_a_a @ G @ ( pi_a_a @ A @ B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_cong
% 0.27/0.49 thf(fact_127_Pi__cong,axiom,
% 0.27/0.49 ! [A: set_a,F: a > a > c > d,G: a > a > c > d,B: a > set_a_c_d] :
% 0.27/0.49 ( ! [W: a] :
% 0.27/0.49 ( ( member_a @ W @ A )
% 0.27/0.49 => ( ( F @ W )
% 0.27/0.49 = ( G @ W ) ) )
% 0.27/0.49 => ( ( member_a_a_c_d2 @ F @ ( pi_a_a_c_d2 @ A @ B ) )
% 0.27/0.49 = ( member_a_a_c_d2 @ G @ ( pi_a_a_c_d2 @ A @ B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_cong
% 0.27/0.49 thf(fact_128_Pi__cong,axiom,
% 0.27/0.49 ! [A: set_a,F: a > a > a > a > a,G: a > a > a > a > a,B: a > set_a_a_a_a] :
% 0.27/0.49 ( ! [W: a] :
% 0.27/0.49 ( ( member_a @ W @ A )
% 0.27/0.49 => ( ( F @ W )
% 0.27/0.49 = ( G @ W ) ) )
% 0.27/0.49 => ( ( member_a_a_a_a_a3 @ F @ ( pi_a_a_a_a_a2 @ A @ B ) )
% 0.27/0.49 = ( member_a_a_a_a_a3 @ G @ ( pi_a_a_a_a_a2 @ A @ B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_cong
% 0.27/0.49 thf(fact_129_Pi__cong,axiom,
% 0.27/0.49 ! [A: set_a,F: a > a > a > a,G: a > a > a > a,B: a > set_a_a_a] :
% 0.27/0.49 ( ! [W: a] :
% 0.27/0.49 ( ( member_a @ W @ A )
% 0.27/0.49 => ( ( F @ W )
% 0.27/0.49 = ( G @ W ) ) )
% 0.27/0.49 => ( ( member_a_a_a_a4 @ F @ ( pi_a_a_a_a4 @ A @ B ) )
% 0.27/0.49 = ( member_a_a_a_a4 @ G @ ( pi_a_a_a_a4 @ A @ B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_cong
% 0.27/0.49 thf(fact_130_Pi__cong,axiom,
% 0.27/0.49 ! [A: set_c,F: c > d,G: c > d,B: c > set_d] :
% 0.27/0.49 ( ! [W: c] :
% 0.27/0.49 ( ( member_c @ W @ A )
% 0.27/0.49 => ( ( F @ W )
% 0.27/0.49 = ( G @ W ) ) )
% 0.27/0.49 => ( ( member_c_d @ F @ ( pi_c_d @ A @ B ) )
% 0.27/0.49 = ( member_c_d @ G @ ( pi_c_d @ A @ B ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_cong
% 0.27/0.49 thf(fact_131_Pi__mem,axiom,
% 0.27/0.49 ! [F: a > d,A: set_a,B: a > set_d,X2: a] :
% 0.27/0.49 ( ( member_a_d @ F @ ( pi_a_d @ A @ B ) )
% 0.27/0.49 => ( ( member_a @ X2 @ A )
% 0.27/0.49 => ( member_d @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_mem
% 0.27/0.49 thf(fact_132_Pi__mem,axiom,
% 0.27/0.49 ! [F: a > c,A: set_a,B: a > set_c,X2: a] :
% 0.27/0.49 ( ( member_a_c @ F @ ( pi_a_c @ A @ B ) )
% 0.27/0.49 => ( ( member_a @ X2 @ A )
% 0.27/0.49 => ( member_c @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_mem
% 0.27/0.49 thf(fact_133_Pi__mem,axiom,
% 0.27/0.49 ! [F: d > a,A: set_d,B: d > set_a,X2: d] :
% 0.27/0.49 ( ( member_d_a @ F @ ( pi_d_a @ A @ B ) )
% 0.27/0.49 => ( ( member_d @ X2 @ A )
% 0.27/0.49 => ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_mem
% 0.27/0.49 thf(fact_134_Pi__mem,axiom,
% 0.27/0.49 ! [F: d > d,A: set_d,B: d > set_d,X2: d] :
% 0.27/0.49 ( ( member_d_d @ F @ ( pi_d_d @ A @ B ) )
% 0.27/0.49 => ( ( member_d @ X2 @ A )
% 0.27/0.49 => ( member_d @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_mem
% 0.27/0.49 thf(fact_135_Pi__mem,axiom,
% 0.27/0.49 ! [F: d > c,A: set_d,B: d > set_c,X2: d] :
% 0.27/0.49 ( ( member_d_c @ F @ ( pi_d_c @ A @ B ) )
% 0.27/0.49 => ( ( member_d @ X2 @ A )
% 0.27/0.49 => ( member_c @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_mem
% 0.27/0.49 thf(fact_136_Pi__mem,axiom,
% 0.27/0.49 ! [F: c > a,A: set_c,B: c > set_a,X2: c] :
% 0.27/0.49 ( ( member_c_a @ F @ ( pi_c_a @ A @ B ) )
% 0.27/0.49 => ( ( member_c @ X2 @ A )
% 0.27/0.49 => ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_mem
% 0.27/0.49 thf(fact_137_Pi__mem,axiom,
% 0.27/0.49 ! [F: c > c,A: set_c,B: c > set_c,X2: c] :
% 0.27/0.49 ( ( member_c_c @ F @ ( pi_c_c @ A @ B ) )
% 0.27/0.49 => ( ( member_c @ X2 @ A )
% 0.27/0.49 => ( member_c @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_mem
% 0.27/0.49 thf(fact_138_Pi__mem,axiom,
% 0.27/0.49 ! [F: c > d,A: set_c,B: c > set_d,X2: c] :
% 0.27/0.49 ( ( member_c_d @ F @ ( pi_c_d @ A @ B ) )
% 0.27/0.49 => ( ( member_c @ X2 @ A )
% 0.27/0.49 => ( member_d @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_mem
% 0.27/0.49 thf(fact_139_Pi__mem,axiom,
% 0.27/0.49 ! [F: a > a,A: set_a,B: a > set_a,X2: a] :
% 0.27/0.49 ( ( member_a_a @ F @ ( pi_a_a @ A @ B ) )
% 0.27/0.49 => ( ( member_a @ X2 @ A )
% 0.27/0.49 => ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_mem
% 0.27/0.49 thf(fact_140_Pi__mem,axiom,
% 0.27/0.49 ! [F: ( c > d ) > a,A: set_c_d,B: ( c > d ) > set_a,X2: c > d] :
% 0.27/0.49 ( ( member_c_d_a @ F @ ( pi_c_d_a @ A @ B ) )
% 0.27/0.49 => ( ( member_c_d @ X2 @ A )
% 0.27/0.49 => ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_mem
% 0.27/0.49 thf(fact_141_Pi__iff,axiom,
% 0.27/0.49 ! [F: a > a > ( c > d ) > c > d,I: set_a,X4: a > set_a_c_d_c_d] :
% 0.27/0.49 ( ( member_a_a_c_d_c_d @ F @ ( pi_a_a_c_d_c_d @ I @ X4 ) )
% 0.27/0.49 = ( ! [X3: a] :
% 0.27/0.49 ( ( member_a @ X3 @ I )
% 0.27/0.49 => ( member_a_c_d_c_d @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_iff
% 0.27/0.49 thf(fact_142_Pi__iff,axiom,
% 0.27/0.49 ! [F: ( c > d ) > ( c > d ) > c > d,I: set_c_d,X4: ( c > d ) > set_c_d_c_d] :
% 0.27/0.49 ( ( member_c_d_c_d_c_d @ F @ ( pi_c_d_c_d_c_d @ I @ X4 ) )
% 0.27/0.49 = ( ! [X3: c > d] :
% 0.27/0.49 ( ( member_c_d @ X3 @ I )
% 0.27/0.49 => ( member_c_d_c_d @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_iff
% 0.27/0.49 thf(fact_143_Pi__iff,axiom,
% 0.27/0.49 ! [F: a > c > d,I: set_a,X4: a > set_c_d] :
% 0.27/0.49 ( ( member_a_c_d @ F @ ( pi_a_c_d @ I @ X4 ) )
% 0.27/0.49 = ( ! [X3: a] :
% 0.27/0.49 ( ( member_a @ X3 @ I )
% 0.27/0.49 => ( member_c_d @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_iff
% 0.27/0.49 thf(fact_144_Pi__iff,axiom,
% 0.27/0.49 ! [F: ( c > d ) > c > d,I: set_c_d,X4: ( c > d ) > set_c_d] :
% 0.27/0.49 ( ( member_c_d_c_d @ F @ ( pi_c_d_c_d @ I @ X4 ) )
% 0.27/0.49 = ( ! [X3: c > d] :
% 0.27/0.49 ( ( member_c_d @ X3 @ I )
% 0.27/0.49 => ( member_c_d @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_iff
% 0.27/0.49 thf(fact_145_Pi__iff,axiom,
% 0.27/0.49 ! [F: c > d,I: set_c,X4: c > set_d] :
% 0.27/0.49 ( ( member_c_d @ F @ ( pi_c_d @ I @ X4 ) )
% 0.27/0.49 = ( ! [X3: c] :
% 0.27/0.49 ( ( member_c @ X3 @ I )
% 0.27/0.49 => ( member_d @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_iff
% 0.27/0.49 thf(fact_146_Pi__iff,axiom,
% 0.27/0.49 ! [F: a > a > a,I: set_a,X4: a > set_a_a] :
% 0.27/0.49 ( ( member_a_a_a2 @ F @ ( pi_a_a_a2 @ I @ X4 ) )
% 0.27/0.49 = ( ! [X3: a] :
% 0.27/0.49 ( ( member_a @ X3 @ I )
% 0.27/0.49 => ( member_a_a @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_iff
% 0.27/0.49 thf(fact_147_Pi__iff,axiom,
% 0.27/0.49 ! [F: a > a,I: set_a,X4: a > set_a] :
% 0.27/0.49 ( ( member_a_a @ F @ ( pi_a_a @ I @ X4 ) )
% 0.27/0.49 = ( ! [X3: a] :
% 0.27/0.49 ( ( member_a @ X3 @ I )
% 0.27/0.49 => ( member_a @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_iff
% 0.27/0.49 thf(fact_148_Pi__iff,axiom,
% 0.27/0.49 ! [F: a > a > c > d,I: set_a,X4: a > set_a_c_d] :
% 0.27/0.49 ( ( member_a_a_c_d2 @ F @ ( pi_a_a_c_d2 @ I @ X4 ) )
% 0.27/0.49 = ( ! [X3: a] :
% 0.27/0.49 ( ( member_a @ X3 @ I )
% 0.27/0.49 => ( member_a_c_d @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_iff
% 0.27/0.49 thf(fact_149_Pi__iff,axiom,
% 0.27/0.49 ! [F: a > a > a > a > a,I: set_a,X4: a > set_a_a_a_a] :
% 0.27/0.49 ( ( member_a_a_a_a_a3 @ F @ ( pi_a_a_a_a_a2 @ I @ X4 ) )
% 0.27/0.49 = ( ! [X3: a] :
% 0.27/0.49 ( ( member_a @ X3 @ I )
% 0.27/0.49 => ( member_a_a_a_a4 @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_iff
% 0.27/0.49 thf(fact_150_Pi__iff,axiom,
% 0.27/0.49 ! [F: a > a > a > a,I: set_a,X4: a > set_a_a_a] :
% 0.27/0.49 ( ( member_a_a_a_a4 @ F @ ( pi_a_a_a_a4 @ I @ X4 ) )
% 0.27/0.49 = ( ! [X3: a] :
% 0.27/0.49 ( ( member_a @ X3 @ I )
% 0.27/0.49 => ( member_a_a_a2 @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_iff
% 0.27/0.49 thf(fact_151_Pi__I_H,axiom,
% 0.27/0.49 ! [A: set_a,F: a > a,B: a > set_a] :
% 0.27/0.49 ( ! [X: a] :
% 0.27/0.49 ( ( member_a @ X @ A )
% 0.27/0.49 => ( member_a @ ( F @ X ) @ ( B @ X ) ) )
% 0.27/0.49 => ( member_a_a @ F @ ( pi_a_a @ A @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_I'
% 0.27/0.49 thf(fact_152_Pi__I_H,axiom,
% 0.27/0.49 ! [A: set_a,F: a > d,B: a > set_d] :
% 0.27/0.49 ( ! [X: a] :
% 0.27/0.49 ( ( member_a @ X @ A )
% 0.27/0.49 => ( member_d @ ( F @ X ) @ ( B @ X ) ) )
% 0.27/0.49 => ( member_a_d @ F @ ( pi_a_d @ A @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_I'
% 0.27/0.49 thf(fact_153_Pi__I_H,axiom,
% 0.27/0.49 ! [A: set_a,F: a > c,B: a > set_c] :
% 0.27/0.49 ( ! [X: a] :
% 0.27/0.49 ( ( member_a @ X @ A )
% 0.27/0.49 => ( member_c @ ( F @ X ) @ ( B @ X ) ) )
% 0.27/0.49 => ( member_a_c @ F @ ( pi_a_c @ A @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_I'
% 0.27/0.49 thf(fact_154_Pi__I_H,axiom,
% 0.27/0.49 ! [A: set_d,F: d > a,B: d > set_a] :
% 0.27/0.49 ( ! [X: d] :
% 0.27/0.49 ( ( member_d @ X @ A )
% 0.27/0.49 => ( member_a @ ( F @ X ) @ ( B @ X ) ) )
% 0.27/0.49 => ( member_d_a @ F @ ( pi_d_a @ A @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_I'
% 0.27/0.49 thf(fact_155_Pi__I_H,axiom,
% 0.27/0.49 ! [A: set_d,F: d > d,B: d > set_d] :
% 0.27/0.49 ( ! [X: d] :
% 0.27/0.49 ( ( member_d @ X @ A )
% 0.27/0.49 => ( member_d @ ( F @ X ) @ ( B @ X ) ) )
% 0.27/0.49 => ( member_d_d @ F @ ( pi_d_d @ A @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_I'
% 0.27/0.49 thf(fact_156_Pi__I_H,axiom,
% 0.27/0.49 ! [A: set_d,F: d > c,B: d > set_c] :
% 0.27/0.49 ( ! [X: d] :
% 0.27/0.49 ( ( member_d @ X @ A )
% 0.27/0.49 => ( member_c @ ( F @ X ) @ ( B @ X ) ) )
% 0.27/0.49 => ( member_d_c @ F @ ( pi_d_c @ A @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_I'
% 0.27/0.49 thf(fact_157_Pi__I_H,axiom,
% 0.27/0.49 ! [A: set_c,F: c > a,B: c > set_a] :
% 0.27/0.49 ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ A )
% 0.27/0.49 => ( member_a @ ( F @ X ) @ ( B @ X ) ) )
% 0.27/0.49 => ( member_c_a @ F @ ( pi_c_a @ A @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_I'
% 0.27/0.49 thf(fact_158_Pi__I_H,axiom,
% 0.27/0.49 ! [A: set_c,F: c > d,B: c > set_d] :
% 0.27/0.49 ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ A )
% 0.27/0.49 => ( member_d @ ( F @ X ) @ ( B @ X ) ) )
% 0.27/0.49 => ( member_c_d @ F @ ( pi_c_d @ A @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_I'
% 0.27/0.49 thf(fact_159_Pi__I_H,axiom,
% 0.27/0.49 ! [A: set_c,F: c > c,B: c > set_c] :
% 0.27/0.49 ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ A )
% 0.27/0.49 => ( member_c @ ( F @ X ) @ ( B @ X ) ) )
% 0.27/0.49 => ( member_c_c @ F @ ( pi_c_c @ A @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_I'
% 0.27/0.49 thf(fact_160_Pi__I_H,axiom,
% 0.27/0.49 ! [A: set_c_d,F: ( c > d ) > a,B: ( c > d ) > set_a] :
% 0.27/0.49 ( ! [X: c > d] :
% 0.27/0.49 ( ( member_c_d @ X @ A )
% 0.27/0.49 => ( member_a @ ( F @ X ) @ ( B @ X ) ) )
% 0.27/0.49 => ( member_c_d_a @ F @ ( pi_c_d_a @ A @ B ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Pi_I'
% 0.27/0.49 thf(fact_161_PiE,axiom,
% 0.27/0.49 ! [F: d > a,A: set_d,B: d > set_a,X2: d] :
% 0.27/0.49 ( ( member_d_a @ F @ ( pi_d_a @ A @ B ) )
% 0.27/0.49 => ( ~ ( member_a @ ( F @ X2 ) @ ( B @ X2 ) )
% 0.27/0.49 => ~ ( member_d @ X2 @ A ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % PiE
% 0.27/0.49 thf(fact_162_PiE,axiom,
% 0.27/0.49 ! [F: c > a,A: set_c,B: c > set_a,X2: c] :
% 0.27/0.49 ( ( member_c_a @ F @ ( pi_c_a @ A @ B ) )
% 0.27/0.49 => ( ~ ( member_a @ ( F @ X2 ) @ ( B @ X2 ) )
% 0.27/0.49 => ~ ( member_c @ X2 @ A ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % PiE
% 0.27/0.49 thf(fact_163_PiE,axiom,
% 0.27/0.49 ! [F: a > d,A: set_a,B: a > set_d,X2: a] :
% 0.27/0.49 ( ( member_a_d @ F @ ( pi_a_d @ A @ B ) )
% 0.27/0.49 => ( ~ ( member_d @ ( F @ X2 ) @ ( B @ X2 ) )
% 0.27/0.49 => ~ ( member_a @ X2 @ A ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % PiE
% 0.27/0.49 thf(fact_164_PiE,axiom,
% 0.27/0.49 ! [F: d > d,A: set_d,B: d > set_d,X2: d] :
% 0.27/0.49 ( ( member_d_d @ F @ ( pi_d_d @ A @ B ) )
% 0.27/0.49 => ( ~ ( member_d @ ( F @ X2 ) @ ( B @ X2 ) )
% 0.27/0.49 => ~ ( member_d @ X2 @ A ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % PiE
% 0.27/0.49 thf(fact_165_PiE,axiom,
% 0.27/0.49 ! [F: a > c,A: set_a,B: a > set_c,X2: a] :
% 0.27/0.49 ( ( member_a_c @ F @ ( pi_a_c @ A @ B ) )
% 0.27/0.49 => ( ~ ( member_c @ ( F @ X2 ) @ ( B @ X2 ) )
% 0.27/0.49 => ~ ( member_a @ X2 @ A ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % PiE
% 0.27/0.49 thf(fact_166_PiE,axiom,
% 0.27/0.49 ! [F: d > c,A: set_d,B: d > set_c,X2: d] :
% 0.27/0.49 ( ( member_d_c @ F @ ( pi_d_c @ A @ B ) )
% 0.27/0.49 => ( ~ ( member_c @ ( F @ X2 ) @ ( B @ X2 ) )
% 0.27/0.49 => ~ ( member_d @ X2 @ A ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % PiE
% 0.27/0.49 thf(fact_167_PiE,axiom,
% 0.27/0.49 ! [F: c > c,A: set_c,B: c > set_c,X2: c] :
% 0.27/0.49 ( ( member_c_c @ F @ ( pi_c_c @ A @ B ) )
% 0.27/0.49 => ( ~ ( member_c @ ( F @ X2 ) @ ( B @ X2 ) )
% 0.27/0.49 => ~ ( member_c @ X2 @ A ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % PiE
% 0.27/0.49 thf(fact_168_PiE,axiom,
% 0.27/0.49 ! [F: c > d,A: set_c,B: c > set_d,X2: c] :
% 0.27/0.49 ( ( member_c_d @ F @ ( pi_c_d @ A @ B ) )
% 0.27/0.49 => ( ~ ( member_d @ ( F @ X2 ) @ ( B @ X2 ) )
% 0.27/0.49 => ~ ( member_c @ X2 @ A ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % PiE
% 0.27/0.49 thf(fact_169_PiE,axiom,
% 0.27/0.49 ! [F: a > a,A: set_a,B: a > set_a,X2: a] :
% 0.27/0.49 ( ( member_a_a @ F @ ( pi_a_a @ A @ B ) )
% 0.27/0.49 => ( ~ ( member_a @ ( F @ X2 ) @ ( B @ X2 ) )
% 0.27/0.49 => ~ ( member_a @ X2 @ A ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % PiE
% 0.27/0.49 thf(fact_170_PiE,axiom,
% 0.27/0.49 ! [F: a > c > d,A: set_a,B: a > set_c_d,X2: a] :
% 0.27/0.49 ( ( member_a_c_d @ F @ ( pi_a_c_d @ A @ B ) )
% 0.27/0.49 => ( ~ ( member_c_d @ ( F @ X2 ) @ ( B @ X2 ) )
% 0.27/0.49 => ~ ( member_a @ X2 @ A ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % PiE
% 0.27/0.49 thf(fact_171_mem__subring__mem__ring,axiom,
% 0.27/0.49 ! [S: carrie1105631105xt_a_b,X2: a] :
% 0.27/0.49 ( ( subring_a_b_b @ r @ S )
% 0.27/0.49 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ S ) )
% 0.27/0.49 => ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mem_subring_mem_ring
% 0.27/0.49 thf(fact_172_subring__Ring,axiom,
% 0.27/0.49 ! [S: carrie1105631105xt_a_b] :
% 0.27/0.49 ( ( subring_a_b_b @ r @ S )
% 0.27/0.49 => ( ring_a_b @ S ) ) ).
% 0.27/0.49
% 0.27/0.49 % subring_Ring
% 0.27/0.49 thf(fact_173_prodM__sprod__val,axiom,
% 0.27/0.49 ! [I: set_c_d_c_d,M: ( ( c > d ) > c > d ) > carrie1948989069_d_a_e,A2: a,M2: ( ( c > d ) > c > d ) > d,J: ( c > d ) > c > d] :
% 0.27/0.49 ( ! [X: ( c > d ) > c > d] :
% 0.27/0.49 ( ( member_c_d_c_d @ X @ I )
% 0.27/0.49 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 0.27/0.49 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.49 => ( ( member_c_d_c_d_d @ M2 @ ( carr_p428609062_d_a_e @ I @ M ) )
% 0.27/0.49 => ( ( member_c_d_c_d @ J @ I )
% 0.27/0.49 => ( ( algebr828196337_d_d_e @ r @ I @ M @ A2 @ M2 @ J )
% 0.27/0.49 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodM_sprod_val
% 0.27/0.49 thf(fact_174_prodM__sprod__val,axiom,
% 0.27/0.49 ! [I: set_c_d,M: ( c > d ) > carrie1948989069_d_a_e,A2: a,M2: ( c > d ) > d,J: c > d] :
% 0.27/0.49 ( ! [X: c > d] :
% 0.27/0.49 ( ( member_c_d @ X @ I )
% 0.27/0.49 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 0.27/0.49 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.49 => ( ( member_c_d_d @ M2 @ ( carr_p124510241_d_a_e @ I @ M ) )
% 0.27/0.49 => ( ( member_c_d @ J @ I )
% 0.27/0.49 => ( ( algebr924632044_d_d_e @ r @ I @ M @ A2 @ M2 @ J )
% 0.27/0.49 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodM_sprod_val
% 0.27/0.49 thf(fact_175_prodM__sprod__val,axiom,
% 0.27/0.49 ! [I: set_a_a_a,M: ( a > a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > a ) > d,J: a > a > a] :
% 0.27/0.49 ( ! [X: a > a > a] :
% 0.27/0.49 ( ( member_a_a_a2 @ X @ I )
% 0.27/0.49 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 0.27/0.49 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.49 => ( ( member_a_a_a_d @ M2 @ ( carr_p1364179431_d_a_e @ I @ M ) )
% 0.27/0.49 => ( ( member_a_a_a2 @ J @ I )
% 0.27/0.49 => ( ( algebr1655014672_a_d_e @ r @ I @ M @ A2 @ M2 @ J )
% 0.27/0.49 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodM_sprod_val
% 0.27/0.49 thf(fact_176_prodM__sprod__val,axiom,
% 0.27/0.49 ! [I: set_a_a,M: ( a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a ) > d,J: a > a] :
% 0.27/0.49 ( ! [X: a > a] :
% 0.27/0.49 ( ( member_a_a @ X @ I )
% 0.27/0.49 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 0.27/0.49 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.49 => ( ( member_a_a_d @ M2 @ ( carr_p1646427936_d_a_e @ I @ M ) )
% 0.27/0.49 => ( ( member_a_a @ J @ I )
% 0.27/0.49 => ( ( algebr137561451_a_d_e @ r @ I @ M @ A2 @ M2 @ J )
% 0.27/0.49 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodM_sprod_val
% 0.27/0.49 thf(fact_177_prodM__sprod__val,axiom,
% 0.27/0.49 ! [I: set_a,M: a > carrie1948989069_d_a_e,A2: a,M2: a > d,J: a] :
% 0.27/0.49 ( ! [X: a] :
% 0.27/0.49 ( ( member_a @ X @ I )
% 0.27/0.49 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 0.27/0.49 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.49 => ( ( member_a_d @ M2 @ ( carr_p1084104977_d_a_e @ I @ M ) )
% 0.27/0.49 => ( ( member_a @ J @ I )
% 0.27/0.49 => ( ( algebr2109842362_a_d_e @ r @ I @ M @ A2 @ M2 @ J )
% 0.27/0.49 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodM_sprod_val
% 0.27/0.49 thf(fact_178_prodM__sprod__val,axiom,
% 0.27/0.49 ! [I: set_a_a_c_d,M: ( a > a > c > d ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > c > d ) > d,J: a > a > c > d] :
% 0.27/0.49 ( ! [X: a > a > c > d] :
% 0.27/0.49 ( ( member_a_a_c_d2 @ X @ I )
% 0.27/0.49 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 0.27/0.49 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.49 => ( ( member_a_a_c_d_d @ M2 @ ( carr_p577380087_d_a_e @ I @ M ) )
% 0.27/0.49 => ( ( member_a_a_c_d2 @ J @ I )
% 0.27/0.49 => ( ( algebr1620716610_d_d_e @ r @ I @ M @ A2 @ M2 @ J )
% 0.27/0.49 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodM_sprod_val
% 0.27/0.49 thf(fact_179_prodM__sprod__val,axiom,
% 0.27/0.49 ! [I: set_a_a_a_a_a,M: ( a > a > a > a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > a > a > a ) > d,J: a > a > a > a > a] :
% 0.27/0.49 ( ! [X: a > a > a > a > a] :
% 0.27/0.49 ( ( member_a_a_a_a_a3 @ X @ I )
% 0.27/0.49 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 0.27/0.49 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.49 => ( ( member_a_a_a_a_a_d @ M2 @ ( carr_p1980447933_d_a_e @ I @ M ) )
% 0.27/0.49 => ( ( member_a_a_a_a_a3 @ J @ I )
% 0.27/0.49 => ( ( algebr2122715238_a_d_e @ r @ I @ M @ A2 @ M2 @ J )
% 0.27/0.49 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodM_sprod_val
% 0.27/0.49 thf(fact_180_prodM__sprod__val,axiom,
% 0.27/0.49 ! [I: set_a_a_a_a,M: ( a > a > a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > a > a ) > d,J: a > a > a > a] :
% 0.27/0.49 ( ! [X: a > a > a > a] :
% 0.27/0.49 ( ( member_a_a_a_a4 @ X @ I )
% 0.27/0.49 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 0.27/0.49 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.49 => ( ( member_a_a_a_a_d @ M2 @ ( carr_p305487606_d_a_e @ I @ M ) )
% 0.27/0.49 => ( ( member_a_a_a_a4 @ J @ I )
% 0.27/0.49 => ( ( algebr303206081_a_d_e @ r @ I @ M @ A2 @ M2 @ J )
% 0.27/0.49 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodM_sprod_val
% 0.27/0.49 thf(fact_181_prodM__sprod__val,axiom,
% 0.27/0.49 ! [I: set_d,M: d > carrie1948989069_d_a_e,A2: a,M2: d > d,J: d] :
% 0.27/0.49 ( ! [X: d] :
% 0.27/0.49 ( ( member_d @ X @ I )
% 0.27/0.49 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 0.27/0.49 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.49 => ( ( member_d_d @ M2 @ ( carr_p1912941332_d_a_e @ I @ M ) )
% 0.27/0.49 => ( ( member_d @ J @ I )
% 0.27/0.49 => ( ( algebr417875517_d_d_e @ r @ I @ M @ A2 @ M2 @ J )
% 0.27/0.49 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodM_sprod_val
% 0.27/0.49 thf(fact_182_prodM__sprod__val,axiom,
% 0.27/0.49 ! [I: set_c,M: c > carrie1948989069_d_a_e,A2: a,M2: c > d,J: c] :
% 0.27/0.49 ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 0.27/0.49 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.49 => ( ( member_c_d @ M2 @ ( carr_p1636662547_d_a_e @ I @ M ) )
% 0.27/0.49 => ( ( member_c @ J @ I )
% 0.27/0.49 => ( ( algebr1697692348_c_d_e @ r @ I @ M @ A2 @ M2 @ J )
% 0.27/0.49 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % prodM_sprod_val
% 0.27/0.49 thf(fact_183_Sr__ring,axiom,
% 0.27/0.49 ! [S: set_a] :
% 0.27/0.49 ( ( sr_a_b2 @ r @ S )
% 0.27/0.49 => ( ring_a_b @ ( sr_a_b @ r @ S ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Sr_ring
% 0.27/0.49 thf(fact_184_mHom__func,axiom,
% 0.27/0.49 ! [F: ( c > d ) > c > d,M: carrie1954260683t_unit,N: carrie1954260683t_unit] :
% 0.27/0.49 ( ( member_c_d_c_d @ F @ ( mHom_a1526663829t_unit @ r @ M @ N ) )
% 0.27/0.49 => ( member_c_d_c_d @ F
% 0.27/0.49 @ ( pi_c_d_c_d @ ( carrie20288686t_unit @ M )
% 0.27/0.49 @ ^ [Uu: c > d] : ( carrie20288686t_unit @ N ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % mHom_func
% 0.27/0.49 thf(fact_185_dsum__pOp__func,axiom,
% 0.27/0.49 ! [I: set_c,A: c > carrie1948989069_d_a_e] :
% 0.27/0.49 ( ! [X: c] :
% 0.27/0.49 ( ( member_c @ X @ I )
% 0.27/0.49 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 0.27/0.49 => ( member_c_d_c_d_c_d @ ( prod_p131511076_d_a_e @ I @ A )
% 0.27/0.49 @ ( pi_c_d_c_d_c_d @ ( carr_d56066307_d_a_e @ I @ A )
% 0.27/0.49 @ ^ [Uu: c > d] :
% 0.27/0.49 ( pi_c_d_c_d @ ( carr_d56066307_d_a_e @ I @ A )
% 0.27/0.49 @ ^ [Uv: c > d] : ( carr_d56066307_d_a_e @ I @ A ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % dsum_pOp_func
% 0.27/0.49 thf(fact_186_mem__Collect__eq,axiom,
% 0.27/0.49 ! [A2: a,P: a > $o] :
% 0.27/0.49 ( ( member_a @ A2 @ ( collect_a @ P ) )
% 0.27/0.49 = ( P @ A2 ) ) ).
% 0.27/0.49
% 0.27/0.49 % mem_Collect_eq
% 0.27/0.49 thf(fact_187_mem__Collect__eq,axiom,
% 0.27/0.49 ! [A2: d,P: d > $o] :
% 0.27/0.49 ( ( member_d @ A2 @ ( collect_d @ P ) )
% 0.27/0.49 = ( P @ A2 ) ) ).
% 0.27/0.49
% 0.27/0.49 % mem_Collect_eq
% 0.27/0.49 thf(fact_188_mem__Collect__eq,axiom,
% 0.27/0.49 ! [A2: c,P: c > $o] :
% 0.27/0.49 ( ( member_c @ A2 @ ( collect_c @ P ) )
% 0.27/0.49 = ( P @ A2 ) ) ).
% 0.27/0.49
% 0.27/0.49 % mem_Collect_eq
% 0.27/0.49 thf(fact_189_mem__Collect__eq,axiom,
% 0.27/0.49 ! [A2: nat,P: nat > $o] :
% 0.27/0.49 ( ( member_nat @ A2 @ ( collect_nat @ P ) )
% 0.27/0.49 = ( P @ A2 ) ) ).
% 0.27/0.49
% 0.27/0.49 % mem_Collect_eq
% 0.27/0.49 thf(fact_190_mem__Collect__eq,axiom,
% 0.27/0.49 ! [A2: c > d,P: ( c > d ) > $o] :
% 0.27/0.49 ( ( member_c_d @ A2 @ ( collect_c_d @ P ) )
% 0.27/0.49 = ( P @ A2 ) ) ).
% 0.27/0.49
% 0.27/0.49 % mem_Collect_eq
% 0.27/0.49 thf(fact_191_mem__Collect__eq,axiom,
% 0.27/0.49 ! [A2: a > a,P: ( a > a ) > $o] :
% 0.27/0.49 ( ( member_a_a @ A2 @ ( collect_a_a @ P ) )
% 0.27/0.49 = ( P @ A2 ) ) ).
% 0.27/0.49
% 0.27/0.49 % mem_Collect_eq
% 0.27/0.49 thf(fact_192_mem__Collect__eq,axiom,
% 0.27/0.49 ! [A2: a > a > a,P: ( a > a > a ) > $o] :
% 0.27/0.49 ( ( member_a_a_a2 @ A2 @ ( collect_a_a_a @ P ) )
% 0.27/0.49 = ( P @ A2 ) ) ).
% 0.27/0.49
% 0.27/0.49 % mem_Collect_eq
% 0.27/0.49 thf(fact_193_mem__Collect__eq,axiom,
% 0.27/0.49 ! [A2: ( c > d ) > c > d,P: ( ( c > d ) > c > d ) > $o] :
% 0.27/0.49 ( ( member_c_d_c_d @ A2 @ ( collect_c_d_c_d @ P ) )
% 0.27/0.49 = ( P @ A2 ) ) ).
% 0.27/0.49
% 0.27/0.49 % mem_Collect_eq
% 0.27/0.49 thf(fact_194_mem__Collect__eq,axiom,
% 0.27/0.49 ! [A2: a > a > c > d,P: ( a > a > c > d ) > $o] :
% 0.27/0.49 ( ( member_a_a_c_d2 @ A2 @ ( collect_a_a_c_d @ P ) )
% 0.27/0.49 = ( P @ A2 ) ) ).
% 0.27/0.49
% 0.27/0.49 % mem_Collect_eq
% 0.27/0.49 thf(fact_195_mem__Collect__eq,axiom,
% 0.27/0.49 ! [A2: a > a > a > a,P: ( a > a > a > a ) > $o] :
% 0.27/0.49 ( ( member_a_a_a_a4 @ A2 @ ( collect_a_a_a_a @ P ) )
% 0.27/0.49 = ( P @ A2 ) ) ).
% 0.27/0.49
% 0.27/0.49 % mem_Collect_eq
% 0.27/0.49 thf(fact_196_Collect__mem__eq,axiom,
% 0.27/0.49 ! [A: set_a] :
% 0.27/0.49 ( ( collect_a
% 0.27/0.49 @ ^ [X3: a] : ( member_a @ X3 @ A ) )
% 0.27/0.49 = A ) ).
% 0.27/0.49
% 0.27/0.49 % Collect_mem_eq
% 0.27/0.49 thf(fact_197_Collect__mem__eq,axiom,
% 0.27/0.49 ! [A: set_d] :
% 0.27/0.49 ( ( collect_d
% 0.27/0.49 @ ^ [X3: d] : ( member_d @ X3 @ A ) )
% 0.27/0.49 = A ) ).
% 0.27/0.49
% 0.27/0.49 % Collect_mem_eq
% 0.27/0.49 thf(fact_198_Collect__mem__eq,axiom,
% 0.27/0.49 ! [A: set_c] :
% 0.27/0.49 ( ( collect_c
% 0.27/0.49 @ ^ [X3: c] : ( member_c @ X3 @ A ) )
% 0.27/0.49 = A ) ).
% 0.27/0.49
% 0.27/0.49 % Collect_mem_eq
% 0.27/0.49 thf(fact_199_Collect__mem__eq,axiom,
% 0.27/0.49 ! [A: set_nat] :
% 0.27/0.49 ( ( collect_nat
% 0.27/0.49 @ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
% 0.27/0.49 = A ) ).
% 0.27/0.49
% 0.27/0.49 % Collect_mem_eq
% 0.27/0.49 thf(fact_200_Collect__mem__eq,axiom,
% 0.27/0.49 ! [A: set_c_d] :
% 0.27/0.49 ( ( collect_c_d
% 0.27/0.49 @ ^ [X3: c > d] : ( member_c_d @ X3 @ A ) )
% 0.27/0.49 = A ) ).
% 0.27/0.49
% 0.27/0.49 % Collect_mem_eq
% 0.27/0.49 thf(fact_201_Collect__mem__eq,axiom,
% 0.27/0.49 ! [A: set_a_a] :
% 0.27/0.49 ( ( collect_a_a
% 0.27/0.49 @ ^ [X3: a > a] : ( member_a_a @ X3 @ A ) )
% 0.27/0.49 = A ) ).
% 0.27/0.49
% 0.27/0.49 % Collect_mem_eq
% 0.27/0.49 thf(fact_202_Collect__mem__eq,axiom,
% 0.27/0.49 ! [A: set_a_a_a] :
% 0.27/0.49 ( ( collect_a_a_a
% 0.27/0.49 @ ^ [X3: a > a > a] : ( member_a_a_a2 @ X3 @ A ) )
% 0.27/0.49 = A ) ).
% 0.27/0.49
% 0.27/0.49 % Collect_mem_eq
% 0.27/0.49 thf(fact_203_Collect__mem__eq,axiom,
% 0.27/0.49 ! [A: set_c_d_c_d] :
% 0.27/0.49 ( ( collect_c_d_c_d
% 0.27/0.49 @ ^ [X3: ( c > d ) > c > d] : ( member_c_d_c_d @ X3 @ A ) )
% 0.27/0.49 = A ) ).
% 0.27/0.49
% 0.27/0.49 % Collect_mem_eq
% 0.27/0.49 thf(fact_204_Collect__mem__eq,axiom,
% 0.27/0.49 ! [A: set_a_a_c_d] :
% 0.27/0.49 ( ( collect_a_a_c_d
% 0.27/0.49 @ ^ [X3: a > a > c > d] : ( member_a_a_c_d2 @ X3 @ A ) )
% 0.27/0.49 = A ) ).
% 0.27/0.49
% 0.27/0.49 % Collect_mem_eq
% 0.27/0.49 thf(fact_205_Collect__mem__eq,axiom,
% 0.27/0.49 ! [A: set_a_a_a_a] :
% 0.27/0.49 ( ( collect_a_a_a_a
% 0.27/0.49 @ ^ [X3: a > a > a > a] : ( member_a_a_a_a4 @ X3 @ A ) )
% 0.27/0.49 = A ) ).
% 0.27/0.49
% 0.27/0.49 % Collect_mem_eq
% 0.27/0.49 thf(fact_206_Collect__cong,axiom,
% 0.27/0.49 ! [P: nat > $o,Q: nat > $o] :
% 0.27/0.49 ( ! [X: nat] :
% 0.27/0.49 ( ( P @ X )
% 0.27/0.49 = ( Q @ X ) )
% 0.27/0.49 => ( ( collect_nat @ P )
% 0.27/0.49 = ( collect_nat @ Q ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % Collect_cong
% 0.27/0.49 thf(fact_207_nsEqElm,axiom,
% 0.27/0.49 ! [X2: a,Y2: a,N2: nat] :
% 0.27/0.49 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.49 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.49 => ( ( X2 = Y2 )
% 0.27/0.49 => ( ( nscal_a_b @ r @ X2 @ N2 )
% 0.27/0.49 = ( nscal_a_b @ r @ Y2 @ N2 ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % nsEqElm
% 0.27/0.49 thf(fact_208_nsClose,axiom,
% 0.27/0.49 ! [X2: a,N2: nat] :
% 0.27/0.49 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.49 => ( member_a @ ( nscal_a_b @ r @ X2 @ N2 ) @ ( carrie867757212xt_a_b @ r ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % nsClose
% 0.27/0.49 thf(fact_209_tp__closed,axiom,
% 0.27/0.49 ( member_a_a_a2 @ ( tp_a_b @ r )
% 0.27/0.49 @ ( pi_a_a_a2 @ ( carrie867757212xt_a_b @ r )
% 0.27/0.49 @ ^ [Uu: a] :
% 0.27/0.49 ( pi_a_a @ ( carrie867757212xt_a_b @ r )
% 0.27/0.49 @ ^ [Uv: a] : ( carrie867757212xt_a_b @ r ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % tp_closed
% 0.27/0.49 thf(fact_210_pop__closed,axiom,
% 0.27/0.49 ( member_a_a_a2 @ ( pop_a_Ring_ext_a_b @ r )
% 0.27/0.49 @ ( pi_a_a_a2 @ ( carrie867757212xt_a_b @ r )
% 0.27/0.49 @ ^ [Uu: a] :
% 0.27/0.49 ( pi_a_a @ ( carrie867757212xt_a_b @ r )
% 0.27/0.49 @ ^ [Uv: a] : ( carrie867757212xt_a_b @ r ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % pop_closed
% 0.27/0.49 thf(fact_211_rEQMulR,axiom,
% 0.27/0.49 ! [X2: a,Y2: a,Z: a] :
% 0.27/0.49 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.49 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.49 => ( ( member_a @ Z @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.49 => ( ( X2 = Y2 )
% 0.27/0.49 => ( ( tp_a_b @ r @ X2 @ Z )
% 0.27/0.49 = ( tp_a_b @ r @ Y2 @ Z ) ) ) ) ) ) ).
% 0.27/0.49
% 0.27/0.49 % rEQMulR
% 0.27/0.49 thf(fact_212_rMulLC,axiom,
% 0.27/0.49 ! [X2: a,Y2: a,Z: a] :
% 0.27/0.49 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ Z @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( tp_a_b @ r @ X2 @ ( tp_a_b @ r @ Y2 @ Z ) )
% 0.27/0.50 = ( tp_a_b @ r @ Y2 @ ( tp_a_b @ r @ X2 @ Z ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % rMulLC
% 0.27/0.50 thf(fact_213_ring__tOp__assoc,axiom,
% 0.27/0.50 ! [X2: a,Y2: a,Z: a] :
% 0.27/0.50 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ Z @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( tp_a_b @ r @ ( tp_a_b @ r @ X2 @ Y2 ) @ Z )
% 0.27/0.50 = ( tp_a_b @ r @ X2 @ ( tp_a_b @ r @ Y2 @ Z ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % ring_tOp_assoc
% 0.27/0.50 thf(fact_214_ring__tOp__closed,axiom,
% 0.27/0.50 ! [X2: a,Y2: a] :
% 0.27/0.50 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( member_a @ ( tp_a_b @ r @ X2 @ Y2 ) @ ( carrie867757212xt_a_b @ r ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % ring_tOp_closed
% 0.27/0.50 thf(fact_215_ring__tOp__commute,axiom,
% 0.27/0.50 ! [X2: a,Y2: a] :
% 0.27/0.50 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( tp_a_b @ r @ X2 @ Y2 )
% 0.27/0.50 = ( tp_a_b @ r @ Y2 @ X2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % ring_tOp_commute
% 0.27/0.50 thf(fact_216_ring__tOp__rel,axiom,
% 0.27/0.50 ! [X2: a,Xa: a,Y2: a,Ya: a] :
% 0.27/0.50 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ Xa @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ Ya @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( tp_a_b @ r @ ( tp_a_b @ r @ X2 @ Xa ) @ ( tp_a_b @ r @ Y2 @ Ya ) )
% 0.27/0.50 = ( tp_a_b @ r @ ( tp_a_b @ r @ X2 @ Y2 ) @ ( tp_a_b @ r @ Xa @ Ya ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % ring_tOp_rel
% 0.27/0.50 thf(fact_217_tp__assoc,axiom,
% 0.27/0.50 ! [A2: a,B2: a,C2: a] :
% 0.27/0.50 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( tp_a_b @ r @ ( tp_a_b @ r @ A2 @ B2 ) @ C2 )
% 0.27/0.50 = ( tp_a_b @ r @ A2 @ ( tp_a_b @ r @ B2 @ C2 ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % tp_assoc
% 0.27/0.50 thf(fact_218_tp__commute,axiom,
% 0.27/0.50 ! [A2: a,B2: a] :
% 0.27/0.50 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( tp_a_b @ r @ A2 @ B2 )
% 0.27/0.50 = ( tp_a_b @ r @ B2 @ A2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % tp_commute
% 0.27/0.50 thf(fact_219_pop__aassoc,axiom,
% 0.27/0.50 ! [A2: a,B2: a,C2: a] :
% 0.27/0.50 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( pop_a_Ring_ext_a_b @ r @ ( pop_a_Ring_ext_a_b @ r @ A2 @ B2 ) @ C2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ r @ A2 @ ( pop_a_Ring_ext_a_b @ r @ B2 @ C2 ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % pop_aassoc
% 0.27/0.50 thf(fact_220_pop__commute,axiom,
% 0.27/0.50 ! [A2: a,B2: a] :
% 0.27/0.50 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( pop_a_Ring_ext_a_b @ r @ A2 @ B2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ r @ B2 @ A2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % pop_commute
% 0.27/0.50 thf(fact_221_Sr__tOp__closed,axiom,
% 0.27/0.50 ! [S: set_a,X2: a,Y2: a] :
% 0.27/0.50 ( ( sr_a_b2 @ r @ S )
% 0.27/0.50 => ( ( member_a @ X2 @ S )
% 0.27/0.50 => ( ( member_a @ Y2 @ S )
% 0.27/0.50 => ( member_a @ ( tp_a_b @ r @ X2 @ Y2 ) @ S ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Sr_tOp_closed
% 0.27/0.50 thf(fact_222_Sr__pOp__closed,axiom,
% 0.27/0.50 ! [S: set_a,X2: a,Y2: a] :
% 0.27/0.50 ( ( sr_a_b2 @ r @ S )
% 0.27/0.50 => ( ( member_a @ X2 @ S )
% 0.27/0.50 => ( ( member_a @ Y2 @ S )
% 0.27/0.50 => ( member_a @ ( pop_a_Ring_ext_a_b @ r @ X2 @ Y2 ) @ S ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Sr_pOp_closed
% 0.27/0.50 thf(fact_223_rg__distrib,axiom,
% 0.27/0.50 ! [A2: a,B2: a,C2: a] :
% 0.27/0.50 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( tp_a_b @ r @ A2 @ ( pop_a_Ring_ext_a_b @ r @ B2 @ C2 ) )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ A2 @ B2 ) @ ( tp_a_b @ r @ A2 @ C2 ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % rg_distrib
% 0.27/0.50 thf(fact_224_ring__distrib1,axiom,
% 0.27/0.50 ! [X2: a,Y2: a,Z: a] :
% 0.27/0.50 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ Z @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( tp_a_b @ r @ X2 @ ( pop_a_Ring_ext_a_b @ r @ Y2 @ Z ) )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ X2 @ Y2 ) @ ( tp_a_b @ r @ X2 @ Z ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % ring_distrib1
% 0.27/0.50 thf(fact_225_ring__distrib2,axiom,
% 0.27/0.50 ! [X2: a,Y2: a,Z: a] :
% 0.27/0.50 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ Z @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( tp_a_b @ r @ ( pop_a_Ring_ext_a_b @ r @ Y2 @ Z ) @ X2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ Y2 @ X2 ) @ ( tp_a_b @ r @ Z @ X2 ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % ring_distrib2
% 0.27/0.50 thf(fact_226_ring__distrib3,axiom,
% 0.27/0.50 ! [A2: a,B2: a,X2: a,Y2: a] :
% 0.27/0.50 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( tp_a_b @ r @ ( pop_a_Ring_ext_a_b @ r @ A2 @ B2 ) @ ( pop_a_Ring_ext_a_b @ r @ X2 @ Y2 ) )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ r @ ( pop_a_Ring_ext_a_b @ r @ ( pop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ A2 @ X2 ) @ ( tp_a_b @ r @ A2 @ Y2 ) ) @ ( tp_a_b @ r @ B2 @ X2 ) ) @ ( tp_a_b @ r @ B2 @ Y2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % ring_distrib3
% 0.27/0.50 thf(fact_227_nsMulDistrL,axiom,
% 0.27/0.50 ! [X2: a,Y2: a,N2: nat] :
% 0.27/0.50 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( tp_a_b @ r @ X2 @ ( nscal_a_b @ r @ Y2 @ N2 ) )
% 0.27/0.50 = ( nscal_a_b @ r @ ( tp_a_b @ r @ X2 @ Y2 ) @ N2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % nsMulDistrL
% 0.27/0.50 thf(fact_228_nsMulDistrR,axiom,
% 0.27/0.50 ! [X2: a,Y2: a,N2: nat] :
% 0.27/0.50 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( tp_a_b @ r @ ( nscal_a_b @ r @ Y2 @ N2 ) @ X2 )
% 0.27/0.50 = ( nscal_a_b @ r @ ( tp_a_b @ r @ Y2 @ X2 ) @ N2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % nsMulDistrR
% 0.27/0.50 thf(fact_229_nsDistrL,axiom,
% 0.27/0.50 ! [X2: a,Y2: a,N2: nat] :
% 0.27/0.50 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( pop_a_Ring_ext_a_b @ r @ ( nscal_a_b @ r @ X2 @ N2 ) @ ( nscal_a_b @ r @ Y2 @ N2 ) )
% 0.27/0.50 = ( nscal_a_b @ r @ ( pop_a_Ring_ext_a_b @ r @ X2 @ Y2 ) @ N2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % nsDistrL
% 0.27/0.50 thf(fact_230_Subring__tOp__ring__tOp,axiom,
% 0.27/0.50 ! [S: carrie1105631105xt_a_b,A2: a,B2: a] :
% 0.27/0.50 ( ( subring_a_b_b @ r @ S )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ S ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ S ) )
% 0.27/0.50 => ( ( tp_a_b @ S @ A2 @ B2 )
% 0.27/0.50 = ( tp_a_b @ r @ A2 @ B2 ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Subring_tOp_ring_tOp
% 0.27/0.50 thf(fact_231_Subring__pOp__ring__pOp,axiom,
% 0.27/0.50 ! [S: carrie1105631105xt_a_b,A2: a,B2: a] :
% 0.27/0.50 ( ( subring_a_b_b @ r @ S )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ S ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ S ) )
% 0.27/0.50 => ( ( pop_a_Ring_ext_a_b @ S @ A2 @ B2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ r @ A2 @ B2 ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Subring_pOp_ring_pOp
% 0.27/0.50 thf(fact_232_mul__closed__set__tOp__closed,axiom,
% 0.27/0.50 ! [S: set_a,S2: a,T: a] :
% 0.27/0.50 ( ( mul_closed_set_a_b @ r @ S )
% 0.27/0.50 => ( ( member_a @ S2 @ S )
% 0.27/0.50 => ( ( member_a @ T @ S )
% 0.27/0.50 => ( member_a @ ( tp_a_b @ r @ S2 @ T ) @ S ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mul_closed_set_tOp_closed
% 0.27/0.50 thf(fact_233_Ring_OSr__tOp__closed,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,S: set_a,X2: a,Y2: a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( sr_a_b2 @ R @ S )
% 0.27/0.50 => ( ( member_a @ X2 @ S )
% 0.27/0.50 => ( ( member_a @ Y2 @ S )
% 0.27/0.50 => ( member_a @ ( tp_a_b @ R @ X2 @ Y2 ) @ S ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.Sr_tOp_closed
% 0.27/0.50 thf(fact_234_Ring_OSr__pOp__closed,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,S: set_a,X2: a,Y2: a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( sr_a_b2 @ R @ S )
% 0.27/0.50 => ( ( member_a @ X2 @ S )
% 0.27/0.50 => ( ( member_a @ Y2 @ S )
% 0.27/0.50 => ( member_a @ ( pop_a_Ring_ext_a_b @ R @ X2 @ Y2 ) @ S ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.Sr_pOp_closed
% 0.27/0.50 thf(fact_235_Ring_Oring__distrib3,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,A2: a,B2: a,X2: a,Y2: a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( tp_a_b @ R @ ( pop_a_Ring_ext_a_b @ R @ A2 @ B2 ) @ ( pop_a_Ring_ext_a_b @ R @ X2 @ Y2 ) )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ R @ ( pop_a_Ring_ext_a_b @ R @ ( pop_a_Ring_ext_a_b @ R @ ( tp_a_b @ R @ A2 @ X2 ) @ ( tp_a_b @ R @ A2 @ Y2 ) ) @ ( tp_a_b @ R @ B2 @ X2 ) ) @ ( tp_a_b @ R @ B2 @ Y2 ) ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.ring_distrib3
% 0.27/0.50 thf(fact_236_Ring_Oring__distrib2,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,X2: a,Y2: a,Z: a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ Z @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( tp_a_b @ R @ ( pop_a_Ring_ext_a_b @ R @ Y2 @ Z ) @ X2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ R @ ( tp_a_b @ R @ Y2 @ X2 ) @ ( tp_a_b @ R @ Z @ X2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.ring_distrib2
% 0.27/0.50 thf(fact_237_Ring_Oring__distrib1,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,X2: a,Y2: a,Z: a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ Z @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( tp_a_b @ R @ X2 @ ( pop_a_Ring_ext_a_b @ R @ Y2 @ Z ) )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ R @ ( tp_a_b @ R @ X2 @ Y2 ) @ ( tp_a_b @ R @ X2 @ Z ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.ring_distrib1
% 0.27/0.50 thf(fact_238_Ring_Org__distrib,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( tp_a_b @ R @ A2 @ ( pop_a_Ring_ext_a_b @ R @ B2 @ C2 ) )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ R @ ( tp_a_b @ R @ A2 @ B2 ) @ ( tp_a_b @ R @ A2 @ C2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.rg_distrib
% 0.27/0.50 thf(fact_239_Ring_OnsMulDistrR,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,X2: a,Y2: a,N2: nat] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( tp_a_b @ R @ ( nscal_a_b @ R @ Y2 @ N2 ) @ X2 )
% 0.27/0.50 = ( nscal_a_b @ R @ ( tp_a_b @ R @ Y2 @ X2 ) @ N2 ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.nsMulDistrR
% 0.27/0.50 thf(fact_240_Ring_OnsMulDistrL,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,X2: a,Y2: a,N2: nat] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( tp_a_b @ R @ X2 @ ( nscal_a_b @ R @ Y2 @ N2 ) )
% 0.27/0.50 = ( nscal_a_b @ R @ ( tp_a_b @ R @ X2 @ Y2 ) @ N2 ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.nsMulDistrL
% 0.27/0.50 thf(fact_241_Ring_OnsDistrL,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,X2: a,Y2: a,N2: nat] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( pop_a_Ring_ext_a_b @ R @ ( nscal_a_b @ R @ X2 @ N2 ) @ ( nscal_a_b @ R @ Y2 @ N2 ) )
% 0.27/0.50 = ( nscal_a_b @ R @ ( pop_a_Ring_ext_a_b @ R @ X2 @ Y2 ) @ N2 ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.nsDistrL
% 0.27/0.50 thf(fact_242_Ring_OSr__ring,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,S: set_a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( sr_a_b2 @ R @ S )
% 0.27/0.50 => ( ring_a_b @ ( sr_a_b @ R @ S ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.Sr_ring
% 0.27/0.50 thf(fact_243_aGroup_Oag__pOp__commute,axiom,
% 0.27/0.50 ! [A: carrie1105631105xt_a_b,X2: a,Y2: a] :
% 0.27/0.50 ( ( aGroup2097840802xt_a_b @ A )
% 0.27/0.50 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( pop_a_Ring_ext_a_b @ A @ X2 @ Y2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ A @ Y2 @ X2 ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.ag_pOp_commute
% 0.27/0.50 thf(fact_244_aGroup_Oag__pOp__commute,axiom,
% 0.27/0.50 ! [A: carrie1954260683t_unit,X2: c > d,Y2: c > d] :
% 0.27/0.50 ( ( aGroup2131905830t_unit @ A )
% 0.27/0.50 => ( ( member_c_d @ X2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ Y2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( pop_c_449750139t_unit @ A @ X2 @ Y2 )
% 0.27/0.50 = ( pop_c_449750139t_unit @ A @ Y2 @ X2 ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.ag_pOp_commute
% 0.27/0.50 thf(fact_245_aGroup_Oag__add__commute,axiom,
% 0.27/0.50 ! [A: carrie1105631105xt_a_b,A2: a,B2: a] :
% 0.27/0.50 ( ( aGroup2097840802xt_a_b @ A )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( pop_a_Ring_ext_a_b @ A @ A2 @ B2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ A @ B2 @ A2 ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.ag_add_commute
% 0.27/0.50 thf(fact_246_aGroup_Oag__add__commute,axiom,
% 0.27/0.50 ! [A: carrie1954260683t_unit,A2: c > d,B2: c > d] :
% 0.27/0.50 ( ( aGroup2131905830t_unit @ A )
% 0.27/0.50 => ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( pop_c_449750139t_unit @ A @ A2 @ B2 )
% 0.27/0.50 = ( pop_c_449750139t_unit @ A @ B2 @ A2 ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.ag_add_commute
% 0.27/0.50 thf(fact_247_aGroup_OpOp__assocTr43,axiom,
% 0.27/0.50 ! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a,D: a] :
% 0.27/0.50 ( ( aGroup2097840802xt_a_b @ A )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ D @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ B2 ) @ ( pop_a_Ring_ext_a_b @ A @ C2 @ D ) )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ ( pop_a_Ring_ext_a_b @ A @ B2 @ C2 ) ) @ D ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.pOp_assocTr43
% 0.27/0.50 thf(fact_248_aGroup_OpOp__assocTr43,axiom,
% 0.27/0.50 ! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d,D: c > d] :
% 0.27/0.50 ( ( aGroup2131905830t_unit @ A )
% 0.27/0.50 => ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ D @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ B2 ) @ ( pop_c_449750139t_unit @ A @ C2 @ D ) )
% 0.27/0.50 = ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ ( pop_c_449750139t_unit @ A @ B2 @ C2 ) ) @ D ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.pOp_assocTr43
% 0.27/0.50 thf(fact_249_aGroup_OpOp__assocTr42,axiom,
% 0.27/0.50 ! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a,D: a] :
% 0.27/0.50 ( ( aGroup2097840802xt_a_b @ A )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ D @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ B2 ) @ C2 ) @ D )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ ( pop_a_Ring_ext_a_b @ A @ B2 @ C2 ) ) @ D ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.pOp_assocTr42
% 0.27/0.50 thf(fact_250_aGroup_OpOp__assocTr42,axiom,
% 0.27/0.50 ! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d,D: c > d] :
% 0.27/0.50 ( ( aGroup2131905830t_unit @ A )
% 0.27/0.50 => ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ D @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ B2 ) @ C2 ) @ D )
% 0.27/0.50 = ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ ( pop_c_449750139t_unit @ A @ B2 @ C2 ) ) @ D ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.pOp_assocTr42
% 0.27/0.50 thf(fact_251_aGroup_OpOp__assocTr41,axiom,
% 0.27/0.50 ! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a,D: a] :
% 0.27/0.50 ( ( aGroup2097840802xt_a_b @ A )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ D @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ B2 ) @ C2 ) @ D )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ B2 ) @ ( pop_a_Ring_ext_a_b @ A @ C2 @ D ) ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.pOp_assocTr41
% 0.27/0.50 thf(fact_252_aGroup_OpOp__assocTr41,axiom,
% 0.27/0.50 ! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d,D: c > d] :
% 0.27/0.50 ( ( aGroup2131905830t_unit @ A )
% 0.27/0.50 => ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ D @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ B2 ) @ C2 ) @ D )
% 0.27/0.50 = ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ B2 ) @ ( pop_c_449750139t_unit @ A @ C2 @ D ) ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.pOp_assocTr41
% 0.27/0.50 thf(fact_253_aGroup_Oag__pOp__closed,axiom,
% 0.27/0.50 ! [A: carrie1105631105xt_a_b,X2: a,Y2: a] :
% 0.27/0.50 ( ( aGroup2097840802xt_a_b @ A )
% 0.27/0.50 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( member_a @ ( pop_a_Ring_ext_a_b @ A @ X2 @ Y2 ) @ ( carrie867757212xt_a_b @ A ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.ag_pOp_closed
% 0.27/0.50 thf(fact_254_aGroup_Oag__pOp__closed,axiom,
% 0.27/0.50 ! [A: carrie1954260683t_unit,X2: c > d,Y2: c > d] :
% 0.27/0.50 ( ( aGroup2131905830t_unit @ A )
% 0.27/0.50 => ( ( member_c_d @ X2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ Y2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( member_c_d @ ( pop_c_449750139t_unit @ A @ X2 @ Y2 ) @ ( carrie20288686t_unit @ A ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.ag_pOp_closed
% 0.27/0.50 thf(fact_255_aGroup_OpOp__cancel__r,axiom,
% 0.27/0.50 ! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
% 0.27/0.50 ( ( aGroup2097840802xt_a_b @ A )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( ( pop_a_Ring_ext_a_b @ A @ A2 @ C2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ A @ B2 @ C2 ) )
% 0.27/0.50 => ( A2 = B2 ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.pOp_cancel_r
% 0.27/0.50 thf(fact_256_aGroup_OpOp__cancel__r,axiom,
% 0.27/0.50 ! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d] :
% 0.27/0.50 ( ( aGroup2131905830t_unit @ A )
% 0.27/0.50 => ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( ( pop_c_449750139t_unit @ A @ A2 @ C2 )
% 0.27/0.50 = ( pop_c_449750139t_unit @ A @ B2 @ C2 ) )
% 0.27/0.50 => ( A2 = B2 ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.pOp_cancel_r
% 0.27/0.50 thf(fact_257_aGroup_OpOp__cancel__l,axiom,
% 0.27/0.50 ! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
% 0.27/0.50 ( ( aGroup2097840802xt_a_b @ A )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( ( pop_a_Ring_ext_a_b @ A @ C2 @ A2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ A @ C2 @ B2 ) )
% 0.27/0.50 => ( A2 = B2 ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.pOp_cancel_l
% 0.27/0.50 thf(fact_258_aGroup_OpOp__cancel__l,axiom,
% 0.27/0.50 ! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d] :
% 0.27/0.50 ( ( aGroup2131905830t_unit @ A )
% 0.27/0.50 => ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( ( pop_c_449750139t_unit @ A @ C2 @ A2 )
% 0.27/0.50 = ( pop_c_449750139t_unit @ A @ C2 @ B2 ) )
% 0.27/0.50 => ( A2 = B2 ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.pOp_cancel_l
% 0.27/0.50 thf(fact_259_aGroup_Oag__pOp__assoc,axiom,
% 0.27/0.50 ! [A: carrie1105631105xt_a_b,X2: a,Y2: a,Z: a] :
% 0.27/0.50 ( ( aGroup2097840802xt_a_b @ A )
% 0.27/0.50 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ Z @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ X2 @ Y2 ) @ Z )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ A @ X2 @ ( pop_a_Ring_ext_a_b @ A @ Y2 @ Z ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.ag_pOp_assoc
% 0.27/0.50 thf(fact_260_aGroup_Oag__pOp__assoc,axiom,
% 0.27/0.50 ! [A: carrie1954260683t_unit,X2: c > d,Y2: c > d,Z: c > d] :
% 0.27/0.50 ( ( aGroup2131905830t_unit @ A )
% 0.27/0.50 => ( ( member_c_d @ X2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ Y2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ Z @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ X2 @ Y2 ) @ Z )
% 0.27/0.50 = ( pop_c_449750139t_unit @ A @ X2 @ ( pop_c_449750139t_unit @ A @ Y2 @ Z ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.ag_pOp_assoc
% 0.27/0.50 thf(fact_261_aGroup_Oag__pOp__add__r,axiom,
% 0.27/0.50 ! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
% 0.27/0.50 ( ( aGroup2097840802xt_a_b @ A )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( A2 = B2 )
% 0.27/0.50 => ( ( pop_a_Ring_ext_a_b @ A @ A2 @ C2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ A @ B2 @ C2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.ag_pOp_add_r
% 0.27/0.50 thf(fact_262_aGroup_Oag__pOp__add__r,axiom,
% 0.27/0.50 ! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d] :
% 0.27/0.50 ( ( aGroup2131905830t_unit @ A )
% 0.27/0.50 => ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( A2 = B2 )
% 0.27/0.50 => ( ( pop_c_449750139t_unit @ A @ A2 @ C2 )
% 0.27/0.50 = ( pop_c_449750139t_unit @ A @ B2 @ C2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.ag_pOp_add_r
% 0.27/0.50 thf(fact_263_aGroup_Oag__pOp__add__l,axiom,
% 0.27/0.50 ! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
% 0.27/0.50 ( ( aGroup2097840802xt_a_b @ A )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( A2 = B2 )
% 0.27/0.50 => ( ( pop_a_Ring_ext_a_b @ A @ C2 @ A2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ A @ C2 @ B2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.ag_pOp_add_l
% 0.27/0.50 thf(fact_264_aGroup_Oag__pOp__add__l,axiom,
% 0.27/0.50 ! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d] :
% 0.27/0.50 ( ( aGroup2131905830t_unit @ A )
% 0.27/0.50 => ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( A2 = B2 )
% 0.27/0.50 => ( ( pop_c_449750139t_unit @ A @ C2 @ A2 )
% 0.27/0.50 = ( pop_c_449750139t_unit @ A @ C2 @ B2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.ag_pOp_add_l
% 0.27/0.50 thf(fact_265_aGroup_OgEQAddcross,axiom,
% 0.27/0.50 ! [A: carrie1105631105xt_a_b,L1: a,L2: a,R1: a,R2: a] :
% 0.27/0.50 ( ( aGroup2097840802xt_a_b @ A )
% 0.27/0.50 => ( ( member_a @ L1 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ L2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ R1 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ R1 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( L1 = R2 )
% 0.27/0.50 => ( ( L2 = R1 )
% 0.27/0.50 => ( ( pop_a_Ring_ext_a_b @ A @ L1 @ L2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ A @ R1 @ R2 ) ) ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.gEQAddcross
% 0.27/0.50 thf(fact_266_aGroup_OgEQAddcross,axiom,
% 0.27/0.50 ! [A: carrie1954260683t_unit,L1: c > d,L2: c > d,R1: c > d,R2: c > d] :
% 0.27/0.50 ( ( aGroup2131905830t_unit @ A )
% 0.27/0.50 => ( ( member_c_d @ L1 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ L2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ R1 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ R1 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( L1 = R2 )
% 0.27/0.50 => ( ( L2 = R1 )
% 0.27/0.50 => ( ( pop_c_449750139t_unit @ A @ L1 @ L2 )
% 0.27/0.50 = ( pop_c_449750139t_unit @ A @ R1 @ R2 ) ) ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.gEQAddcross
% 0.27/0.50 thf(fact_267_aGroup_Oag__add4__rel,axiom,
% 0.27/0.50 ! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a,D: a] :
% 0.27/0.50 ( ( aGroup2097840802xt_a_b @ A )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ D @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ B2 ) @ ( pop_a_Ring_ext_a_b @ A @ C2 @ D ) )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ C2 ) @ ( pop_a_Ring_ext_a_b @ A @ B2 @ D ) ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.ag_add4_rel
% 0.27/0.50 thf(fact_268_aGroup_Oag__add4__rel,axiom,
% 0.27/0.50 ! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d,D: c > d] :
% 0.27/0.50 ( ( aGroup2131905830t_unit @ A )
% 0.27/0.50 => ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ D @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ B2 ) @ ( pop_c_449750139t_unit @ A @ C2 @ D ) )
% 0.27/0.50 = ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ C2 ) @ ( pop_c_449750139t_unit @ A @ B2 @ D ) ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.ag_add4_rel
% 0.27/0.50 thf(fact_269_aGroup_Oaassoc,axiom,
% 0.27/0.50 ! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
% 0.27/0.50 ( ( aGroup2097840802xt_a_b @ A )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
% 0.27/0.50 => ( ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ B2 ) @ C2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ A @ A2 @ ( pop_a_Ring_ext_a_b @ A @ B2 @ C2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.aassoc
% 0.27/0.50 thf(fact_270_aGroup_Oaassoc,axiom,
% 0.27/0.50 ! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d] :
% 0.27/0.50 ( ( aGroup2131905830t_unit @ A )
% 0.27/0.50 => ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
% 0.27/0.50 => ( ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ B2 ) @ C2 )
% 0.27/0.50 = ( pop_c_449750139t_unit @ A @ A2 @ ( pop_c_449750139t_unit @ A @ B2 @ C2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.aassoc
% 0.27/0.50 thf(fact_271_Ring_Oring__tOp__commute,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,X2: a,Y2: a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( tp_a_b @ R @ X2 @ Y2 )
% 0.27/0.50 = ( tp_a_b @ R @ Y2 @ X2 ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.ring_tOp_commute
% 0.27/0.50 thf(fact_272_Ring_Oring__tOp__closed,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,X2: a,Y2: a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( member_a @ ( tp_a_b @ R @ X2 @ Y2 ) @ ( carrie867757212xt_a_b @ R ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.ring_tOp_closed
% 0.27/0.50 thf(fact_273_Ring_Oring__tOp__assoc,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,X2: a,Y2: a,Z: a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ Z @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( tp_a_b @ R @ ( tp_a_b @ R @ X2 @ Y2 ) @ Z )
% 0.27/0.50 = ( tp_a_b @ R @ X2 @ ( tp_a_b @ R @ Y2 @ Z ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.ring_tOp_assoc
% 0.27/0.50 thf(fact_274_Ring_Oring__tOp__rel,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,X2: a,Xa: a,Y2: a,Ya: a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ Xa @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ Ya @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( tp_a_b @ R @ ( tp_a_b @ R @ X2 @ Xa ) @ ( tp_a_b @ R @ Y2 @ Ya ) )
% 0.27/0.50 = ( tp_a_b @ R @ ( tp_a_b @ R @ X2 @ Y2 ) @ ( tp_a_b @ R @ Xa @ Ya ) ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.ring_tOp_rel
% 0.27/0.50 thf(fact_275_Ring_Otp__commute,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,A2: a,B2: a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( tp_a_b @ R @ A2 @ B2 )
% 0.27/0.50 = ( tp_a_b @ R @ B2 @ A2 ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.tp_commute
% 0.27/0.50 thf(fact_276_Ring_Otp__assoc,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( tp_a_b @ R @ ( tp_a_b @ R @ A2 @ B2 ) @ C2 )
% 0.27/0.50 = ( tp_a_b @ R @ A2 @ ( tp_a_b @ R @ B2 @ C2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.tp_assoc
% 0.27/0.50 thf(fact_277_Ring_OrEQMulR,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,X2: a,Y2: a,Z: a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ Z @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( X2 = Y2 )
% 0.27/0.50 => ( ( tp_a_b @ R @ X2 @ Z )
% 0.27/0.50 = ( tp_a_b @ R @ Y2 @ Z ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.rEQMulR
% 0.27/0.50 thf(fact_278_Ring_OrMulLC,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,X2: a,Y2: a,Z: a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ Z @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( tp_a_b @ R @ X2 @ ( tp_a_b @ R @ Y2 @ Z ) )
% 0.27/0.50 = ( tp_a_b @ R @ Y2 @ ( tp_a_b @ R @ X2 @ Z ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.rMulLC
% 0.27/0.50 thf(fact_279_Ring_Opop__commute,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,A2: a,B2: a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( pop_a_Ring_ext_a_b @ R @ A2 @ B2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ R @ B2 @ A2 ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.pop_commute
% 0.27/0.50 thf(fact_280_Ring_Opop__aassoc,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( pop_a_Ring_ext_a_b @ R @ ( pop_a_Ring_ext_a_b @ R @ A2 @ B2 ) @ C2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ R @ A2 @ ( pop_a_Ring_ext_a_b @ R @ B2 @ C2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.pop_aassoc
% 0.27/0.50 thf(fact_281_Ring_OnsEqElm,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,X2: a,Y2: a,N2: nat] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( X2 = Y2 )
% 0.27/0.50 => ( ( nscal_a_b @ R @ X2 @ N2 )
% 0.27/0.50 = ( nscal_a_b @ R @ Y2 @ N2 ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.nsEqElm
% 0.27/0.50 thf(fact_282_Ring_OnsClose,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,X2: a,N2: nat] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( member_a @ ( nscal_a_b @ R @ X2 @ N2 ) @ ( carrie867757212xt_a_b @ R ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.nsClose
% 0.27/0.50 thf(fact_283_prod__pOp__mem__i,axiom,
% 0.27/0.50 ! [I: set_d,A: d > carrie1105631105xt_a_b,X4: d > a,Y: d > a,I2: d] :
% 0.27/0.50 ( ! [X: d] :
% 0.27/0.50 ( ( member_d @ X @ I )
% 0.27/0.50 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 0.27/0.50 => ( ( member_d_a @ X4 @ ( carr_p297433675xt_a_b @ I @ A ) )
% 0.27/0.50 => ( ( member_d_a @ Y @ ( carr_p297433675xt_a_b @ I @ A ) )
% 0.27/0.50 => ( ( member_d @ I2 @ I )
% 0.27/0.50 => ( ( prod_p711161210xt_a_b @ I @ A @ X4 @ Y @ I2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % prod_pOp_mem_i
% 0.27/0.50 thf(fact_284_prod__pOp__mem__i,axiom,
% 0.27/0.50 ! [I: set_c,A: c > carrie1105631105xt_a_b,X4: c > a,Y: c > a,I2: c] :
% 0.27/0.50 ( ! [X: c] :
% 0.27/0.50 ( ( member_c @ X @ I )
% 0.27/0.50 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 0.27/0.50 => ( ( member_c_a @ X4 @ ( carr_p295160588xt_a_b @ I @ A ) )
% 0.27/0.50 => ( ( member_c_a @ Y @ ( carr_p295160588xt_a_b @ I @ A ) )
% 0.27/0.50 => ( ( member_c @ I2 @ I )
% 0.27/0.50 => ( ( prod_p708888123xt_a_b @ I @ A @ X4 @ Y @ I2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % prod_pOp_mem_i
% 0.27/0.50 thf(fact_285_prod__pOp__mem__i,axiom,
% 0.27/0.50 ! [I: set_a,A: a > carrie1105631105xt_a_b,X4: a > a,Y: a > a,I2: a] :
% 0.27/0.50 ( ! [X: a] :
% 0.27/0.50 ( ( member_a @ X @ I )
% 0.27/0.50 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 0.27/0.50 => ( ( member_a_a @ X4 @ ( carr_p290614414xt_a_b @ I @ A ) )
% 0.27/0.50 => ( ( member_a_a @ Y @ ( carr_p290614414xt_a_b @ I @ A ) )
% 0.27/0.50 => ( ( member_a @ I2 @ I )
% 0.27/0.50 => ( ( prod_p704341949xt_a_b @ I @ A @ X4 @ Y @ I2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % prod_pOp_mem_i
% 0.27/0.50 thf(fact_286_prod__pOp__mem__i,axiom,
% 0.27/0.50 ! [I: set_c,A: c > carrie1948989069_d_a_e,X4: c > d,Y: c > d,I2: c] :
% 0.27/0.50 ( ! [X: c] :
% 0.27/0.50 ( ( member_c @ X @ I )
% 0.27/0.50 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 0.27/0.50 => ( ( member_c_d @ X4 @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 0.27/0.50 => ( ( member_c_d @ Y @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 0.27/0.50 => ( ( member_c @ I2 @ I )
% 0.27/0.50 => ( ( prod_p131511076_d_a_e @ I @ A @ X4 @ Y @ I2 )
% 0.27/0.50 = ( pop_d_261713874_d_a_e @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % prod_pOp_mem_i
% 0.27/0.50 thf(fact_287_prod__pOp__mem__i,axiom,
% 0.27/0.50 ! [I: set_c_d,A: ( c > d ) > carrie1105631105xt_a_b,X4: ( c > d ) > a,Y: ( c > d ) > a,I2: c > d] :
% 0.27/0.50 ( ! [X: c > d] :
% 0.27/0.50 ( ( member_c_d @ X @ I )
% 0.27/0.50 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 0.27/0.50 => ( ( member_c_d_a @ X4 @ ( carr_p996491646xt_a_b @ I @ A ) )
% 0.27/0.50 => ( ( member_c_d_a @ Y @ ( carr_p996491646xt_a_b @ I @ A ) )
% 0.27/0.50 => ( ( member_c_d @ I2 @ I )
% 0.27/0.50 => ( ( prod_p1466615439xt_a_b @ I @ A @ X4 @ Y @ I2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % prod_pOp_mem_i
% 0.27/0.50 thf(fact_288_prod__pOp__mem__i,axiom,
% 0.27/0.50 ! [I: set_a_a,A: ( a > a ) > carrie1105631105xt_a_b,X4: ( a > a ) > a,Y: ( a > a ) > a,I2: a > a] :
% 0.27/0.50 ( ! [X: a > a] :
% 0.27/0.50 ( ( member_a_a @ X @ I )
% 0.27/0.50 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 0.27/0.50 => ( ( member_a_a_a @ X4 @ ( carr_p740328767xt_a_b @ I @ A ) )
% 0.27/0.50 => ( ( member_a_a_a @ Y @ ( carr_p740328767xt_a_b @ I @ A ) )
% 0.27/0.50 => ( ( member_a_a @ I2 @ I )
% 0.27/0.50 => ( ( prod_p1210452560xt_a_b @ I @ A @ X4 @ Y @ I2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % prod_pOp_mem_i
% 0.27/0.50 thf(fact_289_prod__pOp__mem__i,axiom,
% 0.27/0.50 ! [I: set_a_a_a,A: ( a > a > a ) > carrie1105631105xt_a_b,X4: ( a > a > a ) > a,Y: ( a > a > a ) > a,I2: a > a > a] :
% 0.27/0.50 ( ! [X: a > a > a] :
% 0.27/0.50 ( ( member_a_a_a2 @ X @ I )
% 0.27/0.50 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 0.27/0.50 => ( ( member_a_a_a_a @ X4 @ ( carr_p1481099064xt_a_b @ I @ A ) )
% 0.27/0.50 => ( ( member_a_a_a_a @ Y @ ( carr_p1481099064xt_a_b @ I @ A ) )
% 0.27/0.50 => ( ( member_a_a_a2 @ I2 @ I )
% 0.27/0.50 => ( ( prod_p1460711911xt_a_b @ I @ A @ X4 @ Y @ I2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % prod_pOp_mem_i
% 0.27/0.50 thf(fact_290_prod__pOp__mem__i,axiom,
% 0.27/0.50 ! [I: set_c_d_c_d,A: ( ( c > d ) > c > d ) > carrie1105631105xt_a_b,X4: ( ( c > d ) > c > d ) > a,Y: ( ( c > d ) > c > d ) > a,I2: ( c > d ) > c > d] :
% 0.27/0.50 ( ! [X: ( c > d ) > c > d] :
% 0.27/0.50 ( ( member_c_d_c_d @ X @ I )
% 0.27/0.50 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 0.27/0.50 => ( ( member_c_d_c_d_a @ X4 @ ( carr_p1617094073xt_a_b @ I @ A ) )
% 0.27/0.50 => ( ( member_c_d_c_d_a @ Y @ ( carr_p1617094073xt_a_b @ I @ A ) )
% 0.27/0.50 => ( ( member_c_d_c_d @ I2 @ I )
% 0.27/0.50 => ( ( prod_p1959959370xt_a_b @ I @ A @ X4 @ Y @ I2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % prod_pOp_mem_i
% 0.27/0.50 thf(fact_291_prod__pOp__mem__i,axiom,
% 0.27/0.50 ! [I: set_a_a_c_d,A: ( a > a > c > d ) > carrie1105631105xt_a_b,X4: ( a > a > c > d ) > a,Y: ( a > a > c > d ) > a,I2: a > a > c > d] :
% 0.27/0.50 ( ! [X: a > a > c > d] :
% 0.27/0.50 ( ( member_a_a_c_d2 @ X @ I )
% 0.27/0.50 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 0.27/0.50 => ( ( member_a_a_c_d_a @ X4 @ ( carr_p109409832xt_a_b @ I @ A ) )
% 0.27/0.50 => ( ( member_a_a_c_d_a @ Y @ ( carr_p109409832xt_a_b @ I @ A ) )
% 0.27/0.50 => ( ( member_a_a_c_d2 @ I2 @ I )
% 0.27/0.50 => ( ( prod_p452275129xt_a_b @ I @ A @ X4 @ Y @ I2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % prod_pOp_mem_i
% 0.27/0.50 thf(fact_292_prod__pOp__mem__i,axiom,
% 0.27/0.50 ! [I: set_a_a_a_a,A: ( a > a > a > a ) > carrie1105631105xt_a_b,X4: ( a > a > a > a ) > a,Y: ( a > a > a > a ) > a,I2: a > a > a > a] :
% 0.27/0.50 ( ! [X: a > a > a > a] :
% 0.27/0.50 ( ( member_a_a_a_a4 @ X @ I )
% 0.27/0.50 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 0.27/0.50 => ( ( member_a_a_a_a_a @ X4 @ ( carr_p1850824937xt_a_b @ I @ A ) )
% 0.27/0.50 => ( ( member_a_a_a_a_a @ Y @ ( carr_p1850824937xt_a_b @ I @ A ) )
% 0.27/0.50 => ( ( member_a_a_a_a4 @ I2 @ I )
% 0.27/0.50 => ( ( prod_p46206586xt_a_b @ I @ A @ X4 @ Y @ I2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % prod_pOp_mem_i
% 0.27/0.50 thf(fact_293_dsum__pOp__mem,axiom,
% 0.27/0.50 ! [I: set_a,A: a > carrie1105631105xt_a_b,X4: a > a,Y: a > a] :
% 0.27/0.50 ( ! [X: a] :
% 0.27/0.50 ( ( member_a @ X @ I )
% 0.27/0.50 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 0.27/0.50 => ( ( member_a_a @ X4 @ ( carr_d267359390xt_a_b @ I @ A ) )
% 0.27/0.50 => ( ( member_a_a @ Y @ ( carr_d267359390xt_a_b @ I @ A ) )
% 0.27/0.50 => ( member_a_a @ ( prod_p704341949xt_a_b @ I @ A @ X4 @ Y ) @ ( carr_d267359390xt_a_b @ I @ A ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % dsum_pOp_mem
% 0.27/0.50 thf(fact_294_dsum__pOp__mem,axiom,
% 0.27/0.50 ! [I: set_c,A: c > carrie1948989069_d_a_e,X4: c > d,Y: c > d] :
% 0.27/0.50 ( ! [X: c] :
% 0.27/0.50 ( ( member_c @ X @ I )
% 0.27/0.50 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 0.27/0.50 => ( ( member_c_d @ X4 @ ( carr_d56066307_d_a_e @ I @ A ) )
% 0.27/0.50 => ( ( member_c_d @ Y @ ( carr_d56066307_d_a_e @ I @ A ) )
% 0.27/0.50 => ( member_c_d @ ( prod_p131511076_d_a_e @ I @ A @ X4 @ Y ) @ ( carr_d56066307_d_a_e @ I @ A ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % dsum_pOp_mem
% 0.27/0.50 thf(fact_295_dsum__iOp__mem,axiom,
% 0.27/0.50 ! [I: set_a,A: a > carrie1105631105xt_a_b,X4: a > a] :
% 0.27/0.50 ( ! [X: a] :
% 0.27/0.50 ( ( member_a @ X @ I )
% 0.27/0.50 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 0.27/0.50 => ( ( member_a_a @ X4 @ ( carr_d267359390xt_a_b @ I @ A ) )
% 0.27/0.50 => ( member_a_a @ ( prod_m2022989242xt_a_b @ I @ A @ X4 ) @ ( carr_d267359390xt_a_b @ I @ A ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % dsum_iOp_mem
% 0.27/0.50 thf(fact_296_dsum__iOp__mem,axiom,
% 0.27/0.50 ! [I: set_c,A: c > carrie1948989069_d_a_e,X4: c > d] :
% 0.27/0.50 ( ! [X: c] :
% 0.27/0.50 ( ( member_c @ X @ I )
% 0.27/0.50 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 0.27/0.50 => ( ( member_c_d @ X4 @ ( carr_d56066307_d_a_e @ I @ A ) )
% 0.27/0.50 => ( member_c_d @ ( prod_m854557927_d_a_e @ I @ A @ X4 ) @ ( carr_d56066307_d_a_e @ I @ A ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % dsum_iOp_mem
% 0.27/0.50 thf(fact_297_dsum__zero__func,axiom,
% 0.27/0.50 ! [I: set_a,A: a > carrie1105631105xt_a_b] :
% 0.27/0.50 ( ! [X: a] :
% 0.27/0.50 ( ( member_a @ X @ I )
% 0.27/0.50 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 0.27/0.50 => ( member_a_a @ ( prod_z851058290xt_a_b @ I @ A ) @ ( carr_d267359390xt_a_b @ I @ A ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % dsum_zero_func
% 0.27/0.50 thf(fact_298_dsum__zero__func,axiom,
% 0.27/0.50 ! [I: set_c,A: c > carrie1948989069_d_a_e] :
% 0.27/0.50 ( ! [X: c] :
% 0.27/0.50 ( ( member_c @ X @ I )
% 0.27/0.50 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 0.27/0.50 => ( member_c_d @ ( prod_z1214987823_d_a_e @ I @ A ) @ ( carr_d56066307_d_a_e @ I @ A ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % dsum_zero_func
% 0.27/0.50 thf(fact_299_aGroup_Opop__closed,axiom,
% 0.27/0.50 ! [A: carrie1105631105xt_a_b] :
% 0.27/0.50 ( ( aGroup2097840802xt_a_b @ A )
% 0.27/0.50 => ( member_a_a_a2 @ ( pop_a_Ring_ext_a_b @ A )
% 0.27/0.50 @ ( pi_a_a_a2 @ ( carrie867757212xt_a_b @ A )
% 0.27/0.50 @ ^ [Uu: a] :
% 0.27/0.50 ( pi_a_a @ ( carrie867757212xt_a_b @ A )
% 0.27/0.50 @ ^ [Uv: a] : ( carrie867757212xt_a_b @ A ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.pop_closed
% 0.27/0.50 thf(fact_300_aGroup_Opop__closed,axiom,
% 0.27/0.50 ! [A: carrie1954260683t_unit] :
% 0.27/0.50 ( ( aGroup2131905830t_unit @ A )
% 0.27/0.50 => ( member_c_d_c_d_c_d @ ( pop_c_449750139t_unit @ A )
% 0.27/0.50 @ ( pi_c_d_c_d_c_d @ ( carrie20288686t_unit @ A )
% 0.27/0.50 @ ^ [Uu: c > d] :
% 0.27/0.50 ( pi_c_d_c_d @ ( carrie20288686t_unit @ A )
% 0.27/0.50 @ ^ [Uv: c > d] : ( carrie20288686t_unit @ A ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % aGroup.pop_closed
% 0.27/0.50 thf(fact_301_Ring_OJ__rad__mem,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,X2: a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( member_a @ X2 @ ( j_rad_a_b @ R ) )
% 0.27/0.50 => ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.J_rad_mem
% 0.27/0.50 thf(fact_302_Ring_Otp__closed,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( member_a_a_a2 @ ( tp_a_b @ R )
% 0.27/0.50 @ ( pi_a_a_a2 @ ( carrie867757212xt_a_b @ R )
% 0.27/0.50 @ ^ [Uu: a] :
% 0.27/0.50 ( pi_a_a @ ( carrie867757212xt_a_b @ R )
% 0.27/0.50 @ ^ [Uv: a] : ( carrie867757212xt_a_b @ R ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.tp_closed
% 0.27/0.50 thf(fact_303_Ring_Opop__closed,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( member_a_a_a2 @ ( pop_a_Ring_ext_a_b @ R )
% 0.27/0.50 @ ( pi_a_a_a2 @ ( carrie867757212xt_a_b @ R )
% 0.27/0.50 @ ^ [Uu: a] :
% 0.27/0.50 ( pi_a_a @ ( carrie867757212xt_a_b @ R )
% 0.27/0.50 @ ^ [Uv: a] : ( carrie867757212xt_a_b @ R ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.pop_closed
% 0.27/0.50 thf(fact_304_Ring_OprodM__sprod__val,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,I: set_c_d_c_d,M: ( ( c > d ) > c > d ) > carrie1948989069_d_a_e,A2: a,M2: ( ( c > d ) > c > d ) > d,J: ( c > d ) > c > d] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ! [X: ( c > d ) > c > d] :
% 0.27/0.50 ( ( member_c_d_c_d @ X @ I )
% 0.27/0.50 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_c_d_c_d_d @ M2 @ ( carr_p428609062_d_a_e @ I @ M ) )
% 0.27/0.50 => ( ( member_c_d_c_d @ J @ I )
% 0.27/0.50 => ( ( algebr828196337_d_d_e @ R @ I @ M @ A2 @ M2 @ J )
% 0.27/0.50 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.prodM_sprod_val
% 0.27/0.50 thf(fact_305_Ring_OprodM__sprod__val,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,I: set_c_d,M: ( c > d ) > carrie1948989069_d_a_e,A2: a,M2: ( c > d ) > d,J: c > d] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ! [X: c > d] :
% 0.27/0.50 ( ( member_c_d @ X @ I )
% 0.27/0.50 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_c_d_d @ M2 @ ( carr_p124510241_d_a_e @ I @ M ) )
% 0.27/0.50 => ( ( member_c_d @ J @ I )
% 0.27/0.50 => ( ( algebr924632044_d_d_e @ R @ I @ M @ A2 @ M2 @ J )
% 0.27/0.50 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.prodM_sprod_val
% 0.27/0.50 thf(fact_306_Ring_OprodM__sprod__val,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,I: set_a_a_a,M: ( a > a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > a ) > d,J: a > a > a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ! [X: a > a > a] :
% 0.27/0.50 ( ( member_a_a_a2 @ X @ I )
% 0.27/0.50 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a_a_a_d @ M2 @ ( carr_p1364179431_d_a_e @ I @ M ) )
% 0.27/0.50 => ( ( member_a_a_a2 @ J @ I )
% 0.27/0.50 => ( ( algebr1655014672_a_d_e @ R @ I @ M @ A2 @ M2 @ J )
% 0.27/0.50 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.prodM_sprod_val
% 0.27/0.50 thf(fact_307_Ring_OprodM__sprod__val,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,I: set_a_a,M: ( a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a ) > d,J: a > a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ! [X: a > a] :
% 0.27/0.50 ( ( member_a_a @ X @ I )
% 0.27/0.50 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a_a_d @ M2 @ ( carr_p1646427936_d_a_e @ I @ M ) )
% 0.27/0.50 => ( ( member_a_a @ J @ I )
% 0.27/0.50 => ( ( algebr137561451_a_d_e @ R @ I @ M @ A2 @ M2 @ J )
% 0.27/0.50 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.prodM_sprod_val
% 0.27/0.50 thf(fact_308_Ring_OprodM__sprod__val,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,I: set_a,M: a > carrie1948989069_d_a_e,A2: a,M2: a > d,J: a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ! [X: a] :
% 0.27/0.50 ( ( member_a @ X @ I )
% 0.27/0.50 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a_d @ M2 @ ( carr_p1084104977_d_a_e @ I @ M ) )
% 0.27/0.50 => ( ( member_a @ J @ I )
% 0.27/0.50 => ( ( algebr2109842362_a_d_e @ R @ I @ M @ A2 @ M2 @ J )
% 0.27/0.50 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.prodM_sprod_val
% 0.27/0.50 thf(fact_309_Ring_OprodM__sprod__val,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,I: set_a_a_c_d,M: ( a > a > c > d ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > c > d ) > d,J: a > a > c > d] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ! [X: a > a > c > d] :
% 0.27/0.50 ( ( member_a_a_c_d2 @ X @ I )
% 0.27/0.50 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a_a_c_d_d @ M2 @ ( carr_p577380087_d_a_e @ I @ M ) )
% 0.27/0.50 => ( ( member_a_a_c_d2 @ J @ I )
% 0.27/0.50 => ( ( algebr1620716610_d_d_e @ R @ I @ M @ A2 @ M2 @ J )
% 0.27/0.50 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.prodM_sprod_val
% 0.27/0.50 thf(fact_310_Ring_OprodM__sprod__val,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,I: set_a_a_a_a_a,M: ( a > a > a > a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > a > a > a ) > d,J: a > a > a > a > a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ! [X: a > a > a > a > a] :
% 0.27/0.50 ( ( member_a_a_a_a_a3 @ X @ I )
% 0.27/0.50 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a_a_a_a_a_d @ M2 @ ( carr_p1980447933_d_a_e @ I @ M ) )
% 0.27/0.50 => ( ( member_a_a_a_a_a3 @ J @ I )
% 0.27/0.50 => ( ( algebr2122715238_a_d_e @ R @ I @ M @ A2 @ M2 @ J )
% 0.27/0.50 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.prodM_sprod_val
% 0.27/0.50 thf(fact_311_Ring_OprodM__sprod__val,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,I: set_a_a_a_a,M: ( a > a > a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > a > a ) > d,J: a > a > a > a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ! [X: a > a > a > a] :
% 0.27/0.50 ( ( member_a_a_a_a4 @ X @ I )
% 0.27/0.50 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a_a_a_a_d @ M2 @ ( carr_p305487606_d_a_e @ I @ M ) )
% 0.27/0.50 => ( ( member_a_a_a_a4 @ J @ I )
% 0.27/0.50 => ( ( algebr303206081_a_d_e @ R @ I @ M @ A2 @ M2 @ J )
% 0.27/0.50 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.prodM_sprod_val
% 0.27/0.50 thf(fact_312_Ring_OprodM__sprod__val,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,I: set_d,M: d > carrie1948989069_d_a_e,A2: a,M2: d > d,J: d] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ! [X: d] :
% 0.27/0.50 ( ( member_d @ X @ I )
% 0.27/0.50 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_d_d @ M2 @ ( carr_p1912941332_d_a_e @ I @ M ) )
% 0.27/0.50 => ( ( member_d @ J @ I )
% 0.27/0.50 => ( ( algebr417875517_d_d_e @ R @ I @ M @ A2 @ M2 @ J )
% 0.27/0.50 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.prodM_sprod_val
% 0.27/0.50 thf(fact_313_Ring_OprodM__sprod__val,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,I: set_c,M: c > carrie1948989069_d_a_e,A2: a,M2: c > d,J: c] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ! [X: c] :
% 0.27/0.50 ( ( member_c @ X @ I )
% 0.27/0.50 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_c_d @ M2 @ ( carr_p1636662547_d_a_e @ I @ M ) )
% 0.27/0.50 => ( ( member_c @ J @ I )
% 0.27/0.50 => ( ( algebr1697692348_c_d_e @ R @ I @ M @ A2 @ M2 @ J )
% 0.27/0.50 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.prodM_sprod_val
% 0.27/0.50 thf(fact_314_dsum__iOp__func,axiom,
% 0.27/0.50 ! [I: set_c,A: c > carrie1948989069_d_a_e] :
% 0.27/0.50 ( ! [X: c] :
% 0.27/0.50 ( ( member_c @ X @ I )
% 0.27/0.50 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 0.27/0.50 => ( member_c_d_c_d @ ( prod_m854557927_d_a_e @ I @ A )
% 0.27/0.50 @ ( pi_c_d_c_d @ ( carr_d56066307_d_a_e @ I @ A )
% 0.27/0.50 @ ^ [Uu: c > d] : ( carr_d56066307_d_a_e @ I @ A ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % dsum_iOp_func
% 0.27/0.50 thf(fact_315_zeroring__J__rad__empty,axiom,
% 0.27/0.50 ( ( zeroring_a_b @ r )
% 0.27/0.50 => ( ( j_rad_a_b @ r )
% 0.27/0.50 = ( carrie867757212xt_a_b @ r ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % zeroring_J_rad_empty
% 0.27/0.50 thf(fact_316_Module_OmHom__lin,axiom,
% 0.27/0.50 ! [M: carrie1948989069_d_a_e,R: carrie1105631105xt_a_b,N: carrie1948989069_d_a_e,M2: d,F: d > d,A2: a] :
% 0.27/0.50 ( ( module_d_a_e_b @ M @ R )
% 0.27/0.50 => ( ( module_d_a_e_b @ N @ R )
% 0.27/0.50 => ( ( member_d @ M2 @ ( carrie11369756_d_a_e @ M ) )
% 0.27/0.50 => ( ( member_d_d @ F @ ( mHom_a_b_d_e_d_e @ R @ M @ N ) )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( F @ ( sprod_d_a_e @ M @ A2 @ M2 ) )
% 0.27/0.50 = ( sprod_d_a_e @ N @ A2 @ ( F @ M2 ) ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Module.mHom_lin
% 0.27/0.50 thf(fact_317_Module_OmHom__lin,axiom,
% 0.27/0.50 ! [M: carrie1954260683t_unit,R: carrie1105631105xt_a_b,N: carrie1948989069_d_a_e,M2: c > d,F: ( c > d ) > d,A2: a] :
% 0.27/0.50 ( ( module1440397025unit_b @ M @ R )
% 0.27/0.50 => ( ( module_d_a_e_b @ N @ R )
% 0.27/0.50 => ( ( member_c_d @ M2 @ ( carrie20288686t_unit @ M ) )
% 0.27/0.50 => ( ( member_c_d_d @ F @ ( mHom_a764575953it_d_e @ R @ M @ N ) )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( F @ ( sprod_1929599593t_unit @ M @ A2 @ M2 ) )
% 0.27/0.50 = ( sprod_d_a_e @ N @ A2 @ ( F @ M2 ) ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Module.mHom_lin
% 0.27/0.50 thf(fact_318_Module_Osprod__assoc,axiom,
% 0.27/0.50 ! [M: carrie1948989069_d_a_e,R: carrie1105631105xt_a_b,A2: a,B2: a,M2: d] :
% 0.27/0.50 ( ( module_d_a_e_b @ M @ R )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_d @ M2 @ ( carrie11369756_d_a_e @ M ) )
% 0.27/0.50 => ( ( sprod_d_a_e @ M @ ( tp_a_b @ R @ A2 @ B2 ) @ M2 )
% 0.27/0.50 = ( sprod_d_a_e @ M @ A2 @ ( sprod_d_a_e @ M @ B2 @ M2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Module.sprod_assoc
% 0.27/0.50 thf(fact_319_Module_Osprod__assoc,axiom,
% 0.27/0.50 ! [M: carrie1954260683t_unit,R: carrie1105631105xt_a_b,A2: a,B2: a,M2: c > d] :
% 0.27/0.50 ( ( module1440397025unit_b @ M @ R )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
% 0.27/0.50 => ( ( member_c_d @ M2 @ ( carrie20288686t_unit @ M ) )
% 0.27/0.50 => ( ( sprod_1929599593t_unit @ M @ ( tp_a_b @ R @ A2 @ B2 ) @ M2 )
% 0.27/0.50 = ( sprod_1929599593t_unit @ M @ A2 @ ( sprod_1929599593t_unit @ M @ B2 @ M2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Module.sprod_assoc
% 0.27/0.50 thf(fact_320_Ring_OmHom__func,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,F: ( c > d ) > c > d,M: carrie1954260683t_unit,N: carrie1954260683t_unit] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( member_c_d_c_d @ F @ ( mHom_a1526663829t_unit @ R @ M @ N ) )
% 0.27/0.50 => ( member_c_d_c_d @ F
% 0.27/0.50 @ ( pi_c_d_c_d @ ( carrie20288686t_unit @ M )
% 0.27/0.50 @ ^ [Uu: c > d] : ( carrie20288686t_unit @ N ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.mHom_func
% 0.27/0.50 thf(fact_321_Ring_OSubring__pOp__ring__pOp,axiom,
% 0.27/0.50 ! [R: carrie1105631105xt_a_b,S: carrie1105631105xt_a_b,A2: a,B2: a] :
% 0.27/0.50 ( ( ring_a_b @ R )
% 0.27/0.50 => ( ( subring_a_b_b @ R @ S )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ S ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ S ) )
% 0.27/0.50 => ( ( pop_a_Ring_ext_a_b @ S @ A2 @ B2 )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ R @ A2 @ B2 ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Ring.Subring_pOp_ring_pOp
% 0.27/0.50 thf(fact_322_nsDistr,axiom,
% 0.27/0.50 ! [X2: a,N2: nat,M2: nat] :
% 0.27/0.50 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( pop_a_Ring_ext_a_b @ r @ ( nscal_a_b @ r @ X2 @ N2 ) @ ( nscal_a_b @ r @ X2 @ M2 ) )
% 0.27/0.50 = ( nscal_a_b @ r @ X2 @ ( plus_plus_nat @ N2 @ M2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % nsDistr
% 0.27/0.50 thf(fact_323_add__both,axiom,
% 0.27/0.50 ! [A2: nat,B2: nat,C2: nat] :
% 0.27/0.50 ( ( A2 = B2 )
% 0.27/0.50 => ( ( plus_plus_nat @ A2 @ C2 )
% 0.27/0.50 = ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % add_both
% 0.27/0.50 thf(fact_324_mul__closed__set__sub,axiom,
% 0.27/0.50 ! [S: set_a] :
% 0.27/0.50 ( ( mul_closed_set_a_b @ r @ S )
% 0.27/0.50 => ( ord_less_eq_set_a @ S @ ( carrie867757212xt_a_b @ r ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mul_closed_set_sub
% 0.27/0.50 thf(fact_325_zeroring__no__maximal,axiom,
% 0.27/0.50 ( ( zeroring_a_b @ r )
% 0.27/0.50 => ~ ? [X_1: set_a] : ( maximal_ideal_a_b @ r @ X_1 ) ) ).
% 0.27/0.50
% 0.27/0.50 % zeroring_no_maximal
% 0.27/0.50 thf(fact_326_id__maximal__Exist,axiom,
% 0.27/0.50 ( ~ ( zeroring_a_b @ r )
% 0.27/0.50 => ? [X_12: set_a] : ( maximal_ideal_a_b @ r @ X_12 ) ) ).
% 0.27/0.50
% 0.27/0.50 % id_maximal_Exist
% 0.27/0.50 thf(fact_327_mop__closed,axiom,
% 0.27/0.50 ( member_a_a @ ( mop_a_Ring_ext_a_b @ r )
% 0.27/0.50 @ ( pi_a_a @ ( carrie867757212xt_a_b @ r )
% 0.27/0.50 @ ^ [Uu: a] : ( carrie867757212xt_a_b @ r ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mop_closed
% 0.27/0.50 thf(fact_328_Sr__mOp__closed,axiom,
% 0.27/0.50 ! [S: set_a,X2: a] :
% 0.27/0.50 ( ( sr_a_b2 @ r @ S )
% 0.27/0.50 => ( ( member_a @ X2 @ S )
% 0.27/0.50 => ( member_a @ ( mop_a_Ring_ext_a_b @ r @ X2 ) @ S ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % Sr_mOp_closed
% 0.27/0.50 thf(fact_329_ring__inv1,axiom,
% 0.27/0.50 ! [A2: a,B2: a] :
% 0.27/0.50 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( ( mop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ A2 @ B2 ) )
% 0.27/0.50 = ( tp_a_b @ r @ ( mop_a_Ring_ext_a_b @ r @ A2 ) @ B2 ) )
% 0.27/0.50 & ( ( mop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ A2 @ B2 ) )
% 0.27/0.50 = ( tp_a_b @ r @ A2 @ ( mop_a_Ring_ext_a_b @ r @ B2 ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % ring_inv1
% 0.27/0.50 thf(fact_330_ring__inv1__1,axiom,
% 0.27/0.50 ! [A2: a,B2: a] :
% 0.27/0.50 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( mop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ A2 @ B2 ) )
% 0.27/0.50 = ( tp_a_b @ r @ ( mop_a_Ring_ext_a_b @ r @ A2 ) @ B2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % ring_inv1_1
% 0.27/0.50 thf(fact_331_ring__inv1__2,axiom,
% 0.27/0.50 ! [A2: a,B2: a] :
% 0.27/0.50 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( mop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ A2 @ B2 ) )
% 0.27/0.50 = ( tp_a_b @ r @ A2 @ ( mop_a_Ring_ext_a_b @ r @ B2 ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % ring_inv1_2
% 0.27/0.50 thf(fact_332_ring__inv1__3,axiom,
% 0.27/0.50 ! [A2: a,B2: a] :
% 0.27/0.50 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( tp_a_b @ r @ A2 @ B2 )
% 0.27/0.50 = ( tp_a_b @ r @ ( mop_a_Ring_ext_a_b @ r @ A2 ) @ ( mop_a_Ring_ext_a_b @ r @ B2 ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % ring_inv1_3
% 0.27/0.50 thf(fact_333_ring__distrib4,axiom,
% 0.27/0.50 ! [A2: a,B2: a,X2: a,Y2: a] :
% 0.27/0.50 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( pop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ A2 @ B2 ) @ ( mop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ X2 @ Y2 ) ) )
% 0.27/0.50 = ( pop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ A2 @ ( pop_a_Ring_ext_a_b @ r @ B2 @ ( mop_a_Ring_ext_a_b @ r @ Y2 ) ) ) @ ( tp_a_b @ r @ ( pop_a_Ring_ext_a_b @ r @ A2 @ ( mop_a_Ring_ext_a_b @ r @ X2 ) ) @ Y2 ) ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % ring_distrib4
% 0.27/0.50 thf(fact_334_nonunit__contained__maxid,axiom,
% 0.27/0.50 ! [A2: a] :
% 0.27/0.50 ( ~ ( zeroring_a_b @ r )
% 0.27/0.50 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ~ ( unit_a_b @ r @ A2 )
% 0.27/0.50 => ? [Mx: set_a] :
% 0.27/0.50 ( ( maximal_ideal_a_b @ r @ Mx )
% 0.27/0.50 & ( member_a @ A2 @ Mx ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % nonunit_contained_maxid
% 0.27/0.50 thf(fact_335_set__sum__mem,axiom,
% 0.27/0.50 ! [A2: a,I: set_a,B2: a,J2: set_a] :
% 0.27/0.50 ( ( member_a @ A2 @ I )
% 0.27/0.50 => ( ( member_a @ B2 @ J2 )
% 0.27/0.50 => ( ( ord_less_eq_set_a @ I @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( ord_less_eq_set_a @ J2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( member_a @ ( pop_a_Ring_ext_a_b @ r @ A2 @ B2 ) @ ( aset_s1743553894xt_a_b @ r @ I @ J2 ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % set_sum_mem
% 0.27/0.50 thf(fact_336_sum__mult__pOp__closed,axiom,
% 0.27/0.50 ! [A: set_a,B: set_a,A2: a,B2: a] :
% 0.27/0.50 ( ( ord_less_eq_set_a @ A @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( ord_less_eq_set_a @ B @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ A2 @ ( sum_mult_a_b @ r @ A @ B ) )
% 0.27/0.50 => ( ( member_a @ B2 @ ( sum_mult_a_b @ r @ A @ B ) )
% 0.27/0.50 => ( member_a @ ( pop_a_Ring_ext_a_b @ r @ A2 @ B2 ) @ ( sum_mult_a_b @ r @ A @ B ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sum_mult_pOp_closed
% 0.27/0.50 thf(fact_337_sum__mult__subR,axiom,
% 0.27/0.50 ! [A: set_a,B: set_a] :
% 0.27/0.50 ( ( ord_less_eq_set_a @ A @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( ord_less_eq_set_a @ B @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ord_less_eq_set_a @ ( sum_mult_a_b @ r @ A @ B ) @ ( carrie867757212xt_a_b @ r ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % sum_mult_subR
% 0.27/0.50 thf(fact_338_times__mem__sum__mult,axiom,
% 0.27/0.50 ! [A: set_a,B: set_a,A2: a,B2: a] :
% 0.27/0.50 ( ( ord_less_eq_set_a @ A @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( ord_less_eq_set_a @ B @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( member_a @ A2 @ A )
% 0.27/0.50 => ( ( member_a @ B2 @ B )
% 0.27/0.50 => ( member_a @ ( tp_a_b @ r @ A2 @ B2 ) @ ( sum_mult_a_b @ r @ A @ B ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % times_mem_sum_mult
% 0.27/0.50 thf(fact_339_le__imp__add__int,axiom,
% 0.27/0.50 ! [I2: nat,J: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ I2 @ J )
% 0.27/0.50 => ? [K: nat] :
% 0.27/0.50 ( J
% 0.27/0.50 = ( plus_plus_nat @ I2 @ K ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % le_imp_add_int
% 0.27/0.50 thf(fact_340_J__rad__unit,axiom,
% 0.27/0.50 ! [X2: a] :
% 0.27/0.50 ( ~ ( zeroring_a_b @ r )
% 0.27/0.50 => ( ( member_a @ X2 @ ( j_rad_a_b @ r ) )
% 0.27/0.50 => ! [Y3: a] :
% 0.27/0.50 ( ( member_a @ Y3 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( unit_a_b @ r @ ( pop_a_Ring_ext_a_b @ r @ ( un_a_b @ r ) @ ( tp_a_b @ r @ ( mop_a_Ring_ext_a_b @ r @ X2 ) @ Y3 ) ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % J_rad_unit
% 0.27/0.50 thf(fact_341_un__closed,axiom,
% 0.27/0.50 member_a @ ( un_a_b @ r ) @ ( carrie867757212xt_a_b @ r ) ).
% 0.27/0.50
% 0.27/0.50 % un_closed
% 0.27/0.50 thf(fact_342_maximal__ideal__proper,axiom,
% 0.27/0.50 ! [Mx2: set_a] :
% 0.27/0.50 ( ( maximal_ideal_a_b @ r @ Mx2 )
% 0.27/0.50 => ~ ( member_a @ ( un_a_b @ r ) @ Mx2 ) ) ).
% 0.27/0.50
% 0.27/0.50 % maximal_ideal_proper
% 0.27/0.50 thf(fact_343_Sr__one,axiom,
% 0.27/0.50 ! [S: set_a] :
% 0.27/0.50 ( ( sr_a_b2 @ r @ S )
% 0.27/0.50 => ( member_a @ ( un_a_b @ r ) @ S ) ) ).
% 0.27/0.50
% 0.27/0.50 % Sr_one
% 0.27/0.50 thf(fact_344_rg__l__unit,axiom,
% 0.27/0.50 ! [A2: a] :
% 0.27/0.50 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( tp_a_b @ r @ ( un_a_b @ r ) @ A2 )
% 0.27/0.50 = A2 ) ) ).
% 0.27/0.50
% 0.27/0.50 % rg_l_unit
% 0.27/0.50 thf(fact_345_ring__l__one,axiom,
% 0.27/0.50 ! [X2: a] :
% 0.27/0.50 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( tp_a_b @ r @ ( un_a_b @ r ) @ X2 )
% 0.27/0.50 = X2 ) ) ).
% 0.27/0.50
% 0.27/0.50 % ring_l_one
% 0.27/0.50 thf(fact_346_ring__r__one,axiom,
% 0.27/0.50 ! [X2: a] :
% 0.27/0.50 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( tp_a_b @ r @ X2 @ ( un_a_b @ r ) )
% 0.27/0.50 = X2 ) ) ).
% 0.27/0.50
% 0.27/0.50 % ring_r_one
% 0.27/0.50 thf(fact_347_ring__times__minusr,axiom,
% 0.27/0.50 ! [A2: a] :
% 0.27/0.50 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( mop_a_Ring_ext_a_b @ r @ A2 )
% 0.27/0.50 = ( tp_a_b @ r @ A2 @ ( mop_a_Ring_ext_a_b @ r @ ( un_a_b @ r ) ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % ring_times_minusr
% 0.27/0.50 thf(fact_348_ring__times__minusl,axiom,
% 0.27/0.50 ! [A2: a] :
% 0.27/0.50 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 0.27/0.50 => ( ( mop_a_Ring_ext_a_b @ r @ A2 )
% 0.27/0.50 = ( tp_a_b @ r @ ( mop_a_Ring_ext_a_b @ r @ ( un_a_b @ r ) ) @ A2 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % ring_times_minusl
% 0.27/0.50 thf(fact_349_localring__unit,axiom,
% 0.27/0.50 ! [Mx2: set_a] :
% 0.27/0.50 ( ~ ( zeroring_a_b @ r )
% 0.27/0.50 => ( ( maximal_ideal_a_b @ r @ Mx2 )
% 0.27/0.50 => ( ! [X: a] :
% 0.27/0.50 ( ( member_a @ X @ Mx2 )
% 0.27/0.50 => ( unit_a_b @ r @ ( pop_a_Ring_ext_a_b @ r @ X @ ( un_a_b @ r ) ) ) )
% 0.27/0.50 => ( local_ring_a_b @ r ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % localring_unit
% 0.27/0.50 thf(fact_350_primary__ideal__proper1,axiom,
% 0.27/0.50 ! [Q2: set_a] :
% 0.27/0.50 ( ( primary_ideal_a_b @ r @ Q2 )
% 0.27/0.50 => ~ ( member_a @ ( un_a_b @ r ) @ Q2 ) ) ).
% 0.27/0.50
% 0.27/0.50 % primary_ideal_proper1
% 0.27/0.50 thf(fact_351_self__le,axiom,
% 0.27/0.50 ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% 0.27/0.50
% 0.27/0.50 % self_le
% 0.27/0.50 thf(fact_352_nat__eq__le,axiom,
% 0.27/0.50 ! [M2: nat,N2: nat] :
% 0.27/0.50 ( ( M2 = N2 )
% 0.27/0.50 => ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% 0.27/0.50
% 0.27/0.50 % nat_eq_le
% 0.27/0.50 thf(fact_353_n__in__Nsetn,axiom,
% 0.27/0.50 ! [N2: nat] :
% 0.27/0.50 ( member_nat @ N2
% 0.27/0.50 @ ( collect_nat
% 0.27/0.50 @ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N2 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % n_in_Nsetn
% 0.27/0.50 thf(fact_354_mem__of__Nset,axiom,
% 0.27/0.50 ! [X2: nat,N2: nat] :
% 0.27/0.50 ( ( ord_less_eq_nat @ X2 @ N2 )
% 0.27/0.50 => ( member_nat @ X2
% 0.27/0.50 @ ( collect_nat
% 0.27/0.50 @ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N2 ) ) ) ) ).
% 0.27/0.50
% 0.27/0.50 % mem_of_Nset
% 0.27/0.50
% 0.27/0.50 % Conjectures (3)
% 0.27/0.50 thf(conj_0,hypothesis,
% 0.27/0.50 ! [X5: c] :
% 0.27/0.50 ( ( member_c @ X5 @ i )
% 0.27/0.50 => ( module_d_a_e_b @ ( m @ X5 ) @ r ) ) ).
% 0.27/0.50
% 0.27/0.50 thf(conj_1,hypothesis,
% 0.27/0.50 ( ( member_c_d_c_d_c_d @ ( prod_p131511076_d_a_e @ i @ m )
% 0.27/0.50 @ ( pi_c_d_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
% 0.27/0.50 @ ^ [Uu: c > d] :
% 0.27/0.50 ( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
% 0.27/0.50 @ ^ [Uv: c > d] : ( carr_p1636662547_d_a_e @ i @ m ) ) ) )
% 0.27/0.50 & ! [A3: c > d] :
% 0.27/0.50 ( ( member_c_d @ A3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
% 0.27/0.50 => ! [B3: c > d] :
% 0.27/0.50 ( ( member_c_d @ B3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
% 0.27/0.50 => ! [C3: c > d] :
% 0.27/0.50 ( ( member_c_d @ C3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
% 0.27/0.50 => ( ( prod_p131511076_d_a_e @ i @ m @ ( prod_p131511076_d_a_e @ i @ m @ A3 @ B3 ) @ C3 )
% 0.27/0.50 = ( prod_p131511076_d_a_e @ i @ m @ A3 @ ( prod_p131511076_d_a_e @ i @ m @ B3 @ C3 ) ) ) ) ) )
% 0.27/0.50 & ! [A3: c > d] :
% 0.27/0.50 ( ( member_c_d @ A3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
% 0.27/0.50 => ! [B3: c > d] :
% 0.27/0.50 ( ( member_c_d @ B3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
% 0.27/0.50 => ( ( prod_p131511076_d_a_e @ i @ m @ A3 @ B3 )
% 0.27/0.50 = ( prod_p131511076_d_a_e @ i @ m @ B3 @ A3 ) ) ) )
% 0.27/0.50 & ( member_c_d_c_d @ ( prod_m854557927_d_a_e @ i @ m )
% 0.27/0.50 @ ( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
% 0.27/0.50 @ ^ [Uu: c > d] : ( carr_p1636662547_d_a_e @ i @ m ) ) )
% 0.27/0.50 & ! [A3: c > d] :
% 0.27/0.50 ( ( member_c_d @ A3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
% 0.27/0.50 => ( ( prod_p131511076_d_a_e @ i @ m @ ( prod_m854557927_d_a_e @ i @ m @ A3 ) @ A3 )
% 0.27/0.50 = ( prod_z1214987823_d_a_e @ i @ m ) ) )
% 0.27/0.50 & ( member_c_d @ ( prod_z1214987823_d_a_e @ i @ m ) @ ( carr_p1636662547_d_a_e @ i @ m ) )
% 0.27/0.50 & ! [A3: c > d] :
% 0.27/0.50 ( ( member_c_d @ A3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
% 0.27/0.50 => ( ( prod_p131511076_d_a_e @ i @ m @ ( prod_z1214987823_d_a_e @ i @ m ) @ A3 )
% 0.27/0.50 = A3 ) ) ) ).
% 0.27/0.50
% 0.27/0.50 thf(conj_2,conjecture,
% 0.27/0.50 ! [A4: c > d] :
% 0.27/0.50 ( ~ ( member_c_d @ A4 @ ( carr_p1636662547_d_a_e @ i @ m ) )
% 0.27/0.50 | ( ( prod_p131511076_d_a_e @ i @ m @ A4 @ ( prod_z1214987823_d_a_e @ i @ m ) )
% 0.27/0.50 = A4 ) ) ).
% 0.27/0.50
% 0.27/0.50 %------------------------------------------------------------------------------
% 0.27/0.50 ------------------------
% 3.05/1.54 ---- Embedded in THF ---
% 3.05/1.54 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.05/1.54 %%% Generated on Tue Feb 24 08:24:49 EST 2026
% 3.05/1.54 %%% using '$$dhol' embedding, version 1.3.0.
% 3.05/1.54 %%% Logic specification used:
% 3.05/1.54 %%% thf(spec, logic, $$dhol).
% 3.05/1.54
% 3.05/1.54 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.2el5MKjZyq/DTF2DTF12139.p
% See solution above
% 3.05/1.54 ------------------------
% 3.15/1.63 ---- Cleaned THF ---
% 3.15/1.63 thf(ty_n_t__Algebra1__Ocarrier__Ocarrier____ext_I_062_Itf__c_Mtf__d_J_Mt__Algebra4__OaGroup__OaGroup____ext_I_062_Itf__c_Mtf__d_J_Mt__Algebra7__OModule__OModule____ext_I_062_Itf__c_Mtf__d_J_Mtf__a_Mt__Product____Type__Ounit_J_J_J,type,
% 3.15/1.63 carrie1954260683t_unit: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Algebra1__Ocarrier__Ocarrier____ext_Itf__d_Mt__Algebra4__OaGroup__OaGroup____ext_Itf__d_Mt__Algebra7__OModule__OModule____ext_Itf__d_Mtf__a_Mtf__e_J_J_J,type,
% 3.15/1.63 carrie1948989069_d_a_e: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Algebra1__Ocarrier__Ocarrier____ext_Itf__a_Mt__Algebra4__OaGroup__OaGroup____ext_Itf__a_Mt__Algebra4__ORing__ORing____ext_Itf__a_Mtf__b_J_J_J,type,
% 3.15/1.63 carrie1105631105xt_a_b: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_Itf__a_M_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_J_J_J,type,
% 3.15/1.63 set_a_a_c_d_c_d: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_M_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_J_J,type,
% 3.15/1.63 set_c_d_c_d_c_d: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_M_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_J_Mtf__d_J_J,type,
% 3.15/1.63 set_a_a_a_a_a_d: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_J_J,type,
% 3.15/1.63 set_a_a_a_a_a: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_J_J,type,
% 3.15/1.63 set_a_c_d_c_d: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_J,type,
% 3.15/1.63 set_c_d_a_a_a: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_M_062_I_062_Itf__c_Mtf__d_J_Mtf__a_J_J_J,type,
% 3.15/1.63 set_c_d_c_d_a: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_M_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_J_J,type,
% 3.15/1.63 set_c_d_a_a_a2: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_M_062_Itf__c_Mtf__d_J_J_J,type,
% 3.15/1.63 set_a_a_a_c_d: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_M_062_Itf__a_Mtf__a_J_J_J,type,
% 3.15/1.63 set_a_a_a_a_a2: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_M_062_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_Mtf__d_J_J,type,
% 3.15/1.63 set_a_a_c_d_d: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_M_062_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_Mtf__a_J_J,type,
% 3.15/1.63 set_a_a_c_d_a: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_Mtf__d_J_J,type,
% 3.15/1.63 set_a_a_a_a_d: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_Mtf__a_J_J,type,
% 3.15/1.63 set_a_a_a_a_a3: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_I_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_Mtf__d_J_J,type,
% 3.15/1.63 set_c_d_c_d_d: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_I_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_Mtf__a_J_J,type,
% 3.15/1.63 set_c_d_c_d_a2: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_J,type,
% 3.15/1.63 set_a_a_c_d: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_J,type,
% 3.15/1.63 set_a_a_a_a: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_I_062_Itf__c_Mtf__d_J_Mtf__a_J_J_J,type,
% 3.15/1.63 set_a_c_d_a: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_J_J,type,
% 3.15/1.63 set_a_a_a_a2: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_J,type,
% 3.15/1.63 set_c_d_c_d: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__a_Mtf__a_J_J_J,type,
% 3.15/1.63 set_c_d_a_a: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_Mtf__a_J_M_062_Itf__c_Mtf__d_J_J_J,type,
% 3.15/1.63 set_a_a_c_d2: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_Mtf__a_J_M_062_Itf__a_Mtf__a_J_J_J,type,
% 3.15/1.63 set_a_a_a_a3: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_Mtf__d_J_J,type,
% 3.15/1.63 set_a_a_a_d: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_Mtf__a_J_J,type,
% 3.15/1.63 set_a_a_a_a4: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_Itf__d_M_062_Itf__c_Mtf__d_J_J_J,type,
% 3.15/1.63 set_d_c_d: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_Itf__c_M_062_Itf__c_Mtf__d_J_J_J,type,
% 3.15/1.63 set_c_c_d: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_Itf__c_Mtf__d_J_J_J,type,
% 3.15/1.63 set_a_c_d: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J,type,
% 3.15/1.63 set_a_a_a: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mtf__d_J_J,type,
% 3.15/1.63 set_c_d_d: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__c_Mtf__d_J_Mtf__a_J_J,type,
% 3.15/1.63 set_c_d_a: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_Mtf__a_J_Mtf__d_J_J,type,
% 3.15/1.63 set_a_a_d: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
% 3.15/1.63 set_a_a_a2: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_Itf__d_Mtf__d_J_J,type,
% 3.15/1.63 set_d_d: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_Itf__d_Mtf__c_J_J,type,
% 3.15/1.63 set_d_c: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_Itf__d_Mtf__a_J_J,type,
% 3.15/1.63 set_d_a: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__d_J_J,type,
% 3.15/1.63 set_c_d: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__c_J_J,type,
% 3.15/1.63 set_c_c: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__a_J_J,type,
% 3.15/1.63 set_c_a: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__d_J_J,type,
% 3.15/1.63 set_a_d: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__c_J_J,type,
% 3.15/1.63 set_a_c: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
% 3.15/1.63 set_a_a: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 3.15/1.63 set_nat: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_Itf__d_J,type,
% 3.15/1.63 set_d: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_Itf__c_J,type,
% 3.15/1.63 set_c: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 3.15/1.63 set_a: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_t__Nat__Onat,type,
% 3.15/1.63 nat: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_tf__d,type,
% 3.15/1.63 d: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_tf__c,type,
% 3.15/1.63 c: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(ty_n_tf__a,type,
% 3.15/1.63 a: $tType ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra1_Ocarrier_Ocarrier_001_062_Itf__c_Mtf__d_J_001t__Algebra4__OaGroup__OaGroup____ext_I_062_Itf__c_Mtf__d_J_Mt__Algebra7__OModule__OModule____ext_I_062_Itf__c_Mtf__d_J_Mtf__a_Mt__Product____Type__Ounit_J_J,type,
% 3.15/1.63 carrie20288686t_unit: carrie1954260683t_unit > set_c_d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra1_Ocarrier_Ocarrier_001tf__a_001t__Algebra4__OaGroup__OaGroup____ext_Itf__a_Mt__Algebra4__ORing__ORing____ext_Itf__a_Mtf__b_J_J,type,
% 3.15/1.63 carrie867757212xt_a_b: carrie1105631105xt_a_b > set_a ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra1_Ocarrier_Ocarrier_001tf__d_001t__Algebra4__OaGroup__OaGroup____ext_Itf__d_Mt__Algebra7__OModule__OModule____ext_Itf__d_Mtf__a_Mtf__e_J_J,type,
% 3.15/1.63 carrie11369756_d_a_e: carrie1948989069_d_a_e > set_d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra4_OJ__rad_001tf__a_001tf__b,type,
% 3.15/1.63 j_rad_a_b: carrie1105631105xt_a_b > set_a ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra4_ORing_001tf__a_001tf__b,type,
% 3.15/1.63 ring_a_b: carrie1105631105xt_a_b > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra4_ORing_Otp_001tf__a_001tf__b,type,
% 3.15/1.63 tp_a_b: carrie1105631105xt_a_b > a > a > a ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra4_ORing_Oun_001tf__a_001tf__b,type,
% 3.15/1.63 un_a_b: carrie1105631105xt_a_b > a ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra4_OSr_001tf__a_001tf__b,type,
% 3.15/1.63 sr_a_b: carrie1105631105xt_a_b > set_a > carrie1105631105xt_a_b ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra4_OSubring_001tf__a_001tf__b_001tf__b,type,
% 3.15/1.63 subring_a_b_b: carrie1105631105xt_a_b > carrie1105631105xt_a_b > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra4_OUnit_001tf__a_001tf__b,type,
% 3.15/1.63 unit_a_b: carrie1105631105xt_a_b > a > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra4_OaGroup_001_062_Itf__c_Mtf__d_J_001t__Algebra7__OModule__OModule____ext_I_062_Itf__c_Mtf__d_J_Mtf__a_Mt__Product____Type__Ounit_J,type,
% 3.15/1.63 aGroup2131905830t_unit: carrie1954260683t_unit > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra4_OaGroup_001tf__a_001t__Algebra4__ORing__ORing____ext_Itf__a_Mtf__b_J,type,
% 3.15/1.63 aGroup2097840802xt_a_b: carrie1105631105xt_a_b > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra4_OaGroup_001tf__d_001t__Algebra7__OModule__OModule____ext_Itf__d_Mtf__a_Mtf__e_J,type,
% 3.15/1.63 aGroup1940521469_d_a_e: carrie1948989069_d_a_e > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra4_OaGroup_Omop_001tf__a_001t__Algebra4__ORing__ORing____ext_Itf__a_Mtf__b_J,type,
% 3.15/1.63 mop_a_Ring_ext_a_b: carrie1105631105xt_a_b > a > a ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra4_OaGroup_Opop_001_062_Itf__c_Mtf__d_J_001t__Algebra7__OModule__OModule____ext_I_062_Itf__c_Mtf__d_J_Mtf__a_Mt__Product____Type__Ounit_J,type,
% 3.15/1.63 pop_c_449750139t_unit: carrie1954260683t_unit > ( c > d ) > ( c > d ) > c > d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra4_OaGroup_Opop_001tf__a_001t__Algebra4__ORing__ORing____ext_Itf__a_Mtf__b_J,type,
% 3.15/1.63 pop_a_Ring_ext_a_b: carrie1105631105xt_a_b > a > a > a ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra4_OaGroup_Opop_001tf__d_001t__Algebra7__OModule__OModule____ext_Itf__d_Mtf__a_Mtf__e_J,type,
% 3.15/1.63 pop_d_261713874_d_a_e: carrie1948989069_d_a_e > d > d > d ).
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% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra8__Mirabelle__tltoludslr_OprodM__sprod_001tf__a_001tf__b_001_062_Itf__a_M_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_J_001tf__d_001tf__e,type,
% 3.15/1.63 algebr2122715238_a_d_e: carrie1105631105xt_a_b > set_a_a_a_a_a > ( ( a > a > a > a > a ) > carrie1948989069_d_a_e ) > a > ( ( a > a > a > a > a ) > d ) > ( a > a > a > a > a ) > d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra8__Mirabelle__tltoludslr_OprodM__sprod_001tf__a_001tf__b_001_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_001tf__d_001tf__e,type,
% 3.15/1.63 algebr303206081_a_d_e: carrie1105631105xt_a_b > set_a_a_a_a > ( ( a > a > a > a ) > carrie1948989069_d_a_e ) > a > ( ( a > a > a > a ) > d ) > ( a > a > a > a ) > d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra8__Mirabelle__tltoludslr_OprodM__sprod_001tf__a_001tf__b_001_062_Itf__a_M_062_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_001tf__d_001tf__e,type,
% 3.15/1.63 algebr1620716610_d_d_e: carrie1105631105xt_a_b > set_a_a_c_d > ( ( a > a > c > d ) > carrie1948989069_d_a_e ) > a > ( ( a > a > c > d ) > d ) > ( a > a > c > d ) > d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra8__Mirabelle__tltoludslr_OprodM__sprod_001tf__a_001tf__b_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_001tf__d_001tf__e,type,
% 3.15/1.63 algebr1655014672_a_d_e: carrie1105631105xt_a_b > set_a_a_a > ( ( a > a > a ) > carrie1948989069_d_a_e ) > a > ( ( a > a > a ) > d ) > ( a > a > a ) > d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra8__Mirabelle__tltoludslr_OprodM__sprod_001tf__a_001tf__b_001_062_Itf__a_Mtf__a_J_001tf__d_001tf__e,type,
% 3.15/1.63 algebr137561451_a_d_e: carrie1105631105xt_a_b > set_a_a > ( ( a > a ) > carrie1948989069_d_a_e ) > a > ( ( a > a ) > d ) > ( a > a ) > d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra8__Mirabelle__tltoludslr_OprodM__sprod_001tf__a_001tf__b_001_062_Itf__c_Mtf__d_J_001tf__d_001tf__e,type,
% 3.15/1.63 algebr924632044_d_d_e: carrie1105631105xt_a_b > set_c_d > ( ( c > d ) > carrie1948989069_d_a_e ) > a > ( ( c > d ) > d ) > ( c > d ) > d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra8__Mirabelle__tltoludslr_OprodM__sprod_001tf__a_001tf__b_001tf__a_001tf__d_001tf__e,type,
% 3.15/1.63 algebr2109842362_a_d_e: carrie1105631105xt_a_b > set_a > ( a > carrie1948989069_d_a_e ) > a > ( a > d ) > a > d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra8__Mirabelle__tltoludslr_OprodM__sprod_001tf__a_001tf__b_001tf__c_001tf__d_001tf__e,type,
% 3.15/1.63 algebr1697692348_c_d_e: carrie1105631105xt_a_b > set_c > ( c > carrie1948989069_d_a_e ) > a > ( c > d ) > c > d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Algebra8__Mirabelle__tltoludslr_OprodM__sprod_001tf__a_001tf__b_001tf__d_001tf__d_001tf__e,type,
% 3.15/1.63 algebr417875517_d_d_e: carrie1105631105xt_a_b > set_d > ( d > carrie1948989069_d_a_e ) > a > ( d > d ) > d > d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_001_062_Itf__a_Mtf__a_J,type,
% 3.15/1.63 pi_a_a_a_a_a: set_a_a_a > ( ( a > a > a ) > set_a_a ) > set_a_a_a_a_a2 ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_001_062_Itf__c_Mtf__d_J,type,
% 3.15/1.63 pi_a_a_a_c_d: set_a_a_a > ( ( a > a > a ) > set_c_d ) > set_a_a_a_c_d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_001tf__a,type,
% 3.15/1.63 pi_a_a_a_a: set_a_a_a > ( ( a > a > a ) > set_a ) > set_a_a_a_a4 ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001_062_Itf__a_Mtf__a_J_001_062_Itf__a_Mtf__a_J,type,
% 3.15/1.63 pi_a_a_a_a2: set_a_a > ( ( a > a ) > set_a_a ) > set_a_a_a_a3 ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001_062_Itf__a_Mtf__a_J_001_062_Itf__c_Mtf__d_J,type,
% 3.15/1.63 pi_a_a_c_d: set_a_a > ( ( a > a ) > set_c_d ) > set_a_a_c_d2 ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001_062_Itf__a_Mtf__a_J_001tf__a,type,
% 3.15/1.63 pi_a_a_a: set_a_a > ( ( a > a ) > set_a ) > set_a_a_a2 ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001_062_Itf__c_Mtf__d_J_001_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J,type,
% 3.15/1.63 pi_c_d_a_a_a: set_c_d > ( ( c > d ) > set_a_a_a2 ) > set_c_d_a_a_a2 ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001_062_Itf__c_Mtf__d_J_001_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J,type,
% 3.15/1.63 pi_c_d_c_d_c_d: set_c_d > ( ( c > d ) > set_c_d_c_d ) > set_c_d_c_d_c_d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001_062_Itf__c_Mtf__d_J_001_062_I_062_Itf__c_Mtf__d_J_Mtf__a_J,type,
% 3.15/1.63 pi_c_d_c_d_a: set_c_d > ( ( c > d ) > set_c_d_a ) > set_c_d_c_d_a ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001_062_Itf__c_Mtf__d_J_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J,type,
% 3.15/1.63 pi_c_d_a_a_a2: set_c_d > ( ( c > d ) > set_a_a_a ) > set_c_d_a_a_a ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001_062_Itf__c_Mtf__d_J_001_062_Itf__a_Mtf__a_J,type,
% 3.15/1.63 pi_c_d_a_a: set_c_d > ( ( c > d ) > set_a_a ) > set_c_d_a_a ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001_062_Itf__c_Mtf__d_J_001_062_Itf__c_Mtf__d_J,type,
% 3.15/1.63 pi_c_d_c_d: set_c_d > ( ( c > d ) > set_c_d ) > set_c_d_c_d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001_062_Itf__c_Mtf__d_J_001tf__a,type,
% 3.15/1.63 pi_c_d_a: set_c_d > ( ( c > d ) > set_a ) > set_c_d_a ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001tf__a_001_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J,type,
% 3.15/1.63 pi_a_a_a_a3: set_a > ( a > set_a_a_a2 ) > set_a_a_a_a2 ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001tf__a_001_062_I_062_Itf__c_Mtf__d_J_Mtf__a_J,type,
% 3.15/1.63 pi_a_c_d_a: set_a > ( a > set_c_d_a ) > set_a_c_d_a ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001tf__a_001_062_Itf__a_M_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_J,type,
% 3.15/1.63 pi_a_a_c_d_c_d: set_a > ( a > set_a_c_d_c_d ) > set_a_a_c_d_c_d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001tf__a_001_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J,type,
% 3.15/1.63 pi_a_a_a_a_a2: set_a > ( a > set_a_a_a_a ) > set_a_a_a_a_a ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001tf__a_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J,type,
% 3.15/1.63 pi_a_a_a_a4: set_a > ( a > set_a_a_a ) > set_a_a_a_a ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001tf__a_001_062_Itf__a_M_062_Itf__c_Mtf__d_J_J,type,
% 3.15/1.63 pi_a_a_c_d2: set_a > ( a > set_a_c_d ) > set_a_a_c_d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001tf__a_001_062_Itf__a_Mtf__a_J,type,
% 3.15/1.63 pi_a_a_a2: set_a > ( a > set_a_a ) > set_a_a_a ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001tf__a_001_062_Itf__c_Mtf__d_J,type,
% 3.15/1.63 pi_a_c_d: set_a > ( a > set_c_d ) > set_a_c_d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001tf__a_001tf__a,type,
% 3.15/1.63 pi_a_a: set_a > ( a > set_a ) > set_a_a ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001tf__a_001tf__c,type,
% 3.15/1.63 pi_a_c: set_a > ( a > set_c ) > set_a_c ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001tf__a_001tf__d,type,
% 3.15/1.63 pi_a_d: set_a > ( a > set_d ) > set_a_d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001tf__c_001tf__a,type,
% 3.15/1.63 pi_c_a: set_c > ( c > set_a ) > set_c_a ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001tf__c_001tf__c,type,
% 3.15/1.63 pi_c_c: set_c > ( c > set_c ) > set_c_c ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001tf__c_001tf__d,type,
% 3.15/1.63 pi_c_d: set_c > ( c > set_d ) > set_c_d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001tf__d_001tf__a,type,
% 3.15/1.63 pi_d_a: set_d > ( d > set_a ) > set_d_a ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001tf__d_001tf__c,type,
% 3.15/1.63 pi_d_c: set_d > ( d > set_c ) > set_d_c ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_FuncSet_OPi_001tf__d_001tf__d,type,
% 3.15/1.63 pi_d_d: set_d > ( d > set_d ) > set_d_d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 3.15/1.63 plus_plus_nat: nat > nat > nat ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 3.15/1.63 ord_less_eq_nat: nat > nat > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 3.15/1.63 ord_less_eq_set_a: set_a > set_a > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Set_OCollect_001_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J,type,
% 3.15/1.63 collect_c_d_c_d: ( ( ( c > d ) > c > d ) > $o ) > set_c_d_c_d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Set_OCollect_001_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J,type,
% 3.15/1.63 collect_a_a_a_a: ( ( a > a > a > a ) > $o ) > set_a_a_a_a ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Set_OCollect_001_062_Itf__a_M_062_Itf__a_M_062_Itf__c_Mtf__d_J_J_J,type,
% 3.15/1.63 collect_a_a_c_d: ( ( a > a > c > d ) > $o ) > set_a_a_c_d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Set_OCollect_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J,type,
% 3.15/1.63 collect_a_a_a: ( ( a > a > a ) > $o ) > set_a_a_a ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Set_OCollect_001_062_Itf__a_Mtf__a_J,type,
% 3.15/1.63 collect_a_a: ( ( a > a ) > $o ) > set_a_a ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Set_OCollect_001_062_Itf__c_Mtf__d_J,type,
% 3.15/1.63 collect_c_d: ( ( c > d ) > $o ) > set_c_d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 3.15/1.63 collect_nat: ( nat > $o ) > set_nat ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Set_OCollect_001tf__a,type,
% 3.15/1.63 collect_a: ( a > $o ) > set_a ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Set_OCollect_001tf__c,type,
% 3.15/1.63 collect_c: ( c > $o ) > set_c ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_Set_OCollect_001tf__d,type,
% 3.15/1.63 collect_d: ( d > $o ) > set_d ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_I_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_Mtf__a_J,type,
% 3.15/1.63 member_c_d_c_d_a: ( ( ( c > d ) > c > d ) > a ) > set_c_d_c_d_a2 > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_I_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_Mtf__d_J,type,
% 3.15/1.63 member_c_d_c_d_d: ( ( ( c > d ) > c > d ) > d ) > set_c_d_c_d_d > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_J_Mtf__d_J,type,
% 3.15/1.63 member_a_a_a_a_a_d: ( ( a > a > a > a > a ) > d ) > set_a_a_a_a_a_d > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_Mtf__a_J,type,
% 3.15/1.63 member_a_a_a_a_a: ( ( a > a > a > a ) > a ) > set_a_a_a_a_a3 > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_Mtf__d_J,type,
% 3.15/1.63 member_a_a_a_a_d: ( ( a > a > a > a ) > d ) > set_a_a_a_a_d > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_Mtf__a_J,type,
% 3.15/1.63 member_a_a_c_d_a: ( ( a > a > c > d ) > a ) > set_a_a_c_d_a > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_M_062_Itf__c_Mtf__d_J_J_J_Mtf__d_J,type,
% 3.15/1.63 member_a_a_c_d_d: ( ( a > a > c > d ) > d ) > set_a_a_c_d_d > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_M_062_Itf__a_Mtf__a_J_J,type,
% 3.15/1.63 member_a_a_a_a_a2: ( ( a > a > a ) > a > a ) > set_a_a_a_a_a2 > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_M_062_Itf__c_Mtf__d_J_J,type,
% 3.15/1.63 member_a_a_a_c_d: ( ( a > a > a ) > c > d ) > set_a_a_a_c_d > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_Mtf__a_J,type,
% 3.15/1.63 member_a_a_a_a: ( ( a > a > a ) > a ) > set_a_a_a_a4 > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_I_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_Mtf__d_J,type,
% 3.15/1.63 member_a_a_a_d: ( ( a > a > a ) > d ) > set_a_a_a_d > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_I_062_Itf__a_Mtf__a_J_M_062_Itf__a_Mtf__a_J_J,type,
% 3.15/1.63 member_a_a_a_a2: ( ( a > a ) > a > a ) > set_a_a_a_a3 > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_I_062_Itf__a_Mtf__a_J_M_062_Itf__c_Mtf__d_J_J,type,
% 3.15/1.63 member_a_a_c_d: ( ( a > a ) > c > d ) > set_a_a_c_d2 > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J,type,
% 3.15/1.63 member_a_a_a: ( ( a > a ) > a ) > set_a_a_a2 > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_I_062_Itf__a_Mtf__a_J_Mtf__d_J,type,
% 3.15/1.63 member_a_a_d: ( ( a > a ) > d ) > set_a_a_d > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_M_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
% 3.15/1.63 member_c_d_a_a_a: ( ( c > d ) > ( a > a ) > a ) > set_c_d_a_a_a2 > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_M_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_J,type,
% 3.15/1.63 member_c_d_c_d_c_d: ( ( c > d ) > ( c > d ) > c > d ) > set_c_d_c_d_c_d > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_M_062_I_062_Itf__c_Mtf__d_J_Mtf__a_J_J,type,
% 3.15/1.63 member_c_d_c_d_a2: ( ( c > d ) > ( c > d ) > a ) > set_c_d_c_d_a > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J,type,
% 3.15/1.63 member_c_d_a_a_a2: ( ( c > d ) > a > a > a ) > set_c_d_a_a_a > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__a_Mtf__a_J_J,type,
% 3.15/1.63 member_c_d_a_a: ( ( c > d ) > a > a ) > set_c_d_a_a > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J,type,
% 3.15/1.63 member_c_d_c_d: ( ( c > d ) > c > d ) > set_c_d_c_d > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_Mtf__a_J,type,
% 3.15/1.63 member_c_d_a: ( ( c > d ) > a ) > set_c_d_a > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_I_062_Itf__c_Mtf__d_J_Mtf__d_J,type,
% 3.15/1.63 member_c_d_d: ( ( c > d ) > d ) > set_c_d_d > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_Itf__a_M_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
% 3.15/1.63 member_a_a_a_a3: ( a > ( a > a ) > a ) > set_a_a_a_a2 > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_Itf__a_M_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_J,type,
% 3.15/1.63 member_a_c_d_c_d: ( a > ( c > d ) > c > d ) > set_a_c_d_c_d > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_Itf__a_M_062_I_062_Itf__c_Mtf__d_J_Mtf__a_J_J,type,
% 3.15/1.63 member_a_c_d_a: ( a > ( c > d ) > a ) > set_a_c_d_a > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_Itf__a_M_062_Itf__a_M_062_I_062_Itf__c_Mtf__d_J_M_062_Itf__c_Mtf__d_J_J_J_J,type,
% 3.15/1.63 member_a_a_c_d_c_d: ( a > a > ( c > d ) > c > d ) > set_a_a_c_d_c_d > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_Itf__a_M_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J_J,type,
% 3.15/1.63 member_a_a_a_a_a3: ( a > a > a > a > a ) > set_a_a_a_a_a > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_Itf__a_M_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J,type,
% 3.15/1.63 member_a_a_a_a4: ( a > a > a > a ) > set_a_a_a_a > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_Itf__a_M_062_Itf__a_M_062_Itf__c_Mtf__d_J_J_J,type,
% 3.15/1.63 member_a_a_c_d2: ( a > a > c > d ) > set_a_a_c_d > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J,type,
% 3.15/1.63 member_a_a_a2: ( a > a > a ) > set_a_a_a > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_Itf__a_M_062_Itf__c_Mtf__d_J_J,type,
% 3.15/1.63 member_a_c_d: ( a > c > d ) > set_a_c_d > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
% 3.15/1.63 member_a_a: ( a > a ) > set_a_a > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_Itf__a_Mtf__c_J,type,
% 3.15/1.63 member_a_c: ( a > c ) > set_a_c > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_Itf__a_Mtf__d_J,type,
% 3.15/1.63 member_a_d: ( a > d ) > set_a_d > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_Itf__c_M_062_Itf__c_Mtf__d_J_J,type,
% 3.15/1.63 member_c_c_d: ( c > c > d ) > set_c_c_d > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_Itf__c_Mtf__a_J,type,
% 3.15/1.63 member_c_a: ( c > a ) > set_c_a > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_Itf__c_Mtf__c_J,type,
% 3.15/1.63 member_c_c: ( c > c ) > set_c_c > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_Itf__c_Mtf__d_J,type,
% 3.15/1.63 member_c_d: ( c > d ) > set_c_d > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_Itf__d_M_062_Itf__c_Mtf__d_J_J,type,
% 3.15/1.63 member_d_c_d: ( d > c > d ) > set_d_c_d > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_Itf__d_Mtf__a_J,type,
% 3.15/1.63 member_d_a: ( d > a ) > set_d_a > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_Itf__d_Mtf__c_J,type,
% 3.15/1.63 member_d_c: ( d > c ) > set_d_c > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001_062_Itf__d_Mtf__d_J,type,
% 3.15/1.63 member_d_d: ( d > d ) > set_d_d > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001t__Nat__Onat,type,
% 3.15/1.63 member_nat: nat > set_nat > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001tf__a,type,
% 3.15/1.63 member_a: a > set_a > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001tf__c,type,
% 3.15/1.63 member_c: c > set_c > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_c_member_001tf__d,type,
% 3.15/1.63 member_d: d > set_d > $o ).
% 3.15/1.63
% 3.15/1.63 thf(sy_v_I,type,
% 3.15/1.63 i: set_c ).
% 3.15/1.63
% 3.15/1.63 thf(sy_v_M,type,
% 3.15/1.63 m: c > carrie1948989069_d_a_e ).
% 3.15/1.63
% 3.15/1.63 thf(sy_v_R,type,
% 3.15/1.63 r: carrie1105631105xt_a_b ).
% 3.15/1.63
% 3.15/1.63 thf(fact_0_Ring__axioms,axiom,
% 3.15/1.63 ring_a_b @ r ).
% 3.15/1.63
% 3.15/1.63 thf(fact_1_bivar__fun,axiom,
% 3.15/1.63 ! [F: a > c > d,A: set_a,B: set_c,C: set_d,A2: a] :
% 3.15/1.63 ( ( member_a_c_d @ F
% 3.15/1.63 @ ( pi_a_c_d @ A
% 3.15/1.63 @ ^ [Uu: a] :
% 3.15/1.63 ( pi_c_d @ B
% 3.15/1.63 @ ^ [Uv: c] : C ) ) )
% 3.15/1.63 => ( ( member_a @ A2 @ A )
% 3.15/1.63 => ( member_c_d @ ( F @ A2 )
% 3.15/1.63 @ ( pi_c_d @ B
% 3.15/1.63 @ ^ [Uu: c] : C ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_2_bivar__fun,axiom,
% 3.15/1.63 ! [F: a > a > a,A: set_a,B: set_a,C: set_a,A2: a] :
% 3.15/1.63 ( ( member_a_a_a2 @ F
% 3.15/1.63 @ ( pi_a_a_a2 @ A
% 3.15/1.63 @ ^ [Uu: a] :
% 3.15/1.63 ( pi_a_a @ B
% 3.15/1.63 @ ^ [Uv: a] : C ) ) )
% 3.15/1.63 => ( ( member_a @ A2 @ A )
% 3.15/1.63 => ( member_a_a @ ( F @ A2 )
% 3.15/1.63 @ ( pi_a_a @ B
% 3.15/1.63 @ ^ [Uu: a] : C ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_3_bivar__fun,axiom,
% 3.15/1.63 ! [F: ( c > d ) > a > a,A: set_c_d,B: set_a,C: set_a,A2: c > d] :
% 3.15/1.63 ( ( member_c_d_a_a @ F
% 3.15/1.63 @ ( pi_c_d_a_a @ A
% 3.15/1.63 @ ^ [Uu: c > d] :
% 3.15/1.63 ( pi_a_a @ B
% 3.15/1.63 @ ^ [Uv: a] : C ) ) )
% 3.15/1.63 => ( ( member_c_d @ A2 @ A )
% 3.15/1.63 => ( member_a_a @ ( F @ A2 )
% 3.15/1.63 @ ( pi_a_a @ B
% 3.15/1.63 @ ^ [Uu: a] : C ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_4_bivar__fun,axiom,
% 3.15/1.63 ! [F: ( a > a ) > c > d,A: set_a_a,B: set_c,C: set_d,A2: a > a] :
% 3.15/1.63 ( ( member_a_a_c_d @ F
% 3.15/1.63 @ ( pi_a_a_c_d @ A
% 3.15/1.63 @ ^ [Uu: a > a] :
% 3.15/1.63 ( pi_c_d @ B
% 3.15/1.63 @ ^ [Uv: c] : C ) ) )
% 3.15/1.63 => ( ( member_a_a @ A2 @ A )
% 3.15/1.63 => ( member_c_d @ ( F @ A2 )
% 3.15/1.63 @ ( pi_c_d @ B
% 3.15/1.63 @ ^ [Uu: c] : C ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_5_bivar__fun,axiom,
% 3.15/1.63 ! [F: ( a > a ) > a > a,A: set_a_a,B: set_a,C: set_a,A2: a > a] :
% 3.15/1.63 ( ( member_a_a_a_a2 @ F
% 3.15/1.63 @ ( pi_a_a_a_a2 @ A
% 3.15/1.63 @ ^ [Uu: a > a] :
% 3.15/1.63 ( pi_a_a @ B
% 3.15/1.63 @ ^ [Uv: a] : C ) ) )
% 3.15/1.63 => ( ( member_a_a @ A2 @ A )
% 3.15/1.63 => ( member_a_a @ ( F @ A2 )
% 3.15/1.63 @ ( pi_a_a @ B
% 3.15/1.63 @ ^ [Uu: a] : C ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_6_bivar__fun,axiom,
% 3.15/1.63 ! [F: a > a > a > a,A: set_a,B: set_a,C: set_a_a,A2: a] :
% 3.15/1.63 ( ( member_a_a_a_a4 @ F
% 3.15/1.63 @ ( pi_a_a_a_a4 @ A
% 3.15/1.63 @ ^ [Uu: a] :
% 3.15/1.63 ( pi_a_a_a2 @ B
% 3.15/1.63 @ ^ [Uv: a] : C ) ) )
% 3.15/1.63 => ( ( member_a @ A2 @ A )
% 3.15/1.63 => ( member_a_a_a2 @ ( F @ A2 )
% 3.15/1.63 @ ( pi_a_a_a2 @ B
% 3.15/1.63 @ ^ [Uu: a] : C ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_7_bivar__fun,axiom,
% 3.15/1.63 ! [F: ( c > d ) > c > d,A: set_c_d,B: set_c,C: set_d,A2: c > d] :
% 3.15/1.63 ( ( member_c_d_c_d @ F
% 3.15/1.63 @ ( pi_c_d_c_d @ A
% 3.15/1.63 @ ^ [Uu: c > d] :
% 3.15/1.63 ( pi_c_d @ B
% 3.15/1.63 @ ^ [Uv: c] : C ) ) )
% 3.15/1.63 => ( ( member_c_d @ A2 @ A )
% 3.15/1.63 => ( member_c_d @ ( F @ A2 )
% 3.15/1.63 @ ( pi_c_d @ B
% 3.15/1.63 @ ^ [Uu: c] : C ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_8_bivar__fun,axiom,
% 3.15/1.63 ! [F: ( c > d ) > a > a > a,A: set_c_d,B: set_a,C: set_a_a,A2: c > d] :
% 3.15/1.63 ( ( member_c_d_a_a_a2 @ F
% 3.15/1.63 @ ( pi_c_d_a_a_a2 @ A
% 3.15/1.63 @ ^ [Uu: c > d] :
% 3.15/1.63 ( pi_a_a_a2 @ B
% 3.15/1.63 @ ^ [Uv: a] : C ) ) )
% 3.15/1.63 => ( ( member_c_d @ A2 @ A )
% 3.15/1.63 => ( member_a_a_a2 @ ( F @ A2 )
% 3.15/1.63 @ ( pi_a_a_a2 @ B
% 3.15/1.63 @ ^ [Uu: a] : C ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_9_bivar__fun,axiom,
% 3.15/1.63 ! [F: ( a > a > a ) > c > d,A: set_a_a_a,B: set_c,C: set_d,A2: a > a > a] :
% 3.15/1.63 ( ( member_a_a_a_c_d @ F
% 3.15/1.63 @ ( pi_a_a_a_c_d @ A
% 3.15/1.63 @ ^ [Uu: a > a > a] :
% 3.15/1.63 ( pi_c_d @ B
% 3.15/1.63 @ ^ [Uv: c] : C ) ) )
% 3.15/1.63 => ( ( member_a_a_a2 @ A2 @ A )
% 3.15/1.63 => ( member_c_d @ ( F @ A2 )
% 3.15/1.63 @ ( pi_c_d @ B
% 3.15/1.63 @ ^ [Uu: c] : C ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_10_bivar__fun,axiom,
% 3.15/1.63 ! [F: ( a > a > a ) > a > a,A: set_a_a_a,B: set_a,C: set_a,A2: a > a > a] :
% 3.15/1.63 ( ( member_a_a_a_a_a2 @ F
% 3.15/1.63 @ ( pi_a_a_a_a_a @ A
% 3.15/1.63 @ ^ [Uu: a > a > a] :
% 3.15/1.63 ( pi_a_a @ B
% 3.15/1.63 @ ^ [Uv: a] : C ) ) )
% 3.15/1.63 => ( ( member_a_a_a2 @ A2 @ A )
% 3.15/1.63 => ( member_a_a @ ( F @ A2 )
% 3.15/1.63 @ ( pi_a_a @ B
% 3.15/1.63 @ ^ [Uu: a] : C ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_11_bivar__fun__mem,axiom,
% 3.15/1.63 ! [F: a > a > a,A: set_a,B: set_a,C: set_a,A2: a,B2: a] :
% 3.15/1.63 ( ( member_a_a_a2 @ F
% 3.15/1.63 @ ( pi_a_a_a2 @ A
% 3.15/1.63 @ ^ [Uu: a] :
% 3.15/1.63 ( pi_a_a @ B
% 3.15/1.63 @ ^ [Uv: a] : C ) ) )
% 3.15/1.63 => ( ( member_a @ A2 @ A )
% 3.15/1.63 => ( ( member_a @ B2 @ B )
% 3.15/1.63 => ( member_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% 3.15/1.63
% 3.15/1.63 thf(fact_12_bivar__fun__mem,axiom,
% 3.15/1.63 ! [F: ( c > d ) > a > a,A: set_c_d,B: set_a,C: set_a,A2: c > d,B2: a] :
% 3.15/1.63 ( ( member_c_d_a_a @ F
% 3.15/1.64 @ ( pi_c_d_a_a @ A
% 3.15/1.64 @ ^ [Uu: c > d] :
% 3.15/1.64 ( pi_a_a @ B
% 3.15/1.64 @ ^ [Uv: a] : C ) ) )
% 3.15/1.64 => ( ( member_c_d @ A2 @ A )
% 3.15/1.64 => ( ( member_a @ B2 @ B )
% 3.15/1.64 => ( member_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_13_bivar__fun__mem,axiom,
% 3.15/1.64 ! [F: ( a > a ) > a > a,A: set_a_a,B: set_a,C: set_a,A2: a > a,B2: a] :
% 3.15/1.64 ( ( member_a_a_a_a2 @ F
% 3.15/1.64 @ ( pi_a_a_a_a2 @ A
% 3.15/1.64 @ ^ [Uu: a > a] :
% 3.15/1.64 ( pi_a_a @ B
% 3.15/1.64 @ ^ [Uv: a] : C ) ) )
% 3.15/1.64 => ( ( member_a_a @ A2 @ A )
% 3.15/1.64 => ( ( member_a @ B2 @ B )
% 3.15/1.64 => ( member_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_14_bivar__fun__mem,axiom,
% 3.15/1.64 ! [F: a > ( c > d ) > a,A: set_a,B: set_c_d,C: set_a,A2: a,B2: c > d] :
% 3.15/1.64 ( ( member_a_c_d_a @ F
% 3.15/1.64 @ ( pi_a_c_d_a @ A
% 3.15/1.64 @ ^ [Uu: a] :
% 3.15/1.64 ( pi_c_d_a @ B
% 3.15/1.64 @ ^ [Uv: c > d] : C ) ) )
% 3.15/1.64 => ( ( member_a @ A2 @ A )
% 3.15/1.64 => ( ( member_c_d @ B2 @ B )
% 3.15/1.64 => ( member_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_15_bivar__fun__mem,axiom,
% 3.15/1.64 ! [F: a > ( a > a ) > a,A: set_a,B: set_a_a,C: set_a,A2: a,B2: a > a] :
% 3.15/1.64 ( ( member_a_a_a_a3 @ F
% 3.15/1.64 @ ( pi_a_a_a_a3 @ A
% 3.15/1.64 @ ^ [Uu: a] :
% 3.15/1.64 ( pi_a_a_a @ B
% 3.15/1.64 @ ^ [Uv: a > a] : C ) ) )
% 3.15/1.64 => ( ( member_a @ A2 @ A )
% 3.15/1.64 => ( ( member_a_a @ B2 @ B )
% 3.15/1.64 => ( member_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_16_bivar__fun__mem,axiom,
% 3.15/1.64 ! [F: a > a > c > d,A: set_a,B: set_a,C: set_c_d,A2: a,B2: a] :
% 3.15/1.64 ( ( member_a_a_c_d2 @ F
% 3.15/1.64 @ ( pi_a_a_c_d2 @ A
% 3.15/1.64 @ ^ [Uu: a] :
% 3.15/1.64 ( pi_a_c_d @ B
% 3.15/1.64 @ ^ [Uv: a] : C ) ) )
% 3.15/1.64 => ( ( member_a @ A2 @ A )
% 3.15/1.64 => ( ( member_a @ B2 @ B )
% 3.15/1.64 => ( member_c_d @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_17_bivar__fun__mem,axiom,
% 3.15/1.64 ! [F: a > a > a > a,A: set_a,B: set_a,C: set_a_a,A2: a,B2: a] :
% 3.15/1.64 ( ( member_a_a_a_a4 @ F
% 3.15/1.64 @ ( pi_a_a_a_a4 @ A
% 3.15/1.64 @ ^ [Uu: a] :
% 3.15/1.64 ( pi_a_a_a2 @ B
% 3.15/1.64 @ ^ [Uv: a] : C ) ) )
% 3.15/1.64 => ( ( member_a @ A2 @ A )
% 3.15/1.64 => ( ( member_a @ B2 @ B )
% 3.15/1.64 => ( member_a_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_18_bivar__fun__mem,axiom,
% 3.15/1.64 ! [F: ( c > d ) > c > d,A: set_c_d,B: set_c,C: set_d,A2: c > d,B2: c] :
% 3.15/1.64 ( ( member_c_d_c_d @ F
% 3.15/1.64 @ ( pi_c_d_c_d @ A
% 3.15/1.64 @ ^ [Uu: c > d] :
% 3.15/1.64 ( pi_c_d @ B
% 3.15/1.64 @ ^ [Uv: c] : C ) ) )
% 3.15/1.64 => ( ( member_c_d @ A2 @ A )
% 3.15/1.64 => ( ( member_c @ B2 @ B )
% 3.15/1.64 => ( member_d @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_19_bivar__fun__mem,axiom,
% 3.15/1.64 ! [F: ( c > d ) > ( c > d ) > a,A: set_c_d,B: set_c_d,C: set_a,A2: c > d,B2: c > d] :
% 3.15/1.64 ( ( member_c_d_c_d_a2 @ F
% 3.15/1.64 @ ( pi_c_d_c_d_a @ A
% 3.15/1.64 @ ^ [Uu: c > d] :
% 3.15/1.64 ( pi_c_d_a @ B
% 3.15/1.64 @ ^ [Uv: c > d] : C ) ) )
% 3.15/1.64 => ( ( member_c_d @ A2 @ A )
% 3.15/1.64 => ( ( member_c_d @ B2 @ B )
% 3.15/1.64 => ( member_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_20_bivar__fun__mem,axiom,
% 3.15/1.64 ! [F: ( c > d ) > ( a > a ) > a,A: set_c_d,B: set_a_a,C: set_a,A2: c > d,B2: a > a] :
% 3.15/1.64 ( ( member_c_d_a_a_a @ F
% 3.15/1.64 @ ( pi_c_d_a_a_a @ A
% 3.15/1.64 @ ^ [Uu: c > d] :
% 3.15/1.64 ( pi_a_a_a @ B
% 3.15/1.64 @ ^ [Uv: a > a] : C ) ) )
% 3.15/1.64 => ( ( member_c_d @ A2 @ A )
% 3.15/1.64 => ( ( member_a_a @ B2 @ B )
% 3.15/1.64 => ( member_a @ ( F @ A2 @ B2 ) @ C ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_21_Pi__I,axiom,
% 3.15/1.64 ! [A: set_c,F: c > d,B: c > set_d] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ A )
% 3.15/1.64 => ( member_d @ ( F @ X ) @ ( B @ X ) ) )
% 3.15/1.64 => ( member_c_d @ F @ ( pi_c_d @ A @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_22_Pi__I,axiom,
% 3.15/1.64 ! [A: set_a,F: a > a,B: a > set_a] :
% 3.15/1.64 ( ! [X: a] :
% 3.15/1.64 ( ( member_a @ X @ A )
% 3.15/1.64 => ( member_a @ ( F @ X ) @ ( B @ X ) ) )
% 3.15/1.64 => ( member_a_a @ F @ ( pi_a_a @ A @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_23_Pi__I,axiom,
% 3.15/1.64 ! [A: set_c_d,F: ( c > d ) > a,B: ( c > d ) > set_a] :
% 3.15/1.64 ( ! [X: c > d] :
% 3.15/1.64 ( ( member_c_d @ X @ A )
% 3.15/1.64 => ( member_a @ ( F @ X ) @ ( B @ X ) ) )
% 3.15/1.64 => ( member_c_d_a @ F @ ( pi_c_d_a @ A @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_24_Pi__I,axiom,
% 3.15/1.64 ! [A: set_a_a,F: ( a > a ) > a,B: ( a > a ) > set_a] :
% 3.15/1.64 ( ! [X: a > a] :
% 3.15/1.64 ( ( member_a_a @ X @ A )
% 3.15/1.64 => ( member_a @ ( F @ X ) @ ( B @ X ) ) )
% 3.15/1.64 => ( member_a_a_a @ F @ ( pi_a_a_a @ A @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_25_Pi__I,axiom,
% 3.15/1.64 ! [A: set_a,F: a > c > d,B: a > set_c_d] :
% 3.15/1.64 ( ! [X: a] :
% 3.15/1.64 ( ( member_a @ X @ A )
% 3.15/1.64 => ( member_c_d @ ( F @ X ) @ ( B @ X ) ) )
% 3.15/1.64 => ( member_a_c_d @ F @ ( pi_a_c_d @ A @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_26_Pi__I,axiom,
% 3.15/1.64 ! [A: set_a,F: a > a > a,B: a > set_a_a] :
% 3.15/1.64 ( ! [X: a] :
% 3.15/1.64 ( ( member_a @ X @ A )
% 3.15/1.64 => ( member_a_a @ ( F @ X ) @ ( B @ X ) ) )
% 3.15/1.64 => ( member_a_a_a2 @ F @ ( pi_a_a_a2 @ A @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_27_Pi__I,axiom,
% 3.15/1.64 ! [A: set_c_d,F: ( c > d ) > c > d,B: ( c > d ) > set_c_d] :
% 3.15/1.64 ( ! [X: c > d] :
% 3.15/1.64 ( ( member_c_d @ X @ A )
% 3.15/1.64 => ( member_c_d @ ( F @ X ) @ ( B @ X ) ) )
% 3.15/1.64 => ( member_c_d_c_d @ F @ ( pi_c_d_c_d @ A @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_28_Pi__I,axiom,
% 3.15/1.64 ! [A: set_c_d,F: ( c > d ) > a > a,B: ( c > d ) > set_a_a] :
% 3.15/1.64 ( ! [X: c > d] :
% 3.15/1.64 ( ( member_c_d @ X @ A )
% 3.15/1.64 => ( member_a_a @ ( F @ X ) @ ( B @ X ) ) )
% 3.15/1.64 => ( member_c_d_a_a @ F @ ( pi_c_d_a_a @ A @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_29_Pi__I,axiom,
% 3.15/1.64 ! [A: set_a_a_a,F: ( a > a > a ) > a,B: ( a > a > a ) > set_a] :
% 3.15/1.64 ( ! [X: a > a > a] :
% 3.15/1.64 ( ( member_a_a_a2 @ X @ A )
% 3.15/1.64 => ( member_a @ ( F @ X ) @ ( B @ X ) ) )
% 3.15/1.64 => ( member_a_a_a_a @ F @ ( pi_a_a_a_a @ A @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_30_Pi__I,axiom,
% 3.15/1.64 ! [A: set_a_a,F: ( a > a ) > c > d,B: ( a > a ) > set_c_d] :
% 3.15/1.64 ( ! [X: a > a] :
% 3.15/1.64 ( ( member_a_a @ X @ A )
% 3.15/1.64 => ( member_c_d @ ( F @ X ) @ ( B @ X ) ) )
% 3.15/1.64 => ( member_a_a_c_d @ F @ ( pi_a_a_c_d @ A @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_31_prodM__sprod__mem,axiom,
% 3.15/1.64 ! [I: set_c,M: c > carrie1948989069_d_a_e,A2: a,M2: c > d] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_c_d @ M2 @ ( carr_p1636662547_d_a_e @ I @ M ) )
% 3.15/1.64 => ( member_c_d @ ( algebr1697692348_c_d_e @ r @ I @ M @ A2 @ M2 ) @ ( carr_p1636662547_d_a_e @ I @ M ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_32_prodM__carr,axiom,
% 3.15/1.64 ! [I: set_c,M: c > carrie1948989069_d_a_e] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 3.15/1.64 => ( ( carrie20288686t_unit @ ( algebr1968153176_c_d_e @ r @ I @ M ) )
% 3.15/1.64 = ( carr_p1636662547_d_a_e @ I @ M ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_33_eq__fun,axiom,
% 3.15/1.64 ! [F: a > c > d,A: set_a,B: set_c_d,G: a > c > d] :
% 3.15/1.64 ( ( member_a_c_d @ F
% 3.15/1.64 @ ( pi_a_c_d @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( member_a_c_d @ G
% 3.15/1.64 @ ( pi_a_c_d @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_34_eq__fun,axiom,
% 3.15/1.64 ! [F: a > a > ( c > d ) > c > d,A: set_a,B: set_a_c_d_c_d,G: a > a > ( c > d ) > c > d] :
% 3.15/1.64 ( ( member_a_a_c_d_c_d @ F
% 3.15/1.64 @ ( pi_a_a_c_d_c_d @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( member_a_a_c_d_c_d @ G
% 3.15/1.64 @ ( pi_a_a_c_d_c_d @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_35_eq__fun,axiom,
% 3.15/1.64 ! [F: a > a > c > d,A: set_a,B: set_a_c_d,G: a > a > c > d] :
% 3.15/1.64 ( ( member_a_a_c_d2 @ F
% 3.15/1.64 @ ( pi_a_a_c_d2 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( member_a_a_c_d2 @ G
% 3.15/1.64 @ ( pi_a_a_c_d2 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_36_eq__fun,axiom,
% 3.15/1.64 ! [F: a > a > a > a > a,A: set_a,B: set_a_a_a_a,G: a > a > a > a > a] :
% 3.15/1.64 ( ( member_a_a_a_a_a3 @ F
% 3.15/1.64 @ ( pi_a_a_a_a_a2 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( member_a_a_a_a_a3 @ G
% 3.15/1.64 @ ( pi_a_a_a_a_a2 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_37_eq__fun,axiom,
% 3.15/1.64 ! [F: a > a > a > a,A: set_a,B: set_a_a_a,G: a > a > a > a] :
% 3.15/1.64 ( ( member_a_a_a_a4 @ F
% 3.15/1.64 @ ( pi_a_a_a_a4 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( member_a_a_a_a4 @ G
% 3.15/1.64 @ ( pi_a_a_a_a4 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_38_eq__fun,axiom,
% 3.15/1.64 ! [F: ( c > d ) > ( c > d ) > c > d,A: set_c_d,B: set_c_d_c_d,G: ( c > d ) > ( c > d ) > c > d] :
% 3.15/1.64 ( ( member_c_d_c_d_c_d @ F
% 3.15/1.64 @ ( pi_c_d_c_d_c_d @ A
% 3.15/1.64 @ ^ [Uu: c > d] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( member_c_d_c_d_c_d @ G
% 3.15/1.64 @ ( pi_c_d_c_d_c_d @ A
% 3.15/1.64 @ ^ [Uu: c > d] : B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_39_eq__fun,axiom,
% 3.15/1.64 ! [F: ( c > d ) > c > d,A: set_c_d,B: set_c_d,G: ( c > d ) > c > d] :
% 3.15/1.64 ( ( member_c_d_c_d @ F
% 3.15/1.64 @ ( pi_c_d_c_d @ A
% 3.15/1.64 @ ^ [Uu: c > d] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( member_c_d_c_d @ G
% 3.15/1.64 @ ( pi_c_d_c_d @ A
% 3.15/1.64 @ ^ [Uu: c > d] : B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_40_eq__fun,axiom,
% 3.15/1.64 ! [F: c > d,A: set_c,B: set_d,G: c > d] :
% 3.15/1.64 ( ( member_c_d @ F
% 3.15/1.64 @ ( pi_c_d @ A
% 3.15/1.64 @ ^ [Uu: c] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( member_c_d @ G
% 3.15/1.64 @ ( pi_c_d @ A
% 3.15/1.64 @ ^ [Uu: c] : B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_41_eq__fun,axiom,
% 3.15/1.64 ! [F: a > a > a,A: set_a,B: set_a_a,G: a > a > a] :
% 3.15/1.64 ( ( member_a_a_a2 @ F
% 3.15/1.64 @ ( pi_a_a_a2 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( member_a_a_a2 @ G
% 3.15/1.64 @ ( pi_a_a_a2 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_42_eq__fun,axiom,
% 3.15/1.64 ! [F: a > a,A: set_a,B: set_a,G: a > a] :
% 3.15/1.64 ( ( member_a_a @ F
% 3.15/1.64 @ ( pi_a_a @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( member_a_a @ G
% 3.15/1.64 @ ( pi_a_a @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_43_funcTr,axiom,
% 3.15/1.64 ! [F: a > c > d,A: set_a,B: set_c_d,G: a > c > d,A2: a] :
% 3.15/1.64 ( ( member_a_c_d @ F
% 3.15/1.64 @ ( pi_a_c_d @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( member_a_c_d @ G
% 3.15/1.64 @ ( pi_a_c_d @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( ( member_a @ A2 @ A )
% 3.15/1.64 => ( ( F @ A2 )
% 3.15/1.64 = ( G @ A2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_44_funcTr,axiom,
% 3.15/1.64 ! [F: a > a > ( c > d ) > c > d,A: set_a,B: set_a_c_d_c_d,G: a > a > ( c > d ) > c > d,A2: a] :
% 3.15/1.64 ( ( member_a_a_c_d_c_d @ F
% 3.15/1.64 @ ( pi_a_a_c_d_c_d @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( member_a_a_c_d_c_d @ G
% 3.15/1.64 @ ( pi_a_a_c_d_c_d @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( ( member_a @ A2 @ A )
% 3.15/1.64 => ( ( F @ A2 )
% 3.15/1.64 = ( G @ A2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_45_funcTr,axiom,
% 3.15/1.64 ! [F: a > a > c > d,A: set_a,B: set_a_c_d,G: a > a > c > d,A2: a] :
% 3.15/1.64 ( ( member_a_a_c_d2 @ F
% 3.15/1.64 @ ( pi_a_a_c_d2 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( member_a_a_c_d2 @ G
% 3.15/1.64 @ ( pi_a_a_c_d2 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( ( member_a @ A2 @ A )
% 3.15/1.64 => ( ( F @ A2 )
% 3.15/1.64 = ( G @ A2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_46_funcTr,axiom,
% 3.15/1.64 ! [F: a > a > a > a > a,A: set_a,B: set_a_a_a_a,G: a > a > a > a > a,A2: a] :
% 3.15/1.64 ( ( member_a_a_a_a_a3 @ F
% 3.15/1.64 @ ( pi_a_a_a_a_a2 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( member_a_a_a_a_a3 @ G
% 3.15/1.64 @ ( pi_a_a_a_a_a2 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( ( member_a @ A2 @ A )
% 3.15/1.64 => ( ( F @ A2 )
% 3.15/1.64 = ( G @ A2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_47_funcTr,axiom,
% 3.15/1.64 ! [F: a > a > a > a,A: set_a,B: set_a_a_a,G: a > a > a > a,A2: a] :
% 3.15/1.64 ( ( member_a_a_a_a4 @ F
% 3.15/1.64 @ ( pi_a_a_a_a4 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( member_a_a_a_a4 @ G
% 3.15/1.64 @ ( pi_a_a_a_a4 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( ( member_a @ A2 @ A )
% 3.15/1.64 => ( ( F @ A2 )
% 3.15/1.64 = ( G @ A2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_48_funcTr,axiom,
% 3.15/1.64 ! [F: ( c > d ) > ( c > d ) > c > d,A: set_c_d,B: set_c_d_c_d,G: ( c > d ) > ( c > d ) > c > d,A2: c > d] :
% 3.15/1.64 ( ( member_c_d_c_d_c_d @ F
% 3.15/1.64 @ ( pi_c_d_c_d_c_d @ A
% 3.15/1.64 @ ^ [Uu: c > d] : B ) )
% 3.15/1.64 => ( ( member_c_d_c_d_c_d @ G
% 3.15/1.64 @ ( pi_c_d_c_d_c_d @ A
% 3.15/1.64 @ ^ [Uu: c > d] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( ( member_c_d @ A2 @ A )
% 3.15/1.64 => ( ( F @ A2 )
% 3.15/1.64 = ( G @ A2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_49_funcTr,axiom,
% 3.15/1.64 ! [F: ( c > d ) > c > d,A: set_c_d,B: set_c_d,G: ( c > d ) > c > d,A2: c > d] :
% 3.15/1.64 ( ( member_c_d_c_d @ F
% 3.15/1.64 @ ( pi_c_d_c_d @ A
% 3.15/1.64 @ ^ [Uu: c > d] : B ) )
% 3.15/1.64 => ( ( member_c_d_c_d @ G
% 3.15/1.64 @ ( pi_c_d_c_d @ A
% 3.15/1.64 @ ^ [Uu: c > d] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( ( member_c_d @ A2 @ A )
% 3.15/1.64 => ( ( F @ A2 )
% 3.15/1.64 = ( G @ A2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_50_funcTr,axiom,
% 3.15/1.64 ! [F: c > d,A: set_c,B: set_d,G: c > d,A2: c] :
% 3.15/1.64 ( ( member_c_d @ F
% 3.15/1.64 @ ( pi_c_d @ A
% 3.15/1.64 @ ^ [Uu: c] : B ) )
% 3.15/1.64 => ( ( member_c_d @ G
% 3.15/1.64 @ ( pi_c_d @ A
% 3.15/1.64 @ ^ [Uu: c] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( ( member_c @ A2 @ A )
% 3.15/1.64 => ( ( F @ A2 )
% 3.15/1.64 = ( G @ A2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_51_funcTr,axiom,
% 3.15/1.64 ! [F: a > a > a,A: set_a,B: set_a_a,G: a > a > a,A2: a] :
% 3.15/1.64 ( ( member_a_a_a2 @ F
% 3.15/1.64 @ ( pi_a_a_a2 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( member_a_a_a2 @ G
% 3.15/1.64 @ ( pi_a_a_a2 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( ( member_a @ A2 @ A )
% 3.15/1.64 => ( ( F @ A2 )
% 3.15/1.64 = ( G @ A2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_52_funcTr,axiom,
% 3.15/1.64 ! [F: a > a,A: set_a,B: set_a,G: a > a,A2: a] :
% 3.15/1.64 ( ( member_a_a @ F
% 3.15/1.64 @ ( pi_a_a @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( member_a_a @ G
% 3.15/1.64 @ ( pi_a_a @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( ( member_a @ A2 @ A )
% 3.15/1.64 => ( ( F @ A2 )
% 3.15/1.64 = ( G @ A2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_53_funcsetI,axiom,
% 3.15/1.64 ! [A: set_c,F: c > d,B: set_d] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ A )
% 3.15/1.64 => ( member_d @ ( F @ X ) @ B ) )
% 3.15/1.64 => ( member_c_d @ F
% 3.15/1.64 @ ( pi_c_d @ A
% 3.15/1.64 @ ^ [Uu: c] : B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_54_funcsetI,axiom,
% 3.15/1.64 ! [A: set_a,F: a > a,B: set_a] :
% 3.15/1.64 ( ! [X: a] :
% 3.15/1.64 ( ( member_a @ X @ A )
% 3.15/1.64 => ( member_a @ ( F @ X ) @ B ) )
% 3.15/1.64 => ( member_a_a @ F
% 3.15/1.64 @ ( pi_a_a @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_55_funcsetI,axiom,
% 3.15/1.64 ! [A: set_c_d,F: ( c > d ) > a,B: set_a] :
% 3.15/1.64 ( ! [X: c > d] :
% 3.15/1.64 ( ( member_c_d @ X @ A )
% 3.15/1.64 => ( member_a @ ( F @ X ) @ B ) )
% 3.15/1.64 => ( member_c_d_a @ F
% 3.15/1.64 @ ( pi_c_d_a @ A
% 3.15/1.64 @ ^ [Uu: c > d] : B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_56_funcsetI,axiom,
% 3.15/1.64 ! [A: set_a_a,F: ( a > a ) > a,B: set_a] :
% 3.15/1.64 ( ! [X: a > a] :
% 3.15/1.64 ( ( member_a_a @ X @ A )
% 3.15/1.64 => ( member_a @ ( F @ X ) @ B ) )
% 3.15/1.64 => ( member_a_a_a @ F
% 3.15/1.64 @ ( pi_a_a_a @ A
% 3.15/1.64 @ ^ [Uu: a > a] : B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_57_funcsetI,axiom,
% 3.15/1.64 ! [A: set_a,F: a > c > d,B: set_c_d] :
% 3.15/1.64 ( ! [X: a] :
% 3.15/1.64 ( ( member_a @ X @ A )
% 3.15/1.64 => ( member_c_d @ ( F @ X ) @ B ) )
% 3.15/1.64 => ( member_a_c_d @ F
% 3.15/1.64 @ ( pi_a_c_d @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_58_funcsetI,axiom,
% 3.15/1.64 ! [A: set_a,F: a > a > a,B: set_a_a] :
% 3.15/1.64 ( ! [X: a] :
% 3.15/1.64 ( ( member_a @ X @ A )
% 3.15/1.64 => ( member_a_a @ ( F @ X ) @ B ) )
% 3.15/1.64 => ( member_a_a_a2 @ F
% 3.15/1.64 @ ( pi_a_a_a2 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_59_funcsetI,axiom,
% 3.15/1.64 ! [A: set_c_d,F: ( c > d ) > c > d,B: set_c_d] :
% 3.15/1.64 ( ! [X: c > d] :
% 3.15/1.64 ( ( member_c_d @ X @ A )
% 3.15/1.64 => ( member_c_d @ ( F @ X ) @ B ) )
% 3.15/1.64 => ( member_c_d_c_d @ F
% 3.15/1.64 @ ( pi_c_d_c_d @ A
% 3.15/1.64 @ ^ [Uu: c > d] : B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_60_funcsetI,axiom,
% 3.15/1.64 ! [A: set_c_d,F: ( c > d ) > a > a,B: set_a_a] :
% 3.15/1.64 ( ! [X: c > d] :
% 3.15/1.64 ( ( member_c_d @ X @ A )
% 3.15/1.64 => ( member_a_a @ ( F @ X ) @ B ) )
% 3.15/1.64 => ( member_c_d_a_a @ F
% 3.15/1.64 @ ( pi_c_d_a_a @ A
% 3.15/1.64 @ ^ [Uu: c > d] : B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_61_funcsetI,axiom,
% 3.15/1.64 ! [A: set_a_a_a,F: ( a > a > a ) > a,B: set_a] :
% 3.15/1.64 ( ! [X: a > a > a] :
% 3.15/1.64 ( ( member_a_a_a2 @ X @ A )
% 3.15/1.64 => ( member_a @ ( F @ X ) @ B ) )
% 3.15/1.64 => ( member_a_a_a_a @ F
% 3.15/1.64 @ ( pi_a_a_a_a @ A
% 3.15/1.64 @ ^ [Uu: a > a > a] : B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_62_funcsetI,axiom,
% 3.15/1.64 ! [A: set_a_a,F: ( a > a ) > c > d,B: set_c_d] :
% 3.15/1.64 ( ! [X: a > a] :
% 3.15/1.64 ( ( member_a_a @ X @ A )
% 3.15/1.64 => ( member_c_d @ ( F @ X ) @ B ) )
% 3.15/1.64 => ( member_a_a_c_d @ F
% 3.15/1.64 @ ( pi_a_a_c_d @ A
% 3.15/1.64 @ ^ [Uu: a > a] : B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_63_eq__funcs,axiom,
% 3.15/1.64 ! [F: a > c > d,A: set_a,B: set_c_d,G: a > c > d,X2: a] :
% 3.15/1.64 ( ( member_a_c_d @ F
% 3.15/1.64 @ ( pi_a_c_d @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( member_a_c_d @ G
% 3.15/1.64 @ ( pi_a_c_d @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( ( member_a @ X2 @ A )
% 3.15/1.64 => ( ( F @ X2 )
% 3.15/1.64 = ( G @ X2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_64_eq__funcs,axiom,
% 3.15/1.64 ! [F: a > a > ( c > d ) > c > d,A: set_a,B: set_a_c_d_c_d,G: a > a > ( c > d ) > c > d,X2: a] :
% 3.15/1.64 ( ( member_a_a_c_d_c_d @ F
% 3.15/1.64 @ ( pi_a_a_c_d_c_d @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( member_a_a_c_d_c_d @ G
% 3.15/1.64 @ ( pi_a_a_c_d_c_d @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( ( member_a @ X2 @ A )
% 3.15/1.64 => ( ( F @ X2 )
% 3.15/1.64 = ( G @ X2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_65_eq__funcs,axiom,
% 3.15/1.64 ! [F: a > a > c > d,A: set_a,B: set_a_c_d,G: a > a > c > d,X2: a] :
% 3.15/1.64 ( ( member_a_a_c_d2 @ F
% 3.15/1.64 @ ( pi_a_a_c_d2 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( member_a_a_c_d2 @ G
% 3.15/1.64 @ ( pi_a_a_c_d2 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( ( member_a @ X2 @ A )
% 3.15/1.64 => ( ( F @ X2 )
% 3.15/1.64 = ( G @ X2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_66_eq__funcs,axiom,
% 3.15/1.64 ! [F: a > a > a > a > a,A: set_a,B: set_a_a_a_a,G: a > a > a > a > a,X2: a] :
% 3.15/1.64 ( ( member_a_a_a_a_a3 @ F
% 3.15/1.64 @ ( pi_a_a_a_a_a2 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( member_a_a_a_a_a3 @ G
% 3.15/1.64 @ ( pi_a_a_a_a_a2 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( ( member_a @ X2 @ A )
% 3.15/1.64 => ( ( F @ X2 )
% 3.15/1.64 = ( G @ X2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_67_eq__funcs,axiom,
% 3.15/1.64 ! [F: a > a > a > a,A: set_a,B: set_a_a_a,G: a > a > a > a,X2: a] :
% 3.15/1.64 ( ( member_a_a_a_a4 @ F
% 3.15/1.64 @ ( pi_a_a_a_a4 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( member_a_a_a_a4 @ G
% 3.15/1.64 @ ( pi_a_a_a_a4 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( ( member_a @ X2 @ A )
% 3.15/1.64 => ( ( F @ X2 )
% 3.15/1.64 = ( G @ X2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_68_eq__funcs,axiom,
% 3.15/1.64 ! [F: ( c > d ) > ( c > d ) > c > d,A: set_c_d,B: set_c_d_c_d,G: ( c > d ) > ( c > d ) > c > d,X2: c > d] :
% 3.15/1.64 ( ( member_c_d_c_d_c_d @ F
% 3.15/1.64 @ ( pi_c_d_c_d_c_d @ A
% 3.15/1.64 @ ^ [Uu: c > d] : B ) )
% 3.15/1.64 => ( ( member_c_d_c_d_c_d @ G
% 3.15/1.64 @ ( pi_c_d_c_d_c_d @ A
% 3.15/1.64 @ ^ [Uu: c > d] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( ( member_c_d @ X2 @ A )
% 3.15/1.64 => ( ( F @ X2 )
% 3.15/1.64 = ( G @ X2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_69_eq__funcs,axiom,
% 3.15/1.64 ! [F: ( c > d ) > c > d,A: set_c_d,B: set_c_d,G: ( c > d ) > c > d,X2: c > d] :
% 3.15/1.64 ( ( member_c_d_c_d @ F
% 3.15/1.64 @ ( pi_c_d_c_d @ A
% 3.15/1.64 @ ^ [Uu: c > d] : B ) )
% 3.15/1.64 => ( ( member_c_d_c_d @ G
% 3.15/1.64 @ ( pi_c_d_c_d @ A
% 3.15/1.64 @ ^ [Uu: c > d] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( ( member_c_d @ X2 @ A )
% 3.15/1.64 => ( ( F @ X2 )
% 3.15/1.64 = ( G @ X2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_70_eq__funcs,axiom,
% 3.15/1.64 ! [F: c > d,A: set_c,B: set_d,G: c > d,X2: c] :
% 3.15/1.64 ( ( member_c_d @ F
% 3.15/1.64 @ ( pi_c_d @ A
% 3.15/1.64 @ ^ [Uu: c] : B ) )
% 3.15/1.64 => ( ( member_c_d @ G
% 3.15/1.64 @ ( pi_c_d @ A
% 3.15/1.64 @ ^ [Uu: c] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( ( member_c @ X2 @ A )
% 3.15/1.64 => ( ( F @ X2 )
% 3.15/1.64 = ( G @ X2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_71_eq__funcs,axiom,
% 3.15/1.64 ! [F: a > a > a,A: set_a,B: set_a_a,G: a > a > a,X2: a] :
% 3.15/1.64 ( ( member_a_a_a2 @ F
% 3.15/1.64 @ ( pi_a_a_a2 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( member_a_a_a2 @ G
% 3.15/1.64 @ ( pi_a_a_a2 @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( ( member_a @ X2 @ A )
% 3.15/1.64 => ( ( F @ X2 )
% 3.15/1.64 = ( G @ X2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_72_eq__funcs,axiom,
% 3.15/1.64 ! [F: a > a,A: set_a,B: set_a,G: a > a,X2: a] :
% 3.15/1.64 ( ( member_a_a @ F
% 3.15/1.64 @ ( pi_a_a @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( member_a_a @ G
% 3.15/1.64 @ ( pi_a_a @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( F = G )
% 3.15/1.64 => ( ( member_a @ X2 @ A )
% 3.15/1.64 => ( ( F @ X2 )
% 3.15/1.64 = ( G @ X2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_73_funcset__id,axiom,
% 3.15/1.64 ! [A: set_c_d] :
% 3.15/1.64 ( member_c_d_c_d
% 3.15/1.64 @ ^ [X3: c > d] : X3
% 3.15/1.64 @ ( pi_c_d_c_d @ A
% 3.15/1.64 @ ^ [Uu: c > d] : A ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_74_funcset__id,axiom,
% 3.15/1.64 ! [A: set_a] :
% 3.15/1.64 ( member_a_a
% 3.15/1.64 @ ^ [X3: a] : X3
% 3.15/1.64 @ ( pi_a_a @ A
% 3.15/1.64 @ ^ [Uu: a] : A ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_75_Algebra1_Ofuncset__mem,axiom,
% 3.15/1.64 ! [F: a > d,A: set_a,B: set_d,X2: a] :
% 3.15/1.64 ( ( member_a_d @ F
% 3.15/1.64 @ ( pi_a_d @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( member_a @ X2 @ A )
% 3.15/1.64 => ( member_d @ ( F @ X2 ) @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_76_Algebra1_Ofuncset__mem,axiom,
% 3.15/1.64 ! [F: a > c,A: set_a,B: set_c,X2: a] :
% 3.15/1.64 ( ( member_a_c @ F
% 3.15/1.64 @ ( pi_a_c @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( member_a @ X2 @ A )
% 3.15/1.64 => ( member_c @ ( F @ X2 ) @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_77_Algebra1_Ofuncset__mem,axiom,
% 3.15/1.64 ! [F: d > a,A: set_d,B: set_a,X2: d] :
% 3.15/1.64 ( ( member_d_a @ F
% 3.15/1.64 @ ( pi_d_a @ A
% 3.15/1.64 @ ^ [Uu: d] : B ) )
% 3.15/1.64 => ( ( member_d @ X2 @ A )
% 3.15/1.64 => ( member_a @ ( F @ X2 ) @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_78_Algebra1_Ofuncset__mem,axiom,
% 3.15/1.64 ! [F: d > d,A: set_d,B: set_d,X2: d] :
% 3.15/1.64 ( ( member_d_d @ F
% 3.15/1.64 @ ( pi_d_d @ A
% 3.15/1.64 @ ^ [Uu: d] : B ) )
% 3.15/1.64 => ( ( member_d @ X2 @ A )
% 3.15/1.64 => ( member_d @ ( F @ X2 ) @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_79_Algebra1_Ofuncset__mem,axiom,
% 3.15/1.64 ! [F: d > c,A: set_d,B: set_c,X2: d] :
% 3.15/1.64 ( ( member_d_c @ F
% 3.15/1.64 @ ( pi_d_c @ A
% 3.15/1.64 @ ^ [Uu: d] : B ) )
% 3.15/1.64 => ( ( member_d @ X2 @ A )
% 3.15/1.64 => ( member_c @ ( F @ X2 ) @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_80_Algebra1_Ofuncset__mem,axiom,
% 3.15/1.64 ! [F: c > a,A: set_c,B: set_a,X2: c] :
% 3.15/1.64 ( ( member_c_a @ F
% 3.15/1.64 @ ( pi_c_a @ A
% 3.15/1.64 @ ^ [Uu: c] : B ) )
% 3.15/1.64 => ( ( member_c @ X2 @ A )
% 3.15/1.64 => ( member_a @ ( F @ X2 ) @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_81_Algebra1_Ofuncset__mem,axiom,
% 3.15/1.64 ! [F: c > c,A: set_c,B: set_c,X2: c] :
% 3.15/1.64 ( ( member_c_c @ F
% 3.15/1.64 @ ( pi_c_c @ A
% 3.15/1.64 @ ^ [Uu: c] : B ) )
% 3.15/1.64 => ( ( member_c @ X2 @ A )
% 3.15/1.64 => ( member_c @ ( F @ X2 ) @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_82_Algebra1_Ofuncset__mem,axiom,
% 3.15/1.64 ! [F: c > d,A: set_c,B: set_d,X2: c] :
% 3.15/1.64 ( ( member_c_d @ F
% 3.15/1.64 @ ( pi_c_d @ A
% 3.15/1.64 @ ^ [Uu: c] : B ) )
% 3.15/1.64 => ( ( member_c @ X2 @ A )
% 3.15/1.64 => ( member_d @ ( F @ X2 ) @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_83_Algebra1_Ofuncset__mem,axiom,
% 3.15/1.64 ! [F: a > a,A: set_a,B: set_a,X2: a] :
% 3.15/1.64 ( ( member_a_a @ F
% 3.15/1.64 @ ( pi_a_a @ A
% 3.15/1.64 @ ^ [Uu: a] : B ) )
% 3.15/1.64 => ( ( member_a @ X2 @ A )
% 3.15/1.64 => ( member_a @ ( F @ X2 ) @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_84_Algebra1_Ofuncset__mem,axiom,
% 3.15/1.64 ! [F: ( c > d ) > a,A: set_c_d,B: set_a,X2: c > d] :
% 3.15/1.64 ( ( member_c_d_a @ F
% 3.15/1.64 @ ( pi_c_d_a @ A
% 3.15/1.64 @ ^ [Uu: c > d] : B ) )
% 3.15/1.64 => ( ( member_c_d @ X2 @ A )
% 3.15/1.64 => ( member_a @ ( F @ X2 ) @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_85_prod__pOp__func,axiom,
% 3.15/1.64 ! [I: set_c,A: c > carrie1948989069_d_a_e] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 3.15/1.64 => ( member_c_d_c_d_c_d @ ( prod_p131511076_d_a_e @ I @ A )
% 3.15/1.64 @ ( pi_c_d_c_d_c_d @ ( carr_p1636662547_d_a_e @ I @ A )
% 3.15/1.64 @ ^ [Uu: c > d] :
% 3.15/1.64 ( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ I @ A )
% 3.15/1.64 @ ^ [Uv: c > d] : ( carr_p1636662547_d_a_e @ I @ A ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_86_ring__is__ag,axiom,
% 3.15/1.64 aGroup2097840802xt_a_b @ r ).
% 3.15/1.64
% 3.15/1.64 thf(fact_87_prodM__mem__eq,axiom,
% 3.15/1.64 ! [I: set_c,M: c > carrie1948989069_d_a_e,M1: c > d,M22: c > d] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 3.15/1.64 => ( ( member_c_d @ M1 @ ( carrie20288686t_unit @ ( algebr1968153176_c_d_e @ r @ I @ M ) ) )
% 3.15/1.64 => ( ( member_c_d @ M22 @ ( carrie20288686t_unit @ ( algebr1968153176_c_d_e @ r @ I @ M ) ) )
% 3.15/1.64 => ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( ( M1 @ X )
% 3.15/1.64 = ( M22 @ X ) ) )
% 3.15/1.64 => ( M1 = M22 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_88_J__rad__mem,axiom,
% 3.15/1.64 ! [X2: a] :
% 3.15/1.64 ( ( member_a @ X2 @ ( j_rad_a_b @ r ) )
% 3.15/1.64 => ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_89_Ring_ORing,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ring_a_b @ R ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_90_Ring_Oring__is__ag,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( aGroup2097840802xt_a_b @ R ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_91_Ring_OprodM__mem__eq,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,I: set_c,M: c > carrie1948989069_d_a_e,M1: c > d,M22: c > d] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 3.15/1.64 => ( ( member_c_d @ M1 @ ( carrie20288686t_unit @ ( algebr1968153176_c_d_e @ R @ I @ M ) ) )
% 3.15/1.64 => ( ( member_c_d @ M22 @ ( carrie20288686t_unit @ ( algebr1968153176_c_d_e @ R @ I @ M ) ) )
% 3.15/1.64 => ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( ( M1 @ X )
% 3.15/1.64 = ( M22 @ X ) ) )
% 3.15/1.64 => ( M1 = M22 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_92_carr__prodag__mem__eq,axiom,
% 3.15/1.64 ! [I: set_a,A: a > carrie1105631105xt_a_b,X4: a > a,Y: a > a] :
% 3.15/1.64 ( ! [X: a] :
% 3.15/1.64 ( ( member_a @ X @ I )
% 3.15/1.64 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_a_a @ X4 @ ( carr_p290614414xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_a_a @ Y @ ( carr_p290614414xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ! [X: a] :
% 3.15/1.64 ( ( member_a @ X @ I )
% 3.15/1.64 => ( ( X4 @ X )
% 3.15/1.64 = ( Y @ X ) ) )
% 3.15/1.64 => ( X4 = Y ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_93_carr__prodag__mem__eq,axiom,
% 3.15/1.64 ! [I: set_c,A: c > carrie1948989069_d_a_e,X4: c > d,Y: c > d] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_c_d @ X4 @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 3.15/1.64 => ( ( member_c_d @ Y @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 3.15/1.64 => ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( ( X4 @ X )
% 3.15/1.64 = ( Y @ X ) ) )
% 3.15/1.64 => ( X4 = Y ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_94_prodag__sameTr1,axiom,
% 3.15/1.64 ! [I: set_c,A: c > carrie1948989069_d_a_e,B: c > carrie1948989069_d_a_e] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 3.15/1.64 => ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( ( A @ X )
% 3.15/1.64 = ( B @ X ) ) )
% 3.15/1.64 => ( ( carr_p1636662547_d_a_e @ I @ A )
% 3.15/1.64 = ( carr_p1636662547_d_a_e @ I @ B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_95_prodag__comp__i,axiom,
% 3.15/1.64 ! [A2: d > a,I: set_d,A: d > carrie1105631105xt_a_b,I2: d] :
% 3.15/1.64 ( ( member_d_a @ A2 @ ( carr_p297433675xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_d @ I2 @ I )
% 3.15/1.64 => ( member_a @ ( A2 @ I2 ) @ ( carrie867757212xt_a_b @ ( A @ I2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_96_prodag__comp__i,axiom,
% 3.15/1.64 ! [A2: c > a,I: set_c,A: c > carrie1105631105xt_a_b,I2: c] :
% 3.15/1.64 ( ( member_c_a @ A2 @ ( carr_p295160588xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_c @ I2 @ I )
% 3.15/1.64 => ( member_a @ ( A2 @ I2 ) @ ( carrie867757212xt_a_b @ ( A @ I2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_97_prodag__comp__i,axiom,
% 3.15/1.64 ! [A2: a > a,I: set_a,A: a > carrie1105631105xt_a_b,I2: a] :
% 3.15/1.64 ( ( member_a_a @ A2 @ ( carr_p290614414xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_a @ I2 @ I )
% 3.15/1.64 => ( member_a @ ( A2 @ I2 ) @ ( carrie867757212xt_a_b @ ( A @ I2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_98_prodag__comp__i,axiom,
% 3.15/1.64 ! [A2: c > d,I: set_c,A: c > carrie1948989069_d_a_e,I2: c] :
% 3.15/1.64 ( ( member_c_d @ A2 @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 3.15/1.64 => ( ( member_c @ I2 @ I )
% 3.15/1.64 => ( member_d @ ( A2 @ I2 ) @ ( carrie11369756_d_a_e @ ( A @ I2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_99_prodag__comp__i,axiom,
% 3.15/1.64 ! [A2: ( c > d ) > a,I: set_c_d,A: ( c > d ) > carrie1105631105xt_a_b,I2: c > d] :
% 3.15/1.64 ( ( member_c_d_a @ A2 @ ( carr_p996491646xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_c_d @ I2 @ I )
% 3.15/1.64 => ( member_a @ ( A2 @ I2 ) @ ( carrie867757212xt_a_b @ ( A @ I2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_100_prodag__comp__i,axiom,
% 3.15/1.64 ! [A2: ( a > a ) > a,I: set_a_a,A: ( a > a ) > carrie1105631105xt_a_b,I2: a > a] :
% 3.15/1.64 ( ( member_a_a_a @ A2 @ ( carr_p740328767xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_a_a @ I2 @ I )
% 3.15/1.64 => ( member_a @ ( A2 @ I2 ) @ ( carrie867757212xt_a_b @ ( A @ I2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_101_prodag__comp__i,axiom,
% 3.15/1.64 ! [A2: ( a > a > a ) > a,I: set_a_a_a,A: ( a > a > a ) > carrie1105631105xt_a_b,I2: a > a > a] :
% 3.15/1.64 ( ( member_a_a_a_a @ A2 @ ( carr_p1481099064xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_a_a_a2 @ I2 @ I )
% 3.15/1.64 => ( member_a @ ( A2 @ I2 ) @ ( carrie867757212xt_a_b @ ( A @ I2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_102_prodag__comp__i,axiom,
% 3.15/1.64 ! [A2: a > c > d,I: set_a,A: a > carrie1954260683t_unit,I2: a] :
% 3.15/1.64 ( ( member_a_c_d @ A2 @ ( carr_p923087418t_unit @ I @ A ) )
% 3.15/1.64 => ( ( member_a @ I2 @ I )
% 3.15/1.64 => ( member_c_d @ ( A2 @ I2 ) @ ( carrie20288686t_unit @ ( A @ I2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_103_prodag__comp__i,axiom,
% 3.15/1.64 ! [A2: d > c > d,I: set_d,A: d > carrie1954260683t_unit,I2: d] :
% 3.15/1.64 ( ( member_d_c_d @ A2 @ ( carr_p1159644861t_unit @ I @ A ) )
% 3.15/1.64 => ( ( member_d @ I2 @ I )
% 3.15/1.64 => ( member_c_d @ ( A2 @ I2 ) @ ( carrie20288686t_unit @ ( A @ I2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_104_prodag__comp__i,axiom,
% 3.15/1.64 ! [A2: c > c > d,I: set_c,A: c > carrie1954260683t_unit,I2: c] :
% 3.15/1.64 ( ( member_c_c_d @ A2 @ ( carr_p1080792380t_unit @ I @ A ) )
% 3.15/1.64 => ( ( member_c @ I2 @ I )
% 3.15/1.64 => ( member_c_d @ ( A2 @ I2 ) @ ( carrie20288686t_unit @ ( A @ I2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_105_prodag__sameTr2,axiom,
% 3.15/1.64 ! [I: set_c,A: c > carrie1948989069_d_a_e,B: c > carrie1948989069_d_a_e] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 3.15/1.64 => ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( ( A @ X )
% 3.15/1.64 = ( B @ X ) ) )
% 3.15/1.64 => ( ( prod_p131511076_d_a_e @ I @ A )
% 3.15/1.64 = ( prod_p131511076_d_a_e @ I @ B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_106_prodag__sameTr3,axiom,
% 3.15/1.64 ! [I: set_c,A: c > carrie1948989069_d_a_e,B: c > carrie1948989069_d_a_e] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 3.15/1.64 => ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( ( A @ X )
% 3.15/1.64 = ( B @ X ) ) )
% 3.15/1.64 => ( ( prod_m854557927_d_a_e @ I @ A )
% 3.15/1.64 = ( prod_m854557927_d_a_e @ I @ B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_107_prodag__sameTr4,axiom,
% 3.15/1.64 ! [I: set_c,A: c > carrie1948989069_d_a_e,B: c > carrie1948989069_d_a_e] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 3.15/1.64 => ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( ( A @ X )
% 3.15/1.64 = ( B @ X ) ) )
% 3.15/1.64 => ( ( prod_z1214987823_d_a_e @ I @ A )
% 3.15/1.64 = ( prod_z1214987823_d_a_e @ I @ B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_108_Ring_OprodM__carr,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,I: set_c,M: c > carrie1948989069_d_a_e] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 3.15/1.64 => ( ( carrie20288686t_unit @ ( algebr1968153176_c_d_e @ R @ I @ M ) )
% 3.15/1.64 = ( carr_p1636662547_d_a_e @ I @ M ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_109_Ring_OprodM__sprod__mem,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,I: set_c,M: c > carrie1948989069_d_a_e,A2: a,M2: c > d] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_c_d @ M2 @ ( carr_p1636662547_d_a_e @ I @ M ) )
% 3.15/1.64 => ( member_c_d @ ( algebr1697692348_c_d_e @ R @ I @ M @ A2 @ M2 ) @ ( carr_p1636662547_d_a_e @ I @ M ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_110_prod__pOp__commute,axiom,
% 3.15/1.64 ! [I: set_a,A: a > carrie1105631105xt_a_b,A2: a > a,B2: a > a] :
% 3.15/1.64 ( ! [X: a] :
% 3.15/1.64 ( ( member_a @ X @ I )
% 3.15/1.64 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_a_a @ A2 @ ( carr_p290614414xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_a_a @ B2 @ ( carr_p290614414xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( prod_p704341949xt_a_b @ I @ A @ A2 @ B2 )
% 3.15/1.64 = ( prod_p704341949xt_a_b @ I @ A @ B2 @ A2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_111_prod__pOp__commute,axiom,
% 3.15/1.64 ! [I: set_c,A: c > carrie1948989069_d_a_e,A2: c > d,B2: c > d] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_c_d @ A2 @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 3.15/1.64 => ( ( member_c_d @ B2 @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 3.15/1.64 => ( ( prod_p131511076_d_a_e @ I @ A @ A2 @ B2 )
% 3.15/1.64 = ( prod_p131511076_d_a_e @ I @ A @ B2 @ A2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_112_prod__pOp__assoc,axiom,
% 3.15/1.64 ! [I: set_a,A: a > carrie1105631105xt_a_b,A2: a > a,B2: a > a,C2: a > a] :
% 3.15/1.64 ( ! [X: a] :
% 3.15/1.64 ( ( member_a @ X @ I )
% 3.15/1.64 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_a_a @ A2 @ ( carr_p290614414xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_a_a @ B2 @ ( carr_p290614414xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_a_a @ C2 @ ( carr_p290614414xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( prod_p704341949xt_a_b @ I @ A @ ( prod_p704341949xt_a_b @ I @ A @ A2 @ B2 ) @ C2 )
% 3.15/1.64 = ( prod_p704341949xt_a_b @ I @ A @ A2 @ ( prod_p704341949xt_a_b @ I @ A @ B2 @ C2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_113_prod__pOp__assoc,axiom,
% 3.15/1.64 ! [I: set_c,A: c > carrie1948989069_d_a_e,A2: c > d,B2: c > d,C2: c > d] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_c_d @ A2 @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 3.15/1.64 => ( ( member_c_d @ B2 @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 3.15/1.64 => ( ( member_c_d @ C2 @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 3.15/1.64 => ( ( prod_p131511076_d_a_e @ I @ A @ ( prod_p131511076_d_a_e @ I @ A @ A2 @ B2 ) @ C2 )
% 3.15/1.64 = ( prod_p131511076_d_a_e @ I @ A @ A2 @ ( prod_p131511076_d_a_e @ I @ A @ B2 @ C2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_114_prod__pOp__mem,axiom,
% 3.15/1.64 ! [I: set_a,A: a > carrie1105631105xt_a_b,X4: a > a,Y: a > a] :
% 3.15/1.64 ( ! [X: a] :
% 3.15/1.64 ( ( member_a @ X @ I )
% 3.15/1.64 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_a_a @ X4 @ ( carr_p290614414xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_a_a @ Y @ ( carr_p290614414xt_a_b @ I @ A ) )
% 3.15/1.64 => ( member_a_a @ ( prod_p704341949xt_a_b @ I @ A @ X4 @ Y ) @ ( carr_p290614414xt_a_b @ I @ A ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_115_prod__pOp__mem,axiom,
% 3.15/1.64 ! [I: set_c,A: c > carrie1948989069_d_a_e,X4: c > d,Y: c > d] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_c_d @ X4 @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 3.15/1.64 => ( ( member_c_d @ Y @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 3.15/1.64 => ( member_c_d @ ( prod_p131511076_d_a_e @ I @ A @ X4 @ Y ) @ ( carr_p1636662547_d_a_e @ I @ A ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_116_prod__mOp__mem,axiom,
% 3.15/1.64 ! [I: set_a,A: a > carrie1105631105xt_a_b,X4: a > a] :
% 3.15/1.64 ( ! [X: a] :
% 3.15/1.64 ( ( member_a @ X @ I )
% 3.15/1.64 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_a_a @ X4 @ ( carr_p290614414xt_a_b @ I @ A ) )
% 3.15/1.64 => ( member_a_a @ ( prod_m2022989242xt_a_b @ I @ A @ X4 ) @ ( carr_p290614414xt_a_b @ I @ A ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_117_prod__mOp__mem,axiom,
% 3.15/1.64 ! [I: set_c,A: c > carrie1948989069_d_a_e,X4: c > d] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_c_d @ X4 @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 3.15/1.64 => ( member_c_d @ ( prod_m854557927_d_a_e @ I @ A @ X4 ) @ ( carr_p1636662547_d_a_e @ I @ A ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_118_prod__zero__func,axiom,
% 3.15/1.64 ! [I: set_a,A: a > carrie1105631105xt_a_b] :
% 3.15/1.64 ( ! [X: a] :
% 3.15/1.64 ( ( member_a @ X @ I )
% 3.15/1.64 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 3.15/1.64 => ( member_a_a @ ( prod_z851058290xt_a_b @ I @ A ) @ ( carr_p290614414xt_a_b @ I @ A ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_119_prod__zero__func,axiom,
% 3.15/1.64 ! [I: set_c,A: c > carrie1948989069_d_a_e] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 3.15/1.64 => ( member_c_d @ ( prod_z1214987823_d_a_e @ I @ A ) @ ( carr_p1636662547_d_a_e @ I @ A ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_120_prod__mOp__func,axiom,
% 3.15/1.64 ! [I: set_c,A: c > carrie1948989069_d_a_e] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 3.15/1.64 => ( member_c_d_c_d @ ( prod_m854557927_d_a_e @ I @ A )
% 3.15/1.64 @ ( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ I @ A )
% 3.15/1.64 @ ^ [Uu: c > d] : ( carr_p1636662547_d_a_e @ I @ A ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_121_Pi__cong,axiom,
% 3.15/1.64 ! [A: set_c_d,F: ( c > d ) > ( c > d ) > c > d,G: ( c > d ) > ( c > d ) > c > d,B: ( c > d ) > set_c_d_c_d] :
% 3.15/1.64 ( ! [W: c > d] :
% 3.15/1.64 ( ( member_c_d @ W @ A )
% 3.15/1.64 => ( ( F @ W )
% 3.15/1.64 = ( G @ W ) ) )
% 3.15/1.64 => ( ( member_c_d_c_d_c_d @ F @ ( pi_c_d_c_d_c_d @ A @ B ) )
% 3.15/1.64 = ( member_c_d_c_d_c_d @ G @ ( pi_c_d_c_d_c_d @ A @ B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_122_Pi__cong,axiom,
% 3.15/1.64 ! [A: set_c_d,F: ( c > d ) > c > d,G: ( c > d ) > c > d,B: ( c > d ) > set_c_d] :
% 3.15/1.64 ( ! [W: c > d] :
% 3.15/1.64 ( ( member_c_d @ W @ A )
% 3.15/1.64 => ( ( F @ W )
% 3.15/1.64 = ( G @ W ) ) )
% 3.15/1.64 => ( ( member_c_d_c_d @ F @ ( pi_c_d_c_d @ A @ B ) )
% 3.15/1.64 = ( member_c_d_c_d @ G @ ( pi_c_d_c_d @ A @ B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_123_Pi__cong,axiom,
% 3.15/1.64 ! [A: set_a,F: a > c > d,G: a > c > d,B: a > set_c_d] :
% 3.15/1.64 ( ! [W: a] :
% 3.15/1.64 ( ( member_a @ W @ A )
% 3.15/1.64 => ( ( F @ W )
% 3.15/1.64 = ( G @ W ) ) )
% 3.15/1.64 => ( ( member_a_c_d @ F @ ( pi_a_c_d @ A @ B ) )
% 3.15/1.64 = ( member_a_c_d @ G @ ( pi_a_c_d @ A @ B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_124_Pi__cong,axiom,
% 3.15/1.64 ! [A: set_a,F: a > a > ( c > d ) > c > d,G: a > a > ( c > d ) > c > d,B: a > set_a_c_d_c_d] :
% 3.15/1.64 ( ! [W: a] :
% 3.15/1.64 ( ( member_a @ W @ A )
% 3.15/1.64 => ( ( F @ W )
% 3.15/1.64 = ( G @ W ) ) )
% 3.15/1.64 => ( ( member_a_a_c_d_c_d @ F @ ( pi_a_a_c_d_c_d @ A @ B ) )
% 3.15/1.64 = ( member_a_a_c_d_c_d @ G @ ( pi_a_a_c_d_c_d @ A @ B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_125_Pi__cong,axiom,
% 3.15/1.64 ! [A: set_a,F: a > a > a,G: a > a > a,B: a > set_a_a] :
% 3.15/1.64 ( ! [W: a] :
% 3.15/1.64 ( ( member_a @ W @ A )
% 3.15/1.64 => ( ( F @ W )
% 3.15/1.64 = ( G @ W ) ) )
% 3.15/1.64 => ( ( member_a_a_a2 @ F @ ( pi_a_a_a2 @ A @ B ) )
% 3.15/1.64 = ( member_a_a_a2 @ G @ ( pi_a_a_a2 @ A @ B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_126_Pi__cong,axiom,
% 3.15/1.64 ! [A: set_a,F: a > a,G: a > a,B: a > set_a] :
% 3.15/1.64 ( ! [W: a] :
% 3.15/1.64 ( ( member_a @ W @ A )
% 3.15/1.64 => ( ( F @ W )
% 3.15/1.64 = ( G @ W ) ) )
% 3.15/1.64 => ( ( member_a_a @ F @ ( pi_a_a @ A @ B ) )
% 3.15/1.64 = ( member_a_a @ G @ ( pi_a_a @ A @ B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_127_Pi__cong,axiom,
% 3.15/1.64 ! [A: set_a,F: a > a > c > d,G: a > a > c > d,B: a > set_a_c_d] :
% 3.15/1.64 ( ! [W: a] :
% 3.15/1.64 ( ( member_a @ W @ A )
% 3.15/1.64 => ( ( F @ W )
% 3.15/1.64 = ( G @ W ) ) )
% 3.15/1.64 => ( ( member_a_a_c_d2 @ F @ ( pi_a_a_c_d2 @ A @ B ) )
% 3.15/1.64 = ( member_a_a_c_d2 @ G @ ( pi_a_a_c_d2 @ A @ B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_128_Pi__cong,axiom,
% 3.15/1.64 ! [A: set_a,F: a > a > a > a > a,G: a > a > a > a > a,B: a > set_a_a_a_a] :
% 3.15/1.64 ( ! [W: a] :
% 3.15/1.64 ( ( member_a @ W @ A )
% 3.15/1.64 => ( ( F @ W )
% 3.15/1.64 = ( G @ W ) ) )
% 3.15/1.64 => ( ( member_a_a_a_a_a3 @ F @ ( pi_a_a_a_a_a2 @ A @ B ) )
% 3.15/1.64 = ( member_a_a_a_a_a3 @ G @ ( pi_a_a_a_a_a2 @ A @ B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_129_Pi__cong,axiom,
% 3.15/1.64 ! [A: set_a,F: a > a > a > a,G: a > a > a > a,B: a > set_a_a_a] :
% 3.15/1.64 ( ! [W: a] :
% 3.15/1.64 ( ( member_a @ W @ A )
% 3.15/1.64 => ( ( F @ W )
% 3.15/1.64 = ( G @ W ) ) )
% 3.15/1.64 => ( ( member_a_a_a_a4 @ F @ ( pi_a_a_a_a4 @ A @ B ) )
% 3.15/1.64 = ( member_a_a_a_a4 @ G @ ( pi_a_a_a_a4 @ A @ B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_130_Pi__cong,axiom,
% 3.15/1.64 ! [A: set_c,F: c > d,G: c > d,B: c > set_d] :
% 3.15/1.64 ( ! [W: c] :
% 3.15/1.64 ( ( member_c @ W @ A )
% 3.15/1.64 => ( ( F @ W )
% 3.15/1.64 = ( G @ W ) ) )
% 3.15/1.64 => ( ( member_c_d @ F @ ( pi_c_d @ A @ B ) )
% 3.15/1.64 = ( member_c_d @ G @ ( pi_c_d @ A @ B ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_131_Pi__mem,axiom,
% 3.15/1.64 ! [F: a > d,A: set_a,B: a > set_d,X2: a] :
% 3.15/1.64 ( ( member_a_d @ F @ ( pi_a_d @ A @ B ) )
% 3.15/1.64 => ( ( member_a @ X2 @ A )
% 3.15/1.64 => ( member_d @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_132_Pi__mem,axiom,
% 3.15/1.64 ! [F: a > c,A: set_a,B: a > set_c,X2: a] :
% 3.15/1.64 ( ( member_a_c @ F @ ( pi_a_c @ A @ B ) )
% 3.15/1.64 => ( ( member_a @ X2 @ A )
% 3.15/1.64 => ( member_c @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_133_Pi__mem,axiom,
% 3.15/1.64 ! [F: d > a,A: set_d,B: d > set_a,X2: d] :
% 3.15/1.64 ( ( member_d_a @ F @ ( pi_d_a @ A @ B ) )
% 3.15/1.64 => ( ( member_d @ X2 @ A )
% 3.15/1.64 => ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_134_Pi__mem,axiom,
% 3.15/1.64 ! [F: d > d,A: set_d,B: d > set_d,X2: d] :
% 3.15/1.64 ( ( member_d_d @ F @ ( pi_d_d @ A @ B ) )
% 3.15/1.64 => ( ( member_d @ X2 @ A )
% 3.15/1.64 => ( member_d @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_135_Pi__mem,axiom,
% 3.15/1.64 ! [F: d > c,A: set_d,B: d > set_c,X2: d] :
% 3.15/1.64 ( ( member_d_c @ F @ ( pi_d_c @ A @ B ) )
% 3.15/1.64 => ( ( member_d @ X2 @ A )
% 3.15/1.64 => ( member_c @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_136_Pi__mem,axiom,
% 3.15/1.64 ! [F: c > a,A: set_c,B: c > set_a,X2: c] :
% 3.15/1.64 ( ( member_c_a @ F @ ( pi_c_a @ A @ B ) )
% 3.15/1.64 => ( ( member_c @ X2 @ A )
% 3.15/1.64 => ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_137_Pi__mem,axiom,
% 3.15/1.64 ! [F: c > c,A: set_c,B: c > set_c,X2: c] :
% 3.15/1.64 ( ( member_c_c @ F @ ( pi_c_c @ A @ B ) )
% 3.15/1.64 => ( ( member_c @ X2 @ A )
% 3.15/1.64 => ( member_c @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_138_Pi__mem,axiom,
% 3.15/1.64 ! [F: c > d,A: set_c,B: c > set_d,X2: c] :
% 3.15/1.64 ( ( member_c_d @ F @ ( pi_c_d @ A @ B ) )
% 3.15/1.64 => ( ( member_c @ X2 @ A )
% 3.15/1.64 => ( member_d @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_139_Pi__mem,axiom,
% 3.15/1.64 ! [F: a > a,A: set_a,B: a > set_a,X2: a] :
% 3.15/1.64 ( ( member_a_a @ F @ ( pi_a_a @ A @ B ) )
% 3.15/1.64 => ( ( member_a @ X2 @ A )
% 3.15/1.64 => ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_140_Pi__mem,axiom,
% 3.15/1.64 ! [F: ( c > d ) > a,A: set_c_d,B: ( c > d ) > set_a,X2: c > d] :
% 3.15/1.64 ( ( member_c_d_a @ F @ ( pi_c_d_a @ A @ B ) )
% 3.15/1.64 => ( ( member_c_d @ X2 @ A )
% 3.15/1.64 => ( member_a @ ( F @ X2 ) @ ( B @ X2 ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_141_Pi__iff,axiom,
% 3.15/1.64 ! [F: a > a > ( c > d ) > c > d,I: set_a,X4: a > set_a_c_d_c_d] :
% 3.15/1.64 ( ( member_a_a_c_d_c_d @ F @ ( pi_a_a_c_d_c_d @ I @ X4 ) )
% 3.15/1.64 = ( ! [X3: a] :
% 3.15/1.64 ( ( member_a @ X3 @ I )
% 3.15/1.64 => ( member_a_c_d_c_d @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_142_Pi__iff,axiom,
% 3.15/1.64 ! [F: ( c > d ) > ( c > d ) > c > d,I: set_c_d,X4: ( c > d ) > set_c_d_c_d] :
% 3.15/1.64 ( ( member_c_d_c_d_c_d @ F @ ( pi_c_d_c_d_c_d @ I @ X4 ) )
% 3.15/1.64 = ( ! [X3: c > d] :
% 3.15/1.64 ( ( member_c_d @ X3 @ I )
% 3.15/1.64 => ( member_c_d_c_d @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_143_Pi__iff,axiom,
% 3.15/1.64 ! [F: a > c > d,I: set_a,X4: a > set_c_d] :
% 3.15/1.64 ( ( member_a_c_d @ F @ ( pi_a_c_d @ I @ X4 ) )
% 3.15/1.64 = ( ! [X3: a] :
% 3.15/1.64 ( ( member_a @ X3 @ I )
% 3.15/1.64 => ( member_c_d @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_144_Pi__iff,axiom,
% 3.15/1.64 ! [F: ( c > d ) > c > d,I: set_c_d,X4: ( c > d ) > set_c_d] :
% 3.15/1.64 ( ( member_c_d_c_d @ F @ ( pi_c_d_c_d @ I @ X4 ) )
% 3.15/1.64 = ( ! [X3: c > d] :
% 3.15/1.64 ( ( member_c_d @ X3 @ I )
% 3.15/1.64 => ( member_c_d @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_145_Pi__iff,axiom,
% 3.15/1.64 ! [F: c > d,I: set_c,X4: c > set_d] :
% 3.15/1.64 ( ( member_c_d @ F @ ( pi_c_d @ I @ X4 ) )
% 3.15/1.64 = ( ! [X3: c] :
% 3.15/1.64 ( ( member_c @ X3 @ I )
% 3.15/1.64 => ( member_d @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_146_Pi__iff,axiom,
% 3.15/1.64 ! [F: a > a > a,I: set_a,X4: a > set_a_a] :
% 3.15/1.64 ( ( member_a_a_a2 @ F @ ( pi_a_a_a2 @ I @ X4 ) )
% 3.15/1.64 = ( ! [X3: a] :
% 3.15/1.64 ( ( member_a @ X3 @ I )
% 3.15/1.64 => ( member_a_a @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_147_Pi__iff,axiom,
% 3.15/1.64 ! [F: a > a,I: set_a,X4: a > set_a] :
% 3.15/1.64 ( ( member_a_a @ F @ ( pi_a_a @ I @ X4 ) )
% 3.15/1.64 = ( ! [X3: a] :
% 3.15/1.64 ( ( member_a @ X3 @ I )
% 3.15/1.64 => ( member_a @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_148_Pi__iff,axiom,
% 3.15/1.64 ! [F: a > a > c > d,I: set_a,X4: a > set_a_c_d] :
% 3.15/1.64 ( ( member_a_a_c_d2 @ F @ ( pi_a_a_c_d2 @ I @ X4 ) )
% 3.15/1.64 = ( ! [X3: a] :
% 3.15/1.64 ( ( member_a @ X3 @ I )
% 3.15/1.64 => ( member_a_c_d @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_149_Pi__iff,axiom,
% 3.15/1.64 ! [F: a > a > a > a > a,I: set_a,X4: a > set_a_a_a_a] :
% 3.15/1.64 ( ( member_a_a_a_a_a3 @ F @ ( pi_a_a_a_a_a2 @ I @ X4 ) )
% 3.15/1.64 = ( ! [X3: a] :
% 3.15/1.64 ( ( member_a @ X3 @ I )
% 3.15/1.64 => ( member_a_a_a_a4 @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_150_Pi__iff,axiom,
% 3.15/1.64 ! [F: a > a > a > a,I: set_a,X4: a > set_a_a_a] :
% 3.15/1.64 ( ( member_a_a_a_a4 @ F @ ( pi_a_a_a_a4 @ I @ X4 ) )
% 3.15/1.64 = ( ! [X3: a] :
% 3.15/1.64 ( ( member_a @ X3 @ I )
% 3.15/1.64 => ( member_a_a_a2 @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_151_Pi__I_H,axiom,
% 3.15/1.64 ! [A: set_a,F: a > a,B: a > set_a] :
% 3.15/1.64 ( ! [X: a] :
% 3.15/1.64 ( ( member_a @ X @ A )
% 3.15/1.64 => ( member_a @ ( F @ X ) @ ( B @ X ) ) )
% 3.15/1.64 => ( member_a_a @ F @ ( pi_a_a @ A @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_152_Pi__I_H,axiom,
% 3.15/1.64 ! [A: set_a,F: a > d,B: a > set_d] :
% 3.15/1.64 ( ! [X: a] :
% 3.15/1.64 ( ( member_a @ X @ A )
% 3.15/1.64 => ( member_d @ ( F @ X ) @ ( B @ X ) ) )
% 3.15/1.64 => ( member_a_d @ F @ ( pi_a_d @ A @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_153_Pi__I_H,axiom,
% 3.15/1.64 ! [A: set_a,F: a > c,B: a > set_c] :
% 3.15/1.64 ( ! [X: a] :
% 3.15/1.64 ( ( member_a @ X @ A )
% 3.15/1.64 => ( member_c @ ( F @ X ) @ ( B @ X ) ) )
% 3.15/1.64 => ( member_a_c @ F @ ( pi_a_c @ A @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_154_Pi__I_H,axiom,
% 3.15/1.64 ! [A: set_d,F: d > a,B: d > set_a] :
% 3.15/1.64 ( ! [X: d] :
% 3.15/1.64 ( ( member_d @ X @ A )
% 3.15/1.64 => ( member_a @ ( F @ X ) @ ( B @ X ) ) )
% 3.15/1.64 => ( member_d_a @ F @ ( pi_d_a @ A @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_155_Pi__I_H,axiom,
% 3.15/1.64 ! [A: set_d,F: d > d,B: d > set_d] :
% 3.15/1.64 ( ! [X: d] :
% 3.15/1.64 ( ( member_d @ X @ A )
% 3.15/1.64 => ( member_d @ ( F @ X ) @ ( B @ X ) ) )
% 3.15/1.64 => ( member_d_d @ F @ ( pi_d_d @ A @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_156_Pi__I_H,axiom,
% 3.15/1.64 ! [A: set_d,F: d > c,B: d > set_c] :
% 3.15/1.64 ( ! [X: d] :
% 3.15/1.64 ( ( member_d @ X @ A )
% 3.15/1.64 => ( member_c @ ( F @ X ) @ ( B @ X ) ) )
% 3.15/1.64 => ( member_d_c @ F @ ( pi_d_c @ A @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_157_Pi__I_H,axiom,
% 3.15/1.64 ! [A: set_c,F: c > a,B: c > set_a] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ A )
% 3.15/1.64 => ( member_a @ ( F @ X ) @ ( B @ X ) ) )
% 3.15/1.64 => ( member_c_a @ F @ ( pi_c_a @ A @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_158_Pi__I_H,axiom,
% 3.15/1.64 ! [A: set_c,F: c > d,B: c > set_d] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ A )
% 3.15/1.64 => ( member_d @ ( F @ X ) @ ( B @ X ) ) )
% 3.15/1.64 => ( member_c_d @ F @ ( pi_c_d @ A @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_159_Pi__I_H,axiom,
% 3.15/1.64 ! [A: set_c,F: c > c,B: c > set_c] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ A )
% 3.15/1.64 => ( member_c @ ( F @ X ) @ ( B @ X ) ) )
% 3.15/1.64 => ( member_c_c @ F @ ( pi_c_c @ A @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_160_Pi__I_H,axiom,
% 3.15/1.64 ! [A: set_c_d,F: ( c > d ) > a,B: ( c > d ) > set_a] :
% 3.15/1.64 ( ! [X: c > d] :
% 3.15/1.64 ( ( member_c_d @ X @ A )
% 3.15/1.64 => ( member_a @ ( F @ X ) @ ( B @ X ) ) )
% 3.15/1.64 => ( member_c_d_a @ F @ ( pi_c_d_a @ A @ B ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_161_PiE,axiom,
% 3.15/1.64 ! [F: d > a,A: set_d,B: d > set_a,X2: d] :
% 3.15/1.64 ( ( member_d_a @ F @ ( pi_d_a @ A @ B ) )
% 3.15/1.64 => ( ~ ( member_a @ ( F @ X2 ) @ ( B @ X2 ) )
% 3.15/1.64 => ~ ( member_d @ X2 @ A ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_162_PiE,axiom,
% 3.15/1.64 ! [F: c > a,A: set_c,B: c > set_a,X2: c] :
% 3.15/1.64 ( ( member_c_a @ F @ ( pi_c_a @ A @ B ) )
% 3.15/1.64 => ( ~ ( member_a @ ( F @ X2 ) @ ( B @ X2 ) )
% 3.15/1.64 => ~ ( member_c @ X2 @ A ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_163_PiE,axiom,
% 3.15/1.64 ! [F: a > d,A: set_a,B: a > set_d,X2: a] :
% 3.15/1.64 ( ( member_a_d @ F @ ( pi_a_d @ A @ B ) )
% 3.15/1.64 => ( ~ ( member_d @ ( F @ X2 ) @ ( B @ X2 ) )
% 3.15/1.64 => ~ ( member_a @ X2 @ A ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_164_PiE,axiom,
% 3.15/1.64 ! [F: d > d,A: set_d,B: d > set_d,X2: d] :
% 3.15/1.64 ( ( member_d_d @ F @ ( pi_d_d @ A @ B ) )
% 3.15/1.64 => ( ~ ( member_d @ ( F @ X2 ) @ ( B @ X2 ) )
% 3.15/1.64 => ~ ( member_d @ X2 @ A ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_165_PiE,axiom,
% 3.15/1.64 ! [F: a > c,A: set_a,B: a > set_c,X2: a] :
% 3.15/1.64 ( ( member_a_c @ F @ ( pi_a_c @ A @ B ) )
% 3.15/1.64 => ( ~ ( member_c @ ( F @ X2 ) @ ( B @ X2 ) )
% 3.15/1.64 => ~ ( member_a @ X2 @ A ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_166_PiE,axiom,
% 3.15/1.64 ! [F: d > c,A: set_d,B: d > set_c,X2: d] :
% 3.15/1.64 ( ( member_d_c @ F @ ( pi_d_c @ A @ B ) )
% 3.15/1.64 => ( ~ ( member_c @ ( F @ X2 ) @ ( B @ X2 ) )
% 3.15/1.64 => ~ ( member_d @ X2 @ A ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_167_PiE,axiom,
% 3.15/1.64 ! [F: c > c,A: set_c,B: c > set_c,X2: c] :
% 3.15/1.64 ( ( member_c_c @ F @ ( pi_c_c @ A @ B ) )
% 3.15/1.64 => ( ~ ( member_c @ ( F @ X2 ) @ ( B @ X2 ) )
% 3.15/1.64 => ~ ( member_c @ X2 @ A ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_168_PiE,axiom,
% 3.15/1.64 ! [F: c > d,A: set_c,B: c > set_d,X2: c] :
% 3.15/1.64 ( ( member_c_d @ F @ ( pi_c_d @ A @ B ) )
% 3.15/1.64 => ( ~ ( member_d @ ( F @ X2 ) @ ( B @ X2 ) )
% 3.15/1.64 => ~ ( member_c @ X2 @ A ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_169_PiE,axiom,
% 3.15/1.64 ! [F: a > a,A: set_a,B: a > set_a,X2: a] :
% 3.15/1.64 ( ( member_a_a @ F @ ( pi_a_a @ A @ B ) )
% 3.15/1.64 => ( ~ ( member_a @ ( F @ X2 ) @ ( B @ X2 ) )
% 3.15/1.64 => ~ ( member_a @ X2 @ A ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_170_PiE,axiom,
% 3.15/1.64 ! [F: a > c > d,A: set_a,B: a > set_c_d,X2: a] :
% 3.15/1.64 ( ( member_a_c_d @ F @ ( pi_a_c_d @ A @ B ) )
% 3.15/1.64 => ( ~ ( member_c_d @ ( F @ X2 ) @ ( B @ X2 ) )
% 3.15/1.64 => ~ ( member_a @ X2 @ A ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_171_mem__subring__mem__ring,axiom,
% 3.15/1.64 ! [S: carrie1105631105xt_a_b,X2: a] :
% 3.15/1.64 ( ( subring_a_b_b @ r @ S )
% 3.15/1.64 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ S ) )
% 3.15/1.64 => ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_172_subring__Ring,axiom,
% 3.15/1.64 ! [S: carrie1105631105xt_a_b] :
% 3.15/1.64 ( ( subring_a_b_b @ r @ S )
% 3.15/1.64 => ( ring_a_b @ S ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_173_prodM__sprod__val,axiom,
% 3.15/1.64 ! [I: set_c_d_c_d,M: ( ( c > d ) > c > d ) > carrie1948989069_d_a_e,A2: a,M2: ( ( c > d ) > c > d ) > d,J: ( c > d ) > c > d] :
% 3.15/1.64 ( ! [X: ( c > d ) > c > d] :
% 3.15/1.64 ( ( member_c_d_c_d @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_c_d_c_d_d @ M2 @ ( carr_p428609062_d_a_e @ I @ M ) )
% 3.15/1.64 => ( ( member_c_d_c_d @ J @ I )
% 3.15/1.64 => ( ( algebr828196337_d_d_e @ r @ I @ M @ A2 @ M2 @ J )
% 3.15/1.64 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_174_prodM__sprod__val,axiom,
% 3.15/1.64 ! [I: set_c_d,M: ( c > d ) > carrie1948989069_d_a_e,A2: a,M2: ( c > d ) > d,J: c > d] :
% 3.15/1.64 ( ! [X: c > d] :
% 3.15/1.64 ( ( member_c_d @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_c_d_d @ M2 @ ( carr_p124510241_d_a_e @ I @ M ) )
% 3.15/1.64 => ( ( member_c_d @ J @ I )
% 3.15/1.64 => ( ( algebr924632044_d_d_e @ r @ I @ M @ A2 @ M2 @ J )
% 3.15/1.64 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_175_prodM__sprod__val,axiom,
% 3.15/1.64 ! [I: set_a_a_a,M: ( a > a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > a ) > d,J: a > a > a] :
% 3.15/1.64 ( ! [X: a > a > a] :
% 3.15/1.64 ( ( member_a_a_a2 @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a_a_a_d @ M2 @ ( carr_p1364179431_d_a_e @ I @ M ) )
% 3.15/1.64 => ( ( member_a_a_a2 @ J @ I )
% 3.15/1.64 => ( ( algebr1655014672_a_d_e @ r @ I @ M @ A2 @ M2 @ J )
% 3.15/1.64 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_176_prodM__sprod__val,axiom,
% 3.15/1.64 ! [I: set_a_a,M: ( a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a ) > d,J: a > a] :
% 3.15/1.64 ( ! [X: a > a] :
% 3.15/1.64 ( ( member_a_a @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a_a_d @ M2 @ ( carr_p1646427936_d_a_e @ I @ M ) )
% 3.15/1.64 => ( ( member_a_a @ J @ I )
% 3.15/1.64 => ( ( algebr137561451_a_d_e @ r @ I @ M @ A2 @ M2 @ J )
% 3.15/1.64 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_177_prodM__sprod__val,axiom,
% 3.15/1.64 ! [I: set_a,M: a > carrie1948989069_d_a_e,A2: a,M2: a > d,J: a] :
% 3.15/1.64 ( ! [X: a] :
% 3.15/1.64 ( ( member_a @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a_d @ M2 @ ( carr_p1084104977_d_a_e @ I @ M ) )
% 3.15/1.64 => ( ( member_a @ J @ I )
% 3.15/1.64 => ( ( algebr2109842362_a_d_e @ r @ I @ M @ A2 @ M2 @ J )
% 3.15/1.64 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_178_prodM__sprod__val,axiom,
% 3.15/1.64 ! [I: set_a_a_c_d,M: ( a > a > c > d ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > c > d ) > d,J: a > a > c > d] :
% 3.15/1.64 ( ! [X: a > a > c > d] :
% 3.15/1.64 ( ( member_a_a_c_d2 @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a_a_c_d_d @ M2 @ ( carr_p577380087_d_a_e @ I @ M ) )
% 3.15/1.64 => ( ( member_a_a_c_d2 @ J @ I )
% 3.15/1.64 => ( ( algebr1620716610_d_d_e @ r @ I @ M @ A2 @ M2 @ J )
% 3.15/1.64 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_179_prodM__sprod__val,axiom,
% 3.15/1.64 ! [I: set_a_a_a_a_a,M: ( a > a > a > a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > a > a > a ) > d,J: a > a > a > a > a] :
% 3.15/1.64 ( ! [X: a > a > a > a > a] :
% 3.15/1.64 ( ( member_a_a_a_a_a3 @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a_a_a_a_a_d @ M2 @ ( carr_p1980447933_d_a_e @ I @ M ) )
% 3.15/1.64 => ( ( member_a_a_a_a_a3 @ J @ I )
% 3.15/1.64 => ( ( algebr2122715238_a_d_e @ r @ I @ M @ A2 @ M2 @ J )
% 3.15/1.64 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_180_prodM__sprod__val,axiom,
% 3.15/1.64 ! [I: set_a_a_a_a,M: ( a > a > a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > a > a ) > d,J: a > a > a > a] :
% 3.15/1.64 ( ! [X: a > a > a > a] :
% 3.15/1.64 ( ( member_a_a_a_a4 @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a_a_a_a_d @ M2 @ ( carr_p305487606_d_a_e @ I @ M ) )
% 3.15/1.64 => ( ( member_a_a_a_a4 @ J @ I )
% 3.15/1.64 => ( ( algebr303206081_a_d_e @ r @ I @ M @ A2 @ M2 @ J )
% 3.15/1.64 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_181_prodM__sprod__val,axiom,
% 3.15/1.64 ! [I: set_d,M: d > carrie1948989069_d_a_e,A2: a,M2: d > d,J: d] :
% 3.15/1.64 ( ! [X: d] :
% 3.15/1.64 ( ( member_d @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_d_d @ M2 @ ( carr_p1912941332_d_a_e @ I @ M ) )
% 3.15/1.64 => ( ( member_d @ J @ I )
% 3.15/1.64 => ( ( algebr417875517_d_d_e @ r @ I @ M @ A2 @ M2 @ J )
% 3.15/1.64 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_182_prodM__sprod__val,axiom,
% 3.15/1.64 ! [I: set_c,M: c > carrie1948989069_d_a_e,A2: a,M2: c > d,J: c] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ r ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_c_d @ M2 @ ( carr_p1636662547_d_a_e @ I @ M ) )
% 3.15/1.64 => ( ( member_c @ J @ I )
% 3.15/1.64 => ( ( algebr1697692348_c_d_e @ r @ I @ M @ A2 @ M2 @ J )
% 3.15/1.64 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_183_Sr__ring,axiom,
% 3.15/1.64 ! [S: set_a] :
% 3.15/1.64 ( ( sr_a_b2 @ r @ S )
% 3.15/1.64 => ( ring_a_b @ ( sr_a_b @ r @ S ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_184_mHom__func,axiom,
% 3.15/1.64 ! [F: ( c > d ) > c > d,M: carrie1954260683t_unit,N: carrie1954260683t_unit] :
% 3.15/1.64 ( ( member_c_d_c_d @ F @ ( mHom_a1526663829t_unit @ r @ M @ N ) )
% 3.15/1.64 => ( member_c_d_c_d @ F
% 3.15/1.64 @ ( pi_c_d_c_d @ ( carrie20288686t_unit @ M )
% 3.15/1.64 @ ^ [Uu: c > d] : ( carrie20288686t_unit @ N ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_185_dsum__pOp__func,axiom,
% 3.15/1.64 ! [I: set_c,A: c > carrie1948989069_d_a_e] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 3.15/1.64 => ( member_c_d_c_d_c_d @ ( prod_p131511076_d_a_e @ I @ A )
% 3.15/1.64 @ ( pi_c_d_c_d_c_d @ ( carr_d56066307_d_a_e @ I @ A )
% 3.15/1.64 @ ^ [Uu: c > d] :
% 3.15/1.64 ( pi_c_d_c_d @ ( carr_d56066307_d_a_e @ I @ A )
% 3.15/1.64 @ ^ [Uv: c > d] : ( carr_d56066307_d_a_e @ I @ A ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_186_mem__Collect__eq,axiom,
% 3.15/1.64 ! [A2: a,P: a > $o] :
% 3.15/1.64 ( ( member_a @ A2 @ ( collect_a @ P ) )
% 3.15/1.64 = ( P @ A2 ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_187_mem__Collect__eq,axiom,
% 3.15/1.64 ! [A2: d,P: d > $o] :
% 3.15/1.64 ( ( member_d @ A2 @ ( collect_d @ P ) )
% 3.15/1.64 = ( P @ A2 ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_188_mem__Collect__eq,axiom,
% 3.15/1.64 ! [A2: c,P: c > $o] :
% 3.15/1.64 ( ( member_c @ A2 @ ( collect_c @ P ) )
% 3.15/1.64 = ( P @ A2 ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_189_mem__Collect__eq,axiom,
% 3.15/1.64 ! [A2: nat,P: nat > $o] :
% 3.15/1.64 ( ( member_nat @ A2 @ ( collect_nat @ P ) )
% 3.15/1.64 = ( P @ A2 ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_190_mem__Collect__eq,axiom,
% 3.15/1.64 ! [A2: c > d,P: ( c > d ) > $o] :
% 3.15/1.64 ( ( member_c_d @ A2 @ ( collect_c_d @ P ) )
% 3.15/1.64 = ( P @ A2 ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_191_mem__Collect__eq,axiom,
% 3.15/1.64 ! [A2: a > a,P: ( a > a ) > $o] :
% 3.15/1.64 ( ( member_a_a @ A2 @ ( collect_a_a @ P ) )
% 3.15/1.64 = ( P @ A2 ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_192_mem__Collect__eq,axiom,
% 3.15/1.64 ! [A2: a > a > a,P: ( a > a > a ) > $o] :
% 3.15/1.64 ( ( member_a_a_a2 @ A2 @ ( collect_a_a_a @ P ) )
% 3.15/1.64 = ( P @ A2 ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_193_mem__Collect__eq,axiom,
% 3.15/1.64 ! [A2: ( c > d ) > c > d,P: ( ( c > d ) > c > d ) > $o] :
% 3.15/1.64 ( ( member_c_d_c_d @ A2 @ ( collect_c_d_c_d @ P ) )
% 3.15/1.64 = ( P @ A2 ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_194_mem__Collect__eq,axiom,
% 3.15/1.64 ! [A2: a > a > c > d,P: ( a > a > c > d ) > $o] :
% 3.15/1.64 ( ( member_a_a_c_d2 @ A2 @ ( collect_a_a_c_d @ P ) )
% 3.15/1.64 = ( P @ A2 ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_195_mem__Collect__eq,axiom,
% 3.15/1.64 ! [A2: a > a > a > a,P: ( a > a > a > a ) > $o] :
% 3.15/1.64 ( ( member_a_a_a_a4 @ A2 @ ( collect_a_a_a_a @ P ) )
% 3.15/1.64 = ( P @ A2 ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_196_Collect__mem__eq,axiom,
% 3.15/1.64 ! [A: set_a] :
% 3.15/1.64 ( ( collect_a
% 3.15/1.64 @ ^ [X3: a] : ( member_a @ X3 @ A ) )
% 3.15/1.64 = A ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_197_Collect__mem__eq,axiom,
% 3.15/1.64 ! [A: set_d] :
% 3.15/1.64 ( ( collect_d
% 3.15/1.64 @ ^ [X3: d] : ( member_d @ X3 @ A ) )
% 3.15/1.64 = A ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_198_Collect__mem__eq,axiom,
% 3.15/1.64 ! [A: set_c] :
% 3.15/1.64 ( ( collect_c
% 3.15/1.64 @ ^ [X3: c] : ( member_c @ X3 @ A ) )
% 3.15/1.64 = A ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_199_Collect__mem__eq,axiom,
% 3.15/1.64 ! [A: set_nat] :
% 3.15/1.64 ( ( collect_nat
% 3.15/1.64 @ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
% 3.15/1.64 = A ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_200_Collect__mem__eq,axiom,
% 3.15/1.64 ! [A: set_c_d] :
% 3.15/1.64 ( ( collect_c_d
% 3.15/1.64 @ ^ [X3: c > d] : ( member_c_d @ X3 @ A ) )
% 3.15/1.64 = A ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_201_Collect__mem__eq,axiom,
% 3.15/1.64 ! [A: set_a_a] :
% 3.15/1.64 ( ( collect_a_a
% 3.15/1.64 @ ^ [X3: a > a] : ( member_a_a @ X3 @ A ) )
% 3.15/1.64 = A ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_202_Collect__mem__eq,axiom,
% 3.15/1.64 ! [A: set_a_a_a] :
% 3.15/1.64 ( ( collect_a_a_a
% 3.15/1.64 @ ^ [X3: a > a > a] : ( member_a_a_a2 @ X3 @ A ) )
% 3.15/1.64 = A ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_203_Collect__mem__eq,axiom,
% 3.15/1.64 ! [A: set_c_d_c_d] :
% 3.15/1.64 ( ( collect_c_d_c_d
% 3.15/1.64 @ ^ [X3: ( c > d ) > c > d] : ( member_c_d_c_d @ X3 @ A ) )
% 3.15/1.64 = A ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_204_Collect__mem__eq,axiom,
% 3.15/1.64 ! [A: set_a_a_c_d] :
% 3.15/1.64 ( ( collect_a_a_c_d
% 3.15/1.64 @ ^ [X3: a > a > c > d] : ( member_a_a_c_d2 @ X3 @ A ) )
% 3.15/1.64 = A ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_205_Collect__mem__eq,axiom,
% 3.15/1.64 ! [A: set_a_a_a_a] :
% 3.15/1.64 ( ( collect_a_a_a_a
% 3.15/1.64 @ ^ [X3: a > a > a > a] : ( member_a_a_a_a4 @ X3 @ A ) )
% 3.15/1.64 = A ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_206_Collect__cong,axiom,
% 3.15/1.64 ! [P: nat > $o,Q: nat > $o] :
% 3.15/1.64 ( ! [X: nat] :
% 3.15/1.64 ( ( P @ X )
% 3.15/1.64 = ( Q @ X ) )
% 3.15/1.64 => ( ( collect_nat @ P )
% 3.15/1.64 = ( collect_nat @ Q ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_207_nsEqElm,axiom,
% 3.15/1.64 ! [X2: a,Y2: a,N2: nat] :
% 3.15/1.64 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( X2 = Y2 )
% 3.15/1.64 => ( ( nscal_a_b @ r @ X2 @ N2 )
% 3.15/1.64 = ( nscal_a_b @ r @ Y2 @ N2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_208_nsClose,axiom,
% 3.15/1.64 ! [X2: a,N2: nat] :
% 3.15/1.64 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( member_a @ ( nscal_a_b @ r @ X2 @ N2 ) @ ( carrie867757212xt_a_b @ r ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_209_tp__closed,axiom,
% 3.15/1.64 ( member_a_a_a2 @ ( tp_a_b @ r )
% 3.15/1.64 @ ( pi_a_a_a2 @ ( carrie867757212xt_a_b @ r )
% 3.15/1.64 @ ^ [Uu: a] :
% 3.15/1.64 ( pi_a_a @ ( carrie867757212xt_a_b @ r )
% 3.15/1.64 @ ^ [Uv: a] : ( carrie867757212xt_a_b @ r ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_210_pop__closed,axiom,
% 3.15/1.64 ( member_a_a_a2 @ ( pop_a_Ring_ext_a_b @ r )
% 3.15/1.64 @ ( pi_a_a_a2 @ ( carrie867757212xt_a_b @ r )
% 3.15/1.64 @ ^ [Uu: a] :
% 3.15/1.64 ( pi_a_a @ ( carrie867757212xt_a_b @ r )
% 3.15/1.64 @ ^ [Uv: a] : ( carrie867757212xt_a_b @ r ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_211_rEQMulR,axiom,
% 3.15/1.64 ! [X2: a,Y2: a,Z: a] :
% 3.15/1.64 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ Z @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( X2 = Y2 )
% 3.15/1.64 => ( ( tp_a_b @ r @ X2 @ Z )
% 3.15/1.64 = ( tp_a_b @ r @ Y2 @ Z ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_212_rMulLC,axiom,
% 3.15/1.64 ! [X2: a,Y2: a,Z: a] :
% 3.15/1.64 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ Z @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( tp_a_b @ r @ X2 @ ( tp_a_b @ r @ Y2 @ Z ) )
% 3.15/1.64 = ( tp_a_b @ r @ Y2 @ ( tp_a_b @ r @ X2 @ Z ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_213_ring__tOp__assoc,axiom,
% 3.15/1.64 ! [X2: a,Y2: a,Z: a] :
% 3.15/1.64 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ Z @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( tp_a_b @ r @ ( tp_a_b @ r @ X2 @ Y2 ) @ Z )
% 3.15/1.64 = ( tp_a_b @ r @ X2 @ ( tp_a_b @ r @ Y2 @ Z ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_214_ring__tOp__closed,axiom,
% 3.15/1.64 ! [X2: a,Y2: a] :
% 3.15/1.64 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( member_a @ ( tp_a_b @ r @ X2 @ Y2 ) @ ( carrie867757212xt_a_b @ r ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_215_ring__tOp__commute,axiom,
% 3.15/1.64 ! [X2: a,Y2: a] :
% 3.15/1.64 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( tp_a_b @ r @ X2 @ Y2 )
% 3.15/1.64 = ( tp_a_b @ r @ Y2 @ X2 ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_216_ring__tOp__rel,axiom,
% 3.15/1.64 ! [X2: a,Xa: a,Y2: a,Ya: a] :
% 3.15/1.64 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ Xa @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ Ya @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( tp_a_b @ r @ ( tp_a_b @ r @ X2 @ Xa ) @ ( tp_a_b @ r @ Y2 @ Ya ) )
% 3.15/1.64 = ( tp_a_b @ r @ ( tp_a_b @ r @ X2 @ Y2 ) @ ( tp_a_b @ r @ Xa @ Ya ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_217_tp__assoc,axiom,
% 3.15/1.64 ! [A2: a,B2: a,C2: a] :
% 3.15/1.64 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( tp_a_b @ r @ ( tp_a_b @ r @ A2 @ B2 ) @ C2 )
% 3.15/1.64 = ( tp_a_b @ r @ A2 @ ( tp_a_b @ r @ B2 @ C2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_218_tp__commute,axiom,
% 3.15/1.64 ! [A2: a,B2: a] :
% 3.15/1.64 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( tp_a_b @ r @ A2 @ B2 )
% 3.15/1.64 = ( tp_a_b @ r @ B2 @ A2 ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_219_pop__aassoc,axiom,
% 3.15/1.64 ! [A2: a,B2: a,C2: a] :
% 3.15/1.64 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( pop_a_Ring_ext_a_b @ r @ ( pop_a_Ring_ext_a_b @ r @ A2 @ B2 ) @ C2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ r @ A2 @ ( pop_a_Ring_ext_a_b @ r @ B2 @ C2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_220_pop__commute,axiom,
% 3.15/1.64 ! [A2: a,B2: a] :
% 3.15/1.64 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( pop_a_Ring_ext_a_b @ r @ A2 @ B2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ r @ B2 @ A2 ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_221_Sr__tOp__closed,axiom,
% 3.15/1.64 ! [S: set_a,X2: a,Y2: a] :
% 3.15/1.64 ( ( sr_a_b2 @ r @ S )
% 3.15/1.64 => ( ( member_a @ X2 @ S )
% 3.15/1.64 => ( ( member_a @ Y2 @ S )
% 3.15/1.64 => ( member_a @ ( tp_a_b @ r @ X2 @ Y2 ) @ S ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_222_Sr__pOp__closed,axiom,
% 3.15/1.64 ! [S: set_a,X2: a,Y2: a] :
% 3.15/1.64 ( ( sr_a_b2 @ r @ S )
% 3.15/1.64 => ( ( member_a @ X2 @ S )
% 3.15/1.64 => ( ( member_a @ Y2 @ S )
% 3.15/1.64 => ( member_a @ ( pop_a_Ring_ext_a_b @ r @ X2 @ Y2 ) @ S ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_223_rg__distrib,axiom,
% 3.15/1.64 ! [A2: a,B2: a,C2: a] :
% 3.15/1.64 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( tp_a_b @ r @ A2 @ ( pop_a_Ring_ext_a_b @ r @ B2 @ C2 ) )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ A2 @ B2 ) @ ( tp_a_b @ r @ A2 @ C2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_224_ring__distrib1,axiom,
% 3.15/1.64 ! [X2: a,Y2: a,Z: a] :
% 3.15/1.64 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ Z @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( tp_a_b @ r @ X2 @ ( pop_a_Ring_ext_a_b @ r @ Y2 @ Z ) )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ X2 @ Y2 ) @ ( tp_a_b @ r @ X2 @ Z ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_225_ring__distrib2,axiom,
% 3.15/1.64 ! [X2: a,Y2: a,Z: a] :
% 3.15/1.64 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ Z @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( tp_a_b @ r @ ( pop_a_Ring_ext_a_b @ r @ Y2 @ Z ) @ X2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ Y2 @ X2 ) @ ( tp_a_b @ r @ Z @ X2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_226_ring__distrib3,axiom,
% 3.15/1.64 ! [A2: a,B2: a,X2: a,Y2: a] :
% 3.15/1.64 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( tp_a_b @ r @ ( pop_a_Ring_ext_a_b @ r @ A2 @ B2 ) @ ( pop_a_Ring_ext_a_b @ r @ X2 @ Y2 ) )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ r @ ( pop_a_Ring_ext_a_b @ r @ ( pop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ A2 @ X2 ) @ ( tp_a_b @ r @ A2 @ Y2 ) ) @ ( tp_a_b @ r @ B2 @ X2 ) ) @ ( tp_a_b @ r @ B2 @ Y2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_227_nsMulDistrL,axiom,
% 3.15/1.64 ! [X2: a,Y2: a,N2: nat] :
% 3.15/1.64 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( tp_a_b @ r @ X2 @ ( nscal_a_b @ r @ Y2 @ N2 ) )
% 3.15/1.64 = ( nscal_a_b @ r @ ( tp_a_b @ r @ X2 @ Y2 ) @ N2 ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_228_nsMulDistrR,axiom,
% 3.15/1.64 ! [X2: a,Y2: a,N2: nat] :
% 3.15/1.64 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( tp_a_b @ r @ ( nscal_a_b @ r @ Y2 @ N2 ) @ X2 )
% 3.15/1.64 = ( nscal_a_b @ r @ ( tp_a_b @ r @ Y2 @ X2 ) @ N2 ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_229_nsDistrL,axiom,
% 3.15/1.64 ! [X2: a,Y2: a,N2: nat] :
% 3.15/1.64 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( pop_a_Ring_ext_a_b @ r @ ( nscal_a_b @ r @ X2 @ N2 ) @ ( nscal_a_b @ r @ Y2 @ N2 ) )
% 3.15/1.64 = ( nscal_a_b @ r @ ( pop_a_Ring_ext_a_b @ r @ X2 @ Y2 ) @ N2 ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_230_Subring__tOp__ring__tOp,axiom,
% 3.15/1.64 ! [S: carrie1105631105xt_a_b,A2: a,B2: a] :
% 3.15/1.64 ( ( subring_a_b_b @ r @ S )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ S ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ S ) )
% 3.15/1.64 => ( ( tp_a_b @ S @ A2 @ B2 )
% 3.15/1.64 = ( tp_a_b @ r @ A2 @ B2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_231_Subring__pOp__ring__pOp,axiom,
% 3.15/1.64 ! [S: carrie1105631105xt_a_b,A2: a,B2: a] :
% 3.15/1.64 ( ( subring_a_b_b @ r @ S )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ S ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ S ) )
% 3.15/1.64 => ( ( pop_a_Ring_ext_a_b @ S @ A2 @ B2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ r @ A2 @ B2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_232_mul__closed__set__tOp__closed,axiom,
% 3.15/1.64 ! [S: set_a,S2: a,T: a] :
% 3.15/1.64 ( ( mul_closed_set_a_b @ r @ S )
% 3.15/1.64 => ( ( member_a @ S2 @ S )
% 3.15/1.64 => ( ( member_a @ T @ S )
% 3.15/1.64 => ( member_a @ ( tp_a_b @ r @ S2 @ T ) @ S ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_233_Ring_OSr__tOp__closed,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,S: set_a,X2: a,Y2: a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( sr_a_b2 @ R @ S )
% 3.15/1.64 => ( ( member_a @ X2 @ S )
% 3.15/1.64 => ( ( member_a @ Y2 @ S )
% 3.15/1.64 => ( member_a @ ( tp_a_b @ R @ X2 @ Y2 ) @ S ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_234_Ring_OSr__pOp__closed,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,S: set_a,X2: a,Y2: a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( sr_a_b2 @ R @ S )
% 3.15/1.64 => ( ( member_a @ X2 @ S )
% 3.15/1.64 => ( ( member_a @ Y2 @ S )
% 3.15/1.64 => ( member_a @ ( pop_a_Ring_ext_a_b @ R @ X2 @ Y2 ) @ S ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_235_Ring_Oring__distrib3,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,A2: a,B2: a,X2: a,Y2: a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( tp_a_b @ R @ ( pop_a_Ring_ext_a_b @ R @ A2 @ B2 ) @ ( pop_a_Ring_ext_a_b @ R @ X2 @ Y2 ) )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ R @ ( pop_a_Ring_ext_a_b @ R @ ( pop_a_Ring_ext_a_b @ R @ ( tp_a_b @ R @ A2 @ X2 ) @ ( tp_a_b @ R @ A2 @ Y2 ) ) @ ( tp_a_b @ R @ B2 @ X2 ) ) @ ( tp_a_b @ R @ B2 @ Y2 ) ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_236_Ring_Oring__distrib2,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,X2: a,Y2: a,Z: a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ Z @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( tp_a_b @ R @ ( pop_a_Ring_ext_a_b @ R @ Y2 @ Z ) @ X2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ R @ ( tp_a_b @ R @ Y2 @ X2 ) @ ( tp_a_b @ R @ Z @ X2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_237_Ring_Oring__distrib1,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,X2: a,Y2: a,Z: a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ Z @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( tp_a_b @ R @ X2 @ ( pop_a_Ring_ext_a_b @ R @ Y2 @ Z ) )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ R @ ( tp_a_b @ R @ X2 @ Y2 ) @ ( tp_a_b @ R @ X2 @ Z ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_238_Ring_Org__distrib,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( tp_a_b @ R @ A2 @ ( pop_a_Ring_ext_a_b @ R @ B2 @ C2 ) )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ R @ ( tp_a_b @ R @ A2 @ B2 ) @ ( tp_a_b @ R @ A2 @ C2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_239_Ring_OnsMulDistrR,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,X2: a,Y2: a,N2: nat] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( tp_a_b @ R @ ( nscal_a_b @ R @ Y2 @ N2 ) @ X2 )
% 3.15/1.64 = ( nscal_a_b @ R @ ( tp_a_b @ R @ Y2 @ X2 ) @ N2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_240_Ring_OnsMulDistrL,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,X2: a,Y2: a,N2: nat] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( tp_a_b @ R @ X2 @ ( nscal_a_b @ R @ Y2 @ N2 ) )
% 3.15/1.64 = ( nscal_a_b @ R @ ( tp_a_b @ R @ X2 @ Y2 ) @ N2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_241_Ring_OnsDistrL,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,X2: a,Y2: a,N2: nat] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( pop_a_Ring_ext_a_b @ R @ ( nscal_a_b @ R @ X2 @ N2 ) @ ( nscal_a_b @ R @ Y2 @ N2 ) )
% 3.15/1.64 = ( nscal_a_b @ R @ ( pop_a_Ring_ext_a_b @ R @ X2 @ Y2 ) @ N2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_242_Ring_OSr__ring,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,S: set_a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( sr_a_b2 @ R @ S )
% 3.15/1.64 => ( ring_a_b @ ( sr_a_b @ R @ S ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_243_aGroup_Oag__pOp__commute,axiom,
% 3.15/1.64 ! [A: carrie1105631105xt_a_b,X2: a,Y2: a] :
% 3.15/1.64 ( ( aGroup2097840802xt_a_b @ A )
% 3.15/1.64 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( pop_a_Ring_ext_a_b @ A @ X2 @ Y2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ A @ Y2 @ X2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_244_aGroup_Oag__pOp__commute,axiom,
% 3.15/1.64 ! [A: carrie1954260683t_unit,X2: c > d,Y2: c > d] :
% 3.15/1.64 ( ( aGroup2131905830t_unit @ A )
% 3.15/1.64 => ( ( member_c_d @ X2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ Y2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( pop_c_449750139t_unit @ A @ X2 @ Y2 )
% 3.15/1.64 = ( pop_c_449750139t_unit @ A @ Y2 @ X2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_245_aGroup_Oag__add__commute,axiom,
% 3.15/1.64 ! [A: carrie1105631105xt_a_b,A2: a,B2: a] :
% 3.15/1.64 ( ( aGroup2097840802xt_a_b @ A )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( pop_a_Ring_ext_a_b @ A @ A2 @ B2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ A @ B2 @ A2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_246_aGroup_Oag__add__commute,axiom,
% 3.15/1.64 ! [A: carrie1954260683t_unit,A2: c > d,B2: c > d] :
% 3.15/1.64 ( ( aGroup2131905830t_unit @ A )
% 3.15/1.64 => ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( pop_c_449750139t_unit @ A @ A2 @ B2 )
% 3.15/1.64 = ( pop_c_449750139t_unit @ A @ B2 @ A2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_247_aGroup_OpOp__assocTr43,axiom,
% 3.15/1.64 ! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a,D: a] :
% 3.15/1.64 ( ( aGroup2097840802xt_a_b @ A )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ D @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ B2 ) @ ( pop_a_Ring_ext_a_b @ A @ C2 @ D ) )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ ( pop_a_Ring_ext_a_b @ A @ B2 @ C2 ) ) @ D ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_248_aGroup_OpOp__assocTr43,axiom,
% 3.15/1.64 ! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d,D: c > d] :
% 3.15/1.64 ( ( aGroup2131905830t_unit @ A )
% 3.15/1.64 => ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ D @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ B2 ) @ ( pop_c_449750139t_unit @ A @ C2 @ D ) )
% 3.15/1.64 = ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ ( pop_c_449750139t_unit @ A @ B2 @ C2 ) ) @ D ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_249_aGroup_OpOp__assocTr42,axiom,
% 3.15/1.64 ! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a,D: a] :
% 3.15/1.64 ( ( aGroup2097840802xt_a_b @ A )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ D @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ B2 ) @ C2 ) @ D )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ ( pop_a_Ring_ext_a_b @ A @ B2 @ C2 ) ) @ D ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_250_aGroup_OpOp__assocTr42,axiom,
% 3.15/1.64 ! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d,D: c > d] :
% 3.15/1.64 ( ( aGroup2131905830t_unit @ A )
% 3.15/1.64 => ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ D @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ B2 ) @ C2 ) @ D )
% 3.15/1.64 = ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ ( pop_c_449750139t_unit @ A @ B2 @ C2 ) ) @ D ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_251_aGroup_OpOp__assocTr41,axiom,
% 3.15/1.64 ! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a,D: a] :
% 3.15/1.64 ( ( aGroup2097840802xt_a_b @ A )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ D @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ B2 ) @ C2 ) @ D )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ B2 ) @ ( pop_a_Ring_ext_a_b @ A @ C2 @ D ) ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_252_aGroup_OpOp__assocTr41,axiom,
% 3.15/1.64 ! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d,D: c > d] :
% 3.15/1.64 ( ( aGroup2131905830t_unit @ A )
% 3.15/1.64 => ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ D @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ B2 ) @ C2 ) @ D )
% 3.15/1.64 = ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ B2 ) @ ( pop_c_449750139t_unit @ A @ C2 @ D ) ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_253_aGroup_Oag__pOp__closed,axiom,
% 3.15/1.64 ! [A: carrie1105631105xt_a_b,X2: a,Y2: a] :
% 3.15/1.64 ( ( aGroup2097840802xt_a_b @ A )
% 3.15/1.64 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( member_a @ ( pop_a_Ring_ext_a_b @ A @ X2 @ Y2 ) @ ( carrie867757212xt_a_b @ A ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_254_aGroup_Oag__pOp__closed,axiom,
% 3.15/1.64 ! [A: carrie1954260683t_unit,X2: c > d,Y2: c > d] :
% 3.15/1.64 ( ( aGroup2131905830t_unit @ A )
% 3.15/1.64 => ( ( member_c_d @ X2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ Y2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( member_c_d @ ( pop_c_449750139t_unit @ A @ X2 @ Y2 ) @ ( carrie20288686t_unit @ A ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_255_aGroup_OpOp__cancel__r,axiom,
% 3.15/1.64 ! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
% 3.15/1.64 ( ( aGroup2097840802xt_a_b @ A )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( ( pop_a_Ring_ext_a_b @ A @ A2 @ C2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ A @ B2 @ C2 ) )
% 3.15/1.64 => ( A2 = B2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_256_aGroup_OpOp__cancel__r,axiom,
% 3.15/1.64 ! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d] :
% 3.15/1.64 ( ( aGroup2131905830t_unit @ A )
% 3.15/1.64 => ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( ( pop_c_449750139t_unit @ A @ A2 @ C2 )
% 3.15/1.64 = ( pop_c_449750139t_unit @ A @ B2 @ C2 ) )
% 3.15/1.64 => ( A2 = B2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_257_aGroup_OpOp__cancel__l,axiom,
% 3.15/1.64 ! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
% 3.15/1.64 ( ( aGroup2097840802xt_a_b @ A )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( ( pop_a_Ring_ext_a_b @ A @ C2 @ A2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ A @ C2 @ B2 ) )
% 3.15/1.64 => ( A2 = B2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_258_aGroup_OpOp__cancel__l,axiom,
% 3.15/1.64 ! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d] :
% 3.15/1.64 ( ( aGroup2131905830t_unit @ A )
% 3.15/1.64 => ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( ( pop_c_449750139t_unit @ A @ C2 @ A2 )
% 3.15/1.64 = ( pop_c_449750139t_unit @ A @ C2 @ B2 ) )
% 3.15/1.64 => ( A2 = B2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_259_aGroup_Oag__pOp__assoc,axiom,
% 3.15/1.64 ! [A: carrie1105631105xt_a_b,X2: a,Y2: a,Z: a] :
% 3.15/1.64 ( ( aGroup2097840802xt_a_b @ A )
% 3.15/1.64 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ Z @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ X2 @ Y2 ) @ Z )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ A @ X2 @ ( pop_a_Ring_ext_a_b @ A @ Y2 @ Z ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_260_aGroup_Oag__pOp__assoc,axiom,
% 3.15/1.64 ! [A: carrie1954260683t_unit,X2: c > d,Y2: c > d,Z: c > d] :
% 3.15/1.64 ( ( aGroup2131905830t_unit @ A )
% 3.15/1.64 => ( ( member_c_d @ X2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ Y2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ Z @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ X2 @ Y2 ) @ Z )
% 3.15/1.64 = ( pop_c_449750139t_unit @ A @ X2 @ ( pop_c_449750139t_unit @ A @ Y2 @ Z ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_261_aGroup_Oag__pOp__add__r,axiom,
% 3.15/1.64 ! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
% 3.15/1.64 ( ( aGroup2097840802xt_a_b @ A )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( A2 = B2 )
% 3.15/1.64 => ( ( pop_a_Ring_ext_a_b @ A @ A2 @ C2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ A @ B2 @ C2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_262_aGroup_Oag__pOp__add__r,axiom,
% 3.15/1.64 ! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d] :
% 3.15/1.64 ( ( aGroup2131905830t_unit @ A )
% 3.15/1.64 => ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( A2 = B2 )
% 3.15/1.64 => ( ( pop_c_449750139t_unit @ A @ A2 @ C2 )
% 3.15/1.64 = ( pop_c_449750139t_unit @ A @ B2 @ C2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_263_aGroup_Oag__pOp__add__l,axiom,
% 3.15/1.64 ! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
% 3.15/1.64 ( ( aGroup2097840802xt_a_b @ A )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( A2 = B2 )
% 3.15/1.64 => ( ( pop_a_Ring_ext_a_b @ A @ C2 @ A2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ A @ C2 @ B2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_264_aGroup_Oag__pOp__add__l,axiom,
% 3.15/1.64 ! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d] :
% 3.15/1.64 ( ( aGroup2131905830t_unit @ A )
% 3.15/1.64 => ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( A2 = B2 )
% 3.15/1.64 => ( ( pop_c_449750139t_unit @ A @ C2 @ A2 )
% 3.15/1.64 = ( pop_c_449750139t_unit @ A @ C2 @ B2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_265_aGroup_OgEQAddcross,axiom,
% 3.15/1.64 ! [A: carrie1105631105xt_a_b,L1: a,L2: a,R1: a,R2: a] :
% 3.15/1.64 ( ( aGroup2097840802xt_a_b @ A )
% 3.15/1.64 => ( ( member_a @ L1 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ L2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ R1 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ R1 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( L1 = R2 )
% 3.15/1.64 => ( ( L2 = R1 )
% 3.15/1.64 => ( ( pop_a_Ring_ext_a_b @ A @ L1 @ L2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ A @ R1 @ R2 ) ) ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_266_aGroup_OgEQAddcross,axiom,
% 3.15/1.64 ! [A: carrie1954260683t_unit,L1: c > d,L2: c > d,R1: c > d,R2: c > d] :
% 3.15/1.64 ( ( aGroup2131905830t_unit @ A )
% 3.15/1.64 => ( ( member_c_d @ L1 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ L2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ R1 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ R1 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( L1 = R2 )
% 3.15/1.64 => ( ( L2 = R1 )
% 3.15/1.64 => ( ( pop_c_449750139t_unit @ A @ L1 @ L2 )
% 3.15/1.64 = ( pop_c_449750139t_unit @ A @ R1 @ R2 ) ) ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_267_aGroup_Oag__add4__rel,axiom,
% 3.15/1.64 ! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a,D: a] :
% 3.15/1.64 ( ( aGroup2097840802xt_a_b @ A )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ D @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ B2 ) @ ( pop_a_Ring_ext_a_b @ A @ C2 @ D ) )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ C2 ) @ ( pop_a_Ring_ext_a_b @ A @ B2 @ D ) ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_268_aGroup_Oag__add4__rel,axiom,
% 3.15/1.64 ! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d,D: c > d] :
% 3.15/1.64 ( ( aGroup2131905830t_unit @ A )
% 3.15/1.64 => ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ D @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ B2 ) @ ( pop_c_449750139t_unit @ A @ C2 @ D ) )
% 3.15/1.64 = ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ C2 ) @ ( pop_c_449750139t_unit @ A @ B2 @ D ) ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_269_aGroup_Oaassoc,axiom,
% 3.15/1.64 ! [A: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
% 3.15/1.64 ( ( aGroup2097840802xt_a_b @ A )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ A ) )
% 3.15/1.64 => ( ( pop_a_Ring_ext_a_b @ A @ ( pop_a_Ring_ext_a_b @ A @ A2 @ B2 ) @ C2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ A @ A2 @ ( pop_a_Ring_ext_a_b @ A @ B2 @ C2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_270_aGroup_Oaassoc,axiom,
% 3.15/1.64 ! [A: carrie1954260683t_unit,A2: c > d,B2: c > d,C2: c > d] :
% 3.15/1.64 ( ( aGroup2131905830t_unit @ A )
% 3.15/1.64 => ( ( member_c_d @ A2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ B2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( member_c_d @ C2 @ ( carrie20288686t_unit @ A ) )
% 3.15/1.64 => ( ( pop_c_449750139t_unit @ A @ ( pop_c_449750139t_unit @ A @ A2 @ B2 ) @ C2 )
% 3.15/1.64 = ( pop_c_449750139t_unit @ A @ A2 @ ( pop_c_449750139t_unit @ A @ B2 @ C2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_271_Ring_Oring__tOp__commute,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,X2: a,Y2: a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( tp_a_b @ R @ X2 @ Y2 )
% 3.15/1.64 = ( tp_a_b @ R @ Y2 @ X2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_272_Ring_Oring__tOp__closed,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,X2: a,Y2: a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( member_a @ ( tp_a_b @ R @ X2 @ Y2 ) @ ( carrie867757212xt_a_b @ R ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_273_Ring_Oring__tOp__assoc,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,X2: a,Y2: a,Z: a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ Z @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( tp_a_b @ R @ ( tp_a_b @ R @ X2 @ Y2 ) @ Z )
% 3.15/1.64 = ( tp_a_b @ R @ X2 @ ( tp_a_b @ R @ Y2 @ Z ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_274_Ring_Oring__tOp__rel,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,X2: a,Xa: a,Y2: a,Ya: a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ Xa @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ Ya @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( tp_a_b @ R @ ( tp_a_b @ R @ X2 @ Xa ) @ ( tp_a_b @ R @ Y2 @ Ya ) )
% 3.15/1.64 = ( tp_a_b @ R @ ( tp_a_b @ R @ X2 @ Y2 ) @ ( tp_a_b @ R @ Xa @ Ya ) ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_275_Ring_Otp__commute,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,A2: a,B2: a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( tp_a_b @ R @ A2 @ B2 )
% 3.15/1.64 = ( tp_a_b @ R @ B2 @ A2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_276_Ring_Otp__assoc,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( tp_a_b @ R @ ( tp_a_b @ R @ A2 @ B2 ) @ C2 )
% 3.15/1.64 = ( tp_a_b @ R @ A2 @ ( tp_a_b @ R @ B2 @ C2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_277_Ring_OrEQMulR,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,X2: a,Y2: a,Z: a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ Z @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( X2 = Y2 )
% 3.15/1.64 => ( ( tp_a_b @ R @ X2 @ Z )
% 3.15/1.64 = ( tp_a_b @ R @ Y2 @ Z ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_278_Ring_OrMulLC,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,X2: a,Y2: a,Z: a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ Z @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( tp_a_b @ R @ X2 @ ( tp_a_b @ R @ Y2 @ Z ) )
% 3.15/1.64 = ( tp_a_b @ R @ Y2 @ ( tp_a_b @ R @ X2 @ Z ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_279_Ring_Opop__commute,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,A2: a,B2: a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( pop_a_Ring_ext_a_b @ R @ A2 @ B2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ R @ B2 @ A2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_280_Ring_Opop__aassoc,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,A2: a,B2: a,C2: a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ C2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( pop_a_Ring_ext_a_b @ R @ ( pop_a_Ring_ext_a_b @ R @ A2 @ B2 ) @ C2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ R @ A2 @ ( pop_a_Ring_ext_a_b @ R @ B2 @ C2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_281_Ring_OnsEqElm,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,X2: a,Y2: a,N2: nat] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( X2 = Y2 )
% 3.15/1.64 => ( ( nscal_a_b @ R @ X2 @ N2 )
% 3.15/1.64 = ( nscal_a_b @ R @ Y2 @ N2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_282_Ring_OnsClose,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,X2: a,N2: nat] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( member_a @ ( nscal_a_b @ R @ X2 @ N2 ) @ ( carrie867757212xt_a_b @ R ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_283_prod__pOp__mem__i,axiom,
% 3.15/1.64 ! [I: set_d,A: d > carrie1105631105xt_a_b,X4: d > a,Y: d > a,I2: d] :
% 3.15/1.64 ( ! [X: d] :
% 3.15/1.64 ( ( member_d @ X @ I )
% 3.15/1.64 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_d_a @ X4 @ ( carr_p297433675xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_d_a @ Y @ ( carr_p297433675xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_d @ I2 @ I )
% 3.15/1.64 => ( ( prod_p711161210xt_a_b @ I @ A @ X4 @ Y @ I2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_284_prod__pOp__mem__i,axiom,
% 3.15/1.64 ! [I: set_c,A: c > carrie1105631105xt_a_b,X4: c > a,Y: c > a,I2: c] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_c_a @ X4 @ ( carr_p295160588xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_c_a @ Y @ ( carr_p295160588xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_c @ I2 @ I )
% 3.15/1.64 => ( ( prod_p708888123xt_a_b @ I @ A @ X4 @ Y @ I2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_285_prod__pOp__mem__i,axiom,
% 3.15/1.64 ! [I: set_a,A: a > carrie1105631105xt_a_b,X4: a > a,Y: a > a,I2: a] :
% 3.15/1.64 ( ! [X: a] :
% 3.15/1.64 ( ( member_a @ X @ I )
% 3.15/1.64 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_a_a @ X4 @ ( carr_p290614414xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_a_a @ Y @ ( carr_p290614414xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_a @ I2 @ I )
% 3.15/1.64 => ( ( prod_p704341949xt_a_b @ I @ A @ X4 @ Y @ I2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_286_prod__pOp__mem__i,axiom,
% 3.15/1.64 ! [I: set_c,A: c > carrie1948989069_d_a_e,X4: c > d,Y: c > d,I2: c] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_c_d @ X4 @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 3.15/1.64 => ( ( member_c_d @ Y @ ( carr_p1636662547_d_a_e @ I @ A ) )
% 3.15/1.64 => ( ( member_c @ I2 @ I )
% 3.15/1.64 => ( ( prod_p131511076_d_a_e @ I @ A @ X4 @ Y @ I2 )
% 3.15/1.64 = ( pop_d_261713874_d_a_e @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_287_prod__pOp__mem__i,axiom,
% 3.15/1.64 ! [I: set_c_d,A: ( c > d ) > carrie1105631105xt_a_b,X4: ( c > d ) > a,Y: ( c > d ) > a,I2: c > d] :
% 3.15/1.64 ( ! [X: c > d] :
% 3.15/1.64 ( ( member_c_d @ X @ I )
% 3.15/1.64 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_c_d_a @ X4 @ ( carr_p996491646xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_c_d_a @ Y @ ( carr_p996491646xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_c_d @ I2 @ I )
% 3.15/1.64 => ( ( prod_p1466615439xt_a_b @ I @ A @ X4 @ Y @ I2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_288_prod__pOp__mem__i,axiom,
% 3.15/1.64 ! [I: set_a_a,A: ( a > a ) > carrie1105631105xt_a_b,X4: ( a > a ) > a,Y: ( a > a ) > a,I2: a > a] :
% 3.15/1.64 ( ! [X: a > a] :
% 3.15/1.64 ( ( member_a_a @ X @ I )
% 3.15/1.64 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_a_a_a @ X4 @ ( carr_p740328767xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_a_a_a @ Y @ ( carr_p740328767xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_a_a @ I2 @ I )
% 3.15/1.64 => ( ( prod_p1210452560xt_a_b @ I @ A @ X4 @ Y @ I2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_289_prod__pOp__mem__i,axiom,
% 3.15/1.64 ! [I: set_a_a_a,A: ( a > a > a ) > carrie1105631105xt_a_b,X4: ( a > a > a ) > a,Y: ( a > a > a ) > a,I2: a > a > a] :
% 3.15/1.64 ( ! [X: a > a > a] :
% 3.15/1.64 ( ( member_a_a_a2 @ X @ I )
% 3.15/1.64 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_a_a_a_a @ X4 @ ( carr_p1481099064xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_a_a_a_a @ Y @ ( carr_p1481099064xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_a_a_a2 @ I2 @ I )
% 3.15/1.64 => ( ( prod_p1460711911xt_a_b @ I @ A @ X4 @ Y @ I2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_290_prod__pOp__mem__i,axiom,
% 3.15/1.64 ! [I: set_c_d_c_d,A: ( ( c > d ) > c > d ) > carrie1105631105xt_a_b,X4: ( ( c > d ) > c > d ) > a,Y: ( ( c > d ) > c > d ) > a,I2: ( c > d ) > c > d] :
% 3.15/1.64 ( ! [X: ( c > d ) > c > d] :
% 3.15/1.64 ( ( member_c_d_c_d @ X @ I )
% 3.15/1.64 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_c_d_c_d_a @ X4 @ ( carr_p1617094073xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_c_d_c_d_a @ Y @ ( carr_p1617094073xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_c_d_c_d @ I2 @ I )
% 3.15/1.64 => ( ( prod_p1959959370xt_a_b @ I @ A @ X4 @ Y @ I2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_291_prod__pOp__mem__i,axiom,
% 3.15/1.64 ! [I: set_a_a_c_d,A: ( a > a > c > d ) > carrie1105631105xt_a_b,X4: ( a > a > c > d ) > a,Y: ( a > a > c > d ) > a,I2: a > a > c > d] :
% 3.15/1.64 ( ! [X: a > a > c > d] :
% 3.15/1.64 ( ( member_a_a_c_d2 @ X @ I )
% 3.15/1.64 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_a_a_c_d_a @ X4 @ ( carr_p109409832xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_a_a_c_d_a @ Y @ ( carr_p109409832xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_a_a_c_d2 @ I2 @ I )
% 3.15/1.64 => ( ( prod_p452275129xt_a_b @ I @ A @ X4 @ Y @ I2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_292_prod__pOp__mem__i,axiom,
% 3.15/1.64 ! [I: set_a_a_a_a,A: ( a > a > a > a ) > carrie1105631105xt_a_b,X4: ( a > a > a > a ) > a,Y: ( a > a > a > a ) > a,I2: a > a > a > a] :
% 3.15/1.64 ( ! [X: a > a > a > a] :
% 3.15/1.64 ( ( member_a_a_a_a4 @ X @ I )
% 3.15/1.64 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_a_a_a_a_a @ X4 @ ( carr_p1850824937xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_a_a_a_a_a @ Y @ ( carr_p1850824937xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_a_a_a_a4 @ I2 @ I )
% 3.15/1.64 => ( ( prod_p46206586xt_a_b @ I @ A @ X4 @ Y @ I2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ ( A @ I2 ) @ ( X4 @ I2 ) @ ( Y @ I2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_293_dsum__pOp__mem,axiom,
% 3.15/1.64 ! [I: set_a,A: a > carrie1105631105xt_a_b,X4: a > a,Y: a > a] :
% 3.15/1.64 ( ! [X: a] :
% 3.15/1.64 ( ( member_a @ X @ I )
% 3.15/1.64 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_a_a @ X4 @ ( carr_d267359390xt_a_b @ I @ A ) )
% 3.15/1.64 => ( ( member_a_a @ Y @ ( carr_d267359390xt_a_b @ I @ A ) )
% 3.15/1.64 => ( member_a_a @ ( prod_p704341949xt_a_b @ I @ A @ X4 @ Y ) @ ( carr_d267359390xt_a_b @ I @ A ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_294_dsum__pOp__mem,axiom,
% 3.15/1.64 ! [I: set_c,A: c > carrie1948989069_d_a_e,X4: c > d,Y: c > d] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_c_d @ X4 @ ( carr_d56066307_d_a_e @ I @ A ) )
% 3.15/1.64 => ( ( member_c_d @ Y @ ( carr_d56066307_d_a_e @ I @ A ) )
% 3.15/1.64 => ( member_c_d @ ( prod_p131511076_d_a_e @ I @ A @ X4 @ Y ) @ ( carr_d56066307_d_a_e @ I @ A ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_295_dsum__iOp__mem,axiom,
% 3.15/1.64 ! [I: set_a,A: a > carrie1105631105xt_a_b,X4: a > a] :
% 3.15/1.64 ( ! [X: a] :
% 3.15/1.64 ( ( member_a @ X @ I )
% 3.15/1.64 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_a_a @ X4 @ ( carr_d267359390xt_a_b @ I @ A ) )
% 3.15/1.64 => ( member_a_a @ ( prod_m2022989242xt_a_b @ I @ A @ X4 ) @ ( carr_d267359390xt_a_b @ I @ A ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_296_dsum__iOp__mem,axiom,
% 3.15/1.64 ! [I: set_c,A: c > carrie1948989069_d_a_e,X4: c > d] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 3.15/1.64 => ( ( member_c_d @ X4 @ ( carr_d56066307_d_a_e @ I @ A ) )
% 3.15/1.64 => ( member_c_d @ ( prod_m854557927_d_a_e @ I @ A @ X4 ) @ ( carr_d56066307_d_a_e @ I @ A ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_297_dsum__zero__func,axiom,
% 3.15/1.64 ! [I: set_a,A: a > carrie1105631105xt_a_b] :
% 3.15/1.64 ( ! [X: a] :
% 3.15/1.64 ( ( member_a @ X @ I )
% 3.15/1.64 => ( aGroup2097840802xt_a_b @ ( A @ X ) ) )
% 3.15/1.64 => ( member_a_a @ ( prod_z851058290xt_a_b @ I @ A ) @ ( carr_d267359390xt_a_b @ I @ A ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_298_dsum__zero__func,axiom,
% 3.15/1.64 ! [I: set_c,A: c > carrie1948989069_d_a_e] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 3.15/1.64 => ( member_c_d @ ( prod_z1214987823_d_a_e @ I @ A ) @ ( carr_d56066307_d_a_e @ I @ A ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_299_aGroup_Opop__closed,axiom,
% 3.15/1.64 ! [A: carrie1105631105xt_a_b] :
% 3.15/1.64 ( ( aGroup2097840802xt_a_b @ A )
% 3.15/1.64 => ( member_a_a_a2 @ ( pop_a_Ring_ext_a_b @ A )
% 3.15/1.64 @ ( pi_a_a_a2 @ ( carrie867757212xt_a_b @ A )
% 3.15/1.64 @ ^ [Uu: a] :
% 3.15/1.64 ( pi_a_a @ ( carrie867757212xt_a_b @ A )
% 3.15/1.64 @ ^ [Uv: a] : ( carrie867757212xt_a_b @ A ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_300_aGroup_Opop__closed,axiom,
% 3.15/1.64 ! [A: carrie1954260683t_unit] :
% 3.15/1.64 ( ( aGroup2131905830t_unit @ A )
% 3.15/1.64 => ( member_c_d_c_d_c_d @ ( pop_c_449750139t_unit @ A )
% 3.15/1.64 @ ( pi_c_d_c_d_c_d @ ( carrie20288686t_unit @ A )
% 3.15/1.64 @ ^ [Uu: c > d] :
% 3.15/1.64 ( pi_c_d_c_d @ ( carrie20288686t_unit @ A )
% 3.15/1.64 @ ^ [Uv: c > d] : ( carrie20288686t_unit @ A ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_301_Ring_OJ__rad__mem,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,X2: a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( member_a @ X2 @ ( j_rad_a_b @ R ) )
% 3.15/1.64 => ( member_a @ X2 @ ( carrie867757212xt_a_b @ R ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_302_Ring_Otp__closed,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( member_a_a_a2 @ ( tp_a_b @ R )
% 3.15/1.64 @ ( pi_a_a_a2 @ ( carrie867757212xt_a_b @ R )
% 3.15/1.64 @ ^ [Uu: a] :
% 3.15/1.64 ( pi_a_a @ ( carrie867757212xt_a_b @ R )
% 3.15/1.64 @ ^ [Uv: a] : ( carrie867757212xt_a_b @ R ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_303_Ring_Opop__closed,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( member_a_a_a2 @ ( pop_a_Ring_ext_a_b @ R )
% 3.15/1.64 @ ( pi_a_a_a2 @ ( carrie867757212xt_a_b @ R )
% 3.15/1.64 @ ^ [Uu: a] :
% 3.15/1.64 ( pi_a_a @ ( carrie867757212xt_a_b @ R )
% 3.15/1.64 @ ^ [Uv: a] : ( carrie867757212xt_a_b @ R ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_304_Ring_OprodM__sprod__val,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,I: set_c_d_c_d,M: ( ( c > d ) > c > d ) > carrie1948989069_d_a_e,A2: a,M2: ( ( c > d ) > c > d ) > d,J: ( c > d ) > c > d] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ! [X: ( c > d ) > c > d] :
% 3.15/1.64 ( ( member_c_d_c_d @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_c_d_c_d_d @ M2 @ ( carr_p428609062_d_a_e @ I @ M ) )
% 3.15/1.64 => ( ( member_c_d_c_d @ J @ I )
% 3.15/1.64 => ( ( algebr828196337_d_d_e @ R @ I @ M @ A2 @ M2 @ J )
% 3.15/1.64 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_305_Ring_OprodM__sprod__val,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,I: set_c_d,M: ( c > d ) > carrie1948989069_d_a_e,A2: a,M2: ( c > d ) > d,J: c > d] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ! [X: c > d] :
% 3.15/1.64 ( ( member_c_d @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_c_d_d @ M2 @ ( carr_p124510241_d_a_e @ I @ M ) )
% 3.15/1.64 => ( ( member_c_d @ J @ I )
% 3.15/1.64 => ( ( algebr924632044_d_d_e @ R @ I @ M @ A2 @ M2 @ J )
% 3.15/1.64 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_306_Ring_OprodM__sprod__val,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,I: set_a_a_a,M: ( a > a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > a ) > d,J: a > a > a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ! [X: a > a > a] :
% 3.15/1.64 ( ( member_a_a_a2 @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a_a_a_d @ M2 @ ( carr_p1364179431_d_a_e @ I @ M ) )
% 3.15/1.64 => ( ( member_a_a_a2 @ J @ I )
% 3.15/1.64 => ( ( algebr1655014672_a_d_e @ R @ I @ M @ A2 @ M2 @ J )
% 3.15/1.64 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_307_Ring_OprodM__sprod__val,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,I: set_a_a,M: ( a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a ) > d,J: a > a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ! [X: a > a] :
% 3.15/1.64 ( ( member_a_a @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a_a_d @ M2 @ ( carr_p1646427936_d_a_e @ I @ M ) )
% 3.15/1.64 => ( ( member_a_a @ J @ I )
% 3.15/1.64 => ( ( algebr137561451_a_d_e @ R @ I @ M @ A2 @ M2 @ J )
% 3.15/1.64 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_308_Ring_OprodM__sprod__val,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,I: set_a,M: a > carrie1948989069_d_a_e,A2: a,M2: a > d,J: a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ! [X: a] :
% 3.15/1.64 ( ( member_a @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a_d @ M2 @ ( carr_p1084104977_d_a_e @ I @ M ) )
% 3.15/1.64 => ( ( member_a @ J @ I )
% 3.15/1.64 => ( ( algebr2109842362_a_d_e @ R @ I @ M @ A2 @ M2 @ J )
% 3.15/1.64 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_309_Ring_OprodM__sprod__val,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,I: set_a_a_c_d,M: ( a > a > c > d ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > c > d ) > d,J: a > a > c > d] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ! [X: a > a > c > d] :
% 3.15/1.64 ( ( member_a_a_c_d2 @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a_a_c_d_d @ M2 @ ( carr_p577380087_d_a_e @ I @ M ) )
% 3.15/1.64 => ( ( member_a_a_c_d2 @ J @ I )
% 3.15/1.64 => ( ( algebr1620716610_d_d_e @ R @ I @ M @ A2 @ M2 @ J )
% 3.15/1.64 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_310_Ring_OprodM__sprod__val,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,I: set_a_a_a_a_a,M: ( a > a > a > a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > a > a > a ) > d,J: a > a > a > a > a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ! [X: a > a > a > a > a] :
% 3.15/1.64 ( ( member_a_a_a_a_a3 @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a_a_a_a_a_d @ M2 @ ( carr_p1980447933_d_a_e @ I @ M ) )
% 3.15/1.64 => ( ( member_a_a_a_a_a3 @ J @ I )
% 3.15/1.64 => ( ( algebr2122715238_a_d_e @ R @ I @ M @ A2 @ M2 @ J )
% 3.15/1.64 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_311_Ring_OprodM__sprod__val,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,I: set_a_a_a_a,M: ( a > a > a > a ) > carrie1948989069_d_a_e,A2: a,M2: ( a > a > a > a ) > d,J: a > a > a > a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ! [X: a > a > a > a] :
% 3.15/1.64 ( ( member_a_a_a_a4 @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a_a_a_a_d @ M2 @ ( carr_p305487606_d_a_e @ I @ M ) )
% 3.15/1.64 => ( ( member_a_a_a_a4 @ J @ I )
% 3.15/1.64 => ( ( algebr303206081_a_d_e @ R @ I @ M @ A2 @ M2 @ J )
% 3.15/1.64 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_312_Ring_OprodM__sprod__val,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,I: set_d,M: d > carrie1948989069_d_a_e,A2: a,M2: d > d,J: d] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ! [X: d] :
% 3.15/1.64 ( ( member_d @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_d_d @ M2 @ ( carr_p1912941332_d_a_e @ I @ M ) )
% 3.15/1.64 => ( ( member_d @ J @ I )
% 3.15/1.64 => ( ( algebr417875517_d_d_e @ R @ I @ M @ A2 @ M2 @ J )
% 3.15/1.64 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_313_Ring_OprodM__sprod__val,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,I: set_c,M: c > carrie1948989069_d_a_e,A2: a,M2: c > d,J: c] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( module_d_a_e_b @ ( M @ X ) @ R ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_c_d @ M2 @ ( carr_p1636662547_d_a_e @ I @ M ) )
% 3.15/1.64 => ( ( member_c @ J @ I )
% 3.15/1.64 => ( ( algebr1697692348_c_d_e @ R @ I @ M @ A2 @ M2 @ J )
% 3.15/1.64 = ( sprod_d_a_e @ ( M @ J ) @ A2 @ ( M2 @ J ) ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_314_dsum__iOp__func,axiom,
% 3.15/1.64 ! [I: set_c,A: c > carrie1948989069_d_a_e] :
% 3.15/1.64 ( ! [X: c] :
% 3.15/1.64 ( ( member_c @ X @ I )
% 3.15/1.64 => ( aGroup1940521469_d_a_e @ ( A @ X ) ) )
% 3.15/1.64 => ( member_c_d_c_d @ ( prod_m854557927_d_a_e @ I @ A )
% 3.15/1.64 @ ( pi_c_d_c_d @ ( carr_d56066307_d_a_e @ I @ A )
% 3.15/1.64 @ ^ [Uu: c > d] : ( carr_d56066307_d_a_e @ I @ A ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_315_zeroring__J__rad__empty,axiom,
% 3.15/1.64 ( ( zeroring_a_b @ r )
% 3.15/1.64 => ( ( j_rad_a_b @ r )
% 3.15/1.64 = ( carrie867757212xt_a_b @ r ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_316_Module_OmHom__lin,axiom,
% 3.15/1.64 ! [M: carrie1948989069_d_a_e,R: carrie1105631105xt_a_b,N: carrie1948989069_d_a_e,M2: d,F: d > d,A2: a] :
% 3.15/1.64 ( ( module_d_a_e_b @ M @ R )
% 3.15/1.64 => ( ( module_d_a_e_b @ N @ R )
% 3.15/1.64 => ( ( member_d @ M2 @ ( carrie11369756_d_a_e @ M ) )
% 3.15/1.64 => ( ( member_d_d @ F @ ( mHom_a_b_d_e_d_e @ R @ M @ N ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( F @ ( sprod_d_a_e @ M @ A2 @ M2 ) )
% 3.15/1.64 = ( sprod_d_a_e @ N @ A2 @ ( F @ M2 ) ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_317_Module_OmHom__lin,axiom,
% 3.15/1.64 ! [M: carrie1954260683t_unit,R: carrie1105631105xt_a_b,N: carrie1948989069_d_a_e,M2: c > d,F: ( c > d ) > d,A2: a] :
% 3.15/1.64 ( ( module1440397025unit_b @ M @ R )
% 3.15/1.64 => ( ( module_d_a_e_b @ N @ R )
% 3.15/1.64 => ( ( member_c_d @ M2 @ ( carrie20288686t_unit @ M ) )
% 3.15/1.64 => ( ( member_c_d_d @ F @ ( mHom_a764575953it_d_e @ R @ M @ N ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( F @ ( sprod_1929599593t_unit @ M @ A2 @ M2 ) )
% 3.15/1.64 = ( sprod_d_a_e @ N @ A2 @ ( F @ M2 ) ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_318_Module_Osprod__assoc,axiom,
% 3.15/1.64 ! [M: carrie1948989069_d_a_e,R: carrie1105631105xt_a_b,A2: a,B2: a,M2: d] :
% 3.15/1.64 ( ( module_d_a_e_b @ M @ R )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_d @ M2 @ ( carrie11369756_d_a_e @ M ) )
% 3.15/1.64 => ( ( sprod_d_a_e @ M @ ( tp_a_b @ R @ A2 @ B2 ) @ M2 )
% 3.15/1.64 = ( sprod_d_a_e @ M @ A2 @ ( sprod_d_a_e @ M @ B2 @ M2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_319_Module_Osprod__assoc,axiom,
% 3.15/1.64 ! [M: carrie1954260683t_unit,R: carrie1105631105xt_a_b,A2: a,B2: a,M2: c > d] :
% 3.15/1.64 ( ( module1440397025unit_b @ M @ R )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ R ) )
% 3.15/1.64 => ( ( member_c_d @ M2 @ ( carrie20288686t_unit @ M ) )
% 3.15/1.64 => ( ( sprod_1929599593t_unit @ M @ ( tp_a_b @ R @ A2 @ B2 ) @ M2 )
% 3.15/1.64 = ( sprod_1929599593t_unit @ M @ A2 @ ( sprod_1929599593t_unit @ M @ B2 @ M2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_320_Ring_OmHom__func,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,F: ( c > d ) > c > d,M: carrie1954260683t_unit,N: carrie1954260683t_unit] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( member_c_d_c_d @ F @ ( mHom_a1526663829t_unit @ R @ M @ N ) )
% 3.15/1.64 => ( member_c_d_c_d @ F
% 3.15/1.64 @ ( pi_c_d_c_d @ ( carrie20288686t_unit @ M )
% 3.15/1.64 @ ^ [Uu: c > d] : ( carrie20288686t_unit @ N ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_321_Ring_OSubring__pOp__ring__pOp,axiom,
% 3.15/1.64 ! [R: carrie1105631105xt_a_b,S: carrie1105631105xt_a_b,A2: a,B2: a] :
% 3.15/1.64 ( ( ring_a_b @ R )
% 3.15/1.64 => ( ( subring_a_b_b @ R @ S )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ S ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ S ) )
% 3.15/1.64 => ( ( pop_a_Ring_ext_a_b @ S @ A2 @ B2 )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ R @ A2 @ B2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_322_nsDistr,axiom,
% 3.15/1.64 ! [X2: a,N2: nat,M2: nat] :
% 3.15/1.64 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( pop_a_Ring_ext_a_b @ r @ ( nscal_a_b @ r @ X2 @ N2 ) @ ( nscal_a_b @ r @ X2 @ M2 ) )
% 3.15/1.64 = ( nscal_a_b @ r @ X2 @ ( plus_plus_nat @ N2 @ M2 ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_323_add__both,axiom,
% 3.15/1.64 ! [A2: nat,B2: nat,C2: nat] :
% 3.15/1.64 ( ( A2 = B2 )
% 3.15/1.64 => ( ( plus_plus_nat @ A2 @ C2 )
% 3.15/1.64 = ( plus_plus_nat @ B2 @ C2 ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_324_mul__closed__set__sub,axiom,
% 3.15/1.64 ! [S: set_a] :
% 3.15/1.64 ( ( mul_closed_set_a_b @ r @ S )
% 3.15/1.64 => ( ord_less_eq_set_a @ S @ ( carrie867757212xt_a_b @ r ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_325_zeroring__no__maximal,axiom,
% 3.15/1.64 ( ( zeroring_a_b @ r )
% 3.15/1.64 => ~ ? [X_1: set_a] : ( maximal_ideal_a_b @ r @ X_1 ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_326_id__maximal__Exist,axiom,
% 3.15/1.64 ( ~ ( zeroring_a_b @ r )
% 3.15/1.64 => ? [X_12: set_a] : ( maximal_ideal_a_b @ r @ X_12 ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_327_mop__closed,axiom,
% 3.15/1.64 ( member_a_a @ ( mop_a_Ring_ext_a_b @ r )
% 3.15/1.64 @ ( pi_a_a @ ( carrie867757212xt_a_b @ r )
% 3.15/1.64 @ ^ [Uu: a] : ( carrie867757212xt_a_b @ r ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_328_Sr__mOp__closed,axiom,
% 3.15/1.64 ! [S: set_a,X2: a] :
% 3.15/1.64 ( ( sr_a_b2 @ r @ S )
% 3.15/1.64 => ( ( member_a @ X2 @ S )
% 3.15/1.64 => ( member_a @ ( mop_a_Ring_ext_a_b @ r @ X2 ) @ S ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_329_ring__inv1,axiom,
% 3.15/1.64 ! [A2: a,B2: a] :
% 3.15/1.64 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( ( mop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ A2 @ B2 ) )
% 3.15/1.64 = ( tp_a_b @ r @ ( mop_a_Ring_ext_a_b @ r @ A2 ) @ B2 ) )
% 3.15/1.64 & ( ( mop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ A2 @ B2 ) )
% 3.15/1.64 = ( tp_a_b @ r @ A2 @ ( mop_a_Ring_ext_a_b @ r @ B2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_330_ring__inv1__1,axiom,
% 3.15/1.64 ! [A2: a,B2: a] :
% 3.15/1.64 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( mop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ A2 @ B2 ) )
% 3.15/1.64 = ( tp_a_b @ r @ ( mop_a_Ring_ext_a_b @ r @ A2 ) @ B2 ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_331_ring__inv1__2,axiom,
% 3.15/1.64 ! [A2: a,B2: a] :
% 3.15/1.64 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( mop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ A2 @ B2 ) )
% 3.15/1.64 = ( tp_a_b @ r @ A2 @ ( mop_a_Ring_ext_a_b @ r @ B2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_332_ring__inv1__3,axiom,
% 3.15/1.64 ! [A2: a,B2: a] :
% 3.15/1.64 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( tp_a_b @ r @ A2 @ B2 )
% 3.15/1.64 = ( tp_a_b @ r @ ( mop_a_Ring_ext_a_b @ r @ A2 ) @ ( mop_a_Ring_ext_a_b @ r @ B2 ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_333_ring__distrib4,axiom,
% 3.15/1.64 ! [A2: a,B2: a,X2: a,Y2: a] :
% 3.15/1.64 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ Y2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( pop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ A2 @ B2 ) @ ( mop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ X2 @ Y2 ) ) )
% 3.15/1.64 = ( pop_a_Ring_ext_a_b @ r @ ( tp_a_b @ r @ A2 @ ( pop_a_Ring_ext_a_b @ r @ B2 @ ( mop_a_Ring_ext_a_b @ r @ Y2 ) ) ) @ ( tp_a_b @ r @ ( pop_a_Ring_ext_a_b @ r @ A2 @ ( mop_a_Ring_ext_a_b @ r @ X2 ) ) @ Y2 ) ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_334_nonunit__contained__maxid,axiom,
% 3.15/1.64 ! [A2: a] :
% 3.15/1.64 ( ~ ( zeroring_a_b @ r )
% 3.15/1.64 => ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ~ ( unit_a_b @ r @ A2 )
% 3.15/1.64 => ? [Mx: set_a] :
% 3.15/1.64 ( ( maximal_ideal_a_b @ r @ Mx )
% 3.15/1.64 & ( member_a @ A2 @ Mx ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_335_set__sum__mem,axiom,
% 3.15/1.64 ! [A2: a,I: set_a,B2: a,J2: set_a] :
% 3.15/1.64 ( ( member_a @ A2 @ I )
% 3.15/1.64 => ( ( member_a @ B2 @ J2 )
% 3.15/1.64 => ( ( ord_less_eq_set_a @ I @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( ord_less_eq_set_a @ J2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( member_a @ ( pop_a_Ring_ext_a_b @ r @ A2 @ B2 ) @ ( aset_s1743553894xt_a_b @ r @ I @ J2 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_336_sum__mult__pOp__closed,axiom,
% 3.15/1.64 ! [A: set_a,B: set_a,A2: a,B2: a] :
% 3.15/1.64 ( ( ord_less_eq_set_a @ A @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( ord_less_eq_set_a @ B @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ A2 @ ( sum_mult_a_b @ r @ A @ B ) )
% 3.15/1.64 => ( ( member_a @ B2 @ ( sum_mult_a_b @ r @ A @ B ) )
% 3.15/1.64 => ( member_a @ ( pop_a_Ring_ext_a_b @ r @ A2 @ B2 ) @ ( sum_mult_a_b @ r @ A @ B ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_337_sum__mult__subR,axiom,
% 3.15/1.64 ! [A: set_a,B: set_a] :
% 3.15/1.64 ( ( ord_less_eq_set_a @ A @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( ord_less_eq_set_a @ B @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ord_less_eq_set_a @ ( sum_mult_a_b @ r @ A @ B ) @ ( carrie867757212xt_a_b @ r ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_338_times__mem__sum__mult,axiom,
% 3.15/1.64 ! [A: set_a,B: set_a,A2: a,B2: a] :
% 3.15/1.64 ( ( ord_less_eq_set_a @ A @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( ord_less_eq_set_a @ B @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( member_a @ A2 @ A )
% 3.15/1.64 => ( ( member_a @ B2 @ B )
% 3.15/1.64 => ( member_a @ ( tp_a_b @ r @ A2 @ B2 ) @ ( sum_mult_a_b @ r @ A @ B ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_339_le__imp__add__int,axiom,
% 3.15/1.64 ! [I2: nat,J: nat] :
% 3.15/1.64 ( ( ord_less_eq_nat @ I2 @ J )
% 3.15/1.64 => ? [K: nat] :
% 3.15/1.64 ( J
% 3.15/1.64 = ( plus_plus_nat @ I2 @ K ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_340_J__rad__unit,axiom,
% 3.15/1.64 ! [X2: a] :
% 3.15/1.64 ( ~ ( zeroring_a_b @ r )
% 3.15/1.64 => ( ( member_a @ X2 @ ( j_rad_a_b @ r ) )
% 3.15/1.64 => ! [Y3: a] :
% 3.15/1.64 ( ( member_a @ Y3 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( unit_a_b @ r @ ( pop_a_Ring_ext_a_b @ r @ ( un_a_b @ r ) @ ( tp_a_b @ r @ ( mop_a_Ring_ext_a_b @ r @ X2 ) @ Y3 ) ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_341_un__closed,axiom,
% 3.15/1.64 member_a @ ( un_a_b @ r ) @ ( carrie867757212xt_a_b @ r ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_342_maximal__ideal__proper,axiom,
% 3.15/1.64 ! [Mx2: set_a] :
% 3.15/1.64 ( ( maximal_ideal_a_b @ r @ Mx2 )
% 3.15/1.64 => ~ ( member_a @ ( un_a_b @ r ) @ Mx2 ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_343_Sr__one,axiom,
% 3.15/1.64 ! [S: set_a] :
% 3.15/1.64 ( ( sr_a_b2 @ r @ S )
% 3.15/1.64 => ( member_a @ ( un_a_b @ r ) @ S ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_344_rg__l__unit,axiom,
% 3.15/1.64 ! [A2: a] :
% 3.15/1.64 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( tp_a_b @ r @ ( un_a_b @ r ) @ A2 )
% 3.15/1.64 = A2 ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_345_ring__l__one,axiom,
% 3.15/1.64 ! [X2: a] :
% 3.15/1.64 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( tp_a_b @ r @ ( un_a_b @ r ) @ X2 )
% 3.15/1.64 = X2 ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_346_ring__r__one,axiom,
% 3.15/1.64 ! [X2: a] :
% 3.15/1.64 ( ( member_a @ X2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( tp_a_b @ r @ X2 @ ( un_a_b @ r ) )
% 3.15/1.64 = X2 ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_347_ring__times__minusr,axiom,
% 3.15/1.64 ! [A2: a] :
% 3.15/1.64 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( mop_a_Ring_ext_a_b @ r @ A2 )
% 3.15/1.64 = ( tp_a_b @ r @ A2 @ ( mop_a_Ring_ext_a_b @ r @ ( un_a_b @ r ) ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_348_ring__times__minusl,axiom,
% 3.15/1.64 ! [A2: a] :
% 3.15/1.64 ( ( member_a @ A2 @ ( carrie867757212xt_a_b @ r ) )
% 3.15/1.64 => ( ( mop_a_Ring_ext_a_b @ r @ A2 )
% 3.15/1.64 = ( tp_a_b @ r @ ( mop_a_Ring_ext_a_b @ r @ ( un_a_b @ r ) ) @ A2 ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_349_localring__unit,axiom,
% 3.15/1.64 ! [Mx2: set_a] :
% 3.15/1.64 ( ~ ( zeroring_a_b @ r )
% 3.15/1.64 => ( ( maximal_ideal_a_b @ r @ Mx2 )
% 3.15/1.64 => ( ! [X: a] :
% 3.15/1.64 ( ( member_a @ X @ Mx2 )
% 3.15/1.64 => ( unit_a_b @ r @ ( pop_a_Ring_ext_a_b @ r @ X @ ( un_a_b @ r ) ) ) )
% 3.15/1.64 => ( local_ring_a_b @ r ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_350_primary__ideal__proper1,axiom,
% 3.15/1.64 ! [Q2: set_a] :
% 3.15/1.64 ( ( primary_ideal_a_b @ r @ Q2 )
% 3.15/1.64 => ~ ( member_a @ ( un_a_b @ r ) @ Q2 ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_351_self__le,axiom,
% 3.15/1.64 ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_352_nat__eq__le,axiom,
% 3.15/1.64 ! [M2: nat,N2: nat] :
% 3.15/1.64 ( ( M2 = N2 )
% 3.15/1.64 => ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_353_n__in__Nsetn,axiom,
% 3.15/1.64 ! [N2: nat] :
% 3.15/1.64 ( member_nat @ N2
% 3.15/1.64 @ ( collect_nat
% 3.15/1.64 @ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N2 ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(fact_354_mem__of__Nset,axiom,
% 3.15/1.64 ! [X2: nat,N2: nat] :
% 3.15/1.64 ( ( ord_less_eq_nat @ X2 @ N2 )
% 3.15/1.64 => ( member_nat @ X2
% 3.15/1.64 @ ( collect_nat
% 3.15/1.64 @ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N2 ) ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(conj_0,hypothesis,
% 3.15/1.64 ! [X5: c] :
% 3.15/1.64 ( ( member_c @ X5 @ i )
% 3.15/1.64 => ( module_d_a_e_b @ ( m @ X5 ) @ r ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(conj_1,hypothesis,
% 3.15/1.64 ( ( member_c_d_c_d_c_d @ ( prod_p131511076_d_a_e @ i @ m )
% 3.15/1.64 @ ( pi_c_d_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
% 3.15/1.64 @ ^ [Uu: c > d] :
% 3.15/1.64 ( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
% 3.15/1.64 @ ^ [Uv: c > d] : ( carr_p1636662547_d_a_e @ i @ m ) ) ) )
% 3.15/1.64 & ! [A3: c > d] :
% 3.15/1.64 ( ( member_c_d @ A3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
% 3.15/1.64 => ! [B3: c > d] :
% 3.15/1.64 ( ( member_c_d @ B3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
% 3.15/1.64 => ! [C3: c > d] :
% 3.15/1.64 ( ( member_c_d @ C3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
% 3.15/1.64 => ( ( prod_p131511076_d_a_e @ i @ m @ ( prod_p131511076_d_a_e @ i @ m @ A3 @ B3 ) @ C3 )
% 3.15/1.64 = ( prod_p131511076_d_a_e @ i @ m @ A3 @ ( prod_p131511076_d_a_e @ i @ m @ B3 @ C3 ) ) ) ) ) )
% 3.15/1.64 & ! [A3: c > d] :
% 3.15/1.64 ( ( member_c_d @ A3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
% 3.15/1.64 => ! [B3: c > d] :
% 3.15/1.64 ( ( member_c_d @ B3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
% 3.15/1.64 => ( ( prod_p131511076_d_a_e @ i @ m @ A3 @ B3 )
% 3.15/1.64 = ( prod_p131511076_d_a_e @ i @ m @ B3 @ A3 ) ) ) )
% 3.15/1.64 & ( member_c_d_c_d @ ( prod_m854557927_d_a_e @ i @ m )
% 3.15/1.64 @ ( pi_c_d_c_d @ ( carr_p1636662547_d_a_e @ i @ m )
% 3.15/1.64 @ ^ [Uu: c > d] : ( carr_p1636662547_d_a_e @ i @ m ) ) )
% 3.15/1.64 & ! [A3: c > d] :
% 3.15/1.64 ( ( member_c_d @ A3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
% 3.15/1.64 => ( ( prod_p131511076_d_a_e @ i @ m @ ( prod_m854557927_d_a_e @ i @ m @ A3 ) @ A3 )
% 3.15/1.64 = ( prod_z1214987823_d_a_e @ i @ m ) ) )
% 3.15/1.64 & ( member_c_d @ ( prod_z1214987823_d_a_e @ i @ m ) @ ( carr_p1636662547_d_a_e @ i @ m ) )
% 3.15/1.64 & ! [A3: c > d] :
% 3.15/1.64 ( ( member_c_d @ A3 @ ( carr_p1636662547_d_a_e @ i @ m ) )
% 3.15/1.64 => ( ( prod_p131511076_d_a_e @ i @ m @ ( prod_z1214987823_d_a_e @ i @ m ) @ A3 )
% 3.15/1.64 = A3 ) ) ) ).
% 3.15/1.64
% 3.15/1.64 thf(conj_2,conjecture,
% 3.15/1.64 ! [A4: c > d] :
% 3.15/1.64 ( ~ ( member_c_d @ A4 @ ( carr_p1636662547_d_a_e @ i @ m ) )
% 3.15/1.64 | ( ( prod_p131511076_d_a_e @ i @ m @ A4 @ ( prod_z1214987823_d_a_e @ i @ m ) )
% 3.15/1.64 = A4 ) ) ).
% 3.15/1.64 ------------------------
% 3.23/1.70 % (12429)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_12139 for (2999ds/183Mi)
% 3.23/1.71 % (12432)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_12139 for (2999ds/2Mi)
% 3.23/1.71 % (12430)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_12139 for (2999ds/4Mi)
% 3.23/1.71 % (12431)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_12139 for (2999ds/27Mi)
% 3.23/1.71 % (12434)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_12139 for (2999ds/275Mi)
% 3.23/1.71 % (12435)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_12139 for (2999ds/18Mi)
% 3.23/1.71 % (12433)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_12139 for (2999ds/2Mi)
% 3.23/1.71 % (12432)Instruction limit reached!
% 3.23/1.71 % (12432)------------------------------
% 3.23/1.71 % (12432)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.71 % (12432)Termination reason: Unknown
% 3.23/1.71 % (12432)Termination phase: shuffling
% 3.23/1.71
% 3.23/1.71 % (12432)Memory used [KB]: 1791
% 3.23/1.71 % (12432)Time elapsed: 0.003 s
% 3.23/1.71 % (12432)Instructions burned: 2 (million)
% 3.23/1.71 % (12432)------------------------------
% 3.23/1.71 % (12432)------------------------------
% 3.23/1.71 % (12433)Instruction limit reached!
% 3.23/1.71 % (12433)------------------------------
% 3.23/1.71 % (12433)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.71 % (12433)Termination reason: Unknown
% 3.23/1.71 % (12433)Termination phase: shuffling
% 3.23/1.71
% 3.23/1.71 % (12433)Memory used [KB]: 1791
% 3.23/1.71 % (12433)Time elapsed: 0.003 s
% 3.23/1.71 % (12433)Instructions burned: 2 (million)
% 3.23/1.71 % (12433)------------------------------
% 3.23/1.71 % (12433)------------------------------
% 3.23/1.71 % (12430)Instruction limit reached!
% 3.23/1.71 % (12430)------------------------------
% 3.23/1.71 % (12430)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.71 % (12430)Termination reason: Unknown
% 3.23/1.71 % (12430)Termination phase: shuffling
% 3.23/1.71
% 3.23/1.71 % (12430)Memory used [KB]: 1791
% 3.23/1.71 % (12430)Time elapsed: 0.005 s
% 3.23/1.71 % (12430)Instructions burned: 4 (million)
% 3.23/1.71 % (12430)------------------------------
% 3.23/1.71 % (12430)------------------------------
% 3.23/1.72 % (12435)Instruction limit reached!
% 3.23/1.72 % (12435)------------------------------
% 3.23/1.72 % (12435)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.72 % (12435)Termination reason: Unknown
% 3.23/1.72 % (12435)Termination phase: shuffling
% 3.23/1.72
% 3.23/1.72 % (12435)Memory used [KB]: 2046
% 3.23/1.72 % (12435)Time elapsed: 0.014 s
% 3.23/1.72 % (12435)Instructions burned: 19 (million)
% 3.23/1.72 % (12435)------------------------------
% 3.23/1.72 % (12435)------------------------------
% 3.23/1.72 % (12440)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_12139 for (2999ds/3Mi)
% 3.23/1.72 % (12431)Instruction limit reached!
% 3.23/1.72 % (12431)------------------------------
% 3.23/1.72 % (12431)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.72 % (12431)Termination reason: Unknown
% 3.23/1.72 % (12431)Termination phase: shuffling
% 3.23/1.72
% 3.23/1.72 % (12431)Memory used [KB]: 2046
% 3.23/1.72 % (12431)Time elapsed: 0.019 s
% 3.23/1.72 % (12431)Instructions burned: 27 (million)
% 3.23/1.72 % (12431)------------------------------
% 3.23/1.72 % (12431)------------------------------
% 3.23/1.72 % (12440)Instruction limit reached!
% 3.23/1.72 % (12440)------------------------------
% 3.23/1.72 % (12440)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.72 % (12440)Termination reason: Unknown
% 3.23/1.72 % (12440)Termination phase: shuffling
% 3.23/1.72
% 3.23/1.72 % (12440)Memory used [KB]: 1791
% 3.23/1.72 % (12440)Time elapsed: 0.003 s
% 3.23/1.72 % (12440)Instructions burned: 4 (million)
% 3.23/1.72 % (12440)------------------------------
% 3.23/1.72 % (12440)------------------------------
% 3.23/1.72 % (12442)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_12139 for (2999ds/15Mi)
% 3.23/1.73 % (12441)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_12139 for (2999ds/37Mi)
% 3.23/1.73 % (12442)Instruction limit reached!
% 3.23/1.73 % (12442)------------------------------
% 3.23/1.73 % (12442)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.73 % (12442)Termination reason: Unknown
% 3.23/1.73 % (12442)Termination phase: shuffling
% 3.23/1.73
% 3.23/1.73 % (12442)Memory used [KB]: 1918
% 3.23/1.73 % (12442)Time elapsed: 0.009 s
% 3.23/1.73 % (12442)Instructions burned: 17 (million)
% 3.23/1.73 % (12442)------------------------------
% 3.23/1.73 % (12442)------------------------------
% 3.23/1.73 % (12454)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_12139 for (2999ds/7Mi)
% 3.23/1.74 % (12449)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_12139 for (2999ds/3Mi)
% 3.23/1.74 % (12454)Instruction limit reached!
% 3.23/1.74 % (12454)------------------------------
% 3.23/1.74 % (12454)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.74 % (12454)Termination reason: Unknown
% 3.23/1.74 % (12454)Termination phase: shuffling
% 3.23/1.74
% 3.23/1.74 % (12454)Memory used [KB]: 1918
% 3.23/1.74 % (12454)Time elapsed: 0.004 s
% 3.23/1.74 % (12454)Instructions burned: 7 (million)
% 3.23/1.74 % (12454)------------------------------
% 3.23/1.74 % (12454)------------------------------
% 3.23/1.74 % (12449)Instruction limit reached!
% 3.23/1.74 % (12449)------------------------------
% 3.23/1.74 % (12449)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.74 % (12449)Termination reason: Unknown
% 3.23/1.74 % (12449)Termination phase: shuffling
% 3.23/1.74
% 3.23/1.74 % (12449)Memory used [KB]: 1791
% 3.23/1.74 % (12449)Time elapsed: 0.005 s
% 3.23/1.74 % (12449)Instructions burned: 4 (million)
% 3.23/1.74 % (12449)------------------------------
% 3.23/1.74 % (12449)------------------------------
% 3.23/1.74 % (12452)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_12139 for (2999ds/1041Mi)
% 3.23/1.74 % (12459)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_12139 for (2999ds/16Mi)
% 3.23/1.75 % (12441)Instruction limit reached!
% 3.23/1.75 % (12441)------------------------------
% 3.23/1.75 % (12441)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.75 % (12441)Termination reason: Unknown
% 3.23/1.75 % (12441)Termination phase: shuffling
% 3.23/1.75
% 3.23/1.75 % (12441)Memory used [KB]: 2430
% 3.23/1.75 % (12441)Time elapsed: 0.022 s
% 3.23/1.75 % (12441)Instructions burned: 38 (million)
% 3.23/1.75 % (12441)------------------------------
% 3.23/1.75 % (12441)------------------------------
% 3.23/1.75 % (12463)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_12139 for (2999ds/3Mi)
% 3.23/1.75 % (12459)Instruction limit reached!
% 3.23/1.75 % (12459)------------------------------
% 3.23/1.75 % (12459)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.75 % (12459)Termination reason: Unknown
% 3.23/1.75 % (12459)Termination phase: shuffling
% 3.23/1.75
% 3.23/1.75 % (12459)Memory used [KB]: 2046
% 3.23/1.75 % (12459)Time elapsed: 0.007 s
% 3.23/1.75 % (12459)Instructions burned: 18 (million)
% 3.23/1.75 % (12459)------------------------------
% 3.23/1.75 % (12459)------------------------------
% 3.23/1.75 % (12463)Instruction limit reached!
% 3.23/1.75 % (12463)------------------------------
% 3.23/1.75 % (12463)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.75 % (12463)Termination reason: Unknown
% 3.23/1.75 % (12463)Termination phase: shuffling
% 3.23/1.75
% 3.23/1.75 % (12463)Memory used [KB]: 1791
% 3.23/1.75 % (12463)Time elapsed: 0.003 s
% 3.23/1.75 % (12463)Instructions burned: 5 (million)
% 3.23/1.75 % (12463)------------------------------
% 3.23/1.75 % (12463)------------------------------
% 3.23/1.76 % (12465)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_12139 for (2999ds/3Mi)
% 3.23/1.76 % (12472)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_12139 for (2999ds/3Mi)
% 3.23/1.76 % (12465)Instruction limit reached!
% 3.23/1.76 % (12465)------------------------------
% 3.23/1.76 % (12465)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.76 % (12465)Termination reason: Unknown
% 3.23/1.76 % (12465)Termination phase: shuffling
% 3.23/1.76
% 3.23/1.76 % (12465)Memory used [KB]: 1791
% 3.23/1.76 % (12465)Time elapsed: 0.004 s
% 3.23/1.76 % (12465)Instructions burned: 3 (million)
% 3.23/1.76 % (12465)------------------------------
% 3.23/1.76 % (12465)------------------------------
% 3.23/1.76 % (12472)Instruction limit reached!
% 3.23/1.76 % (12472)------------------------------
% 3.23/1.76 % (12472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.76 % (12472)Termination reason: Unknown
% 3.23/1.76 % (12472)Termination phase: shuffling
% 3.23/1.76
% 3.23/1.76 % (12472)Memory used [KB]: 1791
% 3.23/1.76 % (12472)Time elapsed: 0.003 s
% 3.23/1.76 % (12472)Instructions burned: 5 (million)
% 3.23/1.76 % (12472)------------------------------
% 3.23/1.76 % (12472)------------------------------
% 3.23/1.76 % (12474)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_12139 for (2999ds/4Mi)
% 3.23/1.76 % (12429)Instruction limit reached!
% 3.23/1.76 % (12429)------------------------------
% 3.23/1.76 % (12429)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.76 % (12429)Termination reason: Unknown
% 3.23/1.76 % (12429)Termination phase: Property scanning
% 3.23/1.76
% 3.23/1.76 % (12429)Memory used [KB]: 3326
% 3.23/1.76 % (12429)Time elapsed: 0.061 s
% 3.23/1.76 % (12429)Instructions burned: 183 (million)
% 3.23/1.76 % (12429)------------------------------
% 3.23/1.76 % (12429)------------------------------
% 3.23/1.76 % (12470)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_12139 for (2999ds/7Mi)
% 3.23/1.76 % (12474)Instruction limit reached!
% 3.23/1.76 % (12474)------------------------------
% 3.23/1.76 % (12474)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.76 % (12474)Termination reason: Unknown
% 3.23/1.76 % (12474)Termination phase: shuffling
% 3.23/1.76
% 3.23/1.76 % (12474)Memory used [KB]: 1791
% 3.23/1.76 % (12474)Time elapsed: 0.003 s
% 3.23/1.76 % (12474)Instructions burned: 5 (million)
% 3.23/1.76 % (12474)------------------------------
% 3.23/1.76 % (12474)------------------------------
% 3.23/1.76 % (12470)Instruction limit reached!
% 3.23/1.76 % (12470)------------------------------
% 3.23/1.76 % (12470)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.76 % (12470)Termination reason: Unknown
% 3.23/1.76 % (12470)Termination phase: shuffling
% 3.23/1.76
% 3.23/1.76 % (12470)Memory used [KB]: 1918
% 3.23/1.76 % (12470)Time elapsed: 0.005 s
% 3.23/1.76 % (12470)Instructions burned: 7 (million)
% 3.23/1.76 % (12470)------------------------------
% 3.23/1.76 % (12470)------------------------------
% 3.23/1.77 % (12481)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_12139 for (2998ds/710Mi)
% 3.23/1.77 % (12483)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_12139 for (2998ds/6Mi)
% 3.23/1.77 % (12485)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_12139 for (2998ds/902Mi)
% 3.23/1.77 % (12483)Instruction limit reached!
% 3.23/1.77 % (12483)------------------------------
% 3.23/1.77 % (12483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.77 % (12483)Termination reason: Unknown
% 3.23/1.77 % (12483)Termination phase: shuffling
% 3.23/1.77
% 3.23/1.77 % (12483)Memory used [KB]: 1918
% 3.23/1.77 % (12483)Time elapsed: 0.004 s
% 3.23/1.77 % (12483)Instructions burned: 8 (million)
% 3.23/1.77 % (12483)------------------------------
% 3.23/1.77 % (12483)------------------------------
% 3.23/1.77 % (12487)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_12139 for (2998ds/21Mi)
% 3.23/1.78 % (12480)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_12139 for (2998ds/18Mi)
% 3.23/1.78 % (12491)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_12139 for (2998ds/5Mi)
% 3.23/1.78 % (12487)Instruction limit reached!
% 3.23/1.78 % (12487)------------------------------
% 3.23/1.78 % (12487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.78 % (12487)Termination reason: Unknown
% 3.23/1.78 % (12487)Termination phase: shuffling
% 3.23/1.78
% 3.23/1.78 % (12487)Memory used [KB]: 2046
% 3.23/1.78 % (12487)Time elapsed: 0.009 s
% 3.23/1.78 % (12487)Instructions burned: 21 (million)
% 3.23/1.78 % (12487)------------------------------
% 3.23/1.78 % (12487)------------------------------
% 3.23/1.78 % (12491)Instruction limit reached!
% 3.23/1.78 % (12491)------------------------------
% 3.23/1.78 % (12491)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.78 % (12491)Termination reason: Unknown
% 3.23/1.78 % (12491)Termination phase: shuffling
% 3.23/1.78
% 3.23/1.78 % (12491)Memory used [KB]: 1918
% 3.23/1.78 % (12491)Time elapsed: 0.003 s
% 3.23/1.78 % (12491)Instructions burned: 6 (million)
% 3.23/1.78 % (12491)------------------------------
% 3.23/1.78 % (12491)------------------------------
% 3.23/1.79 % (12480)Instruction limit reached!
% 3.23/1.79 % (12480)------------------------------
% 3.23/1.79 % (12480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.79 % (12480)Termination reason: Unknown
% 3.23/1.79 % (12480)Termination phase: shuffling
% 3.23/1.79
% 3.23/1.79 % (12480)Memory used [KB]: 2046
% 3.23/1.79 % (12480)Time elapsed: 0.014 s
% 3.23/1.79 % (12480)Instructions burned: 19 (million)
% 3.23/1.79 % (12480)------------------------------
% 3.23/1.79 % (12480)------------------------------
% 3.23/1.79 % (12494)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_12139 for (2998ds/377Mi)
% 3.23/1.80 % (12492)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_12139 for (2998ds/6Mi)
% 3.23/1.80 % (12492)Instruction limit reached!
% 3.23/1.80 % (12492)------------------------------
% 3.23/1.80 % (12492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.80 % (12492)Termination reason: Unknown
% 3.23/1.80 % (12492)Termination phase: shuffling
% 3.23/1.80
% 3.23/1.80 % (12492)Memory used [KB]: 1918
% 3.23/1.80 % (12492)Time elapsed: 0.006 s
% 3.23/1.80 % (12492)Instructions burned: 6 (million)
% 3.23/1.80 % (12492)------------------------------
% 3.23/1.80 % (12492)------------------------------
% 3.23/1.80 % (12496)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_12139 for (2998ds/779Mi)
% 3.23/1.80 % (12434)Instruction limit reached!
% 3.23/1.80 % (12434)------------------------------
% 3.23/1.80 % (12434)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.80 % (12434)Termination reason: Unknown
% 3.23/1.80 % (12434)Termination phase: Saturation
% 3.23/1.80
% 3.23/1.80 % (12434)Memory used [KB]: 7803
% 3.23/1.80 % (12434)Time elapsed: 0.101 s
% 3.23/1.80 % (12434)Instructions burned: 278 (million)
% 3.23/1.80 % (12434)------------------------------
% 3.23/1.80 % (12434)------------------------------
% 3.23/1.82 % (12500)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_12139 for (2998ds/879Mi)
% 3.23/1.82 % (12499)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_12139 for (2998ds/19Mi)
% 3.23/1.83 % (12499)Instruction limit reached!
% 3.23/1.83 % (12499)------------------------------
% 3.23/1.83 % (12499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.83 % (12499)Termination reason: Unknown
% 3.23/1.83 % (12499)Termination phase: shuffling
% 3.23/1.83
% 3.23/1.83 % (12499)Memory used [KB]: 2174
% 3.23/1.83 % (12499)Time elapsed: 0.013 s
% 3.23/1.83 % (12499)Instructions burned: 20 (million)
% 3.23/1.83 % (12499)------------------------------
% 3.23/1.83 % (12499)------------------------------
% 3.23/1.85 % (12501)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_12139 for (2998ds/17Mi)
% 3.23/1.85 % (12501)Instruction limit reached!
% 3.23/1.85 % (12501)------------------------------
% 3.23/1.85 % (12501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.85 % (12501)Termination reason: Unknown
% 3.23/1.85 % (12501)Termination phase: shuffling
% 3.23/1.85
% 3.23/1.85 % (12501)Memory used [KB]: 2046
% 3.23/1.85 % (12501)Time elapsed: 0.007 s
% 3.23/1.85 % (12501)Instructions burned: 18 (million)
% 3.23/1.85 % (12501)------------------------------
% 3.23/1.85 % (12501)------------------------------
% 3.23/1.86 % (12502)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_12139 for (2998ds/3Mi)
% 3.23/1.86 % (12502)Instruction limit reached!
% 3.23/1.86 % (12502)------------------------------
% 3.23/1.86 % (12502)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.86 % (12502)Termination reason: Unknown
% 3.23/1.86 % (12502)Termination phase: shuffling
% 3.23/1.86
% 3.23/1.86 % (12502)Memory used [KB]: 1791
% 3.23/1.86 % (12502)Time elapsed: 0.003 s
% 3.23/1.86 % (12502)Instructions burned: 5 (million)
% 3.23/1.86 % (12502)------------------------------
% 3.23/1.86 % (12502)------------------------------
% 3.23/1.87 % (12503)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_12139 for (2997ds/30Mi)
% 3.23/1.88 % (12503)Instruction limit reached!
% 3.23/1.88 % (12503)------------------------------
% 3.23/1.88 % (12503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.88 % (12503)Termination reason: Unknown
% 3.23/1.88 % (12503)Termination phase: shuffling
% 3.23/1.88
% 3.23/1.88 % (12503)Memory used [KB]: 2302
% 3.23/1.88 % (12503)Time elapsed: 0.011 s
% 3.23/1.88 % (12503)Instructions burned: 31 (million)
% 3.23/1.88 % (12503)------------------------------
% 3.23/1.88 % (12503)------------------------------
% 3.23/1.89 % (12504)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_12139 for (2997ds/127Mi)
% 3.23/1.91 % (12494)Instruction limit reached!
% 3.23/1.91 % (12494)------------------------------
% 3.23/1.91 % (12494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.91 % (12494)Termination reason: Unknown
% 3.23/1.91 % (12494)Termination phase: Saturation
% 3.23/1.91
% 3.23/1.91 % (12494)Memory used [KB]: 9466
% 3.23/1.91 % (12494)Time elapsed: 0.114 s
% 3.23/1.91 % (12494)Instructions burned: 377 (million)
% 3.23/1.91 % (12494)------------------------------
% 3.23/1.91 % (12494)------------------------------
% 3.23/1.92 % (12505)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_12139 for (2997ds/100Mi)
% 3.23/1.93 % (12504)Instruction limit reached!
% 3.23/1.93 % (12504)------------------------------
% 3.23/1.93 % (12504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.93 % (12504)Termination reason: Unknown
% 3.23/1.93 % (12504)Termination phase: Preprocessing 3
% 3.23/1.93
% 3.23/1.93 % (12504)Memory used [KB]: 3070
% 3.23/1.93 % (12504)Time elapsed: 0.040 s
% 3.23/1.93 % (12504)Instructions burned: 129 (million)
% 3.23/1.93 % (12504)------------------------------
% 3.23/1.93 % (12504)------------------------------
% 3.23/1.94 % (12506)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_12139 for (2997ds/3Mi)
% 3.23/1.94 % (12506)Instruction limit reached!
% 3.23/1.94 % (12506)------------------------------
% 3.23/1.94 % (12506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.94 % (12506)Termination reason: Unknown
% 3.23/1.94 % (12506)Termination phase: shuffling
% 3.23/1.94
% 3.23/1.94 % (12506)Memory used [KB]: 1791
% 3.23/1.94 % (12506)Time elapsed: 0.002 s
% 3.23/1.94 % (12506)Instructions burned: 3 (million)
% 3.23/1.94 % (12506)------------------------------
% 3.23/1.94 % (12506)------------------------------
% 3.23/1.94 % (12505)Instruction limit reached!
% 3.23/1.94 % (12505)------------------------------
% 3.23/1.94 % (12505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.94 % (12505)Termination reason: Unknown
% 3.23/1.94 % (12505)Termination phase: Property scanning
% 3.23/1.94
% 3.23/1.94 % (12505)Memory used [KB]: 3326
% 3.23/1.94 % (12505)Time elapsed: 0.028 s
% 3.23/1.94 % (12505)Instructions burned: 100 (million)
% 3.23/1.94 % (12505)------------------------------
% 3.23/1.94 % (12505)------------------------------
% 3.23/1.95 % (12500)First to succeed.
% 3.23/1.95 % (12500)Refutation found. Thanks to Tanya!
% 3.23/1.95 % SZS status Theorem for DTF2THF_12139
% 3.23/1.95 % SZS output start Proof for DTF2THF_12139
% 3.23/1.95 thf(type_def_5, type, carrie1954260683t_unit: $tType).
% 3.23/1.95 thf(type_def_6, type, set_c_d: $tType).
% 3.23/1.95 thf(type_def_8, type, carrie1105631105xt_a_b: $tType).
% 3.23/1.95 thf(type_def_9, type, set_a: $tType).
% 3.23/1.95 thf(type_def_10, type, carrie1948989069_d_a_e: $tType).
% 3.23/1.95 thf(type_def_11, type, set_d: $tType).
% 3.23/1.95 thf(type_def_12, type, a: $tType).
% 3.23/1.95 thf(type_def_13, type, c: $tType).
% 3.23/1.95 thf(type_def_14, type, d: $tType).
% 3.23/1.95 thf(type_def_15, type, set_a_a: $tType).
% 3.23/1.95 thf(type_def_16, type, set_c: $tType).
% 3.23/1.95 thf(type_def_17, type, set_c_d_c_d: $tType).
% 3.23/1.95 thf(type_def_18, type, set_c_d_c_d_a2: $tType).
% 3.23/1.95 thf(type_def_19, type, set_c_d_c_d_d: $tType).
% 3.23/1.95 thf(type_def_20, type, set_a_a_a_a_a: $tType).
% 3.23/1.95 thf(type_def_21, type, set_a_a_a_a_a_d: $tType).
% 3.23/1.95 thf(type_def_22, type, set_a_a_a_a: $tType).
% 3.23/1.95 thf(type_def_23, type, set_a_a_a_a_a3: $tType).
% 3.23/1.95 thf(type_def_24, type, set_a_a_a_a_d: $tType).
% 3.23/1.95 thf(type_def_25, type, set_a_a_c_d: $tType).
% 3.23/1.95 thf(type_def_26, type, set_a_a_c_d_a: $tType).
% 3.23/1.95 thf(type_def_27, type, set_a_a_c_d_d: $tType).
% 3.23/1.95 thf(type_def_28, type, set_a_a_a: $tType).
% 3.23/1.95 thf(type_def_29, type, set_a_a_a_a4: $tType).
% 3.23/1.95 thf(type_def_30, type, set_a_a_a_d: $tType).
% 3.23/1.95 thf(type_def_31, type, set_a_a_a2: $tType).
% 3.23/1.95 thf(type_def_32, type, set_a_a_d: $tType).
% 3.23/1.95 thf(type_def_33, type, set_c_d_a: $tType).
% 3.23/1.95 thf(type_def_34, type, set_c_d_d: $tType).
% 3.23/1.95 thf(type_def_35, type, set_a_c_d: $tType).
% 3.23/1.95 thf(type_def_36, type, set_a_d: $tType).
% 3.23/1.95 thf(type_def_37, type, set_c_c_d: $tType).
% 3.23/1.95 thf(type_def_38, type, set_c_a: $tType).
% 3.23/1.95 thf(type_def_39, type, set_d_c_d: $tType).
% 3.23/1.95 thf(type_def_40, type, set_d_a: $tType).
% 3.23/1.95 thf(type_def_41, type, set_d_d: $tType).
% 3.23/1.95 thf(type_def_42, type, nat: $tType).
% 3.23/1.95 thf(type_def_43, type, set_a_a_a_a_a2: $tType).
% 3.23/1.95 thf(type_def_44, type, set_a_a_a_c_d: $tType).
% 3.23/1.95 thf(type_def_45, type, set_a_a_a_a3: $tType).
% 3.23/1.95 thf(type_def_46, type, set_a_a_c_d2: $tType).
% 3.23/1.95 thf(type_def_47, type, set_c_d_a_a_a2: $tType).
% 3.23/1.95 thf(type_def_48, type, set_c_d_c_d_c_d: $tType).
% 3.23/1.95 thf(type_def_49, type, set_c_d_c_d_a: $tType).
% 3.23/1.95 thf(type_def_50, type, set_c_d_a_a_a: $tType).
% 3.23/1.95 thf(type_def_51, type, set_c_d_a_a: $tType).
% 3.23/1.95 thf(type_def_52, type, set_a_a_a_a2: $tType).
% 3.23/1.95 thf(type_def_53, type, set_a_c_d_a: $tType).
% 3.23/1.95 thf(type_def_54, type, set_a_c_d_c_d: $tType).
% 3.23/1.95 thf(type_def_55, type, set_a_a_c_d_c_d: $tType).
% 3.23/1.95 thf(type_def_56, type, set_a_c: $tType).
% 3.23/1.95 thf(type_def_57, type, set_c_c: $tType).
% 3.23/1.95 thf(type_def_58, type, set_d_c: $tType).
% 3.23/1.95 thf(type_def_59, type, set_nat: $tType).
% 3.23/1.95 thf(func_def_0, type, carrie1954260683t_unit: $tType).
% 3.23/1.95 thf(func_def_1, type, carrie1948989069_d_a_e: $tType).
% 3.23/1.95 thf(func_def_2, type, carrie1105631105xt_a_b: $tType).
% 3.23/1.95 thf(func_def_3, type, set_a_a_c_d_c_d: $tType).
% 3.23/1.95 thf(func_def_4, type, set_c_d_c_d_c_d: $tType).
% 3.23/1.95 thf(func_def_5, type, set_a_a_a_a_a_d: $tType).
% 3.23/1.95 thf(func_def_6, type, set_a_a_a_a_a: $tType).
% 3.23/1.95 thf(func_def_7, type, set_a_c_d_c_d: $tType).
% 3.23/1.95 thf(func_def_8, type, set_c_d_a_a_a: $tType).
% 3.23/1.95 thf(func_def_9, type, set_c_d_c_d_a: $tType).
% 3.23/1.95 thf(func_def_10, type, set_c_d_a_a_a2: $tType).
% 3.23/1.95 thf(func_def_11, type, set_a_a_a_c_d: $tType).
% 3.23/1.95 thf(func_def_12, type, set_a_a_a_a_a2: $tType).
% 3.23/1.95 thf(func_def_13, type, set_a_a_c_d_d: $tType).
% 3.23/1.95 thf(func_def_14, type, set_a_a_c_d_a: $tType).
% 3.23/1.95 thf(func_def_15, type, set_a_a_a_a_d: $tType).
% 3.23/1.95 thf(func_def_16, type, set_a_a_a_a_a3: $tType).
% 3.23/1.95 thf(func_def_17, type, set_c_d_c_d_d: $tType).
% 3.23/1.95 thf(func_def_18, type, set_c_d_c_d_a2: $tType).
% 3.23/1.95 thf(func_def_19, type, set_a_a_c_d: $tType).
% 3.23/1.95 thf(func_def_20, type, set_a_a_a_a: $tType).
% 3.23/1.95 thf(func_def_21, type, set_a_c_d_a: $tType).
% 3.23/1.95 thf(func_def_22, type, set_a_a_a_a2: $tType).
% 3.23/1.95 thf(func_def_23, type, set_c_d_c_d: $tType).
% 3.23/1.95 thf(func_def_24, type, set_c_d_a_a: $tType).
% 3.23/1.95 thf(func_def_25, type, set_a_a_c_d2: $tType).
% 3.23/1.95 thf(func_def_26, type, set_a_a_a_a3: $tType).
% 3.23/1.95 thf(func_def_27, type, set_a_a_a_d: $tType).
% 3.23/1.95 thf(func_def_28, type, set_a_a_a_a4: $tType).
% 3.23/1.95 thf(func_def_29, type, set_d_c_d: $tType).
% 3.23/1.95 thf(func_def_30, type, set_c_c_d: $tType).
% 3.23/1.95 thf(func_def_31, type, set_a_c_d: $tType).
% 3.23/1.95 thf(func_def_32, type, set_a_a_a: $tType).
% 3.23/1.95 thf(func_def_33, type, set_c_d_d: $tType).
% 3.23/1.95 thf(func_def_34, type, set_c_d_a: $tType).
% 3.23/1.95 thf(func_def_35, type, set_a_a_d: $tType).
% 3.23/1.95 thf(func_def_36, type, set_a_a_a2: $tType).
% 3.23/1.95 thf(func_def_37, type, set_d_d: $tType).
% 3.23/1.95 thf(func_def_38, type, set_d_c: $tType).
% 3.23/1.95 thf(func_def_39, type, set_d_a: $tType).
% 3.23/1.95 thf(func_def_40, type, set_c_d: $tType).
% 3.23/1.95 thf(func_def_41, type, set_c_c: $tType).
% 3.23/1.95 thf(func_def_42, type, set_c_a: $tType).
% 3.23/1.95 thf(func_def_43, type, set_a_d: $tType).
% 3.23/1.95 thf(func_def_44, type, set_a_c: $tType).
% 3.23/1.95 thf(func_def_45, type, set_a_a: $tType).
% 3.23/1.95 thf(func_def_46, type, set_nat: $tType).
% 3.23/1.95 thf(func_def_47, type, set_d: $tType).
% 3.23/1.95 thf(func_def_48, type, set_c: $tType).
% 3.23/1.95 thf(func_def_49, type, set_a: $tType).
% 3.23/1.95 thf(func_def_50, type, nat: $tType).
% 3.23/1.95 thf(func_def_51, type, d: $tType).
% 3.23/1.95 thf(func_def_52, type, c: $tType).
% 3.23/1.95 thf(func_def_53, type, a: $tType).
% 3.23/1.95 thf(func_def_54, type, carrie20288686t_unit: carrie1954260683t_unit > set_c_d).
% 3.23/1.95 thf(func_def_55, type, carrie867757212xt_a_b: carrie1105631105xt_a_b > set_a).
% 3.23/1.95 thf(func_def_56, type, carrie11369756_d_a_e: carrie1948989069_d_a_e > set_d).
% 3.23/1.95 thf(func_def_57, type, j_rad_a_b: carrie1105631105xt_a_b > set_a).
% 3.23/1.95 thf(func_def_58, type, ring_a_b: carrie1105631105xt_a_b > $o).
% 3.23/1.95 thf(func_def_59, type, tp_a_b: carrie1105631105xt_a_b > a > a > a).
% 3.23/1.95 thf(func_def_60, type, un_a_b: carrie1105631105xt_a_b > a).
% 3.23/1.95 thf(func_def_61, type, sr_a_b: carrie1105631105xt_a_b > set_a > carrie1105631105xt_a_b).
% 3.23/1.95 thf(func_def_62, type, subring_a_b_b: carrie1105631105xt_a_b > carrie1105631105xt_a_b > $o).
% 3.23/1.95 thf(func_def_63, type, unit_a_b: carrie1105631105xt_a_b > a > $o).
% 3.23/1.95 thf(func_def_64, type, aGroup2131905830t_unit: carrie1954260683t_unit > $o).
% 3.23/1.95 thf(func_def_65, type, aGroup2097840802xt_a_b: carrie1105631105xt_a_b > $o).
% 3.23/1.95 thf(func_def_66, type, aGroup1940521469_d_a_e: carrie1948989069_d_a_e > $o).
% 3.23/1.95 thf(func_def_67, type, mop_a_Ring_ext_a_b: carrie1105631105xt_a_b > a > a).
% 3.23/1.95 thf(func_def_68, type, pop_c_449750139t_unit: carrie1954260683t_unit > (c > d) > (c > d) > c > d).
% 3.23/1.95 thf(func_def_69, type, pop_a_Ring_ext_a_b: carrie1105631105xt_a_b > a > a > a).
% 3.23/1.95 thf(func_def_70, type, pop_d_261713874_d_a_e: carrie1948989069_d_a_e > d > d > d).
% 3.23/1.95 thf(func_def_71, type, aset_s1743553894xt_a_b: carrie1105631105xt_a_b > set_a > set_a > set_a).
% 3.23/1.95 thf(func_def_72, type, carr_d267359390xt_a_b: set_a > (a > carrie1105631105xt_a_b) > set_a_a).
% 3.23/1.95 thf(func_def_73, type, carr_d56066307_d_a_e: set_c > (c > carrie1948989069_d_a_e) > set_c_d).
% 3.23/1.95 thf(func_def_74, type, carr_p1617094073xt_a_b: set_c_d_c_d > (((c > d) > c > d) > carrie1105631105xt_a_b) > set_c_d_c_d_a2).
% 3.23/1.95 thf(func_def_75, type, carr_p428609062_d_a_e: set_c_d_c_d > (((c > d) > c > d) > carrie1948989069_d_a_e) > set_c_d_c_d_d).
% 3.23/1.95 thf(func_def_76, type, carr_p1980447933_d_a_e: set_a_a_a_a_a > ((a > a > a > a > a) > carrie1948989069_d_a_e) > set_a_a_a_a_a_d).
% 3.23/1.95 thf(func_def_77, type, carr_p1850824937xt_a_b: set_a_a_a_a > ((a > a > a > a) > carrie1105631105xt_a_b) > set_a_a_a_a_a3).
% 3.23/1.95 thf(func_def_78, type, carr_p305487606_d_a_e: set_a_a_a_a > ((a > a > a > a) > carrie1948989069_d_a_e) > set_a_a_a_a_d).
% 3.23/1.95 thf(func_def_79, type, carr_p109409832xt_a_b: set_a_a_c_d > ((a > a > c > d) > carrie1105631105xt_a_b) > set_a_a_c_d_a).
% 3.23/1.95 thf(func_def_80, type, carr_p577380087_d_a_e: set_a_a_c_d > ((a > a > c > d) > carrie1948989069_d_a_e) > set_a_a_c_d_d).
% 3.23/1.95 thf(func_def_81, type, carr_p1481099064xt_a_b: set_a_a_a > ((a > a > a) > carrie1105631105xt_a_b) > set_a_a_a_a4).
% 3.23/1.95 thf(func_def_82, type, carr_p1364179431_d_a_e: set_a_a_a > ((a > a > a) > carrie1948989069_d_a_e) > set_a_a_a_d).
% 3.23/1.95 thf(func_def_83, type, carr_p740328767xt_a_b: set_a_a > ((a > a) > carrie1105631105xt_a_b) > set_a_a_a2).
% 3.23/1.95 thf(func_def_84, type, carr_p1646427936_d_a_e: set_a_a > ((a > a) > carrie1948989069_d_a_e) > set_a_a_d).
% 3.23/1.95 thf(func_def_85, type, carr_p996491646xt_a_b: set_c_d > ((c > d) > carrie1105631105xt_a_b) > set_c_d_a).
% 3.23/1.95 thf(func_def_86, type, carr_p124510241_d_a_e: set_c_d > ((c > d) > carrie1948989069_d_a_e) > set_c_d_d).
% 3.23/1.95 thf(func_def_87, type, carr_p923087418t_unit: set_a > (a > carrie1954260683t_unit) > set_a_c_d).
% 3.23/1.95 thf(func_def_88, type, carr_p290614414xt_a_b: set_a > (a > carrie1105631105xt_a_b) > set_a_a).
% 3.23/1.95 thf(func_def_89, type, carr_p1084104977_d_a_e: set_a > (a > carrie1948989069_d_a_e) > set_a_d).
% 3.23/1.95 thf(func_def_90, type, carr_p1080792380t_unit: set_c > (c > carrie1954260683t_unit) > set_c_c_d).
% 3.23/1.95 thf(func_def_91, type, carr_p295160588xt_a_b: set_c > (c > carrie1105631105xt_a_b) > set_c_a).
% 3.23/1.95 thf(func_def_92, type, carr_p1636662547_d_a_e: set_c > (c > carrie1948989069_d_a_e) > set_c_d).
% 3.23/1.95 thf(func_def_93, type, carr_p1159644861t_unit: set_d > (d > carrie1954260683t_unit) > set_d_c_d).
% 3.23/1.95 thf(func_def_94, type, carr_p297433675xt_a_b: set_d > (d > carrie1105631105xt_a_b) > set_d_a).
% 3.23/1.95 thf(func_def_95, type, carr_p1912941332_d_a_e: set_d > (d > carrie1948989069_d_a_e) > set_d_d).
% 3.23/1.95 thf(func_def_96, type, local_ring_a_b: carrie1105631105xt_a_b > $o).
% 3.23/1.95 thf(func_def_97, type, maximal_ideal_a_b: carrie1105631105xt_a_b > set_a > $o).
% 3.23/1.95 thf(func_def_98, type, mul_closed_set_a_b: carrie1105631105xt_a_b > set_a > $o).
% 3.23/1.95 thf(func_def_99, type, nscal_a_b: carrie1105631105xt_a_b > a > nat > a).
% 3.23/1.95 thf(func_def_100, type, primary_ideal_a_b: carrie1105631105xt_a_b > set_a > $o).
% 3.23/1.95 thf(func_def_101, type, prod_m2022989242xt_a_b: set_a > (a > carrie1105631105xt_a_b) > (a > a) > a > a).
% 3.23/1.95 thf(func_def_102, type, prod_m854557927_d_a_e: set_c > (c > carrie1948989069_d_a_e) > (c > d) > c > d).
% 3.23/1.95 thf(func_def_103, type, prod_p1959959370xt_a_b: set_c_d_c_d > (((c > d) > c > d) > carrie1105631105xt_a_b) > (((c > d) > c > d) > a) > (((c > d) > c > d) > a) > ((c > d) > c > d) > a).
% 3.23/1.95 thf(func_def_104, type, prod_p46206586xt_a_b: set_a_a_a_a > ((a > a > a > a) > carrie1105631105xt_a_b) > ((a > a > a > a) > a) > ((a > a > a > a) > a) > (a > a > a > a) > a).
% 3.23/1.95 thf(func_def_105, type, prod_p452275129xt_a_b: set_a_a_c_d > ((a > a > c > d) > carrie1105631105xt_a_b) > ((a > a > c > d) > a) > ((a > a > c > d) > a) > (a > a > c > d) > a).
% 3.23/1.95 thf(func_def_106, type, prod_p1460711911xt_a_b: set_a_a_a > ((a > a > a) > carrie1105631105xt_a_b) > ((a > a > a) > a) > ((a > a > a) > a) > (a > a > a) > a).
% 3.23/1.95 thf(func_def_107, type, prod_p1210452560xt_a_b: set_a_a > ((a > a) > carrie1105631105xt_a_b) > ((a > a) > a) > ((a > a) > a) > (a > a) > a).
% 3.23/1.95 thf(func_def_108, type, prod_p1466615439xt_a_b: set_c_d > ((c > d) > carrie1105631105xt_a_b) > ((c > d) > a) > ((c > d) > a) > (c > d) > a).
% 3.23/1.95 thf(func_def_109, type, prod_p704341949xt_a_b: set_a > (a > carrie1105631105xt_a_b) > (a > a) > (a > a) > a > a).
% 3.23/1.95 thf(func_def_110, type, prod_p708888123xt_a_b: set_c > (c > carrie1105631105xt_a_b) > (c > a) > (c > a) > c > a).
% 3.23/1.95 thf(func_def_111, type, prod_p131511076_d_a_e: set_c > (c > carrie1948989069_d_a_e) > (c > d) > (c > d) > c > d).
% 3.23/1.95 thf(func_def_112, type, prod_p711161210xt_a_b: set_d > (d > carrie1105631105xt_a_b) > (d > a) > (d > a) > d > a).
% 3.23/1.95 thf(func_def_113, type, prod_z851058290xt_a_b: set_a > (a > carrie1105631105xt_a_b) > a > a).
% 3.23/1.95 thf(func_def_114, type, prod_z1214987823_d_a_e: set_c > (c > carrie1948989069_d_a_e) > c > d).
% 3.23/1.95 thf(func_def_115, type, sr_a_b2: carrie1105631105xt_a_b > set_a > $o).
% 3.23/1.95 thf(func_def_116, type, zeroring_a_b: carrie1105631105xt_a_b > $o).
% 3.23/1.95 thf(func_def_117, type, sum_mult_a_b: carrie1105631105xt_a_b > set_a > set_a > set_a).
% 3.23/1.95 thf(func_def_118, type, module1440397025unit_b: carrie1954260683t_unit > carrie1105631105xt_a_b > $o).
% 3.23/1.95 thf(func_def_119, type, module_d_a_e_b: carrie1948989069_d_a_e > carrie1105631105xt_a_b > $o).
% 3.23/1.95 thf(func_def_120, type, sprod_1929599593t_unit: carrie1954260683t_unit > a > (c > d) > c > d).
% 3.23/1.95 thf(func_def_121, type, sprod_d_a_e: carrie1948989069_d_a_e > a > d > d).
% 3.23/1.95 thf(func_def_122, type, mHom_a1526663829t_unit: carrie1105631105xt_a_b > carrie1954260683t_unit > carrie1954260683t_unit > set_c_d_c_d).
% 3.23/1.95 thf(func_def_123, type, mHom_a764575953it_d_e: carrie1105631105xt_a_b > carrie1954260683t_unit > carrie1948989069_d_a_e > set_c_d_d).
% 3.23/1.95 thf(func_def_124, type, mHom_a_b_d_e_d_e: carrie1105631105xt_a_b > carrie1948989069_d_a_e > carrie1948989069_d_a_e > set_d_d).
% 3.23/1.95 thf(func_def_125, type, algebr1968153176_c_d_e: carrie1105631105xt_a_b > set_c > (c > carrie1948989069_d_a_e) > carrie1954260683t_unit).
% 3.23/1.95 thf(func_def_126, type, algebr828196337_d_d_e: carrie1105631105xt_a_b > set_c_d_c_d > (((c > d) > c > d) > carrie1948989069_d_a_e) > a > (((c > d) > c > d) > d) > ((c > d) > c > d) > d).
% 3.23/1.95 thf(func_def_127, type, algebr2122715238_a_d_e: carrie1105631105xt_a_b > set_a_a_a_a_a > ((a > a > a > a > a) > carrie1948989069_d_a_e) > a > ((a > a > a > a > a) > d) > (a > a > a > a > a) > d).
% 3.23/1.95 thf(func_def_128, type, algebr303206081_a_d_e: carrie1105631105xt_a_b > set_a_a_a_a > ((a > a > a > a) > carrie1948989069_d_a_e) > a > ((a > a > a > a) > d) > (a > a > a > a) > d).
% 3.23/1.95 thf(func_def_129, type, algebr1620716610_d_d_e: carrie1105631105xt_a_b > set_a_a_c_d > ((a > a > c > d) > carrie1948989069_d_a_e) > a > ((a > a > c > d) > d) > (a > a > c > d) > d).
% 3.23/1.95 thf(func_def_130, type, algebr1655014672_a_d_e: carrie1105631105xt_a_b > set_a_a_a > ((a > a > a) > carrie1948989069_d_a_e) > a > ((a > a > a) > d) > (a > a > a) > d).
% 3.23/1.95 thf(func_def_131, type, algebr137561451_a_d_e: carrie1105631105xt_a_b > set_a_a > ((a > a) > carrie1948989069_d_a_e) > a > ((a > a) > d) > (a > a) > d).
% 3.23/1.95 thf(func_def_132, type, algebr924632044_d_d_e: carrie1105631105xt_a_b > set_c_d > ((c > d) > carrie1948989069_d_a_e) > a > ((c > d) > d) > (c > d) > d).
% 3.23/1.95 thf(func_def_133, type, algebr2109842362_a_d_e: carrie1105631105xt_a_b > set_a > (a > carrie1948989069_d_a_e) > a > (a > d) > a > d).
% 3.23/1.95 thf(func_def_134, type, algebr1697692348_c_d_e: carrie1105631105xt_a_b > set_c > (c > carrie1948989069_d_a_e) > a > (c > d) > c > d).
% 3.23/1.95 thf(func_def_135, type, algebr417875517_d_d_e: carrie1105631105xt_a_b > set_d > (d > carrie1948989069_d_a_e) > a > (d > d) > d > d).
% 3.23/1.95 thf(func_def_136, type, pi_a_a_a_a_a: set_a_a_a > ((a > a > a) > set_a_a) > set_a_a_a_a_a2).
% 3.23/1.95 thf(func_def_137, type, pi_a_a_a_c_d: set_a_a_a > ((a > a > a) > set_c_d) > set_a_a_a_c_d).
% 3.23/1.95 thf(func_def_138, type, pi_a_a_a_a: set_a_a_a > ((a > a > a) > set_a) > set_a_a_a_a4).
% 3.23/1.95 thf(func_def_139, type, pi_a_a_a_a2: set_a_a > ((a > a) > set_a_a) > set_a_a_a_a3).
% 3.23/1.95 thf(func_def_140, type, pi_a_a_c_d: set_a_a > ((a > a) > set_c_d) > set_a_a_c_d2).
% 3.23/1.95 thf(func_def_141, type, pi_a_a_a: set_a_a > ((a > a) > set_a) > set_a_a_a2).
% 3.23/1.95 thf(func_def_142, type, pi_c_d_a_a_a: set_c_d > ((c > d) > set_a_a_a2) > set_c_d_a_a_a2).
% 3.23/1.95 thf(func_def_143, type, pi_c_d_c_d_c_d: set_c_d > ((c > d) > set_c_d_c_d) > set_c_d_c_d_c_d).
% 3.23/1.95 thf(func_def_144, type, pi_c_d_c_d_a: set_c_d > ((c > d) > set_c_d_a) > set_c_d_c_d_a).
% 3.23/1.95 thf(func_def_145, type, pi_c_d_a_a_a2: set_c_d > ((c > d) > set_a_a_a) > set_c_d_a_a_a).
% 3.23/1.95 thf(func_def_146, type, pi_c_d_a_a: set_c_d > ((c > d) > set_a_a) > set_c_d_a_a).
% 3.23/1.95 thf(func_def_147, type, pi_c_d_c_d: set_c_d > ((c > d) > set_c_d) > set_c_d_c_d).
% 3.23/1.95 thf(func_def_148, type, pi_c_d_a: set_c_d > ((c > d) > set_a) > set_c_d_a).
% 3.23/1.95 thf(func_def_149, type, pi_a_a_a_a3: set_a > (a > set_a_a_a2) > set_a_a_a_a2).
% 3.23/1.95 thf(func_def_150, type, pi_a_c_d_a: set_a > (a > set_c_d_a) > set_a_c_d_a).
% 3.23/1.95 thf(func_def_151, type, pi_a_a_c_d_c_d: set_a > (a > set_a_c_d_c_d) > set_a_a_c_d_c_d).
% 3.23/1.95 thf(func_def_152, type, pi_a_a_a_a_a2: set_a > (a > set_a_a_a_a) > set_a_a_a_a_a).
% 3.23/1.95 thf(func_def_153, type, pi_a_a_a_a4: set_a > (a > set_a_a_a) > set_a_a_a_a).
% 3.23/1.95 thf(func_def_154, type, pi_a_a_c_d2: set_a > (a > set_a_c_d) > set_a_a_c_d).
% 3.23/1.95 thf(func_def_155, type, pi_a_a_a2: set_a > (a > set_a_a) > set_a_a_a).
% 3.23/1.95 thf(func_def_156, type, pi_a_c_d: set_a > (a > set_c_d) > set_a_c_d).
% 3.23/1.95 thf(func_def_157, type, pi_a_a: set_a > (a > set_a) > set_a_a).
% 3.23/1.95 thf(func_def_158, type, pi_a_c: set_a > (a > set_c) > set_a_c).
% 3.23/1.95 thf(func_def_159, type, pi_a_d: set_a > (a > set_d) > set_a_d).
% 3.23/1.95 thf(func_def_160, type, pi_c_a: set_c > (c > set_a) > set_c_a).
% 3.23/1.95 thf(func_def_161, type, pi_c_c: set_c > (c > set_c) > set_c_c).
% 3.23/1.95 thf(func_def_162, type, pi_c_d: set_c > (c > set_d) > set_c_d).
% 3.23/1.95 thf(func_def_163, type, pi_d_a: set_d > (d > set_a) > set_d_a).
% 3.23/1.95 thf(func_def_164, type, pi_d_c: set_d > (d > set_c) > set_d_c).
% 3.23/1.95 thf(func_def_165, type, pi_d_d: set_d > (d > set_d) > set_d_d).
% 3.23/1.95 thf(func_def_166, type, plus_plus_nat: nat > nat > nat).
% 3.23/1.95 thf(func_def_167, type, ord_less_eq_nat: nat > nat > $o).
% 3.23/1.95 thf(func_def_168, type, ord_less_eq_set_a: set_a > set_a > $o).
% 3.23/1.95 thf(func_def_169, type, collect_c_d_c_d: (((c > d) > c > d) > $o) > set_c_d_c_d).
% 3.23/1.95 thf(func_def_170, type, collect_a_a_a_a: ((a > a > a > a) > $o) > set_a_a_a_a).
% 3.23/1.95 thf(func_def_171, type, collect_a_a_c_d: ((a > a > c > d) > $o) > set_a_a_c_d).
% 3.23/1.95 thf(func_def_172, type, collect_a_a_a: ((a > a > a) > $o) > set_a_a_a).
% 3.23/1.95 thf(func_def_173, type, collect_a_a: ((a > a) > $o) > set_a_a).
% 3.23/1.95 thf(func_def_174, type, collect_c_d: ((c > d) > $o) > set_c_d).
% 3.23/1.95 thf(func_def_175, type, collect_nat: (nat > $o) > set_nat).
% 3.23/1.95 thf(func_def_176, type, collect_a: (a > $o) > set_a).
% 3.23/1.95 thf(func_def_177, type, collect_c: (c > $o) > set_c).
% 3.23/1.95 thf(func_def_178, type, collect_d: (d > $o) > set_d).
% 3.23/1.95 thf(func_def_179, type, member_c_d_c_d_a: (((c > d) > c > d) > a) > set_c_d_c_d_a2 > $o).
% 3.23/1.95 thf(func_def_180, type, member_c_d_c_d_d: (((c > d) > c > d) > d) > set_c_d_c_d_d > $o).
% 3.23/1.95 thf(func_def_181, type, member_a_a_a_a_a_d: ((a > a > a > a > a) > d) > set_a_a_a_a_a_d > $o).
% 3.23/1.95 thf(func_def_182, type, member_a_a_a_a_a: ((a > a > a > a) > a) > set_a_a_a_a_a3 > $o).
% 3.23/1.95 thf(func_def_183, type, member_a_a_a_a_d: ((a > a > a > a) > d) > set_a_a_a_a_d > $o).
% 3.23/1.95 thf(func_def_184, type, member_a_a_c_d_a: ((a > a > c > d) > a) > set_a_a_c_d_a > $o).
% 3.23/1.95 thf(func_def_185, type, member_a_a_c_d_d: ((a > a > c > d) > d) > set_a_a_c_d_d > $o).
% 3.23/1.95 thf(func_def_186, type, member_a_a_a_a_a2: ((a > a > a) > a > a) > set_a_a_a_a_a2 > $o).
% 3.23/1.95 thf(func_def_187, type, member_a_a_a_c_d: ((a > a > a) > c > d) > set_a_a_a_c_d > $o).
% 3.23/1.95 thf(func_def_188, type, member_a_a_a_a: ((a > a > a) > a) > set_a_a_a_a4 > $o).
% 3.23/1.95 thf(func_def_189, type, member_a_a_a_d: ((a > a > a) > d) > set_a_a_a_d > $o).
% 3.23/1.95 thf(func_def_190, type, member_a_a_a_a2: ((a > a) > a > a) > set_a_a_a_a3 > $o).
% 3.23/1.95 thf(func_def_191, type, member_a_a_c_d: ((a > a) > c > d) > set_a_a_c_d2 > $o).
% 3.23/1.95 thf(func_def_192, type, member_a_a_a: ((a > a) > a) > set_a_a_a2 > $o).
% 3.23/1.95 thf(func_def_193, type, member_a_a_d: ((a > a) > d) > set_a_a_d > $o).
% 3.23/1.95 thf(func_def_194, type, member_c_d_a_a_a: ((c > d) > (a > a) > a) > set_c_d_a_a_a2 > $o).
% 3.23/1.95 thf(func_def_195, type, member_c_d_c_d_c_d: ((c > d) > (c > d) > c > d) > set_c_d_c_d_c_d > $o).
% 3.23/1.95 thf(func_def_196, type, member_c_d_c_d_a2: ((c > d) > (c > d) > a) > set_c_d_c_d_a > $o).
% 3.23/1.95 thf(func_def_197, type, member_c_d_a_a_a2: ((c > d) > a > a > a) > set_c_d_a_a_a > $o).
% 3.23/1.95 thf(func_def_198, type, member_c_d_a_a: ((c > d) > a > a) > set_c_d_a_a > $o).
% 3.23/1.95 thf(func_def_199, type, member_c_d_c_d: ((c > d) > c > d) > set_c_d_c_d > $o).
% 3.23/1.95 thf(func_def_200, type, member_c_d_a: ((c > d) > a) > set_c_d_a > $o).
% 3.23/1.95 thf(func_def_201, type, member_c_d_d: ((c > d) > d) > set_c_d_d > $o).
% 3.23/1.95 thf(func_def_202, type, member_a_a_a_a3: (a > (a > a) > a) > set_a_a_a_a2 > $o).
% 3.23/1.95 thf(func_def_203, type, member_a_c_d_c_d: (a > (c > d) > c > d) > set_a_c_d_c_d > $o).
% 3.23/1.95 thf(func_def_204, type, member_a_c_d_a: (a > (c > d) > a) > set_a_c_d_a > $o).
% 3.23/1.95 thf(func_def_205, type, member_a_a_c_d_c_d: (a > a > (c > d) > c > d) > set_a_a_c_d_c_d > $o).
% 3.23/1.95 thf(func_def_206, type, member_a_a_a_a_a3: (a > a > a > a > a) > set_a_a_a_a_a > $o).
% 3.23/1.95 thf(func_def_207, type, member_a_a_a_a4: (a > a > a > a) > set_a_a_a_a > $o).
% 3.23/1.95 thf(func_def_208, type, member_a_a_c_d2: (a > a > c > d) > set_a_a_c_d > $o).
% 3.23/1.95 thf(func_def_209, type, member_a_a_a2: (a > a > a) > set_a_a_a > $o).
% 3.23/1.95 thf(func_def_210, type, member_a_c_d: (a > c > d) > set_a_c_d > $o).
% 3.23/1.95 thf(func_def_211, type, member_a_a: (a > a) > set_a_a > $o).
% 3.23/1.95 thf(func_def_212, type, member_a_c: (a > c) > set_a_c > $o).
% 3.23/1.95 thf(func_def_213, type, member_a_d: (a > d) > set_a_d > $o).
% 3.23/1.95 thf(func_def_214, type, member_c_c_d: (c > c > d) > set_c_c_d > $o).
% 3.23/1.95 thf(func_def_215, type, member_c_a: (c > a) > set_c_a > $o).
% 3.23/1.95 thf(func_def_216, type, member_c_c: (c > c) > set_c_c > $o).
% 3.23/1.95 thf(func_def_217, type, member_c_d: (c > d) > set_c_d > $o).
% 3.23/1.95 thf(func_def_218, type, member_d_c_d: (d > c > d) > set_d_c_d > $o).
% 3.23/1.95 thf(func_def_219, type, member_d_a: (d > a) > set_d_a > $o).
% 3.23/1.95 thf(func_def_220, type, member_d_c: (d > c) > set_d_c > $o).
% 3.23/1.95 thf(func_def_221, type, member_d_d: (d > d) > set_d_d > $o).
% 3.23/1.95 thf(func_def_222, type, member_nat: nat > set_nat > $o).
% 3.23/1.95 thf(func_def_223, type, member_a: a > set_a > $o).
% 3.23/1.95 thf(func_def_224, type, member_c: c > set_c > $o).
% 3.23/1.95 thf(func_def_225, type, member_d: d > set_d > $o).
% 3.23/1.95 thf(func_def_226, type, i: set_c).
% 3.23/1.95 thf(func_def_227, type, m: c > carrie1948989069_d_a_e).
% 3.23/1.95 thf(func_def_228, type, r: carrie1105631105xt_a_b).
% 3.23/1.95 thf(func_def_236, type, sK0: set_c > (c > carrie1948989069_d_a_e) > c).
% 3.23/1.95 thf(func_def_237, type, sK1: (c > d) > set_c > (c > d) > c).
% 3.23/1.95 thf(func_def_238, type, sK2: set_c > (c > carrie1948989069_d_a_e) > c).
% 3.23/1.95 thf(func_def_239, type, sK3: (c > carrie1948989069_d_a_e) > set_c > carrie1105631105xt_a_b > c).
% 3.23/1.95 thf(func_def_240, type, sK4: ((c > d) > (c > d) > c > d) > ((c > d) > (c > d) > c > d) > set_c_d > c > d).
% 3.23/1.95 thf(func_def_241, type, sK5: c > d).
% 3.23/1.95 thf(func_def_242, type, sK6: (c > carrie1948989069_d_a_e) > carrie1105631105xt_a_b > set_c > c).
% 3.23/1.95 thf(func_def_243, type, sK7: (c > carrie1948989069_d_a_e) > set_c > c).
% 3.23/1.95 thf(func_def_244, type, sK8: (c > carrie1948989069_d_a_e) > set_c > c).
% 3.23/1.95 thf(func_def_245, type, sK9: set_a_a > ((a > a) > carrie1948989069_d_a_e) > a > a).
% 3.23/1.95 thf(func_def_246, type, sK10: set_a_a > ((a > a) > carrie1948989069_d_a_e) > carrie1105631105xt_a_b > a > a).
% 3.23/1.95 thf(func_def_247, type, sK11: (d > carrie1948989069_d_a_e) > carrie1105631105xt_a_b > set_d > d).
% 3.23/1.95 thf(func_def_248, type, sK12: ((a > a > a > a > a) > carrie1948989069_d_a_e) > set_a_a_a_a_a > a > a > a > a > a).
% 3.23/1.95 thf(func_def_249, type, sK13: (c > carrie1948989069_d_a_e) > set_c > c).
% 3.23/1.95 thf(func_def_250, type, sK14: ((c > d) > carrie1948989069_d_a_e) > set_c_d > c > d).
% 3.23/1.95 thf(func_def_251, type, sK15: (((c > d) > c > d) > carrie1948989069_d_a_e) > set_c_d_c_d > (c > d) > c > d).
% 3.23/1.95 thf(func_def_252, type, sK16: set_a > (a > carrie1948989069_d_a_e) > a).
% 3.23/1.95 thf(func_def_253, type, sK17: set_c > (c > d) > (c > d) > c).
% 3.23/1.95 thf(func_def_254, type, sK18: set_c > (c > carrie1948989069_d_a_e) > c).
% 3.23/1.95 thf(func_def_255, type, sK19: set_a_a_a_a > ((a > a > a > a) > carrie1948989069_d_a_e) > a > a > a > a).
% 3.23/1.95 thf(func_def_256, type, sK20: ((a > a > a > a) > carrie1948989069_d_a_e) > set_a_a_a_a > carrie1105631105xt_a_b > a > a > a > a).
% 3.23/1.95 thf(func_def_257, type, sK21: ((c > d) > c > d) > set_c_d > set_c_d > c > d).
% 3.23/1.95 thf(func_def_258, type, sK22: set_c > (c > carrie1948989069_d_a_e) > c).
% 3.23/1.95 thf(func_def_259, type, sK23: set_c > (c > carrie1948989069_d_a_e) > (c > carrie1948989069_d_a_e) > c).
% 3.23/1.95 thf(func_def_260, type, sK24: (c > carrie1948989069_d_a_e) > set_c > carrie1105631105xt_a_b > c).
% 3.23/1.95 thf(func_def_261, type, sK25: set_c > (c > carrie1948989069_d_a_e) > c).
% 3.23/1.95 thf(func_def_262, type, sK26: (((c > d) > c > d) > carrie1105631105xt_a_b) > set_c_d_c_d > (c > d) > c > d).
% 3.23/1.95 thf(func_def_263, type, sK27: set_a > (a > carrie1948989069_d_a_e) > carrie1105631105xt_a_b > a).
% 3.23/1.95 thf(func_def_264, type, sK28: set_c > (c > carrie1948989069_d_a_e) > (c > carrie1948989069_d_a_e) > c).
% 3.23/1.95 thf(func_def_265, type, sK29: set_c > (c > carrie1948989069_d_a_e) > c).
% 3.23/1.95 thf(func_def_266, type, sK30: (c > carrie1948989069_d_a_e) > set_c > c).
% 3.23/1.95 thf(func_def_267, type, sK31: set_c > (c > carrie1948989069_d_a_e) > c).
% 3.23/1.95 thf(func_def_268, type, sK32: set_c > (c > carrie1948989069_d_a_e) > c).
% 3.23/1.95 thf(func_def_269, type, sK33: ((c > d) > set_c_d) > set_c_d > ((c > d) > c > d) > c > d).
% 3.23/1.95 thf(func_def_270, type, sK34: (d > carrie1948989069_d_a_e) > set_d > d).
% 3.23/1.95 thf(func_def_271, type, sK35: set_c_d > ((c > d) > c > d) > ((c > d) > c > d) > c > d).
% 3.23/1.95 thf(func_def_272, type, sK36: set_a_a_a > ((a > a > a) > carrie1948989069_d_a_e) > carrie1105631105xt_a_b > a > a > a).
% 3.23/1.95 thf(func_def_273, type, sK37: (c > carrie1948989069_d_a_e) > set_c > c).
% 3.23/1.95 thf(func_def_274, type, sK38: set_a_a_a > ((a > a > a) > carrie1948989069_d_a_e) > a > a > a).
% 3.23/1.95 thf(func_def_275, type, sK39: (c > carrie1948989069_d_a_e) > set_c > c).
% 3.23/1.95 thf(func_def_276, type, sK40: set_c > (c > carrie1948989069_d_a_e) > c).
% 3.23/1.95 thf(func_def_277, type, sK41: (c > carrie1948989069_d_a_e) > set_c > c).
% 3.23/1.95 thf(func_def_278, type, sK42: set_c > (c > carrie1948989069_d_a_e) > c).
% 3.23/1.95 thf(func_def_279, type, sK43: set_c > (c > carrie1948989069_d_a_e) > (c > carrie1948989069_d_a_e) > c).
% 3.23/1.95 thf(func_def_280, type, sK44: set_c > (c > carrie1948989069_d_a_e) > c).
% 3.23/1.95 thf(func_def_281, type, sK45: (c > carrie1948989069_d_a_e) > set_c > c).
% 3.23/1.95 thf(func_def_282, type, sK46: (c > carrie1948989069_d_a_e) > set_c > carrie1105631105xt_a_b > c).
% 3.23/1.95 thf(func_def_283, type, sK47: (c > d) > set_c > (c > d) > c).
% 3.23/1.95 thf(func_def_284, type, sK48: set_a_a_c_d > ((a > a > c > d) > carrie1948989069_d_a_e) > carrie1105631105xt_a_b > a > a > c > d).
% 3.23/1.95 thf(func_def_285, type, sK49: ((a > a > a > a > a) > carrie1948989069_d_a_e) > set_a_a_a_a_a > carrie1105631105xt_a_b > a > a > a > a > a).
% 3.23/1.95 thf(func_def_286, type, sK50: (c > carrie1948989069_d_a_e) > set_c > c).
% 3.23/1.95 thf(func_def_287, type, sK51: carrie1105631105xt_a_b > ((c > d) > carrie1948989069_d_a_e) > set_c_d > c > d).
% 3.23/1.95 thf(func_def_288, type, sK52: (c > carrie1948989069_d_a_e) > set_c > (c > carrie1948989069_d_a_e) > c).
% 3.23/1.95 thf(func_def_289, type, sK53: (c > carrie1948989069_d_a_e) > set_c > c).
% 3.23/1.95 thf(func_def_290, type, sK54: set_c_d_c_d > (((c > d) > c > d) > carrie1948989069_d_a_e) > carrie1105631105xt_a_b > (c > d) > c > d).
% 3.23/1.95 thf(func_def_291, type, sK55: set_a_a_c_d > ((a > a > c > d) > carrie1948989069_d_a_e) > a > a > c > d).
% 3.23/1.95 thf(func_def_292, type, sK56: set_c > (c > carrie1948989069_d_a_e) > c).
% 3.23/1.95 thf(func_def_294, type, sK58: ((c > d) > set_c_d_c_d) > ((c > d) > (c > d) > c > d) > set_c_d > c > d).
% 3.23/1.95 thf(func_def_295, type, sK59: set_c_d > ((c > d) > c > d) > ((c > d) > set_c_d) > c > d).
% 3.23/1.95 thf(func_def_296, type, ph60: !>[X0: $tType]:(X0)).
% 3.23/1.95 thf(func_def_298, type, sK61: c).
% 3.23/1.95 thf(f1791,plain,(
% 3.23/1.95 $false),
% 3.23/1.95 inference(subsumption_resolution,[status(thm)],[f1790,f1299])).
% 3.23/1.95 thf(f1299,plain,(
% 3.23/1.95 (sK5 != (prod_p131511076_d_a_e @ i @ m @ sK5 @ (prod_z1214987823_d_a_e @ i @ m)))),
% 3.23/1.95 inference(cnf_transformation,[status(thm)],[f1185])).
% 3.23/1.95 thf(f1185,plain,(
% 3.23/1.95 (sK5 != (prod_p131511076_d_a_e @ i @ m @ sK5 @ (prod_z1214987823_d_a_e @ i @ m))) & ((member_c_d @ sK5 @ (carr_p1636662547_d_a_e @ i @ m)) = $true)),
% 3.23/1.95 inference(skolemisation,[status(esa),new_symbols(skolem,[sK5])],[f1085,f1184])).
% 3.23/1.95 thf(f1184,plain,(
% 3.23/1.95 ? [X0 : c > d] : (((prod_p131511076_d_a_e @ i @ m @ X0 @ (prod_z1214987823_d_a_e @ i @ m)) != X0) & ((member_c_d @ X0 @ (carr_p1636662547_d_a_e @ i @ m)) = $true)) => ((sK5 != (prod_p131511076_d_a_e @ i @ m @ sK5 @ (prod_z1214987823_d_a_e @ i @ m))) & ((member_c_d @ sK5 @ (carr_p1636662547_d_a_e @ i @ m)) = $true))),
% 3.23/1.95 introduced(definition,[],[choice_axiom])).
% 3.23/1.95 thf(f1085,plain,(
% 3.23/1.95 ? [X0 : c > d] : (((prod_p131511076_d_a_e @ i @ m @ X0 @ (prod_z1214987823_d_a_e @ i @ m)) != X0) & ((member_c_d @ X0 @ (carr_p1636662547_d_a_e @ i @ m)) = $true))),
% 3.23/1.95 inference(ennf_transformation,[status(thm)],[f1054])).
% 3.23/1.95 thf(f1054,plain,(
% 3.23/1.95 ~! [X0 : c > d] : (((prod_p131511076_d_a_e @ i @ m @ X0 @ (prod_z1214987823_d_a_e @ i @ m)) = X0) | ((member_c_d @ X0 @ (carr_p1636662547_d_a_e @ i @ m)) != $true))),
% 3.23/1.95 inference(flattening,[status(thm)],[f914])).
% 3.23/1.95 thf(f914,plain,(
% 3.23/1.95 ~! [X0 : c > d] : (~((member_c_d @ X0 @ (carr_p1636662547_d_a_e @ i @ m)) = $true) | ((prod_p131511076_d_a_e @ i @ m @ X0 @ (prod_z1214987823_d_a_e @ i @ m)) = X0))),
% 3.23/1.95 inference(fool_elimination,[status(thm)],[f913])).
% 3.23/1.95 thf(f913,plain,(
% 3.23/1.95 ~! [X0 : c > d] : (~(member_c_d @ X0 @ (carr_p1636662547_d_a_e @ i @ m)) | ((prod_p131511076_d_a_e @ i @ m @ X0 @ (prod_z1214987823_d_a_e @ i @ m)) = X0))),
% 3.23/1.95 inference(rectify,[status(thm)],[f359])).
% 3.23/1.95 thf(f359,negated_conjecture,(
% 3.23/1.95 ~! [X53 : c > d] : (~(member_c_d @ X53 @ (carr_p1636662547_d_a_e @ i @ m)) | ((prod_p131511076_d_a_e @ i @ m @ X53 @ (prod_z1214987823_d_a_e @ i @ m)) = X53))),
% 3.23/1.95 inference(negated_conjecture,[status(cth)],[f358])).
% 3.23/1.95 thf(f358,conjecture,(
% 3.23/1.95 ! [X53 : c > d] : (~(member_c_d @ X53 @ (carr_p1636662547_d_a_e @ i @ m)) | ((prod_p131511076_d_a_e @ i @ m @ X53 @ (prod_z1214987823_d_a_e @ i @ m)) = X53))),
% 3.23/1.95 file('/export/starexec/sandbox/tmp/tmp.2el5MKjZyq/DTF2THF_12139.p',conj_2)).
% 3.23/1.95 thf(f1790,plain,(
% 3.23/1.95 (sK5 = (prod_p131511076_d_a_e @ i @ m @ sK5 @ (prod_z1214987823_d_a_e @ i @ m)))),
% 3.23/1.95 inference(forward_demodulation,[status(thm)],[f1781,f1492])).
% 3.23/1.95 thf(f1492,plain,(
% 3.23/1.95 (sK5 = (prod_p131511076_d_a_e @ i @ m @ (prod_z1214987823_d_a_e @ i @ m) @ sK5))),
% 3.23/1.95 inference(trivial_inequality_removal,[status(thm)],[f1491])).
% 3.23/1.95 thf(f1491,plain,(
% 3.23/1.95 ($true != $true) | (sK5 = (prod_p131511076_d_a_e @ i @ m @ (prod_z1214987823_d_a_e @ i @ m) @ sK5))),
% 3.23/1.95 inference(superposition,[status(thm)],[f1328,f1298])).
% 3.23/1.95 thf(f1298,plain,(
% 3.23/1.95 ((member_c_d @ sK5 @ (carr_p1636662547_d_a_e @ i @ m)) = $true)),
% 3.23/1.95 inference(cnf_transformation,[status(thm)],[f1185])).
% 3.23/1.95 thf(f1328,plain,(
% 3.23/1.95 ( ! [X3 : c > d] : (((member_c_d @ X3 @ (carr_p1636662547_d_a_e @ i @ m)) != $true) | ((prod_p131511076_d_a_e @ i @ m @ (prod_z1214987823_d_a_e @ i @ m) @ X3) = X3)) )),
% 3.23/1.95 inference(cnf_transformation,[status(thm)],[f1204])).
% 3.23/1.95 thf(f1204,plain,(
% 3.23/1.95 ((member_c_d @ (prod_z1214987823_d_a_e @ i @ m) @ (carr_p1636662547_d_a_e @ i @ m)) = $true) & ! [X0 : c > d] : (! [X1 : c > d] : (((member_c_d @ X1 @ (carr_p1636662547_d_a_e @ i @ m)) != $true) | ! [X2 : c > d] : (((member_c_d @ X2 @ (carr_p1636662547_d_a_e @ i @ m)) != $true) | ((prod_p131511076_d_a_e @ i @ m @ (prod_p131511076_d_a_e @ i @ m @ X0 @ X1) @ X2) = (prod_p131511076_d_a_e @ i @ m @ X0 @ (prod_p131511076_d_a_e @ i @ m @ X1 @ X2))))) | ((member_c_d @ X0 @ (carr_p1636662547_d_a_e @ i @ m)) != $true)) & ! [X3 : c > d] : (((member_c_d @ X3 @ (carr_p1636662547_d_a_e @ i @ m)) != $true) | ((prod_p131511076_d_a_e @ i @ m @ (prod_z1214987823_d_a_e @ i @ m) @ X3) = X3)) & ((member_c_d_c_d_c_d @ (prod_p131511076_d_a_e @ i @ m) @ (pi_c_d_c_d_c_d @ (carr_p1636662547_d_a_e @ i @ m) @ (^[Y0 : c > d]: (pi_c_d_c_d @ (carr_p1636662547_d_a_e @ i @ m) @ (^[Y1 : c > d]: (carr_p1636662547_d_a_e @ i @ m)))))) = $true) & ! [X4 : c > d] : (((prod_z1214987823_d_a_e @ i @ m) = (prod_p131511076_d_a_e @ i @ m @ (prod_m854557927_d_a_e @ i @ m @ X4) @ X4)) | ((member_c_d @ X4 @ (carr_p1636662547_d_a_e @ i @ m)) != $true)) & ! [X5 : c > d] : (((member_c_d @ X5 @ (carr_p1636662547_d_a_e @ i @ m)) != $true) | ! [X6 : c > d] : (((prod_p131511076_d_a_e @ i @ m @ X6 @ X5) = (prod_p131511076_d_a_e @ i @ m @ X5 @ X6)) | ((member_c_d @ X6 @ (carr_p1636662547_d_a_e @ i @ m)) != $true))) & ((member_c_d_c_d @ (prod_m854557927_d_a_e @ i @ m) @ (pi_c_d_c_d @ (carr_p1636662547_d_a_e @ i @ m) @ (^[Y0 : c > d]: (carr_p1636662547_d_a_e @ i @ m)))) = $true)),
% 3.23/1.95 inference(rectify,[status(thm)],[f1080])).
% 3.23/1.95 thf(f1080,plain,(
% 3.23/1.95 ((member_c_d @ (prod_z1214987823_d_a_e @ i @ m) @ (carr_p1636662547_d_a_e @ i @ m)) = $true) & ! [X0 : c > d] : (! [X1 : c > d] : (((member_c_d @ X1 @ (carr_p1636662547_d_a_e @ i @ m)) != $true) | ! [X2 : c > d] : (((member_c_d @ X2 @ (carr_p1636662547_d_a_e @ i @ m)) != $true) | ((prod_p131511076_d_a_e @ i @ m @ (prod_p131511076_d_a_e @ i @ m @ X0 @ X1) @ X2) = (prod_p131511076_d_a_e @ i @ m @ X0 @ (prod_p131511076_d_a_e @ i @ m @ X1 @ X2))))) | ((member_c_d @ X0 @ (carr_p1636662547_d_a_e @ i @ m)) != $true)) & ! [X4 : c > d] : (((member_c_d @ X4 @ (carr_p1636662547_d_a_e @ i @ m)) != $true) | ((prod_p131511076_d_a_e @ i @ m @ (prod_z1214987823_d_a_e @ i @ m) @ X4) = X4)) & ((member_c_d_c_d_c_d @ (prod_p131511076_d_a_e @ i @ m) @ (pi_c_d_c_d_c_d @ (carr_p1636662547_d_a_e @ i @ m) @ (^[Y0 : c > d]: (pi_c_d_c_d @ (carr_p1636662547_d_a_e @ i @ m) @ (^[Y1 : c > d]: (carr_p1636662547_d_a_e @ i @ m)))))) = $true) & ! [X3 : c > d] : (((prod_z1214987823_d_a_e @ i @ m) = (prod_p131511076_d_a_e @ i @ m @ (prod_m854557927_d_a_e @ i @ m @ X3) @ X3)) | ((member_c_d @ X3 @ (carr_p1636662547_d_a_e @ i @ m)) != $true)) & ! [X5 : c > d] : (((member_c_d @ X5 @ (carr_p1636662547_d_a_e @ i @ m)) != $true) | ! [X6 : c > d] : (((prod_p131511076_d_a_e @ i @ m @ X6 @ X5) = (prod_p131511076_d_a_e @ i @ m @ X5 @ X6)) | ((member_c_d @ X6 @ (carr_p1636662547_d_a_e @ i @ m)) != $true))) & ((member_c_d_c_d @ (prod_m854557927_d_a_e @ i @ m) @ (pi_c_d_c_d @ (carr_p1636662547_d_a_e @ i @ m) @ (^[Y0 : c > d]: (carr_p1636662547_d_a_e @ i @ m)))) = $true)),
% 3.23/1.95 inference(ennf_transformation,[status(thm)],[f1053])).
% 3.23/1.95 thf(f1053,plain,(
% 3.23/1.95 ! [X5 : c > d] : (((member_c_d @ X5 @ (carr_p1636662547_d_a_e @ i @ m)) = $true) => ! [X6 : c > d] : (((member_c_d @ X6 @ (carr_p1636662547_d_a_e @ i @ m)) = $true) => ((prod_p131511076_d_a_e @ i @ m @ X6 @ X5) = (prod_p131511076_d_a_e @ i @ m @ X5 @ X6)))) & ! [X0 : c > d] : (((member_c_d @ X0 @ (carr_p1636662547_d_a_e @ i @ m)) = $true) => ! [X1 : c > d] : (((member_c_d @ X1 @ (carr_p1636662547_d_a_e @ i @ m)) = $true) => ! [X2 : c > d] : (((member_c_d @ X2 @ (carr_p1636662547_d_a_e @ i @ m)) = $true) => ((prod_p131511076_d_a_e @ i @ m @ (prod_p131511076_d_a_e @ i @ m @ X0 @ X1) @ X2) = (prod_p131511076_d_a_e @ i @ m @ X0 @ (prod_p131511076_d_a_e @ i @ m @ X1 @ X2)))))) & ! [X4 : c > d] : (((member_c_d @ X4 @ (carr_p1636662547_d_a_e @ i @ m)) = $true) => ((prod_p131511076_d_a_e @ i @ m @ (prod_z1214987823_d_a_e @ i @ m) @ X4) = X4)) & ((member_c_d_c_d @ (prod_m854557927_d_a_e @ i @ m) @ (pi_c_d_c_d @ (carr_p1636662547_d_a_e @ i @ m) @ (^[Y0 : c > d]: (carr_p1636662547_d_a_e @ i @ m)))) = $true) & ! [X3 : c > d] : (((member_c_d @ X3 @ (carr_p1636662547_d_a_e @ i @ m)) = $true) => ((prod_z1214987823_d_a_e @ i @ m) = (prod_p131511076_d_a_e @ i @ m @ (prod_m854557927_d_a_e @ i @ m @ X3) @ X3))) & ((member_c_d @ (prod_z1214987823_d_a_e @ i @ m) @ (carr_p1636662547_d_a_e @ i @ m)) = $true) & ((member_c_d_c_d_c_d @ (prod_p131511076_d_a_e @ i @ m) @ (pi_c_d_c_d_c_d @ (carr_p1636662547_d_a_e @ i @ m) @ (^[Y0 : c > d]: (pi_c_d_c_d @ (carr_p1636662547_d_a_e @ i @ m) @ (^[Y1 : c > d]: (carr_p1636662547_d_a_e @ i @ m)))))) = $true)),
% 3.23/1.95 inference(rectify,[status(thm)],[f388])).
% 3.23/1.95 thf(f388,plain,(
% 3.23/1.95 ! [X0 : c > d] : (((member_c_d @ X0 @ (carr_p1636662547_d_a_e @ i @ m)) = $true) => ! [X1 : c > d] : (((member_c_d @ X1 @ (carr_p1636662547_d_a_e @ i @ m)) = $true) => ! [X2 : c > d] : (((member_c_d @ X2 @ (carr_p1636662547_d_a_e @ i @ m)) = $true) => ((prod_p131511076_d_a_e @ i @ m @ (prod_p131511076_d_a_e @ i @ m @ X0 @ X1) @ X2) = (prod_p131511076_d_a_e @ i @ m @ X0 @ (prod_p131511076_d_a_e @ i @ m @ X1 @ X2)))))) & ((member_c_d_c_d_c_d @ (prod_p131511076_d_a_e @ i @ m) @ (pi_c_d_c_d_c_d @ (carr_p1636662547_d_a_e @ i @ m) @ (^[Y0 : c > d]: (pi_c_d_c_d @ (carr_p1636662547_d_a_e @ i @ m) @ (^[Y1 : c > d]: (carr_p1636662547_d_a_e @ i @ m)))))) = $true) & ((member_c_d @ (prod_z1214987823_d_a_e @ i @ m) @ (carr_p1636662547_d_a_e @ i @ m)) = $true) & ! [X5 : c > d] : (((member_c_d @ X5 @ (carr_p1636662547_d_a_e @ i @ m)) = $true) => ((prod_z1214987823_d_a_e @ i @ m) = (prod_p131511076_d_a_e @ i @ m @ (prod_m854557927_d_a_e @ i @ m @ X5) @ X5))) & ((member_c_d_c_d @ (prod_m854557927_d_a_e @ i @ m) @ (pi_c_d_c_d @ (carr_p1636662547_d_a_e @ i @ m) @ (^[Y0 : c > d]: (carr_p1636662547_d_a_e @ i @ m)))) = $true) & ! [X7 : c > d] : (((member_c_d @ X7 @ (carr_p1636662547_d_a_e @ i @ m)) = $true) => ((prod_p131511076_d_a_e @ i @ m @ (prod_z1214987823_d_a_e @ i @ m) @ X7) = X7)) & ! [X8 : c > d] : (((member_c_d @ X8 @ (carr_p1636662547_d_a_e @ i @ m)) = $true) => ! [X9 : c > d] : (((member_c_d @ X9 @ (carr_p1636662547_d_a_e @ i @ m)) = $true) => ((prod_p131511076_d_a_e @ i @ m @ X8 @ X9) = (prod_p131511076_d_a_e @ i @ m @ X9 @ X8))))),
% 3.23/1.95 inference(fool_elimination,[status(thm)],[f387])).
% 3.23/1.95 thf(f387,plain,(
% 3.23/1.95 ! [X0 : c > d] : ((member_c_d @ X0 @ (carr_p1636662547_d_a_e @ i @ m)) => ! [X1 : c > d] : ((member_c_d @ X1 @ (carr_p1636662547_d_a_e @ i @ m)) => ! [X2 : c > d] : ((member_c_d @ X2 @ (carr_p1636662547_d_a_e @ i @ m)) => ((prod_p131511076_d_a_e @ i @ m @ (prod_p131511076_d_a_e @ i @ m @ X0 @ X1) @ X2) = (prod_p131511076_d_a_e @ i @ m @ X0 @ (prod_p131511076_d_a_e @ i @ m @ X1 @ X2)))))) & (member_c_d_c_d_c_d @ (prod_p131511076_d_a_e @ i @ m) @ (pi_c_d_c_d_c_d @ (carr_p1636662547_d_a_e @ i @ m) @ (^[X3 : c > d] : (pi_c_d_c_d @ (carr_p1636662547_d_a_e @ i @ m) @ (^[X4 : c > d] : (carr_p1636662547_d_a_e @ i @ m)))))) & (member_c_d @ (prod_z1214987823_d_a_e @ i @ m) @ (carr_p1636662547_d_a_e @ i @ m)) & ! [X5 : c > d] : ((member_c_d @ X5 @ (carr_p1636662547_d_a_e @ i @ m)) => ((prod_z1214987823_d_a_e @ i @ m) = (prod_p131511076_d_a_e @ i @ m @ (prod_m854557927_d_a_e @ i @ m @ X5) @ X5))) & (member_c_d_c_d @ (prod_m854557927_d_a_e @ i @ m) @ (pi_c_d_c_d @ (carr_p1636662547_d_a_e @ i @ m) @ (^[X6 : c > d] : (carr_p1636662547_d_a_e @ i @ m)))) & ! [X7 : c > d] : ((member_c_d @ X7 @ (carr_p1636662547_d_a_e @ i @ m)) => ((prod_p131511076_d_a_e @ i @ m @ (prod_z1214987823_d_a_e @ i @ m) @ X7) = X7)) & ! [X8 : c > d] : ((member_c_d @ X8 @ (carr_p1636662547_d_a_e @ i @ m)) => ! [X9 : c > d] : ((member_c_d @ X9 @ (carr_p1636662547_d_a_e @ i @ m)) => ((prod_p131511076_d_a_e @ i @ m @ X8 @ X9) = (prod_p131511076_d_a_e @ i @ m @ X9 @ X8))))),
% 3.23/1.95 inference(rectify,[status(thm)],[f357])).
% 3.23/1.95 thf(f357,axiom,(
% 3.23/1.95 ! [X50 : c > d] : ((member_c_d @ X50 @ (carr_p1636662547_d_a_e @ i @ m)) => ! [X51 : c > d] : ((member_c_d @ X51 @ (carr_p1636662547_d_a_e @ i @ m)) => ! [X52 : c > d] : ((member_c_d @ X52 @ (carr_p1636662547_d_a_e @ i @ m)) => ((prod_p131511076_d_a_e @ i @ m @ (prod_p131511076_d_a_e @ i @ m @ X50 @ X51) @ X52) = (prod_p131511076_d_a_e @ i @ m @ X50 @ (prod_p131511076_d_a_e @ i @ m @ X51 @ X52)))))) & (member_c_d_c_d_c_d @ (prod_p131511076_d_a_e @ i @ m) @ (pi_c_d_c_d_c_d @ (carr_p1636662547_d_a_e @ i @ m) @ (^[X5 : c > d] : (pi_c_d_c_d @ (carr_p1636662547_d_a_e @ i @ m) @ (^[X6 : c > d] : (carr_p1636662547_d_a_e @ i @ m)))))) & (member_c_d @ (prod_z1214987823_d_a_e @ i @ m) @ (carr_p1636662547_d_a_e @ i @ m)) & ! [X50 : c > d] : ((member_c_d @ X50 @ (carr_p1636662547_d_a_e @ i @ m)) => ((prod_p131511076_d_a_e @ i @ m @ (prod_m854557927_d_a_e @ i @ m @ X50) @ X50) = (prod_z1214987823_d_a_e @ i @ m))) & (member_c_d_c_d @ (prod_m854557927_d_a_e @ i @ m) @ (pi_c_d_c_d @ (carr_p1636662547_d_a_e @ i @ m) @ (^[X5 : c > d] : (carr_p1636662547_d_a_e @ i @ m)))) & ! [X50 : c > d] : ((member_c_d @ X50 @ (carr_p1636662547_d_a_e @ i @ m)) => ((prod_p131511076_d_a_e @ i @ m @ (prod_z1214987823_d_a_e @ i @ m) @ X50) = X50)) & ! [X50 : c > d] : ((member_c_d @ X50 @ (carr_p1636662547_d_a_e @ i @ m)) => ! [X51 : c > d] : ((member_c_d @ X51 @ (carr_p1636662547_d_a_e @ i @ m)) => ((prod_p131511076_d_a_e @ i @ m @ X50 @ X51) = (prod_p131511076_d_a_e @ i @ m @ X51 @ X50))))),
% 3.23/1.95 file('/export/starexec/sandbox/tmp/tmp.2el5MKjZyq/DTF2THF_12139.p',conj_1)).
% 3.23/1.95 thf(f1781,plain,(
% 3.23/1.95 ((prod_p131511076_d_a_e @ i @ m @ sK5 @ (prod_z1214987823_d_a_e @ i @ m)) = (prod_p131511076_d_a_e @ i @ m @ (prod_z1214987823_d_a_e @ i @ m) @ sK5))),
% 3.23/1.95 inference(trivial_inequality_removal,[status(thm)],[f1775])).
% 3.23/1.95 thf(f1775,plain,(
% 3.23/1.95 ($true != $true) | ((prod_p131511076_d_a_e @ i @ m @ sK5 @ (prod_z1214987823_d_a_e @ i @ m)) = (prod_p131511076_d_a_e @ i @ m @ (prod_z1214987823_d_a_e @ i @ m) @ sK5))),
% 3.23/1.95 inference(superposition,[status(thm)],[f1509,f1330])).
% 3.23/1.95 thf(f1330,plain,(
% 3.23/1.95 ((member_c_d @ (prod_z1214987823_d_a_e @ i @ m) @ (carr_p1636662547_d_a_e @ i @ m)) = $true)),
% 3.23/1.95 inference(cnf_transformation,[status(thm)],[f1204])).
% 3.23/1.95 thf(f1509,plain,(
% 3.23/1.95 ( ! [X0 : c > d] : (((member_c_d @ X0 @ (carr_p1636662547_d_a_e @ i @ m)) != $true) | ((prod_p131511076_d_a_e @ i @ m @ sK5 @ X0) = (prod_p131511076_d_a_e @ i @ m @ X0 @ sK5))) )),
% 3.23/1.95 inference(trivial_inequality_removal,[status(thm)],[f1507])).
% 3.23/1.95 thf(f1507,plain,(
% 3.23/1.95 ( ! [X0 : c > d] : (((prod_p131511076_d_a_e @ i @ m @ sK5 @ X0) = (prod_p131511076_d_a_e @ i @ m @ X0 @ sK5)) | ($true != $true) | ((member_c_d @ X0 @ (carr_p1636662547_d_a_e @ i @ m)) != $true)) )),
% 3.23/1.95 inference(superposition,[status(thm)],[f1325,f1298])).
% 3.23/1.95 thf(f1325,plain,(
% 3.23/1.95 ( ! [X6 : c > d,X5 : c > d] : (((member_c_d @ X6 @ (carr_p1636662547_d_a_e @ i @ m)) != $true) | ((prod_p131511076_d_a_e @ i @ m @ X6 @ X5) = (prod_p131511076_d_a_e @ i @ m @ X5 @ X6)) | ((member_c_d @ X5 @ (carr_p1636662547_d_a_e @ i @ m)) != $true)) )),
% 3.23/1.95 inference(cnf_transformation,[status(thm)],[f1204])).
% 3.23/1.95 % SZS output end Proof for DTF2THF_12139
% 3.23/1.95 % (12500)------------------------------
% 3.23/1.95 % (12500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.23/1.95 % (12500)Termination reason: Refutation
% 3.23/1.95
% 3.23/1.95 % (12500)Memory used [KB]: 8827
% 3.23/1.95 % (12500)Time elapsed: 0.134 s
% 3.23/1.95 % (12500)Instructions burned: 419 (million)
% 3.23/1.95 % (12500)------------------------------
% 3.23/1.95 % (12500)------------------------------
% 3.23/1.95 % (12419)Success in time 0.306 s
%------------------------------------------------------------------------------