%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP107^1 : 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 : n019.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 08:55:36 AM UTC 2026
% Result : Theorem 3.31s 1.74s
% Output : Refutation 3.31s
% Verified :
% SZS Type : Refutation
% Derivation depth : 7
% Number of leaves : 5
% Syntax : Number of formulae : 16 ( 16 unt; 0 typ; 0 def)
% Number of atoms : 16 ( 15 equ; 0 cnn)
% Maximal formula atoms : 1 ( 1 avg)
% Number of connectives : 258 ( 6 ~; 0 |; 0 &; 252 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 2 ( 1 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 104 ( 102 usr; 12 con; 0-3 aty)
% Number of variables : 0 ( 0 ^; 0 !; 0 ?; 0 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
partia1833973666xt_a_b: $tType ).
thf(type_def_6,type,
set_a: $tType ).
thf(type_def_7,type,
set_set_a: $tType ).
thf(type_def_9,type,
a: $tType ).
thf(type_def_10,type,
partia1993116613t_unit: $tType ).
thf(type_def_11,type,
set_Product_prod_a_a: $tType ).
thf(type_def_12,type,
partia539144763t_unit: $tType ).
thf(type_def_13,type,
set_se1596668135od_a_a: $tType ).
thf(type_def_14,type,
partia96731725t_unit: $tType ).
thf(type_def_15,type,
product_prod_a_a: $tType ).
thf(type_def_16,type,
int: $tType ).
thf(type_def_17,type,
nat: $tType ).
thf(func_def_0,type,
partia539144763t_unit: $tType ).
thf(func_def_1,type,
partia1993116613t_unit: $tType ).
thf(func_def_2,type,
partia1833973666xt_a_b: $tType ).
thf(func_def_3,type,
partia96731725t_unit: $tType ).
thf(func_def_4,type,
set_se1596668135od_a_a: $tType ).
thf(func_def_5,type,
set_Product_prod_a_a: $tType ).
thf(func_def_6,type,
product_prod_a_a: $tType ).
thf(func_def_7,type,
set_set_a: $tType ).
thf(func_def_8,type,
set_a: $tType ).
thf(func_def_9,type,
nat: $tType ).
thf(func_def_10,type,
int: $tType ).
thf(func_def_11,type,
a: $tType ).
thf(func_def_12,type,
a_RCOSETS_a_b: partia1833973666xt_a_b > set_a > set_set_a ).
thf(func_def_13,type,
a_l_coset_a_b: partia1833973666xt_a_b > a > set_a > set_a ).
thf(func_def_14,type,
a_r_coset_a_b: partia1833973666xt_a_b > set_a > a > set_a ).
thf(func_def_15,type,
additi2104487374up_a_b: set_a > partia1833973666xt_a_b > $o ).
thf(func_def_16,type,
set_add_a_b: partia1833973666xt_a_b > set_a > set_a > set_a ).
thf(func_def_17,type,
equiva2136214035t_unit: partia1993116613t_unit > $o ).
thf(func_def_18,type,
partia206007992t_unit: partia1993116613t_unit > set_Product_prod_a_a ).
thf(func_def_19,type,
partia616129906t_unit: partia539144763t_unit > set_se1596668135od_a_a ).
thf(func_def_20,type,
partia1955795460t_unit: partia96731725t_unit > set_a ).
thf(func_def_21,type,
partia1066395285xt_a_b: partia1833973666xt_a_b > set_a ).
thf(func_def_22,type,
units_1734235042t_unit: partia539144763t_unit > set_se1596668135od_a_a ).
thf(func_def_23,type,
units_a_Product_unit: partia96731725t_unit > set_a ).
thf(func_def_24,type,
units_a_ring_ext_a_b: partia1833973666xt_a_b > set_a ).
thf(func_def_25,type,
group_52730354t_unit: partia539144763t_unit > $o ).
thf(func_def_26,type,
group_a_Product_unit: partia96731725t_unit > $o ).
thf(func_def_27,type,
group_a_ring_ext_a_b: partia1833973666xt_a_b > $o ).
thf(func_def_28,type,
monoid1242292471t_unit: partia539144763t_unit > $o ).
thf(func_def_29,type,
monoid1447254976t_unit: partia96731725t_unit > $o ).
thf(func_def_30,type,
monoid1022105667xt_a_b: partia1833973666xt_a_b > $o ).
thf(func_def_31,type,
mult_s977248631t_unit: partia539144763t_unit > set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(func_def_32,type,
mult_a_Product_unit: partia96731725t_unit > a > a > a ).
thf(func_def_33,type,
mult_a_ring_ext_a_b: partia1833973666xt_a_b > a > a > a ).
thf(func_def_34,type,
one_se1488596747t_unit: partia539144763t_unit > set_Product_prod_a_a ).
thf(func_def_35,type,
one_a_Product_unit: partia96731725t_unit > a ).
thf(func_def_36,type,
one_a_ring_ext_a_b: partia1833973666xt_a_b > a ).
thf(func_def_37,type,
units_390215097t_unit: partia539144763t_unit > partia539144763t_unit ).
thf(func_def_38,type,
units_1411277569xt_a_b: partia1833973666xt_a_b > partia96731725t_unit ).
thf(func_def_39,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(func_def_40,type,
locali1170246543t_unit: partia1993116613t_unit > a > a > set_Product_prod_a_a ).
thf(func_def_41,type,
locali1648887798ac_a_b: partia1833973666xt_a_b > set_a > $o ).
thf(func_def_42,type,
locali1944243560ac_a_b: partia1833973666xt_a_b > set_a > partia539144763t_unit ).
thf(func_def_43,type,
locali807230110ng_a_b: partia1833973666xt_a_b > set_a > $o ).
thf(func_def_44,type,
locali880295127ng_a_b: partia1833973666xt_a_b > set_a > $o ).
thf(func_def_45,type,
locali1538946157t_unit: partia1993116613t_unit > set_se1596668135od_a_a ).
thf(func_def_46,type,
locali1488796916t_unit: partia539144763t_unit > set_se1596668135od_a_a > $o ).
thf(func_def_47,type,
locali425460998xt_a_b: partia1833973666xt_a_b > set_a > $o ).
thf(func_def_48,type,
bot_bo2131659635od_a_a: set_Product_prod_a_a ).
thf(func_def_49,type,
bot_bo1424703315od_a_a: set_se1596668135od_a_a ).
thf(func_def_50,type,
bot_bot_set_set_a: set_set_a ).
thf(func_def_51,type,
bot_bot_set_a: set_a ).
thf(func_def_52,type,
ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(func_def_53,type,
ord_le391835783od_a_a: set_se1596668135od_a_a > set_se1596668135od_a_a > $o ).
thf(func_def_54,type,
ord_le318720350_set_a: set_set_a > set_set_a > $o ).
thf(func_def_55,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(func_def_56,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(func_def_57,type,
abelian_monoid_a_b: partia1833973666xt_a_b > $o ).
thf(func_def_58,type,
add_pow_a_b_int: partia1833973666xt_a_b > int > a > a ).
thf(func_def_59,type,
add_pow_a_b_nat: partia1833973666xt_a_b > nat > a > a ).
thf(func_def_60,type,
field_a_b: partia1833973666xt_a_b > $o ).
thf(func_def_61,type,
ring_a_b: partia1833973666xt_a_b > $o ).
thf(func_def_62,type,
zero_a_b: partia1833973666xt_a_b > a ).
thf(func_def_63,type,
semiring_a_b: partia1833973666xt_a_b > $o ).
thf(func_def_64,type,
collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(func_def_65,type,
collec183727474od_a_a: ( set_Product_prod_a_a > $o ) > set_se1596668135od_a_a ).
thf(func_def_66,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(func_def_67,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_68,type,
pow_a: set_a > set_set_a ).
thf(func_def_69,type,
insert1116662519od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(func_def_70,type,
insert1738756823od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a > set_se1596668135od_a_a ).
thf(func_def_71,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(func_def_72,type,
insert_a: a > set_a > set_a ).
thf(func_def_73,type,
member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(func_def_74,type,
member1838126896od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a > $o ).
thf(func_def_75,type,
member_set_a: set_a > set_set_a > $o ).
thf(func_def_76,type,
member_a: a > set_a > $o ).
thf(func_def_77,type,
r: partia1833973666xt_a_b ).
thf(func_def_78,type,
s: set_a ).
thf(func_def_79,type,
r2: a ).
thf(func_def_80,type,
r3: a ).
thf(func_def_81,type,
rel: partia1993116613t_unit ).
thf(func_def_82,type,
s2: a ).
thf(func_def_83,type,
s3: a ).
thf(func_def_96,type,
sP0: partia539144763t_unit > $o ).
thf(func_def_97,type,
sP1: partia539144763t_unit > $o ).
thf(func_def_98,type,
sP2: partia539144763t_unit > $o ).
thf(func_def_99,type,
sP3: partia539144763t_unit > $o ).
thf(func_def_100,type,
sK4: product_prod_a_a > a ).
thf(func_def_101,type,
sK5: product_prod_a_a > a ).
thf(func_def_102,type,
sK6: set_a > set_a > a ).
thf(func_def_103,type,
sK7: partia539144763t_unit > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(func_def_104,type,
sK8: ( product_prod_a_a > $o ) > a ).
thf(func_def_105,type,
sK9: ( product_prod_a_a > $o ) > a ).
thf(func_def_106,type,
sK10: partia539144763t_unit > set_Product_prod_a_a ).
thf(func_def_107,type,
sK11: partia539144763t_unit > set_Product_prod_a_a ).
thf(func_def_108,type,
sK12: partia539144763t_unit > set_Product_prod_a_a ).
thf(func_def_109,type,
sK13: partia539144763t_unit > set_Product_prod_a_a ).
thf(func_def_110,type,
sK14: partia539144763t_unit > set_Product_prod_a_a ).
thf(func_def_111,type,
sK15: partia539144763t_unit > set_Product_prod_a_a ).
thf(func_def_112,type,
sK16: partia539144763t_unit > set_Product_prod_a_a ).
thf(func_def_113,type,
sK17: ( product_prod_a_a > $o ) > a ).
thf(func_def_114,type,
sK18: ( product_prod_a_a > $o ) > a ).
thf(func_def_115,type,
sK19: product_prod_a_a > a ).
thf(func_def_116,type,
sK20: product_prod_a_a > a ).
thf(func_def_117,type,
sK21: partia539144763t_unit > set_Product_prod_a_a ).
thf(func_def_118,type,
sK22: partia539144763t_unit > set_Product_prod_a_a ).
thf(func_def_119,type,
sK23: partia539144763t_unit > set_Product_prod_a_a ).
thf(func_def_120,type,
sK24: partia539144763t_unit > set_Product_prod_a_a ).
thf(func_def_121,type,
sK25: partia539144763t_unit > set_Product_prod_a_a ).
thf(func_def_122,type,
sK26: partia539144763t_unit > set_Product_prod_a_a ).
thf(func_def_123,type,
sK27: partia539144763t_unit > set_Product_prod_a_a ).
thf(func_def_124,type,
sK28: a > a ).
thf(func_def_126,type,
ph30:
!>[X0: $tType] : X0 ).
thf(f1210,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1209,f1147]) ).
thf(f1147,plain,
( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) )
= ( locali1170246543t_unit @ rel @ ( mult_a_ring_ext_a_b @ r @ r2 @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) ),
inference(cnf_transformation,[status(thm)],[f4]) ).
thf(f4,axiom,
( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) )
= ( locali1170246543t_unit @ rel @ ( mult_a_ring_ext_a_b @ r @ r2 @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.7ORhJzyl8w/DTF2THF_6318.p',fact_3_f1) ).
thf(f1209,plain,
( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) )
!= ( locali1170246543t_unit @ rel @ ( mult_a_ring_ext_a_b @ r @ r2 @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) ),
inference(forward_demodulation,[status(thm)],[f1208,f1152]) ).
thf(f1152,plain,
( ( mult_a_ring_ext_a_b @ r @ s2 @ s3 )
= ( mult_a_ring_ext_a_b @ r @ s3 @ s2 ) ),
inference(cnf_transformation,[status(thm)],[f2]) ).
thf(f2,axiom,
( ( mult_a_ring_ext_a_b @ r @ s2 @ s3 )
= ( mult_a_ring_ext_a_b @ r @ s3 @ s2 ) ),
file('/export/starexec/sandbox/tmp/tmp.7ORhJzyl8w/DTF2THF_6318.p',fact_1_f4) ).
thf(f1208,plain,
( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) )
!= ( locali1170246543t_unit @ rel @ ( mult_a_ring_ext_a_b @ r @ r2 @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ s2 ) ) ),
inference(forward_demodulation,[status(thm)],[f1207,f1158]) ).
thf(f1158,plain,
( ( mult_a_ring_ext_a_b @ r @ r2 @ r3 )
= ( mult_a_ring_ext_a_b @ r @ r3 @ r2 ) ),
inference(cnf_transformation,[status(thm)],[f1]) ).
thf(f1,axiom,
( ( mult_a_ring_ext_a_b @ r @ r2 @ r3 )
= ( mult_a_ring_ext_a_b @ r @ r3 @ r2 ) ),
file('/export/starexec/sandbox/tmp/tmp.7ORhJzyl8w/DTF2THF_6318.p',fact_0_f3) ).
thf(f1207,plain,
( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) )
!= ( locali1170246543t_unit @ rel @ ( mult_a_ring_ext_a_b @ r @ r3 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ s2 ) ) ),
inference(forward_demodulation,[status(thm)],[f1182,f1184]) ).
thf(f1184,plain,
( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) )
= ( locali1170246543t_unit @ rel @ ( mult_a_ring_ext_a_b @ r @ r3 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ s2 ) ) ),
inference(cnf_transformation,[status(thm)],[f5]) ).
thf(f5,axiom,
( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) )
= ( locali1170246543t_unit @ rel @ ( mult_a_ring_ext_a_b @ r @ r3 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ s2 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.7ORhJzyl8w/DTF2THF_6318.p',fact_4_f2) ).
thf(f1182,plain,
( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) )
!= ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) ) ),
inference(cnf_transformation,[status(thm)],[f1009]) ).
thf(f1009,plain,
( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) )
!= ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) ) ),
inference(flattening,[status(thm)],[f357]) ).
thf(f357,negated_conjecture,
( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) )
!= ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) ) ),
inference(negated_conjecture,[status(cth)],[f356]) ).
thf(f356,conjecture,
( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) )
= ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.7ORhJzyl8w/DTF2THF_6318.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP107^1 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n019.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:26:09 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.26/0.46 ---- Original DTF file ---
% 0.26/0.47 thf(spec,logic,$$dhol).
% 0.26/0.47 %------------------------------------------------------------------------------
% 0.26/0.47 % File : ITP107^1 : TPTP v9.2.1. Released v7.5.0.
% 0.26/0.47 % Domain : Interactive Theorem Proving
% 0.26/0.47 % Problem : Sledgehammer Localization problem prob_542__8966104_1
% 0.26/0.47 % Version : Especial.
% 0.26/0.47 % English :
% 0.26/0.47
% 0.26/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.26/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.26/0.47 % Source : [Des21]
% 0.26/0.47 % Names : Localization/prob_542__8966104_1 [Des21]
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% 0.26/0.47 member_a: a > set_a > $o ).
% 0.26/0.47
% 0.26/0.47 thf(sy_v_R,type,
% 0.26/0.47 r: partia1833973666xt_a_b ).
% 0.26/0.47
% 0.26/0.47 thf(sy_v_S,type,
% 0.26/0.47 s: set_a ).
% 0.26/0.47
% 0.26/0.47 thf(sy_v_r,type,
% 0.26/0.47 r2: a ).
% 0.26/0.47
% 0.26/0.47 thf(sy_v_r_H,type,
% 0.26/0.47 r3: a ).
% 0.26/0.47
% 0.26/0.47 thf(sy_v_rel,type,
% 0.26/0.47 rel: partia1993116613t_unit ).
% 0.26/0.47
% 0.26/0.47 thf(sy_v_s,type,
% 0.26/0.47 s2: a ).
% 0.26/0.47
% 0.26/0.47 thf(sy_v_s_H,type,
% 0.26/0.47 s3: a ).
% 0.26/0.47
% 0.26/0.47 % Relevant facts (355)
% 0.26/0.47 thf(fact_0_f3,axiom,
% 0.26/0.47 ( ( mult_a_ring_ext_a_b @ r @ r2 @ r3 )
% 0.26/0.47 = ( mult_a_ring_ext_a_b @ r @ r3 @ r2 ) ) ).
% 0.26/0.47
% 0.26/0.47 % f3
% 0.26/0.47 thf(fact_1_f4,axiom,
% 0.26/0.47 ( ( mult_a_ring_ext_a_b @ r @ s2 @ s3 )
% 0.26/0.47 = ( mult_a_ring_ext_a_b @ r @ s3 @ s2 ) ) ).
% 0.26/0.47
% 0.26/0.47 % f4
% 0.26/0.47 thf(fact_2_eq__obj__rng__of__frac__axioms,axiom,
% 0.26/0.47 locali1648887798ac_a_b @ r @ s ).
% 0.26/0.47
% 0.26/0.47 % eq_obj_rng_of_frac_axioms
% 0.26/0.47 thf(fact_3_f1,axiom,
% 0.26/0.47 ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) )
% 0.26/0.47 = ( locali1170246543t_unit @ rel @ ( mult_a_ring_ext_a_b @ r @ r2 @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % f1
% 0.26/0.47 thf(fact_4_f2,axiom,
% 0.26/0.47 ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) )
% 0.26/0.47 = ( locali1170246543t_unit @ rel @ ( mult_a_ring_ext_a_b @ r @ r3 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ s2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % f2
% 0.26/0.47 thf(fact_5_mult__submonoid__of__crng__axioms,axiom,
% 0.26/0.47 locali807230110ng_a_b @ r @ s ).
% 0.26/0.47
% 0.26/0.47 % mult_submonoid_of_crng_axioms
% 0.26/0.47 thf(fact_6_mult__submonoid__of__rng__axioms,axiom,
% 0.26/0.47 locali880295127ng_a_b @ r @ s ).
% 0.26/0.47
% 0.26/0.47 % mult_submonoid_of_rng_axioms
% 0.26/0.47 thf(fact_7_eq__obj__rng__of__frac_Orec__monoid__rng__of__frac_Ocong,axiom,
% 0.26/0.47 locali1944243560ac_a_b = locali1944243560ac_a_b ).
% 0.26/0.47
% 0.26/0.47 % eq_obj_rng_of_frac.rec_monoid_rng_of_frac.cong
% 0.26/0.47 thf(fact_8_submonoid__axioms,axiom,
% 0.26/0.47 locali425460998xt_a_b @ r @ s ).
% 0.26/0.47
% 0.26/0.47 % submonoid_axioms
% 0.26/0.47 thf(fact_9_m__closed,axiom,
% 0.26/0.47 ! [X: a,Y: a] :
% 0.26/0.47 ( ( member_a @ X @ s )
% 0.26/0.47 => ( ( member_a @ Y @ s )
% 0.26/0.47 => ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ s ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % m_closed
% 0.26/0.47 thf(fact_10_monoid__rng__of__frac,axiom,
% 0.26/0.47 monoid1242292471t_unit @ ( locali1944243560ac_a_b @ r @ s ) ).
% 0.26/0.47
% 0.26/0.47 % monoid_rng_of_frac
% 0.26/0.47 thf(fact_11_local_Osemiring__axioms,axiom,
% 0.26/0.47 semiring_a_b @ r ).
% 0.26/0.47
% 0.26/0.47 % local.semiring_axioms
% 0.26/0.47 thf(fact_12_one__closed,axiom,
% 0.26/0.47 member_a @ ( one_a_ring_ext_a_b @ r ) @ s ).
% 0.26/0.47
% 0.26/0.47 % one_closed
% 0.26/0.47 thf(fact_13_equiv__obj__rng__of__frac,axiom,
% 0.26/0.47 equiva2136214035t_unit @ rel ).
% 0.26/0.47
% 0.26/0.47 % equiv_obj_rng_of_frac
% 0.26/0.47 thf(fact_14_assms_I2_J,axiom,
% 0.26/0.47 member449909584od_a_a @ ( product_Pair_a_a @ r3 @ s3 ) @ ( partia206007992t_unit @ rel ) ).
% 0.26/0.47
% 0.26/0.47 % assms(2)
% 0.26/0.47 thf(fact_15_assms_I1_J,axiom,
% 0.26/0.47 member449909584od_a_a @ ( product_Pair_a_a @ r2 @ s2 ) @ ( partia206007992t_unit @ rel ) ).
% 0.26/0.47
% 0.26/0.47 % assms(1)
% 0.26/0.47 thf(fact_16_assoc__mult__rng__of__frac,axiom,
% 0.26/0.47 ! [R: a,S: a,T: a,U: a,V: a,W: a] :
% 0.26/0.47 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
% 0.26/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ T @ U ) @ ( partia206007992t_unit @ rel ) )
% 0.26/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ V @ W ) @ ( partia206007992t_unit @ rel ) )
% 0.26/0.47 => ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ R @ S ) @ ( locali1170246543t_unit @ rel @ T @ U ) ) @ ( locali1170246543t_unit @ rel @ V @ W ) )
% 0.26/0.47 = ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ R @ S ) @ ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ T @ U ) @ ( locali1170246543t_unit @ rel @ V @ W ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % assoc_mult_rng_of_frac
% 0.26/0.47 thf(fact_17_abelian__monoid__axioms,axiom,
% 0.26/0.47 abelian_monoid_a_b @ r ).
% 0.26/0.47
% 0.26/0.47 % abelian_monoid_axioms
% 0.26/0.47 thf(fact_18_mult__rng__of__frac__fundamental__lemma,axiom,
% 0.26/0.47 ! [R: a,S: a,R2: a,S2: a] :
% 0.26/0.47 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
% 0.26/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ R2 @ S2 ) @ ( partia206007992t_unit @ rel ) )
% 0.26/0.47 => ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ R @ S ) @ ( locali1170246543t_unit @ rel @ R2 @ S2 ) )
% 0.26/0.47 = ( locali1170246543t_unit @ rel @ ( mult_a_ring_ext_a_b @ r @ R @ R2 ) @ ( mult_a_ring_ext_a_b @ r @ S @ S2 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % mult_rng_of_frac_fundamental_lemma
% 0.26/0.47 thf(fact_19_local_Oring__axioms,axiom,
% 0.26/0.47 ring_a_b @ r ).
% 0.26/0.47
% 0.26/0.47 % local.ring_axioms
% 0.26/0.47 thf(fact_20_closed__mult__rng__of__frac,axiom,
% 0.26/0.47 ! [R: a,S: a,T: a,U: a] :
% 0.26/0.47 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
% 0.26/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ T @ U ) @ ( partia206007992t_unit @ rel ) )
% 0.26/0.47 => ( member1838126896od_a_a @ ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ R @ S ) @ ( locali1170246543t_unit @ rel @ T @ U ) ) @ ( locali1538946157t_unit @ rel ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % closed_mult_rng_of_frac
% 0.26/0.47 thf(fact_21_submonoid_Oaxioms_I1_J,axiom,
% 0.26/0.47 ! [M: partia539144763t_unit,S3: set_se1596668135od_a_a] :
% 0.26/0.47 ( ( locali1488796916t_unit @ M @ S3 )
% 0.26/0.47 => ( monoid1242292471t_unit @ M ) ) ).
% 0.26/0.47
% 0.26/0.47 % submonoid.axioms(1)
% 0.26/0.47 thf(fact_22_submonoid_Oaxioms_I1_J,axiom,
% 0.26/0.47 ! [M: partia1833973666xt_a_b,S3: set_a] :
% 0.26/0.47 ( ( locali425460998xt_a_b @ M @ S3 )
% 0.26/0.47 => ( monoid1022105667xt_a_b @ M ) ) ).
% 0.26/0.47
% 0.26/0.47 % submonoid.axioms(1)
% 0.26/0.47 thf(fact_23_eq__obj__rng__of__frac_Oaxioms_I2_J,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b,S3: set_a] :
% 0.26/0.47 ( ( locali1648887798ac_a_b @ R3 @ S3 )
% 0.26/0.47 => ( locali807230110ng_a_b @ R3 @ S3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % eq_obj_rng_of_frac.axioms(2)
% 0.26/0.47 thf(fact_24_mult__submonoid__of__rng_Oaxioms_I2_J,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b,S3: set_a] :
% 0.26/0.47 ( ( locali880295127ng_a_b @ R3 @ S3 )
% 0.26/0.47 => ( locali425460998xt_a_b @ R3 @ S3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % mult_submonoid_of_rng.axioms(2)
% 0.26/0.47 thf(fact_25_mult__submonoid__of__rng_Oaxioms_I1_J,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b,S3: set_a] :
% 0.26/0.47 ( ( locali880295127ng_a_b @ R3 @ S3 )
% 0.26/0.47 => ( ring_a_b @ R3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % mult_submonoid_of_rng.axioms(1)
% 0.26/0.47 thf(fact_26_mult__submonoid__of__crng_Oaxioms_I2_J,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b,S3: set_a] :
% 0.26/0.47 ( ( locali807230110ng_a_b @ R3 @ S3 )
% 0.26/0.47 => ( locali880295127ng_a_b @ R3 @ S3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % mult_submonoid_of_crng.axioms(2)
% 0.26/0.47 thf(fact_27_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Oone__closed,axiom,
% 0.26/0.47 ! [M: partia539144763t_unit,S3: set_se1596668135od_a_a] :
% 0.26/0.47 ( ( locali1488796916t_unit @ M @ S3 )
% 0.26/0.47 => ( member1838126896od_a_a @ ( one_se1488596747t_unit @ M ) @ S3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % Localization_Mirabelle_afvchqjmgj.submonoid.one_closed
% 0.26/0.47 thf(fact_28_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Oone__closed,axiom,
% 0.26/0.47 ! [M: partia1833973666xt_a_b,S3: set_a] :
% 0.26/0.47 ( ( locali425460998xt_a_b @ M @ S3 )
% 0.26/0.47 => ( member_a @ ( one_a_ring_ext_a_b @ M ) @ S3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % Localization_Mirabelle_afvchqjmgj.submonoid.one_closed
% 0.26/0.47 thf(fact_29_mult__submonoid__of__rng__def,axiom,
% 0.26/0.47 ( locali880295127ng_a_b
% 0.26/0.47 = ( ^ [R4: partia1833973666xt_a_b,S4: set_a] :
% 0.26/0.47 ( ( ring_a_b @ R4 )
% 0.26/0.47 & ( locali425460998xt_a_b @ R4 @ S4 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % mult_submonoid_of_rng_def
% 0.26/0.47 thf(fact_30_mult__submonoid__of__rng_Ointro,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b,S3: set_a] :
% 0.26/0.47 ( ( ring_a_b @ R3 )
% 0.26/0.47 => ( ( locali425460998xt_a_b @ R3 @ S3 )
% 0.26/0.47 => ( locali880295127ng_a_b @ R3 @ S3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % mult_submonoid_of_rng.intro
% 0.26/0.47 thf(fact_31_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Ois__submonoid,axiom,
% 0.26/0.47 ! [M: partia1833973666xt_a_b,S3: set_a] :
% 0.26/0.47 ( ( locali425460998xt_a_b @ M @ S3 )
% 0.26/0.47 => ( locali425460998xt_a_b @ M @ S3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % Localization_Mirabelle_afvchqjmgj.submonoid.is_submonoid
% 0.26/0.47 thf(fact_32_eq__obj__rng__of__frac_Omonoid__rng__of__frac,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b,S3: set_a] :
% 0.26/0.47 ( ( locali1648887798ac_a_b @ R3 @ S3 )
% 0.26/0.47 => ( monoid1242292471t_unit @ ( locali1944243560ac_a_b @ R3 @ S3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % eq_obj_rng_of_frac.monoid_rng_of_frac
% 0.26/0.47 thf(fact_33_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Om__closed,axiom,
% 0.26/0.47 ! [M: partia539144763t_unit,S3: set_se1596668135od_a_a,X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
% 0.26/0.47 ( ( locali1488796916t_unit @ M @ S3 )
% 0.26/0.47 => ( ( member1838126896od_a_a @ X @ S3 )
% 0.26/0.47 => ( ( member1838126896od_a_a @ Y @ S3 )
% 0.26/0.47 => ( member1838126896od_a_a @ ( mult_s977248631t_unit @ M @ X @ Y ) @ S3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Localization_Mirabelle_afvchqjmgj.submonoid.m_closed
% 0.26/0.47 thf(fact_34_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Om__closed,axiom,
% 0.26/0.47 ! [M: partia1833973666xt_a_b,S3: set_a,X: a,Y: a] :
% 0.26/0.47 ( ( locali425460998xt_a_b @ M @ S3 )
% 0.26/0.47 => ( ( member_a @ X @ S3 )
% 0.26/0.47 => ( ( member_a @ Y @ S3 )
% 0.26/0.47 => ( member_a @ ( mult_a_ring_ext_a_b @ M @ X @ Y ) @ S3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Localization_Mirabelle_afvchqjmgj.submonoid.m_closed
% 0.26/0.47 thf(fact_35_right__unit__mult__rng__of__frac,axiom,
% 0.26/0.47 ! [R: a,S: a] :
% 0.26/0.47 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
% 0.26/0.47 => ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ R @ S ) @ ( one_se1488596747t_unit @ ( locali1944243560ac_a_b @ r @ s ) ) )
% 0.26/0.47 = ( locali1170246543t_unit @ rel @ R @ S ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % right_unit_mult_rng_of_frac
% 0.26/0.47 thf(fact_36_left__unit__mult__rng__of__frac,axiom,
% 0.26/0.47 ! [R: a,S: a] :
% 0.26/0.47 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
% 0.26/0.47 => ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( one_se1488596747t_unit @ ( locali1944243560ac_a_b @ r @ s ) ) @ ( locali1170246543t_unit @ rel @ R @ S ) )
% 0.26/0.47 = ( locali1170246543t_unit @ rel @ R @ S ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % left_unit_mult_rng_of_frac
% 0.26/0.47 thf(fact_37_zero__in__mult__submonoid,axiom,
% 0.26/0.47 ! [R: a,S: a,R2: a,S2: a] :
% 0.26/0.47 ( ( member_a @ ( zero_a_b @ r ) @ s )
% 0.26/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
% 0.26/0.47 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ R2 @ S2 ) @ ( partia206007992t_unit @ rel ) )
% 0.26/0.47 => ( ( locali1170246543t_unit @ rel @ R @ S )
% 0.26/0.47 = ( locali1170246543t_unit @ rel @ R2 @ S2 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % zero_in_mult_submonoid
% 0.26/0.47 thf(fact_38_non__empty__class,axiom,
% 0.26/0.47 ! [R: a,S: a] :
% 0.26/0.47 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
% 0.26/0.47 => ( ( locali1170246543t_unit @ rel @ R @ S )
% 0.26/0.47 != bot_bo2131659635od_a_a ) ) ).
% 0.26/0.47
% 0.26/0.47 % non_empty_class
% 0.26/0.47 thf(fact_39_one__unique,axiom,
% 0.26/0.47 ! [U: a] :
% 0.26/0.47 ( ( member_a @ U @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ! [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ r @ U @ X2 )
% 0.26/0.47 = X2 ) )
% 0.26/0.47 => ( U
% 0.26/0.47 = ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % one_unique
% 0.26/0.47 thf(fact_40_inv__unique,axiom,
% 0.26/0.47 ! [Y: a,X: a,Y2: a] :
% 0.26/0.47 ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
% 0.26/0.47 = ( one_a_ring_ext_a_b @ r ) )
% 0.26/0.47 => ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
% 0.26/0.47 = ( one_a_ring_ext_a_b @ r ) )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( member_a @ Y2 @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( Y = Y2 ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % inv_unique
% 0.26/0.47 thf(fact_41_Units__inv__comm,axiom,
% 0.26/0.47 ! [X: a,Y: a] :
% 0.26/0.47 ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
% 0.26/0.47 = ( one_a_ring_ext_a_b @ r ) )
% 0.26/0.47 => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 0.26/0.47 => ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
% 0.26/0.47 = ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Units_inv_comm
% 0.26/0.47 thf(fact_42_prod_Oinject,axiom,
% 0.26/0.47 ! [X1: a,X22: a,Y1: a,Y22: a] :
% 0.26/0.47 ( ( ( product_Pair_a_a @ X1 @ X22 )
% 0.26/0.47 = ( product_Pair_a_a @ Y1 @ Y22 ) )
% 0.26/0.47 = ( ( X1 = Y1 )
% 0.26/0.47 & ( X22 = Y22 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % prod.inject
% 0.26/0.47 thf(fact_43_old_Oprod_Oinject,axiom,
% 0.26/0.47 ! [A: a,B: a,A2: a,B2: a] :
% 0.26/0.47 ( ( ( product_Pair_a_a @ A @ B )
% 0.26/0.47 = ( product_Pair_a_a @ A2 @ B2 ) )
% 0.26/0.47 = ( ( A = A2 )
% 0.26/0.47 & ( B = B2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % old.prod.inject
% 0.26/0.47 thf(fact_44_semiring_Oaxioms_I1_J,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( semiring_a_b @ R3 )
% 0.26/0.47 => ( abelian_monoid_a_b @ R3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % semiring.axioms(1)
% 0.26/0.47 thf(fact_45_monoid__axioms,axiom,
% 0.26/0.47 monoid1022105667xt_a_b @ r ).
% 0.26/0.47
% 0.26/0.47 % monoid_axioms
% 0.26/0.47 thf(fact_46_m__assoc,axiom,
% 0.26/0.47 ! [X: a,Y: a,Z: a] :
% 0.26/0.47 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ Z )
% 0.26/0.47 = ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % m_assoc
% 0.26/0.47 thf(fact_47_m__comm,axiom,
% 0.26/0.47 ! [X: a,Y: a] :
% 0.26/0.47 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
% 0.26/0.47 = ( mult_a_ring_ext_a_b @ r @ Y @ X ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % m_comm
% 0.26/0.47 thf(fact_48_mem__Collect__eq,axiom,
% 0.26/0.47 ! [A: a,P: a > $o] :
% 0.26/0.47 ( ( member_a @ A @ ( collect_a @ P ) )
% 0.26/0.47 = ( P @ A ) ) ).
% 0.26/0.47
% 0.26/0.47 % mem_Collect_eq
% 0.26/0.47 thf(fact_49_mem__Collect__eq,axiom,
% 0.26/0.47 ! [A: product_prod_a_a,P: product_prod_a_a > $o] :
% 0.26/0.47 ( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P ) )
% 0.26/0.47 = ( P @ A ) ) ).
% 0.26/0.47
% 0.26/0.47 % mem_Collect_eq
% 0.26/0.47 thf(fact_50_mem__Collect__eq,axiom,
% 0.26/0.47 ! [A: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
% 0.26/0.47 ( ( member1838126896od_a_a @ A @ ( collec183727474od_a_a @ P ) )
% 0.26/0.47 = ( P @ A ) ) ).
% 0.26/0.47
% 0.26/0.47 % mem_Collect_eq
% 0.26/0.47 thf(fact_51_mem__Collect__eq,axiom,
% 0.26/0.47 ! [A: set_a,P: set_a > $o] :
% 0.26/0.47 ( ( member_set_a @ A @ ( collect_set_a @ P ) )
% 0.26/0.47 = ( P @ A ) ) ).
% 0.26/0.47
% 0.26/0.47 % mem_Collect_eq
% 0.26/0.47 thf(fact_52_Collect__mem__eq,axiom,
% 0.26/0.47 ! [A3: set_a] :
% 0.26/0.47 ( ( collect_a
% 0.26/0.47 @ ^ [X3: a] : ( member_a @ X3 @ A3 ) )
% 0.26/0.47 = A3 ) ).
% 0.26/0.47
% 0.26/0.47 % Collect_mem_eq
% 0.26/0.47 thf(fact_53_Collect__mem__eq,axiom,
% 0.26/0.47 ! [A3: set_Product_prod_a_a] :
% 0.26/0.47 ( ( collec645855634od_a_a
% 0.26/0.47 @ ^ [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A3 ) )
% 0.26/0.47 = A3 ) ).
% 0.26/0.47
% 0.26/0.47 % Collect_mem_eq
% 0.26/0.47 thf(fact_54_Collect__mem__eq,axiom,
% 0.26/0.47 ! [A3: set_se1596668135od_a_a] :
% 0.26/0.47 ( ( collec183727474od_a_a
% 0.26/0.47 @ ^ [X3: set_Product_prod_a_a] : ( member1838126896od_a_a @ X3 @ A3 ) )
% 0.26/0.47 = A3 ) ).
% 0.26/0.47
% 0.26/0.47 % Collect_mem_eq
% 0.26/0.47 thf(fact_55_Collect__mem__eq,axiom,
% 0.26/0.47 ! [A3: set_set_a] :
% 0.26/0.47 ( ( collect_set_a
% 0.26/0.47 @ ^ [X3: set_a] : ( member_set_a @ X3 @ A3 ) )
% 0.26/0.47 = A3 ) ).
% 0.26/0.47
% 0.26/0.47 % Collect_mem_eq
% 0.26/0.47 thf(fact_56_m__lcomm,axiom,
% 0.26/0.47 ! [X: a,Y: a,Z: a] :
% 0.26/0.47 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
% 0.26/0.47 = ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % m_lcomm
% 0.26/0.47 thf(fact_57_Units__closed,axiom,
% 0.26/0.47 ! [X: a] :
% 0.26/0.47 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 0.26/0.47 => ( member_a @ X @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Units_closed
% 0.26/0.47 thf(fact_58_Units__l__inv__ex,axiom,
% 0.26/0.47 ! [X: a] :
% 0.26/0.47 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 0.26/0.47 => ? [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 & ( ( mult_a_ring_ext_a_b @ r @ X2 @ X )
% 0.26/0.47 = ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Units_l_inv_ex
% 0.26/0.47 thf(fact_59_Units__r__inv__ex,axiom,
% 0.26/0.47 ! [X: a] :
% 0.26/0.47 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 0.26/0.47 => ? [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 & ( ( mult_a_ring_ext_a_b @ r @ X @ X2 )
% 0.26/0.47 = ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Units_r_inv_ex
% 0.26/0.47 thf(fact_60_zero__closed,axiom,
% 0.26/0.47 member_a @ ( zero_a_b @ r ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.26/0.47
% 0.26/0.47 % zero_closed
% 0.26/0.47 thf(fact_61_semiring__simprules_I3_J,axiom,
% 0.26/0.47 ! [X: a,Y: a] :
% 0.26/0.47 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % semiring_simprules(3)
% 0.26/0.47 thf(fact_62_semiring__simprules_I4_J,axiom,
% 0.26/0.47 member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.26/0.47
% 0.26/0.47 % semiring_simprules(4)
% 0.26/0.47 thf(fact_63_Units__m__closed,axiom,
% 0.26/0.47 ! [X: a,Y: a] :
% 0.26/0.47 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 0.26/0.47 => ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
% 0.26/0.47 => ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Units_m_closed
% 0.26/0.47 thf(fact_64_Units__one__closed,axiom,
% 0.26/0.47 member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ) ).
% 0.26/0.47
% 0.26/0.47 % Units_one_closed
% 0.26/0.47 thf(fact_65_l__null,axiom,
% 0.26/0.47 ! [X: a] :
% 0.26/0.47 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X )
% 0.26/0.47 = ( zero_a_b @ r ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % l_null
% 0.26/0.47 thf(fact_66_r__null,axiom,
% 0.26/0.47 ! [X: a] :
% 0.26/0.47 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( zero_a_b @ r ) )
% 0.26/0.47 = ( zero_a_b @ r ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % r_null
% 0.26/0.47 thf(fact_67_l__one,axiom,
% 0.26/0.47 ! [X: a] :
% 0.26/0.47 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X )
% 0.26/0.47 = X ) ) ).
% 0.26/0.47
% 0.26/0.47 % l_one
% 0.26/0.47 thf(fact_68_r__one,axiom,
% 0.26/0.47 ! [X: a] :
% 0.26/0.47 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( one_a_ring_ext_a_b @ r ) )
% 0.26/0.47 = X ) ) ).
% 0.26/0.47
% 0.26/0.47 % r_one
% 0.26/0.47 thf(fact_69_Units__l__cancel,axiom,
% 0.26/0.47 ! [X: a,Y: a,Z: a] :
% 0.26/0.47 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 0.26/0.47 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
% 0.26/0.47 = ( mult_a_ring_ext_a_b @ r @ X @ Z ) )
% 0.26/0.47 = ( Y = Z ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Units_l_cancel
% 0.26/0.47 thf(fact_70_abelian__monoidE_I2_J,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( abelian_monoid_a_b @ R3 )
% 0.26/0.47 => ( member_a @ ( zero_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % abelian_monoidE(2)
% 0.26/0.47 thf(fact_71_ring_Oring__simprules_I2_J,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( ring_a_b @ R3 )
% 0.26/0.47 => ( member_a @ ( zero_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % ring.ring_simprules(2)
% 0.26/0.47 thf(fact_72_semiring_Osemiring__simprules_I2_J,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( semiring_a_b @ R3 )
% 0.26/0.47 => ( member_a @ ( zero_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % semiring.semiring_simprules(2)
% 0.26/0.47 thf(fact_73_abelian__monoid_Ozero__closed,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( abelian_monoid_a_b @ G )
% 0.26/0.47 => ( member_a @ ( zero_a_b @ G ) @ ( partia1066395285xt_a_b @ G ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % abelian_monoid.zero_closed
% 0.26/0.47 thf(fact_74_ring_Ois__monoid,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( ring_a_b @ R3 )
% 0.26/0.47 => ( monoid1022105667xt_a_b @ R3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % ring.is_monoid
% 0.26/0.47 thf(fact_75_semiring_Oaxioms_I2_J,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( semiring_a_b @ R3 )
% 0.26/0.47 => ( monoid1022105667xt_a_b @ R3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % semiring.axioms(2)
% 0.26/0.47 thf(fact_76_ring_Oring__simprules_I24_J,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b,X: a] :
% 0.26/0.47 ( ( ring_a_b @ R3 )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ R3 @ ( zero_a_b @ R3 ) @ X )
% 0.26/0.47 = ( zero_a_b @ R3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % ring.ring_simprules(24)
% 0.26/0.47 thf(fact_77_ring_Oring__simprules_I25_J,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b,X: a] :
% 0.26/0.47 ( ( ring_a_b @ R3 )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ R3 @ X @ ( zero_a_b @ R3 ) )
% 0.26/0.47 = ( zero_a_b @ R3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % ring.ring_simprules(25)
% 0.26/0.47 thf(fact_78_semiring_Or__null,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b,X: a] :
% 0.26/0.47 ( ( semiring_a_b @ R3 )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ R3 @ X @ ( zero_a_b @ R3 ) )
% 0.26/0.47 = ( zero_a_b @ R3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % semiring.r_null
% 0.26/0.47 thf(fact_79_semiring_Ol__null,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b,X: a] :
% 0.26/0.47 ( ( semiring_a_b @ R3 )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ R3 @ ( zero_a_b @ R3 ) @ X )
% 0.26/0.47 = ( zero_a_b @ R3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % semiring.l_null
% 0.26/0.47 thf(fact_80_ring_Oring__simprules_I5_J,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b,X: a,Y: a] :
% 0.26/0.47 ( ( ring_a_b @ R3 )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( member_a @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % ring.ring_simprules(5)
% 0.26/0.47 thf(fact_81_ring_Oring__simprules_I11_J,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.26/0.47 ( ( ring_a_b @ R3 )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ R3 @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) @ Z )
% 0.26/0.47 = ( mult_a_ring_ext_a_b @ R3 @ X @ ( mult_a_ring_ext_a_b @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % ring.ring_simprules(11)
% 0.26/0.47 thf(fact_82_semiring_Osemiring__simprules_I3_J,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b,X: a,Y: a] :
% 0.26/0.47 ( ( semiring_a_b @ R3 )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( member_a @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % semiring.semiring_simprules(3)
% 0.26/0.47 thf(fact_83_semiring_Osemiring__simprules_I8_J,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.26/0.47 ( ( semiring_a_b @ R3 )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ R3 @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) @ Z )
% 0.26/0.47 = ( mult_a_ring_ext_a_b @ R3 @ X @ ( mult_a_ring_ext_a_b @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % semiring.semiring_simprules(8)
% 0.26/0.47 thf(fact_84_ring_Oring__simprules_I6_J,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( ring_a_b @ R3 )
% 0.26/0.47 => ( member_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % ring.ring_simprules(6)
% 0.26/0.47 thf(fact_85_semiring_Osemiring__simprules_I4_J,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( semiring_a_b @ R3 )
% 0.26/0.47 => ( member_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % semiring.semiring_simprules(4)
% 0.26/0.47 thf(fact_86_ring_Oring__simprules_I12_J,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b,X: a] :
% 0.26/0.47 ( ( ring_a_b @ R3 )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ R3 @ ( one_a_ring_ext_a_b @ R3 ) @ X )
% 0.26/0.47 = X ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % ring.ring_simprules(12)
% 0.26/0.47 thf(fact_87_semiring_Osemiring__simprules_I9_J,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b,X: a] :
% 0.26/0.47 ( ( semiring_a_b @ R3 )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ R3 @ ( one_a_ring_ext_a_b @ R3 ) @ X )
% 0.26/0.47 = X ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % semiring.semiring_simprules(9)
% 0.26/0.47 thf(fact_88_old_Oprod_Oinducts,axiom,
% 0.26/0.47 ! [P: product_prod_a_a > $o,Prod: product_prod_a_a] :
% 0.26/0.47 ( ! [A4: a,B3: a] : ( P @ ( product_Pair_a_a @ A4 @ B3 ) )
% 0.26/0.47 => ( P @ Prod ) ) ).
% 0.26/0.47
% 0.26/0.47 % old.prod.inducts
% 0.26/0.47 thf(fact_89_old_Oprod_Oexhaust,axiom,
% 0.26/0.47 ! [Y: product_prod_a_a] :
% 0.26/0.47 ~ ! [A4: a,B3: a] :
% 0.26/0.47 ( Y
% 0.26/0.47 != ( product_Pair_a_a @ A4 @ B3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % old.prod.exhaust
% 0.26/0.47 thf(fact_90_Pair__inject,axiom,
% 0.26/0.47 ! [A: a,B: a,A2: a,B2: a] :
% 0.26/0.47 ( ( ( product_Pair_a_a @ A @ B )
% 0.26/0.47 = ( product_Pair_a_a @ A2 @ B2 ) )
% 0.26/0.47 => ~ ( ( A = A2 )
% 0.26/0.47 => ( B != B2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Pair_inject
% 0.26/0.47 thf(fact_91_prod__cases,axiom,
% 0.26/0.47 ! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
% 0.26/0.47 ( ! [A4: a,B3: a] : ( P @ ( product_Pair_a_a @ A4 @ B3 ) )
% 0.26/0.47 => ( P @ P2 ) ) ).
% 0.26/0.47
% 0.26/0.47 % prod_cases
% 0.26/0.47 thf(fact_92_surj__pair,axiom,
% 0.26/0.47 ! [P2: product_prod_a_a] :
% 0.26/0.47 ? [X2: a,Y3: a] :
% 0.26/0.47 ( P2
% 0.26/0.47 = ( product_Pair_a_a @ X2 @ Y3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % surj_pair
% 0.26/0.47 thf(fact_93_ring_Ois__ring,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( ring_a_b @ R3 )
% 0.26/0.47 => ( ring_a_b @ R3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % ring.is_ring
% 0.26/0.47 thf(fact_94_cring__fieldI2,axiom,
% 0.26/0.47 ( ( ( zero_a_b @ r )
% 0.26/0.47 != ( one_a_ring_ext_a_b @ r ) )
% 0.26/0.47 => ( ! [A4: a] :
% 0.26/0.47 ( ( member_a @ A4 @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( A4
% 0.26/0.47 != ( zero_a_b @ r ) )
% 0.26/0.47 => ? [X4: a] :
% 0.26/0.47 ( ( member_a @ X4 @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 & ( ( mult_a_ring_ext_a_b @ r @ A4 @ X4 )
% 0.26/0.47 = ( one_a_ring_ext_a_b @ r ) ) ) ) )
% 0.26/0.47 => ( field_a_b @ r ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % cring_fieldI2
% 0.26/0.47 thf(fact_95_monoid_Or__one,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit,X: set_Product_prod_a_a] :
% 0.26/0.47 ( ( monoid1242292471t_unit @ G )
% 0.26/0.47 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( mult_s977248631t_unit @ G @ X @ ( one_se1488596747t_unit @ G ) )
% 0.26/0.47 = X ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.r_one
% 0.26/0.47 thf(fact_96_monoid_Or__one,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b,X: a] :
% 0.26/0.47 ( ( monoid1022105667xt_a_b @ G )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ G @ X @ ( one_a_ring_ext_a_b @ G ) )
% 0.26/0.47 = X ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.r_one
% 0.26/0.47 thf(fact_97_monoid_Ol__one,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit,X: set_Product_prod_a_a] :
% 0.26/0.47 ( ( monoid1242292471t_unit @ G )
% 0.26/0.47 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( mult_s977248631t_unit @ G @ ( one_se1488596747t_unit @ G ) @ X )
% 0.26/0.47 = X ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.l_one
% 0.26/0.47 thf(fact_98_monoid_Ol__one,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b,X: a] :
% 0.26/0.47 ( ( monoid1022105667xt_a_b @ G )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X )
% 0.26/0.47 = X ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.l_one
% 0.26/0.47 thf(fact_99_monoid_Oone__closed,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit] :
% 0.26/0.47 ( ( monoid1242292471t_unit @ G )
% 0.26/0.47 => ( member1838126896od_a_a @ ( one_se1488596747t_unit @ G ) @ ( partia616129906t_unit @ G ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.one_closed
% 0.26/0.47 thf(fact_100_monoid_Oone__closed,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( monoid1022105667xt_a_b @ G )
% 0.26/0.47 => ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia1066395285xt_a_b @ G ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.one_closed
% 0.26/0.47 thf(fact_101_monoid_OUnits__r__inv__ex,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit,X: set_Product_prod_a_a] :
% 0.26/0.47 ( ( monoid1242292471t_unit @ G )
% 0.26/0.47 => ( ( member1838126896od_a_a @ X @ ( units_1734235042t_unit @ G ) )
% 0.26/0.47 => ? [X2: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 & ( ( mult_s977248631t_unit @ G @ X @ X2 )
% 0.26/0.47 = ( one_se1488596747t_unit @ G ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.Units_r_inv_ex
% 0.26/0.47 thf(fact_102_monoid_OUnits__r__inv__ex,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b,X: a] :
% 0.26/0.47 ( ( monoid1022105667xt_a_b @ G )
% 0.26/0.47 => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
% 0.26/0.47 => ? [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 & ( ( mult_a_ring_ext_a_b @ G @ X @ X2 )
% 0.26/0.47 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.Units_r_inv_ex
% 0.26/0.47 thf(fact_103_monoid_OUnits__l__inv__ex,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit,X: set_Product_prod_a_a] :
% 0.26/0.47 ( ( monoid1242292471t_unit @ G )
% 0.26/0.47 => ( ( member1838126896od_a_a @ X @ ( units_1734235042t_unit @ G ) )
% 0.26/0.47 => ? [X2: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 & ( ( mult_s977248631t_unit @ G @ X2 @ X )
% 0.26/0.47 = ( one_se1488596747t_unit @ G ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.Units_l_inv_ex
% 0.26/0.47 thf(fact_104_monoid_OUnits__l__inv__ex,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b,X: a] :
% 0.26/0.47 ( ( monoid1022105667xt_a_b @ G )
% 0.26/0.47 => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
% 0.26/0.47 => ? [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 & ( ( mult_a_ring_ext_a_b @ G @ X2 @ X )
% 0.26/0.47 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.Units_l_inv_ex
% 0.26/0.47 thf(fact_105_monoid_OUnits__inv__comm,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit,X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
% 0.26/0.47 ( ( monoid1242292471t_unit @ G )
% 0.26/0.47 => ( ( ( mult_s977248631t_unit @ G @ X @ Y )
% 0.26/0.47 = ( one_se1488596747t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ X @ ( units_1734235042t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ Y @ ( units_1734235042t_unit @ G ) )
% 0.26/0.47 => ( ( mult_s977248631t_unit @ G @ Y @ X )
% 0.26/0.47 = ( one_se1488596747t_unit @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.Units_inv_comm
% 0.26/0.47 thf(fact_106_monoid_OUnits__inv__comm,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 0.26/0.47 ( ( monoid1022105667xt_a_b @ G )
% 0.26/0.47 => ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
% 0.26/0.47 = ( one_a_ring_ext_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ G ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
% 0.26/0.47 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.Units_inv_comm
% 0.26/0.47 thf(fact_107_add__pow__ldistr__int,axiom,
% 0.26/0.47 ! [A: a,B: a,K: int] :
% 0.26/0.47 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_int @ r @ K @ A ) @ B )
% 0.26/0.47 = ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % add_pow_ldistr_int
% 0.26/0.47 thf(fact_108_add__pow__rdistr__int,axiom,
% 0.26/0.47 ! [A: a,B: a,K: int] :
% 0.26/0.47 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_int @ r @ K @ B ) )
% 0.26/0.47 = ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % add_pow_rdistr_int
% 0.26/0.47 thf(fact_109_add_Oint__pow__closed,axiom,
% 0.26/0.47 ! [X: a,I: int] :
% 0.26/0.47 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( member_a @ ( add_pow_a_b_int @ r @ I @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % add.int_pow_closed
% 0.26/0.47 thf(fact_110_add_Oint__pow__one,axiom,
% 0.26/0.47 ! [Z: int] :
% 0.26/0.47 ( ( add_pow_a_b_int @ r @ Z @ ( zero_a_b @ r ) )
% 0.26/0.47 = ( zero_a_b @ r ) ) ).
% 0.26/0.47
% 0.26/0.47 % add.int_pow_one
% 0.26/0.47 thf(fact_111_field_Ois__ring,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( field_a_b @ R3 )
% 0.26/0.47 => ( ring_a_b @ R3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % field.is_ring
% 0.26/0.47 thf(fact_112_one__not__zero,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( field_a_b @ R3 )
% 0.26/0.47 => ( ( one_a_ring_ext_a_b @ R3 )
% 0.26/0.47 != ( zero_a_b @ R3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % one_not_zero
% 0.26/0.47 thf(fact_113_integral,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b,A: a,B: a] :
% 0.26/0.47 ( ( field_a_b @ R3 )
% 0.26/0.47 => ( ( ( mult_a_ring_ext_a_b @ R3 @ A @ B )
% 0.26/0.47 = ( zero_a_b @ R3 ) )
% 0.26/0.47 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( A
% 0.26/0.47 = ( zero_a_b @ R3 ) )
% 0.26/0.47 | ( B
% 0.26/0.47 = ( zero_a_b @ R3 ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % integral
% 0.26/0.47 thf(fact_114_ring_Oadd__pow__ldistr__int,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b,A: a,B: a,K: int] :
% 0.26/0.47 ( ( ring_a_b @ R3 )
% 0.26/0.47 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ R3 @ ( add_pow_a_b_int @ R3 @ K @ A ) @ B )
% 0.26/0.47 = ( add_pow_a_b_int @ R3 @ K @ ( mult_a_ring_ext_a_b @ R3 @ A @ B ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % ring.add_pow_ldistr_int
% 0.26/0.47 thf(fact_115_ring_Oadd__pow__rdistr__int,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b,A: a,B: a,K: int] :
% 0.26/0.47 ( ( ring_a_b @ R3 )
% 0.26/0.47 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ R3 @ A @ ( add_pow_a_b_int @ R3 @ K @ B ) )
% 0.26/0.47 = ( add_pow_a_b_int @ R3 @ K @ ( mult_a_ring_ext_a_b @ R3 @ A @ B ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % ring.add_pow_rdistr_int
% 0.26/0.47 thf(fact_116_monoid_Ocarrier__not__empty,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit] :
% 0.26/0.47 ( ( monoid1242292471t_unit @ G )
% 0.26/0.47 => ( ( partia616129906t_unit @ G )
% 0.26/0.47 != bot_bo1424703315od_a_a ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.carrier_not_empty
% 0.26/0.47 thf(fact_117_monoid_Ocarrier__not__empty,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( monoid1022105667xt_a_b @ G )
% 0.26/0.47 => ( ( partia1066395285xt_a_b @ G )
% 0.26/0.47 != bot_bot_set_a ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.carrier_not_empty
% 0.26/0.47 thf(fact_118_monoid_Om__assoc,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit,X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
% 0.26/0.47 ( ( monoid1242292471t_unit @ G )
% 0.26/0.47 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ Y @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ Z @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( mult_s977248631t_unit @ G @ ( mult_s977248631t_unit @ G @ X @ Y ) @ Z )
% 0.26/0.47 = ( mult_s977248631t_unit @ G @ X @ ( mult_s977248631t_unit @ G @ Y @ Z ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.m_assoc
% 0.26/0.47 thf(fact_119_monoid_Om__assoc,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.26/0.47 ( ( monoid1022105667xt_a_b @ G )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ Z )
% 0.26/0.47 = ( mult_a_ring_ext_a_b @ G @ X @ ( mult_a_ring_ext_a_b @ G @ Y @ Z ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.m_assoc
% 0.26/0.47 thf(fact_120_monoid_Om__closed,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit,X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
% 0.26/0.47 ( ( monoid1242292471t_unit @ G )
% 0.26/0.47 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ Y @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( member1838126896od_a_a @ ( mult_s977248631t_unit @ G @ X @ Y ) @ ( partia616129906t_unit @ G ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.m_closed
% 0.26/0.47 thf(fact_121_monoid_Om__closed,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 0.26/0.47 ( ( monoid1022105667xt_a_b @ G )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( member_a @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ ( partia1066395285xt_a_b @ G ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.m_closed
% 0.26/0.47 thf(fact_122_monoid_OUnits__closed,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit,X: set_Product_prod_a_a] :
% 0.26/0.47 ( ( monoid1242292471t_unit @ G )
% 0.26/0.47 => ( ( member1838126896od_a_a @ X @ ( units_1734235042t_unit @ G ) )
% 0.26/0.47 => ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.Units_closed
% 0.26/0.47 thf(fact_123_monoid_OUnits__closed,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b,X: a] :
% 0.26/0.47 ( ( monoid1022105667xt_a_b @ G )
% 0.26/0.47 => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
% 0.26/0.47 => ( member_a @ X @ ( partia1066395285xt_a_b @ G ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.Units_closed
% 0.26/0.47 thf(fact_124_monoid_OUnits__m__closed,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit,X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
% 0.26/0.47 ( ( monoid1242292471t_unit @ G )
% 0.26/0.47 => ( ( member1838126896od_a_a @ X @ ( units_1734235042t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ Y @ ( units_1734235042t_unit @ G ) )
% 0.26/0.47 => ( member1838126896od_a_a @ ( mult_s977248631t_unit @ G @ X @ Y ) @ ( units_1734235042t_unit @ G ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.Units_m_closed
% 0.26/0.47 thf(fact_125_monoid_OUnits__m__closed,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 0.26/0.47 ( ( monoid1022105667xt_a_b @ G )
% 0.26/0.47 => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ G ) )
% 0.26/0.47 => ( member_a @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.Units_m_closed
% 0.26/0.47 thf(fact_126_monoid_OUnits__one__closed,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( monoid1022105667xt_a_b @ G )
% 0.26/0.47 => ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( units_a_ring_ext_a_b @ G ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.Units_one_closed
% 0.26/0.47 thf(fact_127_monoid_OUnits__one__closed,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit] :
% 0.26/0.47 ( ( monoid1242292471t_unit @ G )
% 0.26/0.47 => ( member1838126896od_a_a @ ( one_se1488596747t_unit @ G ) @ ( units_1734235042t_unit @ G ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.Units_one_closed
% 0.26/0.47 thf(fact_128_monoidI,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit] :
% 0.26/0.47 ( ! [X2: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ Y3 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( member1838126896od_a_a @ ( mult_s977248631t_unit @ G @ X2 @ Y3 ) @ ( partia616129906t_unit @ G ) ) ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ ( one_se1488596747t_unit @ G ) @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ! [X2: set_Product_prod_a_a,Y3: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ Y3 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ Z2 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( mult_s977248631t_unit @ G @ ( mult_s977248631t_unit @ G @ X2 @ Y3 ) @ Z2 )
% 0.26/0.47 = ( mult_s977248631t_unit @ G @ X2 @ ( mult_s977248631t_unit @ G @ Y3 @ Z2 ) ) ) ) ) )
% 0.26/0.47 => ( ! [X2: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( mult_s977248631t_unit @ G @ ( one_se1488596747t_unit @ G ) @ X2 )
% 0.26/0.47 = X2 ) )
% 0.26/0.47 => ( ! [X2: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( mult_s977248631t_unit @ G @ X2 @ ( one_se1488596747t_unit @ G ) )
% 0.26/0.47 = X2 ) )
% 0.26/0.47 => ( monoid1242292471t_unit @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoidI
% 0.26/0.47 thf(fact_129_monoidI,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b] :
% 0.26/0.47 ( ! [X2: a,Y3: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( member_a @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ ( partia1066395285xt_a_b @ G ) ) ) )
% 0.26/0.47 => ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ! [X2: a,Y3: a,Z2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ Z2 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ Z2 )
% 0.26/0.47 = ( mult_a_ring_ext_a_b @ G @ X2 @ ( mult_a_ring_ext_a_b @ G @ Y3 @ Z2 ) ) ) ) ) )
% 0.26/0.47 => ( ! [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X2 )
% 0.26/0.47 = X2 ) )
% 0.26/0.47 => ( ! [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ G @ X2 @ ( one_a_ring_ext_a_b @ G ) )
% 0.26/0.47 = X2 ) )
% 0.26/0.47 => ( monoid1022105667xt_a_b @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoidI
% 0.26/0.47 thf(fact_130_Group_Omonoid__def,axiom,
% 0.26/0.47 ( monoid1242292471t_unit
% 0.26/0.47 = ( ^ [G2: partia539144763t_unit] :
% 0.26/0.47 ( ! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X3 @ ( partia616129906t_unit @ G2 ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ Y4 @ ( partia616129906t_unit @ G2 ) )
% 0.26/0.47 => ( member1838126896od_a_a @ ( mult_s977248631t_unit @ G2 @ X3 @ Y4 ) @ ( partia616129906t_unit @ G2 ) ) ) )
% 0.26/0.47 & ! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X3 @ ( partia616129906t_unit @ G2 ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ Y4 @ ( partia616129906t_unit @ G2 ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ Z3 @ ( partia616129906t_unit @ G2 ) )
% 0.26/0.47 => ( ( mult_s977248631t_unit @ G2 @ ( mult_s977248631t_unit @ G2 @ X3 @ Y4 ) @ Z3 )
% 0.26/0.47 = ( mult_s977248631t_unit @ G2 @ X3 @ ( mult_s977248631t_unit @ G2 @ Y4 @ Z3 ) ) ) ) ) )
% 0.26/0.47 & ( member1838126896od_a_a @ ( one_se1488596747t_unit @ G2 ) @ ( partia616129906t_unit @ G2 ) )
% 0.26/0.47 & ! [X3: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X3 @ ( partia616129906t_unit @ G2 ) )
% 0.26/0.47 => ( ( mult_s977248631t_unit @ G2 @ ( one_se1488596747t_unit @ G2 ) @ X3 )
% 0.26/0.47 = X3 ) )
% 0.26/0.47 & ! [X3: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X3 @ ( partia616129906t_unit @ G2 ) )
% 0.26/0.47 => ( ( mult_s977248631t_unit @ G2 @ X3 @ ( one_se1488596747t_unit @ G2 ) )
% 0.26/0.47 = X3 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Group.monoid_def
% 0.26/0.47 thf(fact_131_Group_Omonoid__def,axiom,
% 0.26/0.47 ( monoid1022105667xt_a_b
% 0.26/0.47 = ( ^ [G2: partia1833973666xt_a_b] :
% 0.26/0.47 ( ! [X3: a,Y4: a] :
% 0.26/0.47 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G2 ) )
% 0.26/0.47 => ( ( member_a @ Y4 @ ( partia1066395285xt_a_b @ G2 ) )
% 0.26/0.47 => ( member_a @ ( mult_a_ring_ext_a_b @ G2 @ X3 @ Y4 ) @ ( partia1066395285xt_a_b @ G2 ) ) ) )
% 0.26/0.47 & ! [X3: a,Y4: a,Z3: a] :
% 0.26/0.47 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G2 ) )
% 0.26/0.47 => ( ( member_a @ Y4 @ ( partia1066395285xt_a_b @ G2 ) )
% 0.26/0.47 => ( ( member_a @ Z3 @ ( partia1066395285xt_a_b @ G2 ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ G2 @ ( mult_a_ring_ext_a_b @ G2 @ X3 @ Y4 ) @ Z3 )
% 0.26/0.47 = ( mult_a_ring_ext_a_b @ G2 @ X3 @ ( mult_a_ring_ext_a_b @ G2 @ Y4 @ Z3 ) ) ) ) ) )
% 0.26/0.47 & ( member_a @ ( one_a_ring_ext_a_b @ G2 ) @ ( partia1066395285xt_a_b @ G2 ) )
% 0.26/0.47 & ! [X3: a] :
% 0.26/0.47 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G2 ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ G2 @ ( one_a_ring_ext_a_b @ G2 ) @ X3 )
% 0.26/0.47 = X3 ) )
% 0.26/0.47 & ! [X3: a] :
% 0.26/0.47 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G2 ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ G2 @ X3 @ ( one_a_ring_ext_a_b @ G2 ) )
% 0.26/0.47 = X3 ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Group.monoid_def
% 0.26/0.47 thf(fact_132_Group_Omonoid_Ointro,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit] :
% 0.26/0.47 ( ! [X2: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ Y3 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( member1838126896od_a_a @ ( mult_s977248631t_unit @ G @ X2 @ Y3 ) @ ( partia616129906t_unit @ G ) ) ) )
% 0.26/0.47 => ( ! [X2: set_Product_prod_a_a,Y3: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ Y3 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ Z2 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( mult_s977248631t_unit @ G @ ( mult_s977248631t_unit @ G @ X2 @ Y3 ) @ Z2 )
% 0.26/0.47 = ( mult_s977248631t_unit @ G @ X2 @ ( mult_s977248631t_unit @ G @ Y3 @ Z2 ) ) ) ) ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ ( one_se1488596747t_unit @ G ) @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ! [X2: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( mult_s977248631t_unit @ G @ ( one_se1488596747t_unit @ G ) @ X2 )
% 0.26/0.47 = X2 ) )
% 0.26/0.47 => ( ! [X2: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( mult_s977248631t_unit @ G @ X2 @ ( one_se1488596747t_unit @ G ) )
% 0.26/0.47 = X2 ) )
% 0.26/0.47 => ( monoid1242292471t_unit @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Group.monoid.intro
% 0.26/0.47 thf(fact_133_Group_Omonoid_Ointro,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b] :
% 0.26/0.47 ( ! [X2: a,Y3: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( member_a @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ ( partia1066395285xt_a_b @ G ) ) ) )
% 0.26/0.47 => ( ! [X2: a,Y3: a,Z2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ Z2 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ Z2 )
% 0.26/0.47 = ( mult_a_ring_ext_a_b @ G @ X2 @ ( mult_a_ring_ext_a_b @ G @ Y3 @ Z2 ) ) ) ) ) )
% 0.26/0.47 => ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ! [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X2 )
% 0.26/0.47 = X2 ) )
% 0.26/0.47 => ( ! [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ G @ X2 @ ( one_a_ring_ext_a_b @ G ) )
% 0.26/0.47 = X2 ) )
% 0.26/0.47 => ( monoid1022105667xt_a_b @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Group.monoid.intro
% 0.26/0.47 thf(fact_134_monoid_Oinv__unique,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit,Y: set_Product_prod_a_a,X: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
% 0.26/0.47 ( ( monoid1242292471t_unit @ G )
% 0.26/0.47 => ( ( ( mult_s977248631t_unit @ G @ Y @ X )
% 0.26/0.47 = ( one_se1488596747t_unit @ G ) )
% 0.26/0.47 => ( ( ( mult_s977248631t_unit @ G @ X @ Y2 )
% 0.26/0.47 = ( one_se1488596747t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ Y @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ Y2 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( Y = Y2 ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.inv_unique
% 0.26/0.47 thf(fact_135_monoid_Oinv__unique,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b,Y: a,X: a,Y2: a] :
% 0.26/0.47 ( ( monoid1022105667xt_a_b @ G )
% 0.26/0.47 => ( ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
% 0.26/0.47 = ( one_a_ring_ext_a_b @ G ) )
% 0.26/0.47 => ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y2 )
% 0.26/0.47 = ( one_a_ring_ext_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ Y2 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( Y = Y2 ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.inv_unique
% 0.26/0.47 thf(fact_136_monoid_Oone__unique,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit,U: set_Product_prod_a_a] :
% 0.26/0.47 ( ( monoid1242292471t_unit @ G )
% 0.26/0.47 => ( ( member1838126896od_a_a @ U @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ! [X2: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( mult_s977248631t_unit @ G @ U @ X2 )
% 0.26/0.47 = X2 ) )
% 0.26/0.47 => ( U
% 0.26/0.47 = ( one_se1488596747t_unit @ G ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.one_unique
% 0.26/0.47 thf(fact_137_monoid_Oone__unique,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b,U: a] :
% 0.26/0.47 ( ( monoid1022105667xt_a_b @ G )
% 0.26/0.47 => ( ( member_a @ U @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ! [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ G @ U @ X2 )
% 0.26/0.47 = X2 ) )
% 0.26/0.47 => ( U
% 0.26/0.47 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.one_unique
% 0.26/0.47 thf(fact_138_monoid_OUnits__l__cancel,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit,X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
% 0.26/0.47 ( ( monoid1242292471t_unit @ G )
% 0.26/0.47 => ( ( member1838126896od_a_a @ X @ ( units_1734235042t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ Y @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ Z @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( ( mult_s977248631t_unit @ G @ X @ Y )
% 0.26/0.47 = ( mult_s977248631t_unit @ G @ X @ Z ) )
% 0.26/0.47 = ( Y = Z ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.Units_l_cancel
% 0.26/0.47 thf(fact_139_monoid_OUnits__l__cancel,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.26/0.47 ( ( monoid1022105667xt_a_b @ G )
% 0.26/0.47 => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
% 0.26/0.47 = ( mult_a_ring_ext_a_b @ G @ X @ Z ) )
% 0.26/0.47 = ( Y = Z ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.Units_l_cancel
% 0.26/0.47 thf(fact_140_subset,axiom,
% 0.26/0.47 ord_less_eq_set_a @ s @ ( partia1066395285xt_a_b @ r ) ).
% 0.26/0.47
% 0.26/0.47 % subset
% 0.26/0.47 thf(fact_141_add__pow__rdistr,axiom,
% 0.26/0.47 ! [A: a,B: a,K: nat] :
% 0.26/0.47 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_nat @ r @ K @ B ) )
% 0.26/0.47 = ( add_pow_a_b_nat @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % add_pow_rdistr
% 0.26/0.47 thf(fact_142_add__pow__ldistr,axiom,
% 0.26/0.47 ! [A: a,B: a,K: nat] :
% 0.26/0.47 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_nat @ r @ K @ A ) @ B )
% 0.26/0.47 = ( add_pow_a_b_nat @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % add_pow_ldistr
% 0.26/0.47 thf(fact_143_group__l__invI,axiom,
% 0.26/0.47 ( ! [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ? [Xa: a] :
% 0.26/0.47 ( ( member_a @ Xa @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 & ( ( mult_a_ring_ext_a_b @ r @ Xa @ X2 )
% 0.26/0.47 = ( one_a_ring_ext_a_b @ r ) ) ) )
% 0.26/0.47 => ( group_a_ring_ext_a_b @ r ) ) ).
% 0.26/0.47
% 0.26/0.47 % group_l_invI
% 0.26/0.47 thf(fact_144_carrier__not__empty,axiom,
% 0.26/0.47 ( ( partia1066395285xt_a_b @ r )
% 0.26/0.47 != bot_bot_set_a ) ).
% 0.26/0.47
% 0.26/0.47 % carrier_not_empty
% 0.26/0.47 thf(fact_145_add_Onat__pow__closed,axiom,
% 0.26/0.47 ! [X: a,N: nat] :
% 0.26/0.47 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( member_a @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % add.nat_pow_closed
% 0.26/0.47 thf(fact_146_add_Onat__pow__one,axiom,
% 0.26/0.47 ! [N: nat] :
% 0.26/0.47 ( ( add_pow_a_b_nat @ r @ N @ ( zero_a_b @ r ) )
% 0.26/0.47 = ( zero_a_b @ r ) ) ).
% 0.26/0.47
% 0.26/0.47 % add.nat_pow_one
% 0.26/0.47 thf(fact_147_group_Ois__group,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( group_a_ring_ext_a_b @ G )
% 0.26/0.47 => ( group_a_ring_ext_a_b @ G ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.is_group
% 0.26/0.47 thf(fact_148_group_Ois__group,axiom,
% 0.26/0.47 ! [G: partia96731725t_unit] :
% 0.26/0.47 ( ( group_a_Product_unit @ G )
% 0.26/0.47 => ( group_a_Product_unit @ G ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.is_group
% 0.26/0.47 thf(fact_149_group_OUnits,axiom,
% 0.26/0.47 ! [G: partia96731725t_unit] :
% 0.26/0.47 ( ( group_a_Product_unit @ G )
% 0.26/0.47 => ( ord_less_eq_set_a @ ( partia1955795460t_unit @ G ) @ ( units_a_Product_unit @ G ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.Units
% 0.26/0.47 thf(fact_150_group_OUnits,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( group_a_ring_ext_a_b @ G )
% 0.26/0.47 => ( ord_less_eq_set_a @ ( partia1066395285xt_a_b @ G ) @ ( units_a_ring_ext_a_b @ G ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.Units
% 0.26/0.47 thf(fact_151_group_Ois__monoid,axiom,
% 0.26/0.47 ! [G: partia96731725t_unit] :
% 0.26/0.47 ( ( group_a_Product_unit @ G )
% 0.26/0.47 => ( monoid1447254976t_unit @ G ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.is_monoid
% 0.26/0.47 thf(fact_152_group_Ois__monoid,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit] :
% 0.26/0.47 ( ( group_52730354t_unit @ G )
% 0.26/0.47 => ( monoid1242292471t_unit @ G ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.is_monoid
% 0.26/0.47 thf(fact_153_group_Ois__monoid,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( group_a_ring_ext_a_b @ G )
% 0.26/0.47 => ( monoid1022105667xt_a_b @ G ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.is_monoid
% 0.26/0.47 thf(fact_154_Group_Ogroup_Oright__cancel,axiom,
% 0.26/0.47 ! [G: partia96731725t_unit,X: a,Y: a,Z: a] :
% 0.26/0.47 ( ( group_a_Product_unit @ G )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ( ( member_a @ Y @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ( ( member_a @ Z @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ( ( ( mult_a_Product_unit @ G @ Y @ X )
% 0.26/0.47 = ( mult_a_Product_unit @ G @ Z @ X ) )
% 0.26/0.47 = ( Y = Z ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Group.group.right_cancel
% 0.26/0.47 thf(fact_155_Group_Ogroup_Oright__cancel,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit,X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
% 0.26/0.47 ( ( group_52730354t_unit @ G )
% 0.26/0.47 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ Y @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ Z @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( ( mult_s977248631t_unit @ G @ Y @ X )
% 0.26/0.47 = ( mult_s977248631t_unit @ G @ Z @ X ) )
% 0.26/0.47 = ( Y = Z ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Group.group.right_cancel
% 0.26/0.47 thf(fact_156_Group_Ogroup_Oright__cancel,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.26/0.47 ( ( group_a_ring_ext_a_b @ G )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
% 0.26/0.47 = ( mult_a_ring_ext_a_b @ G @ Z @ X ) )
% 0.26/0.47 = ( Y = Z ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Group.group.right_cancel
% 0.26/0.47 thf(fact_157_group_OUnits__eq,axiom,
% 0.26/0.47 ! [G: partia96731725t_unit] :
% 0.26/0.47 ( ( group_a_Product_unit @ G )
% 0.26/0.47 => ( ( units_a_Product_unit @ G )
% 0.26/0.47 = ( partia1955795460t_unit @ G ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.Units_eq
% 0.26/0.47 thf(fact_158_group_OUnits__eq,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( group_a_ring_ext_a_b @ G )
% 0.26/0.47 => ( ( units_a_ring_ext_a_b @ G )
% 0.26/0.47 = ( partia1066395285xt_a_b @ G ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.Units_eq
% 0.26/0.47 thf(fact_159_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Osubset,axiom,
% 0.26/0.47 ! [M: partia1833973666xt_a_b,S3: set_a] :
% 0.26/0.47 ( ( locali425460998xt_a_b @ M @ S3 )
% 0.26/0.47 => ( ord_less_eq_set_a @ S3 @ ( partia1066395285xt_a_b @ M ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Localization_Mirabelle_afvchqjmgj.submonoid.subset
% 0.26/0.47 thf(fact_160_group_Or__cancel__one_H,axiom,
% 0.26/0.47 ! [G: partia96731725t_unit,X: a,A: a] :
% 0.26/0.47 ( ( group_a_Product_unit @ G )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ( ( X
% 0.26/0.47 = ( mult_a_Product_unit @ G @ A @ X ) )
% 0.26/0.47 = ( A
% 0.26/0.47 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.r_cancel_one'
% 0.26/0.47 thf(fact_161_group_Or__cancel__one_H,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit,X: set_Product_prod_a_a,A: set_Product_prod_a_a] :
% 0.26/0.47 ( ( group_52730354t_unit @ G )
% 0.26/0.47 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ A @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( X
% 0.26/0.47 = ( mult_s977248631t_unit @ G @ A @ X ) )
% 0.26/0.47 = ( A
% 0.26/0.47 = ( one_se1488596747t_unit @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.r_cancel_one'
% 0.26/0.47 thf(fact_162_group_Or__cancel__one_H,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b,X: a,A: a] :
% 0.26/0.47 ( ( group_a_ring_ext_a_b @ G )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( X
% 0.26/0.47 = ( mult_a_ring_ext_a_b @ G @ A @ X ) )
% 0.26/0.47 = ( A
% 0.26/0.47 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.r_cancel_one'
% 0.26/0.47 thf(fact_163_group_Ol__cancel__one_H,axiom,
% 0.26/0.47 ! [G: partia96731725t_unit,X: a,A: a] :
% 0.26/0.47 ( ( group_a_Product_unit @ G )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ( ( X
% 0.26/0.47 = ( mult_a_Product_unit @ G @ X @ A ) )
% 0.26/0.47 = ( A
% 0.26/0.47 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.l_cancel_one'
% 0.26/0.47 thf(fact_164_group_Ol__cancel__one_H,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit,X: set_Product_prod_a_a,A: set_Product_prod_a_a] :
% 0.26/0.47 ( ( group_52730354t_unit @ G )
% 0.26/0.47 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ A @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( X
% 0.26/0.47 = ( mult_s977248631t_unit @ G @ X @ A ) )
% 0.26/0.47 = ( A
% 0.26/0.47 = ( one_se1488596747t_unit @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.l_cancel_one'
% 0.26/0.47 thf(fact_165_group_Ol__cancel__one_H,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b,X: a,A: a] :
% 0.26/0.47 ( ( group_a_ring_ext_a_b @ G )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( X
% 0.26/0.47 = ( mult_a_ring_ext_a_b @ G @ X @ A ) )
% 0.26/0.47 = ( A
% 0.26/0.47 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.l_cancel_one'
% 0.26/0.47 thf(fact_166_group_Or__cancel__one,axiom,
% 0.26/0.47 ! [G: partia96731725t_unit,X: a,A: a] :
% 0.26/0.47 ( ( group_a_Product_unit @ G )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ( ( ( mult_a_Product_unit @ G @ A @ X )
% 0.26/0.47 = X )
% 0.26/0.47 = ( A
% 0.26/0.47 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.r_cancel_one
% 0.26/0.47 thf(fact_167_group_Or__cancel__one,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit,X: set_Product_prod_a_a,A: set_Product_prod_a_a] :
% 0.26/0.47 ( ( group_52730354t_unit @ G )
% 0.26/0.47 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ A @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( ( mult_s977248631t_unit @ G @ A @ X )
% 0.26/0.47 = X )
% 0.26/0.47 = ( A
% 0.26/0.47 = ( one_se1488596747t_unit @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.r_cancel_one
% 0.26/0.47 thf(fact_168_group_Or__cancel__one,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b,X: a,A: a] :
% 0.26/0.47 ( ( group_a_ring_ext_a_b @ G )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( ( mult_a_ring_ext_a_b @ G @ A @ X )
% 0.26/0.47 = X )
% 0.26/0.47 = ( A
% 0.26/0.47 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.r_cancel_one
% 0.26/0.47 thf(fact_169_group_Ol__cancel__one,axiom,
% 0.26/0.47 ! [G: partia96731725t_unit,X: a,A: a] :
% 0.26/0.47 ( ( group_a_Product_unit @ G )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ( ( ( mult_a_Product_unit @ G @ X @ A )
% 0.26/0.47 = X )
% 0.26/0.47 = ( A
% 0.26/0.47 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.l_cancel_one
% 0.26/0.47 thf(fact_170_group_Ol__cancel__one,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit,X: set_Product_prod_a_a,A: set_Product_prod_a_a] :
% 0.26/0.47 ( ( group_52730354t_unit @ G )
% 0.26/0.47 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ A @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( ( mult_s977248631t_unit @ G @ X @ A )
% 0.26/0.47 = X )
% 0.26/0.47 = ( A
% 0.26/0.47 = ( one_se1488596747t_unit @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.l_cancel_one
% 0.26/0.47 thf(fact_171_group_Ol__cancel__one,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b,X: a,A: a] :
% 0.26/0.47 ( ( group_a_ring_ext_a_b @ G )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( ( mult_a_ring_ext_a_b @ G @ X @ A )
% 0.26/0.47 = X )
% 0.26/0.47 = ( A
% 0.26/0.47 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.l_cancel_one
% 0.26/0.47 thf(fact_172_group_Or__inv__ex,axiom,
% 0.26/0.47 ! [G: partia96731725t_unit,X: a] :
% 0.26/0.47 ( ( group_a_Product_unit @ G )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ? [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 & ( ( mult_a_Product_unit @ G @ X @ X2 )
% 0.26/0.47 = ( one_a_Product_unit @ G ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.r_inv_ex
% 0.26/0.47 thf(fact_173_group_Or__inv__ex,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit,X: set_Product_prod_a_a] :
% 0.26/0.47 ( ( group_52730354t_unit @ G )
% 0.26/0.47 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ? [X2: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 & ( ( mult_s977248631t_unit @ G @ X @ X2 )
% 0.26/0.47 = ( one_se1488596747t_unit @ G ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.r_inv_ex
% 0.26/0.47 thf(fact_174_group_Or__inv__ex,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b,X: a] :
% 0.26/0.47 ( ( group_a_ring_ext_a_b @ G )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ? [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 & ( ( mult_a_ring_ext_a_b @ G @ X @ X2 )
% 0.26/0.47 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.r_inv_ex
% 0.26/0.47 thf(fact_175_group_Ol__inv__ex,axiom,
% 0.26/0.47 ! [G: partia96731725t_unit,X: a] :
% 0.26/0.47 ( ( group_a_Product_unit @ G )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ? [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 & ( ( mult_a_Product_unit @ G @ X2 @ X )
% 0.26/0.47 = ( one_a_Product_unit @ G ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.l_inv_ex
% 0.26/0.47 thf(fact_176_group_Ol__inv__ex,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit,X: set_Product_prod_a_a] :
% 0.26/0.47 ( ( group_52730354t_unit @ G )
% 0.26/0.47 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ? [X2: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 & ( ( mult_s977248631t_unit @ G @ X2 @ X )
% 0.26/0.47 = ( one_se1488596747t_unit @ G ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.l_inv_ex
% 0.26/0.47 thf(fact_177_group_Ol__inv__ex,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b,X: a] :
% 0.26/0.47 ( ( group_a_ring_ext_a_b @ G )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ? [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 & ( ( mult_a_ring_ext_a_b @ G @ X2 @ X )
% 0.26/0.47 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.l_inv_ex
% 0.26/0.47 thf(fact_178_group_Oinv__comm,axiom,
% 0.26/0.47 ! [G: partia96731725t_unit,X: a,Y: a] :
% 0.26/0.47 ( ( group_a_Product_unit @ G )
% 0.26/0.47 => ( ( ( mult_a_Product_unit @ G @ X @ Y )
% 0.26/0.47 = ( one_a_Product_unit @ G ) )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ( ( member_a @ Y @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ( ( mult_a_Product_unit @ G @ Y @ X )
% 0.26/0.47 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.inv_comm
% 0.26/0.47 thf(fact_179_group_Oinv__comm,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit,X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
% 0.26/0.47 ( ( group_52730354t_unit @ G )
% 0.26/0.47 => ( ( ( mult_s977248631t_unit @ G @ X @ Y )
% 0.26/0.47 = ( one_se1488596747t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ Y @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( mult_s977248631t_unit @ G @ Y @ X )
% 0.26/0.47 = ( one_se1488596747t_unit @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.inv_comm
% 0.26/0.47 thf(fact_180_group_Oinv__comm,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 0.26/0.47 ( ( group_a_ring_ext_a_b @ G )
% 0.26/0.47 => ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
% 0.26/0.47 = ( one_a_ring_ext_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
% 0.26/0.47 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % group.inv_comm
% 0.26/0.47 thf(fact_181_groupI,axiom,
% 0.26/0.47 ! [G: partia96731725t_unit] :
% 0.26/0.47 ( ! [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ! [Y3: a] :
% 0.26/0.47 ( ( member_a @ Y3 @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ( member_a @ ( mult_a_Product_unit @ G @ X2 @ Y3 ) @ ( partia1955795460t_unit @ G ) ) ) )
% 0.26/0.47 => ( ( member_a @ ( one_a_Product_unit @ G ) @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ( ! [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ! [Y3: a] :
% 0.26/0.47 ( ( member_a @ Y3 @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ! [Z2: a] :
% 0.26/0.47 ( ( member_a @ Z2 @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ( ( mult_a_Product_unit @ G @ ( mult_a_Product_unit @ G @ X2 @ Y3 ) @ Z2 )
% 0.26/0.47 = ( mult_a_Product_unit @ G @ X2 @ ( mult_a_Product_unit @ G @ Y3 @ Z2 ) ) ) ) ) )
% 0.26/0.47 => ( ! [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ( ( mult_a_Product_unit @ G @ ( one_a_Product_unit @ G ) @ X2 )
% 0.26/0.47 = X2 ) )
% 0.26/0.47 => ( ! [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ? [Xa: a] :
% 0.26/0.47 ( ( member_a @ Xa @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 & ( ( mult_a_Product_unit @ G @ Xa @ X2 )
% 0.26/0.47 = ( one_a_Product_unit @ G ) ) ) )
% 0.26/0.47 => ( group_a_Product_unit @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % groupI
% 0.26/0.47 thf(fact_182_groupI,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit] :
% 0.26/0.47 ( ! [X2: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ! [Y3: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ Y3 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( member1838126896od_a_a @ ( mult_s977248631t_unit @ G @ X2 @ Y3 ) @ ( partia616129906t_unit @ G ) ) ) )
% 0.26/0.47 => ( ( member1838126896od_a_a @ ( one_se1488596747t_unit @ G ) @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ! [X2: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ! [Y3: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ Y3 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ! [Z2: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ Z2 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( mult_s977248631t_unit @ G @ ( mult_s977248631t_unit @ G @ X2 @ Y3 ) @ Z2 )
% 0.26/0.47 = ( mult_s977248631t_unit @ G @ X2 @ ( mult_s977248631t_unit @ G @ Y3 @ Z2 ) ) ) ) ) )
% 0.26/0.47 => ( ! [X2: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ( ( mult_s977248631t_unit @ G @ ( one_se1488596747t_unit @ G ) @ X2 )
% 0.26/0.47 = X2 ) )
% 0.26/0.47 => ( ! [X2: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ? [Xa: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ Xa @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 & ( ( mult_s977248631t_unit @ G @ Xa @ X2 )
% 0.26/0.47 = ( one_se1488596747t_unit @ G ) ) ) )
% 0.26/0.47 => ( group_52730354t_unit @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % groupI
% 0.26/0.47 thf(fact_183_groupI,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b] :
% 0.26/0.47 ( ! [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ! [Y3: a] :
% 0.26/0.47 ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( member_a @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ ( partia1066395285xt_a_b @ G ) ) ) )
% 0.26/0.47 => ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ! [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ! [Y3: a] :
% 0.26/0.47 ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ! [Z2: a] :
% 0.26/0.47 ( ( member_a @ Z2 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ Z2 )
% 0.26/0.47 = ( mult_a_ring_ext_a_b @ G @ X2 @ ( mult_a_ring_ext_a_b @ G @ Y3 @ Z2 ) ) ) ) ) )
% 0.26/0.47 => ( ! [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X2 )
% 0.26/0.47 = X2 ) )
% 0.26/0.47 => ( ! [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ? [Xa: a] :
% 0.26/0.47 ( ( member_a @ Xa @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 & ( ( mult_a_ring_ext_a_b @ G @ Xa @ X2 )
% 0.26/0.47 = ( one_a_ring_ext_a_b @ G ) ) ) )
% 0.26/0.47 => ( group_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % groupI
% 0.26/0.47 thf(fact_184_monoid_Ogroup__l__invI,axiom,
% 0.26/0.47 ! [G: partia96731725t_unit] :
% 0.26/0.47 ( ( monoid1447254976t_unit @ G )
% 0.26/0.47 => ( ! [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 => ? [Xa: a] :
% 0.26/0.47 ( ( member_a @ Xa @ ( partia1955795460t_unit @ G ) )
% 0.26/0.47 & ( ( mult_a_Product_unit @ G @ Xa @ X2 )
% 0.26/0.47 = ( one_a_Product_unit @ G ) ) ) )
% 0.26/0.47 => ( group_a_Product_unit @ G ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.group_l_invI
% 0.26/0.47 thf(fact_185_monoid_Ogroup__l__invI,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit] :
% 0.26/0.47 ( ( monoid1242292471t_unit @ G )
% 0.26/0.47 => ( ! [X2: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 => ? [Xa: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ Xa @ ( partia616129906t_unit @ G ) )
% 0.26/0.47 & ( ( mult_s977248631t_unit @ G @ Xa @ X2 )
% 0.26/0.47 = ( one_se1488596747t_unit @ G ) ) ) )
% 0.26/0.47 => ( group_52730354t_unit @ G ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.group_l_invI
% 0.26/0.47 thf(fact_186_monoid_Ogroup__l__invI,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( monoid1022105667xt_a_b @ G )
% 0.26/0.47 => ( ! [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 => ? [Xa: a] :
% 0.26/0.47 ( ( member_a @ Xa @ ( partia1066395285xt_a_b @ G ) )
% 0.26/0.47 & ( ( mult_a_ring_ext_a_b @ G @ Xa @ X2 )
% 0.26/0.47 = ( one_a_ring_ext_a_b @ G ) ) ) )
% 0.26/0.47 => ( group_a_ring_ext_a_b @ G ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.group_l_invI
% 0.26/0.47 thf(fact_187_semiring_Oadd__pow__rdistr,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b,A: a,B: a,K: nat] :
% 0.26/0.47 ( ( semiring_a_b @ R3 )
% 0.26/0.47 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ R3 @ A @ ( add_pow_a_b_nat @ R3 @ K @ B ) )
% 0.26/0.47 = ( add_pow_a_b_nat @ R3 @ K @ ( mult_a_ring_ext_a_b @ R3 @ A @ B ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % semiring.add_pow_rdistr
% 0.26/0.47 thf(fact_188_semiring_Oadd__pow__ldistr,axiom,
% 0.26/0.47 ! [R3: partia1833973666xt_a_b,A: a,B: a,K: nat] :
% 0.26/0.47 ( ( semiring_a_b @ R3 )
% 0.26/0.47 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
% 0.26/0.47 => ( ( mult_a_ring_ext_a_b @ R3 @ ( add_pow_a_b_nat @ R3 @ K @ A ) @ B )
% 0.26/0.47 = ( add_pow_a_b_nat @ R3 @ K @ ( mult_a_ring_ext_a_b @ R3 @ A @ B ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % semiring.add_pow_ldistr
% 0.26/0.47 thf(fact_189_a__lcos__mult__one,axiom,
% 0.26/0.47 ! [M: set_a] :
% 0.26/0.47 ( ( ord_less_eq_set_a @ M @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M )
% 0.26/0.47 = M ) ) ).
% 0.26/0.47
% 0.26/0.47 % a_lcos_mult_one
% 0.26/0.47 thf(fact_190_set__add__closed,axiom,
% 0.26/0.47 ! [A3: set_a,B4: set_a] :
% 0.26/0.47 ( ( ord_less_eq_set_a @ A3 @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( ord_less_eq_set_a @ B4 @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A3 @ B4 ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % set_add_closed
% 0.26/0.47 thf(fact_191_setadd__subset__G,axiom,
% 0.26/0.47 ! [H: set_a,K2: set_a] :
% 0.26/0.47 ( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( ord_less_eq_set_a @ K2 @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ord_less_eq_set_a @ ( set_add_a_b @ r @ H @ K2 ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % setadd_subset_G
% 0.26/0.47 thf(fact_192_subset__empty,axiom,
% 0.26/0.47 ! [A3: set_Product_prod_a_a] :
% 0.26/0.47 ( ( ord_le1824328871od_a_a @ A3 @ bot_bo2131659635od_a_a )
% 0.26/0.47 = ( A3 = bot_bo2131659635od_a_a ) ) ).
% 0.26/0.47
% 0.26/0.47 % subset_empty
% 0.26/0.47 thf(fact_193_subset__empty,axiom,
% 0.26/0.47 ! [A3: set_a] :
% 0.26/0.47 ( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
% 0.26/0.47 = ( A3 = bot_bot_set_a ) ) ).
% 0.26/0.47
% 0.26/0.47 % subset_empty
% 0.26/0.47 thf(fact_194_subset__empty,axiom,
% 0.26/0.47 ! [A3: set_set_a] :
% 0.26/0.47 ( ( ord_le318720350_set_a @ A3 @ bot_bot_set_set_a )
% 0.26/0.47 = ( A3 = bot_bot_set_set_a ) ) ).
% 0.26/0.47
% 0.26/0.47 % subset_empty
% 0.26/0.47 thf(fact_195_empty__iff,axiom,
% 0.26/0.47 ! [C: set_Product_prod_a_a] :
% 0.26/0.47 ~ ( member1838126896od_a_a @ C @ bot_bo1424703315od_a_a ) ).
% 0.26/0.47
% 0.26/0.47 % empty_iff
% 0.26/0.47 thf(fact_196_empty__iff,axiom,
% 0.26/0.47 ! [C: set_a] :
% 0.26/0.47 ~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% 0.26/0.47
% 0.26/0.47 % empty_iff
% 0.26/0.47 thf(fact_197_empty__iff,axiom,
% 0.26/0.47 ! [C: product_prod_a_a] :
% 0.26/0.47 ~ ( member449909584od_a_a @ C @ bot_bo2131659635od_a_a ) ).
% 0.26/0.47
% 0.26/0.47 % empty_iff
% 0.26/0.47 thf(fact_198_empty__iff,axiom,
% 0.26/0.47 ! [C: a] :
% 0.26/0.47 ~ ( member_a @ C @ bot_bot_set_a ) ).
% 0.26/0.47
% 0.26/0.47 % empty_iff
% 0.26/0.47 thf(fact_199_all__not__in__conv,axiom,
% 0.26/0.47 ! [A3: set_se1596668135od_a_a] :
% 0.26/0.47 ( ( ! [X3: set_Product_prod_a_a] :
% 0.26/0.47 ~ ( member1838126896od_a_a @ X3 @ A3 ) )
% 0.26/0.47 = ( A3 = bot_bo1424703315od_a_a ) ) ).
% 0.26/0.47
% 0.26/0.47 % all_not_in_conv
% 0.26/0.47 thf(fact_200_all__not__in__conv,axiom,
% 0.26/0.47 ! [A3: set_set_a] :
% 0.26/0.47 ( ( ! [X3: set_a] :
% 0.26/0.47 ~ ( member_set_a @ X3 @ A3 ) )
% 0.26/0.47 = ( A3 = bot_bot_set_set_a ) ) ).
% 0.26/0.47
% 0.26/0.47 % all_not_in_conv
% 0.26/0.47 thf(fact_201_all__not__in__conv,axiom,
% 0.26/0.47 ! [A3: set_Product_prod_a_a] :
% 0.26/0.47 ( ( ! [X3: product_prod_a_a] :
% 0.26/0.47 ~ ( member449909584od_a_a @ X3 @ A3 ) )
% 0.26/0.47 = ( A3 = bot_bo2131659635od_a_a ) ) ).
% 0.26/0.47
% 0.26/0.47 % all_not_in_conv
% 0.26/0.47 thf(fact_202_all__not__in__conv,axiom,
% 0.26/0.47 ! [A3: set_a] :
% 0.26/0.47 ( ( ! [X3: a] :
% 0.26/0.47 ~ ( member_a @ X3 @ A3 ) )
% 0.26/0.47 = ( A3 = bot_bot_set_a ) ) ).
% 0.26/0.47
% 0.26/0.47 % all_not_in_conv
% 0.26/0.47 thf(fact_203_Collect__empty__eq,axiom,
% 0.26/0.47 ! [P: product_prod_a_a > $o] :
% 0.26/0.47 ( ( ( collec645855634od_a_a @ P )
% 0.26/0.47 = bot_bo2131659635od_a_a )
% 0.26/0.47 = ( ! [X3: product_prod_a_a] :
% 0.26/0.47 ~ ( P @ X3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Collect_empty_eq
% 0.26/0.47 thf(fact_204_Collect__empty__eq,axiom,
% 0.26/0.47 ! [P: a > $o] :
% 0.26/0.47 ( ( ( collect_a @ P )
% 0.26/0.47 = bot_bot_set_a )
% 0.26/0.47 = ( ! [X3: a] :
% 0.26/0.47 ~ ( P @ X3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Collect_empty_eq
% 0.26/0.47 thf(fact_205_empty__Collect__eq,axiom,
% 0.26/0.47 ! [P: product_prod_a_a > $o] :
% 0.26/0.47 ( ( bot_bo2131659635od_a_a
% 0.26/0.47 = ( collec645855634od_a_a @ P ) )
% 0.26/0.47 = ( ! [X3: product_prod_a_a] :
% 0.26/0.47 ~ ( P @ X3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % empty_Collect_eq
% 0.26/0.47 thf(fact_206_empty__Collect__eq,axiom,
% 0.26/0.47 ! [P: a > $o] :
% 0.26/0.47 ( ( bot_bot_set_a
% 0.26/0.47 = ( collect_a @ P ) )
% 0.26/0.47 = ( ! [X3: a] :
% 0.26/0.47 ~ ( P @ X3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % empty_Collect_eq
% 0.26/0.47 thf(fact_207_subsetI,axiom,
% 0.26/0.47 ! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
% 0.26/0.47 ( ! [X2: product_prod_a_a] :
% 0.26/0.47 ( ( member449909584od_a_a @ X2 @ A3 )
% 0.26/0.47 => ( member449909584od_a_a @ X2 @ B4 ) )
% 0.26/0.47 => ( ord_le1824328871od_a_a @ A3 @ B4 ) ) ).
% 0.26/0.47
% 0.26/0.47 % subsetI
% 0.26/0.47 thf(fact_208_subsetI,axiom,
% 0.26/0.47 ! [A3: set_se1596668135od_a_a,B4: set_se1596668135od_a_a] :
% 0.26/0.47 ( ! [X2: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X2 @ A3 )
% 0.26/0.47 => ( member1838126896od_a_a @ X2 @ B4 ) )
% 0.26/0.47 => ( ord_le391835783od_a_a @ A3 @ B4 ) ) ).
% 0.26/0.47
% 0.26/0.47 % subsetI
% 0.26/0.47 thf(fact_209_subsetI,axiom,
% 0.26/0.47 ! [A3: set_a,B4: set_a] :
% 0.26/0.47 ( ! [X2: a] :
% 0.26/0.47 ( ( member_a @ X2 @ A3 )
% 0.26/0.47 => ( member_a @ X2 @ B4 ) )
% 0.26/0.47 => ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% 0.26/0.47
% 0.26/0.47 % subsetI
% 0.26/0.47 thf(fact_210_subsetI,axiom,
% 0.26/0.47 ! [A3: set_set_a,B4: set_set_a] :
% 0.26/0.47 ( ! [X2: set_a] :
% 0.26/0.47 ( ( member_set_a @ X2 @ A3 )
% 0.26/0.47 => ( member_set_a @ X2 @ B4 ) )
% 0.26/0.47 => ( ord_le318720350_set_a @ A3 @ B4 ) ) ).
% 0.26/0.47
% 0.26/0.47 % subsetI
% 0.26/0.47 thf(fact_211_subset__antisym,axiom,
% 0.26/0.47 ! [A3: set_a,B4: set_a] :
% 0.26/0.47 ( ( ord_less_eq_set_a @ A3 @ B4 )
% 0.26/0.47 => ( ( ord_less_eq_set_a @ B4 @ A3 )
% 0.26/0.47 => ( A3 = B4 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % subset_antisym
% 0.26/0.47 thf(fact_212_subset__antisym,axiom,
% 0.26/0.47 ! [A3: set_set_a,B4: set_set_a] :
% 0.26/0.47 ( ( ord_le318720350_set_a @ A3 @ B4 )
% 0.26/0.47 => ( ( ord_le318720350_set_a @ B4 @ A3 )
% 0.26/0.47 => ( A3 = B4 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % subset_antisym
% 0.26/0.47 thf(fact_213_a__l__coset__subset__G,axiom,
% 0.26/0.47 ! [H: set_a,X: a] :
% 0.26/0.47 ( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X @ H ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % a_l_coset_subset_G
% 0.26/0.47 thf(fact_214_empty__subsetI,axiom,
% 0.26/0.47 ! [A3: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ bot_bo2131659635od_a_a @ A3 ) ).
% 0.26/0.47
% 0.26/0.47 % empty_subsetI
% 0.26/0.47 thf(fact_215_empty__subsetI,axiom,
% 0.26/0.47 ! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% 0.26/0.47
% 0.26/0.47 % empty_subsetI
% 0.26/0.47 thf(fact_216_empty__subsetI,axiom,
% 0.26/0.47 ! [A3: set_set_a] : ( ord_le318720350_set_a @ bot_bot_set_set_a @ A3 ) ).
% 0.26/0.47
% 0.26/0.47 % empty_subsetI
% 0.26/0.47 thf(fact_217_emptyE,axiom,
% 0.26/0.47 ! [A: set_Product_prod_a_a] :
% 0.26/0.47 ~ ( member1838126896od_a_a @ A @ bot_bo1424703315od_a_a ) ).
% 0.26/0.47
% 0.26/0.47 % emptyE
% 0.26/0.47 thf(fact_218_emptyE,axiom,
% 0.26/0.47 ! [A: set_a] :
% 0.26/0.47 ~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% 0.26/0.47
% 0.26/0.47 % emptyE
% 0.26/0.47 thf(fact_219_emptyE,axiom,
% 0.26/0.47 ! [A: product_prod_a_a] :
% 0.26/0.47 ~ ( member449909584od_a_a @ A @ bot_bo2131659635od_a_a ) ).
% 0.26/0.47
% 0.26/0.47 % emptyE
% 0.26/0.47 thf(fact_220_emptyE,axiom,
% 0.26/0.47 ! [A: a] :
% 0.26/0.47 ~ ( member_a @ A @ bot_bot_set_a ) ).
% 0.26/0.47
% 0.26/0.47 % emptyE
% 0.26/0.47 thf(fact_221_equals0D,axiom,
% 0.26/0.47 ! [A3: set_se1596668135od_a_a,A: set_Product_prod_a_a] :
% 0.26/0.47 ( ( A3 = bot_bo1424703315od_a_a )
% 0.26/0.47 => ~ ( member1838126896od_a_a @ A @ A3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % equals0D
% 0.26/0.47 thf(fact_222_equals0D,axiom,
% 0.26/0.47 ! [A3: set_set_a,A: set_a] :
% 0.26/0.47 ( ( A3 = bot_bot_set_set_a )
% 0.26/0.47 => ~ ( member_set_a @ A @ A3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % equals0D
% 0.26/0.47 thf(fact_223_equals0D,axiom,
% 0.26/0.47 ! [A3: set_Product_prod_a_a,A: product_prod_a_a] :
% 0.26/0.47 ( ( A3 = bot_bo2131659635od_a_a )
% 0.26/0.47 => ~ ( member449909584od_a_a @ A @ A3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % equals0D
% 0.26/0.47 thf(fact_224_equals0D,axiom,
% 0.26/0.47 ! [A3: set_a,A: a] :
% 0.26/0.47 ( ( A3 = bot_bot_set_a )
% 0.26/0.47 => ~ ( member_a @ A @ A3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % equals0D
% 0.26/0.47 thf(fact_225_equals0I,axiom,
% 0.26/0.47 ! [A3: set_se1596668135od_a_a] :
% 0.26/0.47 ( ! [Y3: set_Product_prod_a_a] :
% 0.26/0.47 ~ ( member1838126896od_a_a @ Y3 @ A3 )
% 0.26/0.47 => ( A3 = bot_bo1424703315od_a_a ) ) ).
% 0.26/0.47
% 0.26/0.47 % equals0I
% 0.26/0.47 thf(fact_226_equals0I,axiom,
% 0.26/0.47 ! [A3: set_set_a] :
% 0.26/0.47 ( ! [Y3: set_a] :
% 0.26/0.47 ~ ( member_set_a @ Y3 @ A3 )
% 0.26/0.47 => ( A3 = bot_bot_set_set_a ) ) ).
% 0.26/0.47
% 0.26/0.47 % equals0I
% 0.26/0.47 thf(fact_227_equals0I,axiom,
% 0.26/0.47 ! [A3: set_Product_prod_a_a] :
% 0.26/0.47 ( ! [Y3: product_prod_a_a] :
% 0.26/0.47 ~ ( member449909584od_a_a @ Y3 @ A3 )
% 0.26/0.47 => ( A3 = bot_bo2131659635od_a_a ) ) ).
% 0.26/0.47
% 0.26/0.47 % equals0I
% 0.26/0.47 thf(fact_228_equals0I,axiom,
% 0.26/0.47 ! [A3: set_a] :
% 0.26/0.47 ( ! [Y3: a] :
% 0.26/0.47 ~ ( member_a @ Y3 @ A3 )
% 0.26/0.47 => ( A3 = bot_bot_set_a ) ) ).
% 0.26/0.47
% 0.26/0.47 % equals0I
% 0.26/0.47 thf(fact_229_ex__in__conv,axiom,
% 0.26/0.47 ! [A3: set_se1596668135od_a_a] :
% 0.26/0.47 ( ( ? [X3: set_Product_prod_a_a] : ( member1838126896od_a_a @ X3 @ A3 ) )
% 0.26/0.47 = ( A3 != bot_bo1424703315od_a_a ) ) ).
% 0.26/0.47
% 0.26/0.47 % ex_in_conv
% 0.26/0.47 thf(fact_230_ex__in__conv,axiom,
% 0.26/0.47 ! [A3: set_set_a] :
% 0.26/0.47 ( ( ? [X3: set_a] : ( member_set_a @ X3 @ A3 ) )
% 0.26/0.47 = ( A3 != bot_bot_set_set_a ) ) ).
% 0.26/0.47
% 0.26/0.47 % ex_in_conv
% 0.26/0.47 thf(fact_231_ex__in__conv,axiom,
% 0.26/0.47 ! [A3: set_Product_prod_a_a] :
% 0.26/0.47 ( ( ? [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A3 ) )
% 0.26/0.47 = ( A3 != bot_bo2131659635od_a_a ) ) ).
% 0.26/0.47
% 0.26/0.47 % ex_in_conv
% 0.26/0.47 thf(fact_232_ex__in__conv,axiom,
% 0.26/0.47 ! [A3: set_a] :
% 0.26/0.47 ( ( ? [X3: a] : ( member_a @ X3 @ A3 ) )
% 0.26/0.47 = ( A3 != bot_bot_set_a ) ) ).
% 0.26/0.47
% 0.26/0.47 % ex_in_conv
% 0.26/0.47 thf(fact_233_in__mono,axiom,
% 0.26/0.47 ! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,X: product_prod_a_a] :
% 0.26/0.47 ( ( ord_le1824328871od_a_a @ A3 @ B4 )
% 0.26/0.47 => ( ( member449909584od_a_a @ X @ A3 )
% 0.26/0.47 => ( member449909584od_a_a @ X @ B4 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % in_mono
% 0.26/0.47 thf(fact_234_in__mono,axiom,
% 0.26/0.47 ! [A3: set_se1596668135od_a_a,B4: set_se1596668135od_a_a,X: set_Product_prod_a_a] :
% 0.26/0.47 ( ( ord_le391835783od_a_a @ A3 @ B4 )
% 0.26/0.47 => ( ( member1838126896od_a_a @ X @ A3 )
% 0.26/0.47 => ( member1838126896od_a_a @ X @ B4 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % in_mono
% 0.26/0.47 thf(fact_235_in__mono,axiom,
% 0.26/0.47 ! [A3: set_a,B4: set_a,X: a] :
% 0.26/0.47 ( ( ord_less_eq_set_a @ A3 @ B4 )
% 0.26/0.47 => ( ( member_a @ X @ A3 )
% 0.26/0.47 => ( member_a @ X @ B4 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % in_mono
% 0.26/0.47 thf(fact_236_in__mono,axiom,
% 0.26/0.47 ! [A3: set_set_a,B4: set_set_a,X: set_a] :
% 0.26/0.47 ( ( ord_le318720350_set_a @ A3 @ B4 )
% 0.26/0.47 => ( ( member_set_a @ X @ A3 )
% 0.26/0.47 => ( member_set_a @ X @ B4 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % in_mono
% 0.26/0.47 thf(fact_237_subsetD,axiom,
% 0.26/0.47 ! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,C: product_prod_a_a] :
% 0.26/0.47 ( ( ord_le1824328871od_a_a @ A3 @ B4 )
% 0.26/0.47 => ( ( member449909584od_a_a @ C @ A3 )
% 0.26/0.47 => ( member449909584od_a_a @ C @ B4 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % subsetD
% 0.26/0.47 thf(fact_238_subsetD,axiom,
% 0.26/0.47 ! [A3: set_se1596668135od_a_a,B4: set_se1596668135od_a_a,C: set_Product_prod_a_a] :
% 0.26/0.47 ( ( ord_le391835783od_a_a @ A3 @ B4 )
% 0.26/0.47 => ( ( member1838126896od_a_a @ C @ A3 )
% 0.26/0.47 => ( member1838126896od_a_a @ C @ B4 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % subsetD
% 0.26/0.47 thf(fact_239_subsetD,axiom,
% 0.26/0.47 ! [A3: set_a,B4: set_a,C: a] :
% 0.26/0.47 ( ( ord_less_eq_set_a @ A3 @ B4 )
% 0.26/0.47 => ( ( member_a @ C @ A3 )
% 0.26/0.47 => ( member_a @ C @ B4 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % subsetD
% 0.26/0.47 thf(fact_240_subsetD,axiom,
% 0.26/0.47 ! [A3: set_set_a,B4: set_set_a,C: set_a] :
% 0.26/0.47 ( ( ord_le318720350_set_a @ A3 @ B4 )
% 0.26/0.47 => ( ( member_set_a @ C @ A3 )
% 0.26/0.47 => ( member_set_a @ C @ B4 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % subsetD
% 0.26/0.47 thf(fact_241_equalityE,axiom,
% 0.26/0.47 ! [A3: set_a,B4: set_a] :
% 0.26/0.47 ( ( A3 = B4 )
% 0.26/0.47 => ~ ( ( ord_less_eq_set_a @ A3 @ B4 )
% 0.26/0.47 => ~ ( ord_less_eq_set_a @ B4 @ A3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % equalityE
% 0.26/0.47 thf(fact_242_equalityE,axiom,
% 0.26/0.47 ! [A3: set_set_a,B4: set_set_a] :
% 0.26/0.47 ( ( A3 = B4 )
% 0.26/0.47 => ~ ( ( ord_le318720350_set_a @ A3 @ B4 )
% 0.26/0.47 => ~ ( ord_le318720350_set_a @ B4 @ A3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % equalityE
% 0.26/0.47 thf(fact_243_subset__eq,axiom,
% 0.26/0.47 ( ord_le1824328871od_a_a
% 0.26/0.47 = ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
% 0.26/0.47 ! [X3: product_prod_a_a] :
% 0.26/0.47 ( ( member449909584od_a_a @ X3 @ A5 )
% 0.26/0.47 => ( member449909584od_a_a @ X3 @ B5 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % subset_eq
% 0.26/0.47 thf(fact_244_subset__eq,axiom,
% 0.26/0.47 ( ord_le391835783od_a_a
% 0.26/0.47 = ( ^ [A5: set_se1596668135od_a_a,B5: set_se1596668135od_a_a] :
% 0.26/0.47 ! [X3: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X3 @ A5 )
% 0.26/0.47 => ( member1838126896od_a_a @ X3 @ B5 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % subset_eq
% 0.26/0.47 thf(fact_245_subset__eq,axiom,
% 0.26/0.47 ( ord_less_eq_set_a
% 0.26/0.47 = ( ^ [A5: set_a,B5: set_a] :
% 0.26/0.47 ! [X3: a] :
% 0.26/0.47 ( ( member_a @ X3 @ A5 )
% 0.26/0.47 => ( member_a @ X3 @ B5 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % subset_eq
% 0.26/0.47 thf(fact_246_subset__eq,axiom,
% 0.26/0.47 ( ord_le318720350_set_a
% 0.26/0.47 = ( ^ [A5: set_set_a,B5: set_set_a] :
% 0.26/0.47 ! [X3: set_a] :
% 0.26/0.47 ( ( member_set_a @ X3 @ A5 )
% 0.26/0.47 => ( member_set_a @ X3 @ B5 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % subset_eq
% 0.26/0.47 thf(fact_247_equalityD1,axiom,
% 0.26/0.47 ! [A3: set_a,B4: set_a] :
% 0.26/0.47 ( ( A3 = B4 )
% 0.26/0.47 => ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% 0.26/0.47
% 0.26/0.47 % equalityD1
% 0.26/0.47 thf(fact_248_equalityD1,axiom,
% 0.26/0.47 ! [A3: set_set_a,B4: set_set_a] :
% 0.26/0.47 ( ( A3 = B4 )
% 0.26/0.47 => ( ord_le318720350_set_a @ A3 @ B4 ) ) ).
% 0.26/0.47
% 0.26/0.47 % equalityD1
% 0.26/0.47 thf(fact_249_equalityD2,axiom,
% 0.26/0.47 ! [A3: set_a,B4: set_a] :
% 0.26/0.47 ( ( A3 = B4 )
% 0.26/0.47 => ( ord_less_eq_set_a @ B4 @ A3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % equalityD2
% 0.26/0.47 thf(fact_250_equalityD2,axiom,
% 0.26/0.47 ! [A3: set_set_a,B4: set_set_a] :
% 0.26/0.47 ( ( A3 = B4 )
% 0.26/0.47 => ( ord_le318720350_set_a @ B4 @ A3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % equalityD2
% 0.26/0.47 thf(fact_251_subset__iff,axiom,
% 0.26/0.47 ( ord_le1824328871od_a_a
% 0.26/0.47 = ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
% 0.26/0.47 ! [T2: product_prod_a_a] :
% 0.26/0.47 ( ( member449909584od_a_a @ T2 @ A5 )
% 0.26/0.47 => ( member449909584od_a_a @ T2 @ B5 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % subset_iff
% 0.26/0.47 thf(fact_252_subset__iff,axiom,
% 0.26/0.47 ( ord_le391835783od_a_a
% 0.26/0.47 = ( ^ [A5: set_se1596668135od_a_a,B5: set_se1596668135od_a_a] :
% 0.26/0.47 ! [T2: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ T2 @ A5 )
% 0.26/0.47 => ( member1838126896od_a_a @ T2 @ B5 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % subset_iff
% 0.26/0.47 thf(fact_253_subset__iff,axiom,
% 0.26/0.47 ( ord_less_eq_set_a
% 0.26/0.47 = ( ^ [A5: set_a,B5: set_a] :
% 0.26/0.47 ! [T2: a] :
% 0.26/0.47 ( ( member_a @ T2 @ A5 )
% 0.26/0.47 => ( member_a @ T2 @ B5 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % subset_iff
% 0.26/0.47 thf(fact_254_subset__iff,axiom,
% 0.26/0.47 ( ord_le318720350_set_a
% 0.26/0.47 = ( ^ [A5: set_set_a,B5: set_set_a] :
% 0.26/0.47 ! [T2: set_a] :
% 0.26/0.47 ( ( member_set_a @ T2 @ A5 )
% 0.26/0.47 => ( member_set_a @ T2 @ B5 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % subset_iff
% 0.26/0.47 thf(fact_255_subset__refl,axiom,
% 0.26/0.47 ! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% 0.26/0.47
% 0.26/0.47 % subset_refl
% 0.26/0.47 thf(fact_256_subset__refl,axiom,
% 0.26/0.47 ! [A3: set_set_a] : ( ord_le318720350_set_a @ A3 @ A3 ) ).
% 0.26/0.47
% 0.26/0.47 % subset_refl
% 0.26/0.47 thf(fact_257_Collect__mono,axiom,
% 0.26/0.47 ! [P: a > $o,Q: a > $o] :
% 0.26/0.47 ( ! [X2: a] :
% 0.26/0.47 ( ( P @ X2 )
% 0.26/0.47 => ( Q @ X2 ) )
% 0.26/0.47 => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Collect_mono
% 0.26/0.47 thf(fact_258_Collect__mono,axiom,
% 0.26/0.47 ! [P: set_a > $o,Q: set_a > $o] :
% 0.26/0.47 ( ! [X2: set_a] :
% 0.26/0.47 ( ( P @ X2 )
% 0.26/0.47 => ( Q @ X2 ) )
% 0.26/0.47 => ( ord_le318720350_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Collect_mono
% 0.26/0.47 thf(fact_259_subset__trans,axiom,
% 0.26/0.47 ! [A3: set_a,B4: set_a,C2: set_a] :
% 0.26/0.47 ( ( ord_less_eq_set_a @ A3 @ B4 )
% 0.26/0.47 => ( ( ord_less_eq_set_a @ B4 @ C2 )
% 0.26/0.47 => ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % subset_trans
% 0.26/0.47 thf(fact_260_subset__trans,axiom,
% 0.26/0.47 ! [A3: set_set_a,B4: set_set_a,C2: set_set_a] :
% 0.26/0.47 ( ( ord_le318720350_set_a @ A3 @ B4 )
% 0.26/0.47 => ( ( ord_le318720350_set_a @ B4 @ C2 )
% 0.26/0.47 => ( ord_le318720350_set_a @ A3 @ C2 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % subset_trans
% 0.26/0.47 thf(fact_261_set__eq__subset,axiom,
% 0.26/0.47 ( ( ^ [Y5: set_a,Z4: set_a] : ( Y5 = Z4 ) )
% 0.26/0.47 = ( ^ [A5: set_a,B5: set_a] :
% 0.26/0.47 ( ( ord_less_eq_set_a @ A5 @ B5 )
% 0.26/0.47 & ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % set_eq_subset
% 0.26/0.47 thf(fact_262_set__eq__subset,axiom,
% 0.26/0.47 ( ( ^ [Y5: set_set_a,Z4: set_set_a] : ( Y5 = Z4 ) )
% 0.26/0.47 = ( ^ [A5: set_set_a,B5: set_set_a] :
% 0.26/0.47 ( ( ord_le318720350_set_a @ A5 @ B5 )
% 0.26/0.47 & ( ord_le318720350_set_a @ B5 @ A5 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % set_eq_subset
% 0.26/0.47 thf(fact_263_Collect__mono__iff,axiom,
% 0.26/0.47 ! [P: a > $o,Q: a > $o] :
% 0.26/0.47 ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
% 0.26/0.47 = ( ! [X3: a] :
% 0.26/0.47 ( ( P @ X3 )
% 0.26/0.47 => ( Q @ X3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Collect_mono_iff
% 0.26/0.47 thf(fact_264_Collect__mono__iff,axiom,
% 0.26/0.47 ! [P: set_a > $o,Q: set_a > $o] :
% 0.26/0.47 ( ( ord_le318720350_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
% 0.26/0.47 = ( ! [X3: set_a] :
% 0.26/0.47 ( ( P @ X3 )
% 0.26/0.47 => ( Q @ X3 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Collect_mono_iff
% 0.26/0.47 thf(fact_265_units__group,axiom,
% 0.26/0.47 group_a_Product_unit @ ( units_1411277569xt_a_b @ r ) ).
% 0.26/0.47
% 0.26/0.47 % units_group
% 0.26/0.47 thf(fact_266_add__additive__subgroups,axiom,
% 0.26/0.47 ! [H: set_a,K2: set_a] :
% 0.26/0.47 ( ( additi2104487374up_a_b @ H @ r )
% 0.26/0.47 => ( ( additi2104487374up_a_b @ K2 @ r )
% 0.26/0.47 => ( additi2104487374up_a_b @ ( set_add_a_b @ r @ H @ K2 ) @ r ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % add_additive_subgroups
% 0.26/0.47 thf(fact_267_a__rcos__assoc__lcos,axiom,
% 0.26/0.47 ! [H: set_a,K2: set_a,X: a] :
% 0.26/0.47 ( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( ord_less_eq_set_a @ K2 @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( set_add_a_b @ r @ ( a_r_coset_a_b @ r @ H @ X ) @ K2 )
% 0.26/0.47 = ( set_add_a_b @ r @ H @ ( a_l_coset_a_b @ r @ X @ K2 ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % a_rcos_assoc_lcos
% 0.26/0.47 thf(fact_268_a__r__coset__subset__G,axiom,
% 0.26/0.47 ! [H: set_a,X: a] :
% 0.26/0.47 ( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ord_less_eq_set_a @ ( a_r_coset_a_b @ r @ H @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % a_r_coset_subset_G
% 0.26/0.47 thf(fact_269_a__setmult__rcos__assoc,axiom,
% 0.26/0.47 ! [H: set_a,K2: set_a,X: a] :
% 0.26/0.47 ( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( ord_less_eq_set_a @ K2 @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( set_add_a_b @ r @ H @ ( a_r_coset_a_b @ r @ K2 @ X ) )
% 0.26/0.47 = ( a_r_coset_a_b @ r @ ( set_add_a_b @ r @ H @ K2 ) @ X ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % a_setmult_rcos_assoc
% 0.26/0.47 thf(fact_270_units__of__units,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( units_a_Product_unit @ ( units_1411277569xt_a_b @ G ) )
% 0.26/0.47 = ( units_a_ring_ext_a_b @ G ) ) ).
% 0.26/0.47
% 0.26/0.47 % units_of_units
% 0.26/0.47 thf(fact_271_a__coset__add__zero,axiom,
% 0.26/0.47 ! [M: set_a] :
% 0.26/0.47 ( ( ord_less_eq_set_a @ M @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( a_r_coset_a_b @ r @ M @ ( zero_a_b @ r ) )
% 0.26/0.47 = M ) ) ).
% 0.26/0.47
% 0.26/0.47 % a_coset_add_zero
% 0.26/0.47 thf(fact_272_units__of__carrier,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( partia1955795460t_unit @ ( units_1411277569xt_a_b @ G ) )
% 0.26/0.47 = ( units_a_ring_ext_a_b @ G ) ) ).
% 0.26/0.47
% 0.26/0.47 % units_of_carrier
% 0.26/0.47 thf(fact_273_monoid_Ounits__group,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit] :
% 0.26/0.47 ( ( monoid1242292471t_unit @ G )
% 0.26/0.47 => ( group_52730354t_unit @ ( units_390215097t_unit @ G ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.units_group
% 0.26/0.47 thf(fact_274_monoid_Ounits__group,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( monoid1022105667xt_a_b @ G )
% 0.26/0.47 => ( group_a_Product_unit @ ( units_1411277569xt_a_b @ G ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % monoid.units_group
% 0.26/0.47 thf(fact_275_units__of__mult,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( mult_a_Product_unit @ ( units_1411277569xt_a_b @ G ) )
% 0.26/0.47 = ( mult_a_ring_ext_a_b @ G ) ) ).
% 0.26/0.47
% 0.26/0.47 % units_of_mult
% 0.26/0.47 thf(fact_276_units__of__mult,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit] :
% 0.26/0.47 ( ( mult_s977248631t_unit @ ( units_390215097t_unit @ G ) )
% 0.26/0.47 = ( mult_s977248631t_unit @ G ) ) ).
% 0.26/0.47
% 0.26/0.47 % units_of_mult
% 0.26/0.47 thf(fact_277_units__of__one,axiom,
% 0.26/0.47 ! [G: partia1833973666xt_a_b] :
% 0.26/0.47 ( ( one_a_Product_unit @ ( units_1411277569xt_a_b @ G ) )
% 0.26/0.47 = ( one_a_ring_ext_a_b @ G ) ) ).
% 0.26/0.47
% 0.26/0.47 % units_of_one
% 0.26/0.47 thf(fact_278_units__of__one,axiom,
% 0.26/0.47 ! [G: partia539144763t_unit] :
% 0.26/0.47 ( ( one_se1488596747t_unit @ ( units_390215097t_unit @ G ) )
% 0.26/0.47 = ( one_se1488596747t_unit @ G ) ) ).
% 0.26/0.47
% 0.26/0.47 % units_of_one
% 0.26/0.47 thf(fact_279_a__rcosetsI,axiom,
% 0.26/0.47 ! [H: set_a,X: a] :
% 0.26/0.47 ( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.26/0.47 => ( member_set_a @ ( a_r_coset_a_b @ r @ H @ X ) @ ( a_RCOSETS_a_b @ r @ H ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % a_rcosetsI
% 0.26/0.47 thf(fact_280_one__zeroI,axiom,
% 0.26/0.47 ( ( ( partia1066395285xt_a_b @ r )
% 0.26/0.47 = ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
% 0.26/0.47 => ( ( one_a_ring_ext_a_b @ r )
% 0.26/0.47 = ( zero_a_b @ r ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % one_zeroI
% 0.26/0.47 thf(fact_281_one__zeroD,axiom,
% 0.26/0.47 ( ( ( one_a_ring_ext_a_b @ r )
% 0.26/0.47 = ( zero_a_b @ r ) )
% 0.26/0.47 => ( ( partia1066395285xt_a_b @ r )
% 0.26/0.47 = ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % one_zeroD
% 0.26/0.47 thf(fact_282_insert__absorb2,axiom,
% 0.26/0.47 ! [X: a,A3: set_a] :
% 0.26/0.47 ( ( insert_a @ X @ ( insert_a @ X @ A3 ) )
% 0.26/0.47 = ( insert_a @ X @ A3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_absorb2
% 0.26/0.47 thf(fact_283_insert__iff,axiom,
% 0.26/0.47 ! [A: a,B: a,A3: set_a] :
% 0.26/0.47 ( ( member_a @ A @ ( insert_a @ B @ A3 ) )
% 0.26/0.47 = ( ( A = B )
% 0.26/0.47 | ( member_a @ A @ A3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_iff
% 0.26/0.47 thf(fact_284_insert__iff,axiom,
% 0.26/0.47 ! [A: product_prod_a_a,B: product_prod_a_a,A3: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ A3 ) )
% 0.26/0.47 = ( ( A = B )
% 0.26/0.47 | ( member449909584od_a_a @ A @ A3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_iff
% 0.26/0.47 thf(fact_285_insert__iff,axiom,
% 0.26/0.47 ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,A3: set_se1596668135od_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ A @ ( insert1738756823od_a_a @ B @ A3 ) )
% 0.26/0.47 = ( ( A = B )
% 0.26/0.47 | ( member1838126896od_a_a @ A @ A3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_iff
% 0.26/0.47 thf(fact_286_insert__iff,axiom,
% 0.26/0.47 ! [A: set_a,B: set_a,A3: set_set_a] :
% 0.26/0.47 ( ( member_set_a @ A @ ( insert_set_a @ B @ A3 ) )
% 0.26/0.47 = ( ( A = B )
% 0.26/0.47 | ( member_set_a @ A @ A3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_iff
% 0.26/0.47 thf(fact_287_insertCI,axiom,
% 0.26/0.47 ! [A: a,B4: set_a,B: a] :
% 0.26/0.47 ( ( ~ ( member_a @ A @ B4 )
% 0.26/0.47 => ( A = B ) )
% 0.26/0.47 => ( member_a @ A @ ( insert_a @ B @ B4 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insertCI
% 0.26/0.47 thf(fact_288_insertCI,axiom,
% 0.26/0.47 ! [A: product_prod_a_a,B4: set_Product_prod_a_a,B: product_prod_a_a] :
% 0.26/0.47 ( ( ~ ( member449909584od_a_a @ A @ B4 )
% 0.26/0.47 => ( A = B ) )
% 0.26/0.47 => ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ B4 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insertCI
% 0.26/0.47 thf(fact_289_insertCI,axiom,
% 0.26/0.47 ! [A: set_Product_prod_a_a,B4: set_se1596668135od_a_a,B: set_Product_prod_a_a] :
% 0.26/0.47 ( ( ~ ( member1838126896od_a_a @ A @ B4 )
% 0.26/0.47 => ( A = B ) )
% 0.26/0.47 => ( member1838126896od_a_a @ A @ ( insert1738756823od_a_a @ B @ B4 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insertCI
% 0.26/0.47 thf(fact_290_insertCI,axiom,
% 0.26/0.47 ! [A: set_a,B4: set_set_a,B: set_a] :
% 0.26/0.47 ( ( ~ ( member_set_a @ A @ B4 )
% 0.26/0.47 => ( A = B ) )
% 0.26/0.47 => ( member_set_a @ A @ ( insert_set_a @ B @ B4 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insertCI
% 0.26/0.47 thf(fact_291_carrier__one__not__zero,axiom,
% 0.26/0.47 ( ( ( partia1066395285xt_a_b @ r )
% 0.26/0.47 != ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
% 0.26/0.47 = ( ( one_a_ring_ext_a_b @ r )
% 0.26/0.47 != ( zero_a_b @ r ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % carrier_one_not_zero
% 0.26/0.47 thf(fact_292_carrier__one__zero,axiom,
% 0.26/0.47 ( ( ( partia1066395285xt_a_b @ r )
% 0.26/0.47 = ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
% 0.26/0.47 = ( ( one_a_ring_ext_a_b @ r )
% 0.26/0.47 = ( zero_a_b @ r ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % carrier_one_zero
% 0.26/0.47 thf(fact_293_singletonI,axiom,
% 0.26/0.47 ! [A: set_Product_prod_a_a] : ( member1838126896od_a_a @ A @ ( insert1738756823od_a_a @ A @ bot_bo1424703315od_a_a ) ) ).
% 0.26/0.47
% 0.26/0.47 % singletonI
% 0.26/0.47 thf(fact_294_singletonI,axiom,
% 0.26/0.47 ! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% 0.26/0.47
% 0.26/0.47 % singletonI
% 0.26/0.47 thf(fact_295_singletonI,axiom,
% 0.26/0.47 ! [A: product_prod_a_a] : ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a ) ) ).
% 0.26/0.47
% 0.26/0.47 % singletonI
% 0.26/0.47 thf(fact_296_singletonI,axiom,
% 0.26/0.47 ! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% 0.26/0.47
% 0.26/0.47 % singletonI
% 0.26/0.47 thf(fact_297_insert__subset,axiom,
% 0.26/0.47 ! [X: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
% 0.26/0.47 ( ( ord_le1824328871od_a_a @ ( insert1116662519od_a_a @ X @ A3 ) @ B4 )
% 0.26/0.47 = ( ( member449909584od_a_a @ X @ B4 )
% 0.26/0.47 & ( ord_le1824328871od_a_a @ A3 @ B4 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_subset
% 0.26/0.47 thf(fact_298_insert__subset,axiom,
% 0.26/0.47 ! [X: set_Product_prod_a_a,A3: set_se1596668135od_a_a,B4: set_se1596668135od_a_a] :
% 0.26/0.47 ( ( ord_le391835783od_a_a @ ( insert1738756823od_a_a @ X @ A3 ) @ B4 )
% 0.26/0.47 = ( ( member1838126896od_a_a @ X @ B4 )
% 0.26/0.47 & ( ord_le391835783od_a_a @ A3 @ B4 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_subset
% 0.26/0.47 thf(fact_299_insert__subset,axiom,
% 0.26/0.47 ! [X: a,A3: set_a,B4: set_a] :
% 0.26/0.47 ( ( ord_less_eq_set_a @ ( insert_a @ X @ A3 ) @ B4 )
% 0.26/0.47 = ( ( member_a @ X @ B4 )
% 0.26/0.47 & ( ord_less_eq_set_a @ A3 @ B4 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_subset
% 0.26/0.47 thf(fact_300_insert__subset,axiom,
% 0.26/0.47 ! [X: set_a,A3: set_set_a,B4: set_set_a] :
% 0.26/0.47 ( ( ord_le318720350_set_a @ ( insert_set_a @ X @ A3 ) @ B4 )
% 0.26/0.47 = ( ( member_set_a @ X @ B4 )
% 0.26/0.47 & ( ord_le318720350_set_a @ A3 @ B4 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_subset
% 0.26/0.47 thf(fact_301_singleton__insert__inj__eq_H,axiom,
% 0.26/0.47 ! [A: product_prod_a_a,A3: set_Product_prod_a_a,B: product_prod_a_a] :
% 0.26/0.47 ( ( ( insert1116662519od_a_a @ A @ A3 )
% 0.26/0.47 = ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a ) )
% 0.26/0.47 = ( ( A = B )
% 0.26/0.47 & ( ord_le1824328871od_a_a @ A3 @ ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % singleton_insert_inj_eq'
% 0.26/0.47 thf(fact_302_singleton__insert__inj__eq_H,axiom,
% 0.26/0.47 ! [A: a,A3: set_a,B: a] :
% 0.26/0.47 ( ( ( insert_a @ A @ A3 )
% 0.26/0.47 = ( insert_a @ B @ bot_bot_set_a ) )
% 0.26/0.47 = ( ( A = B )
% 0.26/0.47 & ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % singleton_insert_inj_eq'
% 0.26/0.47 thf(fact_303_singleton__insert__inj__eq_H,axiom,
% 0.26/0.47 ! [A: set_a,A3: set_set_a,B: set_a] :
% 0.26/0.47 ( ( ( insert_set_a @ A @ A3 )
% 0.26/0.47 = ( insert_set_a @ B @ bot_bot_set_set_a ) )
% 0.26/0.47 = ( ( A = B )
% 0.26/0.47 & ( ord_le318720350_set_a @ A3 @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % singleton_insert_inj_eq'
% 0.26/0.47 thf(fact_304_singleton__insert__inj__eq,axiom,
% 0.26/0.47 ! [B: product_prod_a_a,A: product_prod_a_a,A3: set_Product_prod_a_a] :
% 0.26/0.47 ( ( ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a )
% 0.26/0.47 = ( insert1116662519od_a_a @ A @ A3 ) )
% 0.26/0.47 = ( ( A = B )
% 0.26/0.47 & ( ord_le1824328871od_a_a @ A3 @ ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % singleton_insert_inj_eq
% 0.26/0.47 thf(fact_305_singleton__insert__inj__eq,axiom,
% 0.26/0.47 ! [B: a,A: a,A3: set_a] :
% 0.26/0.47 ( ( ( insert_a @ B @ bot_bot_set_a )
% 0.26/0.47 = ( insert_a @ A @ A3 ) )
% 0.26/0.47 = ( ( A = B )
% 0.26/0.47 & ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % singleton_insert_inj_eq
% 0.26/0.47 thf(fact_306_singleton__insert__inj__eq,axiom,
% 0.26/0.47 ! [B: set_a,A: set_a,A3: set_set_a] :
% 0.26/0.47 ( ( ( insert_set_a @ B @ bot_bot_set_set_a )
% 0.26/0.47 = ( insert_set_a @ A @ A3 ) )
% 0.26/0.47 = ( ( A = B )
% 0.26/0.47 & ( ord_le318720350_set_a @ A3 @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % singleton_insert_inj_eq
% 0.26/0.47 thf(fact_307_subset__singleton__iff,axiom,
% 0.26/0.47 ! [X5: set_Product_prod_a_a,A: product_prod_a_a] :
% 0.26/0.47 ( ( ord_le1824328871od_a_a @ X5 @ ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a ) )
% 0.26/0.47 = ( ( X5 = bot_bo2131659635od_a_a )
% 0.26/0.47 | ( X5
% 0.26/0.47 = ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % subset_singleton_iff
% 0.26/0.47 thf(fact_308_subset__singleton__iff,axiom,
% 0.26/0.47 ! [X5: set_a,A: a] :
% 0.26/0.47 ( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A @ bot_bot_set_a ) )
% 0.26/0.47 = ( ( X5 = bot_bot_set_a )
% 0.26/0.47 | ( X5
% 0.26/0.47 = ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % subset_singleton_iff
% 0.26/0.47 thf(fact_309_subset__singleton__iff,axiom,
% 0.26/0.47 ! [X5: set_set_a,A: set_a] :
% 0.26/0.47 ( ( ord_le318720350_set_a @ X5 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
% 0.26/0.47 = ( ( X5 = bot_bot_set_set_a )
% 0.26/0.47 | ( X5
% 0.26/0.47 = ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % subset_singleton_iff
% 0.26/0.47 thf(fact_310_subset__singletonD,axiom,
% 0.26/0.47 ! [A3: set_Product_prod_a_a,X: product_prod_a_a] :
% 0.26/0.47 ( ( ord_le1824328871od_a_a @ A3 @ ( insert1116662519od_a_a @ X @ bot_bo2131659635od_a_a ) )
% 0.26/0.47 => ( ( A3 = bot_bo2131659635od_a_a )
% 0.26/0.47 | ( A3
% 0.26/0.47 = ( insert1116662519od_a_a @ X @ bot_bo2131659635od_a_a ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % subset_singletonD
% 0.26/0.47 thf(fact_311_subset__singletonD,axiom,
% 0.26/0.47 ! [A3: set_a,X: a] :
% 0.26/0.47 ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) )
% 0.26/0.47 => ( ( A3 = bot_bot_set_a )
% 0.26/0.47 | ( A3
% 0.26/0.47 = ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % subset_singletonD
% 0.26/0.47 thf(fact_312_subset__singletonD,axiom,
% 0.26/0.47 ! [A3: set_set_a,X: set_a] :
% 0.26/0.47 ( ( ord_le318720350_set_a @ A3 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
% 0.26/0.47 => ( ( A3 = bot_bot_set_set_a )
% 0.26/0.47 | ( A3
% 0.26/0.47 = ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % subset_singletonD
% 0.26/0.47 thf(fact_313_mk__disjoint__insert,axiom,
% 0.26/0.47 ! [A: a,A3: set_a] :
% 0.26/0.47 ( ( member_a @ A @ A3 )
% 0.26/0.47 => ? [B6: set_a] :
% 0.26/0.47 ( ( A3
% 0.26/0.47 = ( insert_a @ A @ B6 ) )
% 0.26/0.47 & ~ ( member_a @ A @ B6 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % mk_disjoint_insert
% 0.26/0.47 thf(fact_314_mk__disjoint__insert,axiom,
% 0.26/0.47 ! [A: product_prod_a_a,A3: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member449909584od_a_a @ A @ A3 )
% 0.26/0.47 => ? [B6: set_Product_prod_a_a] :
% 0.26/0.47 ( ( A3
% 0.26/0.47 = ( insert1116662519od_a_a @ A @ B6 ) )
% 0.26/0.47 & ~ ( member449909584od_a_a @ A @ B6 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % mk_disjoint_insert
% 0.26/0.47 thf(fact_315_mk__disjoint__insert,axiom,
% 0.26/0.47 ! [A: set_Product_prod_a_a,A3: set_se1596668135od_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ A @ A3 )
% 0.26/0.47 => ? [B6: set_se1596668135od_a_a] :
% 0.26/0.47 ( ( A3
% 0.26/0.47 = ( insert1738756823od_a_a @ A @ B6 ) )
% 0.26/0.47 & ~ ( member1838126896od_a_a @ A @ B6 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % mk_disjoint_insert
% 0.26/0.47 thf(fact_316_mk__disjoint__insert,axiom,
% 0.26/0.47 ! [A: set_a,A3: set_set_a] :
% 0.26/0.47 ( ( member_set_a @ A @ A3 )
% 0.26/0.47 => ? [B6: set_set_a] :
% 0.26/0.47 ( ( A3
% 0.26/0.47 = ( insert_set_a @ A @ B6 ) )
% 0.26/0.47 & ~ ( member_set_a @ A @ B6 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % mk_disjoint_insert
% 0.26/0.47 thf(fact_317_insert__commute,axiom,
% 0.26/0.47 ! [X: a,Y: a,A3: set_a] :
% 0.26/0.47 ( ( insert_a @ X @ ( insert_a @ Y @ A3 ) )
% 0.26/0.47 = ( insert_a @ Y @ ( insert_a @ X @ A3 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_commute
% 0.26/0.47 thf(fact_318_insert__eq__iff,axiom,
% 0.26/0.47 ! [A: a,A3: set_a,B: a,B4: set_a] :
% 0.26/0.47 ( ~ ( member_a @ A @ A3 )
% 0.26/0.47 => ( ~ ( member_a @ B @ B4 )
% 0.26/0.47 => ( ( ( insert_a @ A @ A3 )
% 0.26/0.47 = ( insert_a @ B @ B4 ) )
% 0.26/0.47 = ( ( ( A = B )
% 0.26/0.47 => ( A3 = B4 ) )
% 0.26/0.47 & ( ( A != B )
% 0.26/0.47 => ? [C3: set_a] :
% 0.26/0.47 ( ( A3
% 0.26/0.47 = ( insert_a @ B @ C3 ) )
% 0.26/0.47 & ~ ( member_a @ B @ C3 )
% 0.26/0.47 & ( B4
% 0.26/0.47 = ( insert_a @ A @ C3 ) )
% 0.26/0.47 & ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_eq_iff
% 0.26/0.47 thf(fact_319_insert__eq__iff,axiom,
% 0.26/0.47 ! [A: product_prod_a_a,A3: set_Product_prod_a_a,B: product_prod_a_a,B4: set_Product_prod_a_a] :
% 0.26/0.47 ( ~ ( member449909584od_a_a @ A @ A3 )
% 0.26/0.47 => ( ~ ( member449909584od_a_a @ B @ B4 )
% 0.26/0.47 => ( ( ( insert1116662519od_a_a @ A @ A3 )
% 0.26/0.47 = ( insert1116662519od_a_a @ B @ B4 ) )
% 0.26/0.47 = ( ( ( A = B )
% 0.26/0.47 => ( A3 = B4 ) )
% 0.26/0.47 & ( ( A != B )
% 0.26/0.47 => ? [C3: set_Product_prod_a_a] :
% 0.26/0.47 ( ( A3
% 0.26/0.47 = ( insert1116662519od_a_a @ B @ C3 ) )
% 0.26/0.47 & ~ ( member449909584od_a_a @ B @ C3 )
% 0.26/0.47 & ( B4
% 0.26/0.47 = ( insert1116662519od_a_a @ A @ C3 ) )
% 0.26/0.47 & ~ ( member449909584od_a_a @ A @ C3 ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_eq_iff
% 0.26/0.47 thf(fact_320_insert__eq__iff,axiom,
% 0.26/0.47 ! [A: set_Product_prod_a_a,A3: set_se1596668135od_a_a,B: set_Product_prod_a_a,B4: set_se1596668135od_a_a] :
% 0.26/0.47 ( ~ ( member1838126896od_a_a @ A @ A3 )
% 0.26/0.47 => ( ~ ( member1838126896od_a_a @ B @ B4 )
% 0.26/0.47 => ( ( ( insert1738756823od_a_a @ A @ A3 )
% 0.26/0.47 = ( insert1738756823od_a_a @ B @ B4 ) )
% 0.26/0.47 = ( ( ( A = B )
% 0.26/0.47 => ( A3 = B4 ) )
% 0.26/0.47 & ( ( A != B )
% 0.26/0.47 => ? [C3: set_se1596668135od_a_a] :
% 0.26/0.47 ( ( A3
% 0.26/0.47 = ( insert1738756823od_a_a @ B @ C3 ) )
% 0.26/0.47 & ~ ( member1838126896od_a_a @ B @ C3 )
% 0.26/0.47 & ( B4
% 0.26/0.47 = ( insert1738756823od_a_a @ A @ C3 ) )
% 0.26/0.47 & ~ ( member1838126896od_a_a @ A @ C3 ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_eq_iff
% 0.26/0.47 thf(fact_321_insert__eq__iff,axiom,
% 0.26/0.47 ! [A: set_a,A3: set_set_a,B: set_a,B4: set_set_a] :
% 0.26/0.47 ( ~ ( member_set_a @ A @ A3 )
% 0.26/0.47 => ( ~ ( member_set_a @ B @ B4 )
% 0.26/0.47 => ( ( ( insert_set_a @ A @ A3 )
% 0.26/0.47 = ( insert_set_a @ B @ B4 ) )
% 0.26/0.47 = ( ( ( A = B )
% 0.26/0.47 => ( A3 = B4 ) )
% 0.26/0.47 & ( ( A != B )
% 0.26/0.47 => ? [C3: set_set_a] :
% 0.26/0.47 ( ( A3
% 0.26/0.47 = ( insert_set_a @ B @ C3 ) )
% 0.26/0.47 & ~ ( member_set_a @ B @ C3 )
% 0.26/0.47 & ( B4
% 0.26/0.47 = ( insert_set_a @ A @ C3 ) )
% 0.26/0.47 & ~ ( member_set_a @ A @ C3 ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_eq_iff
% 0.26/0.47 thf(fact_322_insert__absorb,axiom,
% 0.26/0.47 ! [A: a,A3: set_a] :
% 0.26/0.47 ( ( member_a @ A @ A3 )
% 0.26/0.47 => ( ( insert_a @ A @ A3 )
% 0.26/0.47 = A3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_absorb
% 0.26/0.47 thf(fact_323_insert__absorb,axiom,
% 0.26/0.47 ! [A: product_prod_a_a,A3: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member449909584od_a_a @ A @ A3 )
% 0.26/0.47 => ( ( insert1116662519od_a_a @ A @ A3 )
% 0.26/0.47 = A3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_absorb
% 0.26/0.47 thf(fact_324_insert__absorb,axiom,
% 0.26/0.47 ! [A: set_Product_prod_a_a,A3: set_se1596668135od_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ A @ A3 )
% 0.26/0.47 => ( ( insert1738756823od_a_a @ A @ A3 )
% 0.26/0.47 = A3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_absorb
% 0.26/0.47 thf(fact_325_insert__absorb,axiom,
% 0.26/0.47 ! [A: set_a,A3: set_set_a] :
% 0.26/0.47 ( ( member_set_a @ A @ A3 )
% 0.26/0.47 => ( ( insert_set_a @ A @ A3 )
% 0.26/0.47 = A3 ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_absorb
% 0.26/0.47 thf(fact_326_insert__ident,axiom,
% 0.26/0.47 ! [X: a,A3: set_a,B4: set_a] :
% 0.26/0.47 ( ~ ( member_a @ X @ A3 )
% 0.26/0.47 => ( ~ ( member_a @ X @ B4 )
% 0.26/0.47 => ( ( ( insert_a @ X @ A3 )
% 0.26/0.47 = ( insert_a @ X @ B4 ) )
% 0.26/0.47 = ( A3 = B4 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_ident
% 0.26/0.47 thf(fact_327_insert__ident,axiom,
% 0.26/0.47 ! [X: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
% 0.26/0.47 ( ~ ( member449909584od_a_a @ X @ A3 )
% 0.26/0.47 => ( ~ ( member449909584od_a_a @ X @ B4 )
% 0.26/0.47 => ( ( ( insert1116662519od_a_a @ X @ A3 )
% 0.26/0.47 = ( insert1116662519od_a_a @ X @ B4 ) )
% 0.26/0.47 = ( A3 = B4 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_ident
% 0.26/0.47 thf(fact_328_insert__ident,axiom,
% 0.26/0.47 ! [X: set_Product_prod_a_a,A3: set_se1596668135od_a_a,B4: set_se1596668135od_a_a] :
% 0.26/0.47 ( ~ ( member1838126896od_a_a @ X @ A3 )
% 0.26/0.47 => ( ~ ( member1838126896od_a_a @ X @ B4 )
% 0.26/0.47 => ( ( ( insert1738756823od_a_a @ X @ A3 )
% 0.26/0.47 = ( insert1738756823od_a_a @ X @ B4 ) )
% 0.26/0.47 = ( A3 = B4 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_ident
% 0.26/0.47 thf(fact_329_insert__ident,axiom,
% 0.26/0.47 ! [X: set_a,A3: set_set_a,B4: set_set_a] :
% 0.26/0.47 ( ~ ( member_set_a @ X @ A3 )
% 0.26/0.47 => ( ~ ( member_set_a @ X @ B4 )
% 0.26/0.47 => ( ( ( insert_set_a @ X @ A3 )
% 0.26/0.47 = ( insert_set_a @ X @ B4 ) )
% 0.26/0.47 = ( A3 = B4 ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % insert_ident
% 0.26/0.47 thf(fact_330_Set_Oset__insert,axiom,
% 0.26/0.47 ! [X: a,A3: set_a] :
% 0.26/0.47 ( ( member_a @ X @ A3 )
% 0.26/0.47 => ~ ! [B6: set_a] :
% 0.26/0.47 ( ( A3
% 0.26/0.47 = ( insert_a @ X @ B6 ) )
% 0.26/0.47 => ( member_a @ X @ B6 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Set.set_insert
% 0.26/0.47 thf(fact_331_Set_Oset__insert,axiom,
% 0.26/0.47 ! [X: product_prod_a_a,A3: set_Product_prod_a_a] :
% 0.26/0.47 ( ( member449909584od_a_a @ X @ A3 )
% 0.26/0.47 => ~ ! [B6: set_Product_prod_a_a] :
% 0.26/0.47 ( ( A3
% 0.26/0.47 = ( insert1116662519od_a_a @ X @ B6 ) )
% 0.26/0.47 => ( member449909584od_a_a @ X @ B6 ) ) ) ).
% 0.26/0.47
% 0.26/0.47 % Set.set_insert
% 0.26/0.47 thf(fact_332_Set_Oset__insert,axiom,
% 0.26/0.47 ! [X: set_Product_prod_a_a,A3: set_se1596668135od_a_a] :
% 0.26/0.47 ( ( member1838126896od_a_a @ X @ A3 )
% 0.26/0.48 => ~ ! [B6: set_se1596668135od_a_a] :
% 0.26/0.48 ( ( A3
% 0.26/0.48 = ( insert1738756823od_a_a @ X @ B6 ) )
% 0.26/0.48 => ( member1838126896od_a_a @ X @ B6 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Set.set_insert
% 0.26/0.48 thf(fact_333_Set_Oset__insert,axiom,
% 0.26/0.48 ! [X: set_a,A3: set_set_a] :
% 0.26/0.48 ( ( member_set_a @ X @ A3 )
% 0.26/0.48 => ~ ! [B6: set_set_a] :
% 0.26/0.48 ( ( A3
% 0.26/0.48 = ( insert_set_a @ X @ B6 ) )
% 0.26/0.48 => ( member_set_a @ X @ B6 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % Set.set_insert
% 0.26/0.48 thf(fact_334_insertI2,axiom,
% 0.26/0.48 ! [A: a,B4: set_a,B: a] :
% 0.26/0.48 ( ( member_a @ A @ B4 )
% 0.26/0.48 => ( member_a @ A @ ( insert_a @ B @ B4 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % insertI2
% 0.26/0.48 thf(fact_335_insertI2,axiom,
% 0.26/0.48 ! [A: product_prod_a_a,B4: set_Product_prod_a_a,B: product_prod_a_a] :
% 0.26/0.48 ( ( member449909584od_a_a @ A @ B4 )
% 0.26/0.48 => ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ B4 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % insertI2
% 0.26/0.48 thf(fact_336_insertI2,axiom,
% 0.26/0.48 ! [A: set_Product_prod_a_a,B4: set_se1596668135od_a_a,B: set_Product_prod_a_a] :
% 0.26/0.48 ( ( member1838126896od_a_a @ A @ B4 )
% 0.26/0.48 => ( member1838126896od_a_a @ A @ ( insert1738756823od_a_a @ B @ B4 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % insertI2
% 0.26/0.48 thf(fact_337_insertI2,axiom,
% 0.26/0.48 ! [A: set_a,B4: set_set_a,B: set_a] :
% 0.26/0.48 ( ( member_set_a @ A @ B4 )
% 0.26/0.48 => ( member_set_a @ A @ ( insert_set_a @ B @ B4 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % insertI2
% 0.26/0.48 thf(fact_338_insertI1,axiom,
% 0.26/0.48 ! [A: a,B4: set_a] : ( member_a @ A @ ( insert_a @ A @ B4 ) ) ).
% 0.26/0.48
% 0.26/0.48 % insertI1
% 0.26/0.48 thf(fact_339_insertI1,axiom,
% 0.26/0.48 ! [A: product_prod_a_a,B4: set_Product_prod_a_a] : ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ A @ B4 ) ) ).
% 0.26/0.48
% 0.26/0.48 % insertI1
% 0.26/0.48 thf(fact_340_insertI1,axiom,
% 0.26/0.48 ! [A: set_Product_prod_a_a,B4: set_se1596668135od_a_a] : ( member1838126896od_a_a @ A @ ( insert1738756823od_a_a @ A @ B4 ) ) ).
% 0.26/0.48
% 0.26/0.48 % insertI1
% 0.26/0.48 thf(fact_341_insertI1,axiom,
% 0.26/0.48 ! [A: set_a,B4: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B4 ) ) ).
% 0.26/0.48
% 0.26/0.48 % insertI1
% 0.26/0.48 thf(fact_342_insertE,axiom,
% 0.26/0.48 ! [A: a,B: a,A3: set_a] :
% 0.26/0.48 ( ( member_a @ A @ ( insert_a @ B @ A3 ) )
% 0.26/0.48 => ( ( A != B )
% 0.26/0.48 => ( member_a @ A @ A3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % insertE
% 0.26/0.48 thf(fact_343_insertE,axiom,
% 0.26/0.48 ! [A: product_prod_a_a,B: product_prod_a_a,A3: set_Product_prod_a_a] :
% 0.26/0.48 ( ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ A3 ) )
% 0.26/0.48 => ( ( A != B )
% 0.26/0.48 => ( member449909584od_a_a @ A @ A3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % insertE
% 0.26/0.48 thf(fact_344_insertE,axiom,
% 0.26/0.48 ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,A3: set_se1596668135od_a_a] :
% 0.26/0.48 ( ( member1838126896od_a_a @ A @ ( insert1738756823od_a_a @ B @ A3 ) )
% 0.26/0.48 => ( ( A != B )
% 0.26/0.48 => ( member1838126896od_a_a @ A @ A3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % insertE
% 0.26/0.48 thf(fact_345_insertE,axiom,
% 0.26/0.48 ! [A: set_a,B: set_a,A3: set_set_a] :
% 0.26/0.48 ( ( member_set_a @ A @ ( insert_set_a @ B @ A3 ) )
% 0.26/0.48 => ( ( A != B )
% 0.26/0.48 => ( member_set_a @ A @ A3 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % insertE
% 0.26/0.48 thf(fact_346_singleton__inject,axiom,
% 0.26/0.48 ! [A: product_prod_a_a,B: product_prod_a_a] :
% 0.26/0.48 ( ( ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a )
% 0.26/0.48 = ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a ) )
% 0.26/0.48 => ( A = B ) ) ).
% 0.26/0.48
% 0.26/0.48 % singleton_inject
% 0.26/0.48 thf(fact_347_singleton__inject,axiom,
% 0.26/0.48 ! [A: a,B: a] :
% 0.26/0.48 ( ( ( insert_a @ A @ bot_bot_set_a )
% 0.26/0.48 = ( insert_a @ B @ bot_bot_set_a ) )
% 0.26/0.48 => ( A = B ) ) ).
% 0.26/0.48
% 0.26/0.48 % singleton_inject
% 0.26/0.48 thf(fact_348_insert__not__empty,axiom,
% 0.26/0.48 ! [A: product_prod_a_a,A3: set_Product_prod_a_a] :
% 0.26/0.48 ( ( insert1116662519od_a_a @ A @ A3 )
% 0.26/0.48 != bot_bo2131659635od_a_a ) ).
% 0.26/0.48
% 0.26/0.48 % insert_not_empty
% 0.26/0.48 thf(fact_349_insert__not__empty,axiom,
% 0.26/0.48 ! [A: a,A3: set_a] :
% 0.26/0.48 ( ( insert_a @ A @ A3 )
% 0.26/0.48 != bot_bot_set_a ) ).
% 0.26/0.48
% 0.26/0.48 % insert_not_empty
% 0.26/0.48 thf(fact_350_doubleton__eq__iff,axiom,
% 0.26/0.48 ! [A: product_prod_a_a,B: product_prod_a_a,C: product_prod_a_a,D: product_prod_a_a] :
% 0.26/0.48 ( ( ( insert1116662519od_a_a @ A @ ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a ) )
% 0.26/0.48 = ( insert1116662519od_a_a @ C @ ( insert1116662519od_a_a @ D @ bot_bo2131659635od_a_a ) ) )
% 0.26/0.48 = ( ( ( A = C )
% 0.26/0.48 & ( B = D ) )
% 0.26/0.48 | ( ( A = D )
% 0.26/0.48 & ( B = C ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % doubleton_eq_iff
% 0.26/0.48 thf(fact_351_doubleton__eq__iff,axiom,
% 0.26/0.48 ! [A: a,B: a,C: a,D: a] :
% 0.26/0.48 ( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
% 0.26/0.48 = ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
% 0.26/0.48 = ( ( ( A = C )
% 0.26/0.48 & ( B = D ) )
% 0.26/0.48 | ( ( A = D )
% 0.26/0.48 & ( B = C ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % doubleton_eq_iff
% 0.26/0.48 thf(fact_352_field__intro2,axiom,
% 0.26/0.48 ( ( ( zero_a_b @ r )
% 0.26/0.48 != ( one_a_ring_ext_a_b @ r ) )
% 0.26/0.48 => ( ! [X2: a] :
% 0.26/0.48 ( ( member_a @ X2 @ ( minus_minus_set_a @ ( partia1066395285xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
% 0.26/0.48 => ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) ) )
% 0.26/0.48 => ( field_a_b @ r ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % field_intro2
% 0.26/0.48 thf(fact_353_cring__fieldI,axiom,
% 0.26/0.48 ( ( ( units_a_ring_ext_a_b @ r )
% 0.26/0.48 = ( minus_minus_set_a @ ( partia1066395285xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
% 0.26/0.48 => ( field_a_b @ r ) ) ).
% 0.26/0.48
% 0.26/0.48 % cring_fieldI
% 0.26/0.48 thf(fact_354_rcosets__subset__PowG,axiom,
% 0.26/0.48 ! [H: set_a] :
% 0.26/0.48 ( ( additi2104487374up_a_b @ H @ r )
% 0.26/0.48 => ( ord_le318720350_set_a @ ( a_RCOSETS_a_b @ r @ H ) @ ( pow_a @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 % rcosets_subset_PowG
% 0.26/0.48
% 0.26/0.48 % Conjectures (1)
% 0.26/0.48 thf(conj_0,conjecture,
% 0.26/0.48 ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) )
% 0.26/0.48 = ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) ) ) ).
% 0.26/0.48
% 0.26/0.48 %------------------------------------------------------------------------------
% 0.26/0.48 ------------------------
% 3.18/1.66 ---- Embedded in THF ---
% 3.18/1.66 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.18/1.66 %%% Generated on Tue Feb 24 08:26:10 EST 2026
% 3.18/1.66 %%% using '$$dhol' embedding, version 1.3.0.
% 3.18/1.66 %%% Logic specification used:
% 3.18/1.66 %%% thf(spec, logic, $$dhol).
% 3.18/1.66
% 3.18/1.66 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.7ORhJzyl8w/DTF2DTF6318.p
% See solution above
% 3.18/1.66 ------------------------
% 3.18/1.69 ---- Cleaned THF ---
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% 3.18/1.69 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
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% 3.18/1.69 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
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% 3.18/1.69
% 3.18/1.69 thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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% 3.18/1.69
% 3.18/1.69 thf(ty_n_t__Set__Oset_Itf__a_J,type,
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% 3.18/1.69
% 3.18/1.69 thf(ty_n_t__Nat__Onat,type,
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% 3.18/1.69
% 3.18/1.69 thf(ty_n_t__Int__Oint,type,
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% 3.18/1.69
% 3.18/1.69 thf(ty_n_tf__a,type,
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% 3.18/1.69 thf(sy_c_AbelCoset_OA__RCOSETS_001tf__a_001tf__b,type,
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% 3.18/1.69 thf(sy_c_AbelCoset_Oa__l__coset_001tf__a_001tf__b,type,
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% 3.18/1.69
% 3.18/1.69 thf(sy_c_AbelCoset_Oa__r__coset_001tf__a_001tf__b,type,
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% 3.18/1.69
% 3.18/1.69 thf(sy_c_AbelCoset_Oadditive__subgroup_001tf__a_001tf__b,type,
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% 3.18/1.69
% 3.18/1.69 thf(sy_c_AbelCoset_Oset__add_001tf__a_001tf__b,type,
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% 3.18/1.69 thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Congruence__Oeq____object__Oeq____object____ext_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Ounit_J,type,
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% 3.18/1.69
% 3.18/1.69 thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Ounit_J,type,
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% 3.18/1.69
% 3.18/1.69 thf(sy_c_Congruence_Opartial__object_Ocarrier_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Product____Type__Ounit_J,type,
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% 3.18/1.69
% 3.18/1.69 thf(sy_c_Congruence_Opartial__object_Ocarrier_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J,type,
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% 3.18/1.69
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% 3.18/1.69
% 3.18/1.69 thf(sy_c_Group_OUnits_001tf__a_001t__Product____Type__Ounit,type,
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% 3.18/1.69
% 3.18/1.69 thf(sy_c_Group_OUnits_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
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% 3.18/1.69
% 3.18/1.69 thf(sy_c_Group_Ogroup_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Product____Type__Ounit,type,
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% 3.18/1.69
% 3.18/1.69 thf(sy_c_Group_Ogroup_001tf__a_001t__Product____Type__Ounit,type,
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% 3.18/1.69 thf(sy_c_Group_Ogroup_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
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% 3.18/1.69 thf(sy_c_Group_Omonoid_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Product____Type__Ounit,type,
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% 3.18/1.69 thf(sy_c_Group_Omonoid_001tf__a_001t__Product____Type__Ounit,type,
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% 3.18/1.69
% 3.18/1.69 thf(sy_c_Group_Omonoid_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
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% 3.18/1.69 thf(sy_c_Group_Omonoid_Omult_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_001t__Product____Type__Ounit,type,
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% 3.18/1.69 thf(sy_c_Group_Omonoid_Omult_001tf__a_001t__Product____Type__Ounit,type,
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% 3.18/1.69 ord_less_eq_set_a: set_a > set_a > $o ).
% 3.18/1.69
% 3.18/1.69 thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
% 3.18/1.69 product_Pair_a_a: a > a > product_prod_a_a ).
% 3.18/1.69
% 3.18/1.69 thf(sy_c_Ring_Oabelian__monoid_001tf__a_001tf__b,type,
% 3.18/1.69 abelian_monoid_a_b: partia1833973666xt_a_b > $o ).
% 3.18/1.69
% 3.18/1.69 thf(sy_c_Ring_Oadd__pow_001tf__a_001tf__b_001t__Int__Oint,type,
% 3.18/1.69 add_pow_a_b_int: partia1833973666xt_a_b > int > a > a ).
% 3.18/1.69
% 3.18/1.69 thf(sy_c_Ring_Oadd__pow_001tf__a_001tf__b_001t__Nat__Onat,type,
% 3.18/1.69 add_pow_a_b_nat: partia1833973666xt_a_b > nat > a > a ).
% 3.18/1.69
% 3.18/1.69 thf(sy_c_Ring_Ofield_001tf__a_001tf__b,type,
% 3.18/1.69 field_a_b: partia1833973666xt_a_b > $o ).
% 3.18/1.69
% 3.18/1.69 thf(sy_c_Ring_Oring_001tf__a_001tf__b,type,
% 3.18/1.69 ring_a_b: partia1833973666xt_a_b > $o ).
% 3.18/1.69
% 3.18/1.69 thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
% 3.18/1.69 zero_a_b: partia1833973666xt_a_b > a ).
% 3.18/1.69
% 3.18/1.69 thf(sy_c_Ring_Osemiring_001tf__a_001tf__b,type,
% 3.18/1.69 semiring_a_b: partia1833973666xt_a_b > $o ).
% 3.18/1.69
% 3.18/1.69 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.18/1.69 collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
% 3.18/1.69
% 3.18/1.69 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.18/1.69 collec183727474od_a_a: ( set_Product_prod_a_a > $o ) > set_se1596668135od_a_a ).
% 3.18/1.69
% 3.18/1.69 thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
% 3.18/1.69 collect_set_a: ( set_a > $o ) > set_set_a ).
% 3.18/1.69
% 3.18/1.69 thf(sy_c_Set_OCollect_001tf__a,type,
% 3.18/1.69 collect_a: ( a > $o ) > set_a ).
% 3.18/1.69
% 3.18/1.69 thf(sy_c_Set_OPow_001tf__a,type,
% 3.18/1.69 pow_a: set_a > set_set_a ).
% 3.18/1.69
% 3.18/1.69 thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.18/1.69 insert1116662519od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
% 3.18/1.69
% 3.18/1.69 thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.18/1.69 insert1738756823od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a > set_se1596668135od_a_a ).
% 3.18/1.69
% 3.18/1.69 thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
% 3.18/1.69 insert_set_a: set_a > set_set_a > set_set_a ).
% 3.18/1.69
% 3.18/1.69 thf(sy_c_Set_Oinsert_001tf__a,type,
% 3.18/1.69 insert_a: a > set_a > set_a ).
% 3.18/1.69
% 3.18/1.69 thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 3.18/1.69 member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
% 3.18/1.69
% 3.18/1.69 thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
% 3.18/1.69 member1838126896od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a > $o ).
% 3.18/1.69
% 3.18/1.69 thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
% 3.18/1.69 member_set_a: set_a > set_set_a > $o ).
% 3.18/1.69
% 3.18/1.69 thf(sy_c_member_001tf__a,type,
% 3.18/1.69 member_a: a > set_a > $o ).
% 3.18/1.69
% 3.18/1.69 thf(sy_v_R,type,
% 3.18/1.69 r: partia1833973666xt_a_b ).
% 3.18/1.69
% 3.18/1.69 thf(sy_v_S,type,
% 3.18/1.69 s: set_a ).
% 3.18/1.69
% 3.18/1.69 thf(sy_v_r,type,
% 3.18/1.69 r2: a ).
% 3.18/1.69
% 3.18/1.69 thf(sy_v_r_H,type,
% 3.18/1.69 r3: a ).
% 3.18/1.69
% 3.18/1.69 thf(sy_v_rel,type,
% 3.18/1.69 rel: partia1993116613t_unit ).
% 3.18/1.69
% 3.18/1.69 thf(sy_v_s,type,
% 3.18/1.69 s2: a ).
% 3.18/1.69
% 3.18/1.69 thf(sy_v_s_H,type,
% 3.18/1.69 s3: a ).
% 3.18/1.69
% 3.18/1.69 thf(fact_0_f3,axiom,
% 3.18/1.69 ( ( mult_a_ring_ext_a_b @ r @ r2 @ r3 )
% 3.18/1.69 = ( mult_a_ring_ext_a_b @ r @ r3 @ r2 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_1_f4,axiom,
% 3.18/1.69 ( ( mult_a_ring_ext_a_b @ r @ s2 @ s3 )
% 3.18/1.69 = ( mult_a_ring_ext_a_b @ r @ s3 @ s2 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_2_eq__obj__rng__of__frac__axioms,axiom,
% 3.18/1.69 locali1648887798ac_a_b @ r @ s ).
% 3.18/1.69
% 3.18/1.69 thf(fact_3_f1,axiom,
% 3.18/1.69 ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) )
% 3.18/1.69 = ( locali1170246543t_unit @ rel @ ( mult_a_ring_ext_a_b @ r @ r2 @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_4_f2,axiom,
% 3.18/1.69 ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) )
% 3.18/1.69 = ( locali1170246543t_unit @ rel @ ( mult_a_ring_ext_a_b @ r @ r3 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ s2 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_5_mult__submonoid__of__crng__axioms,axiom,
% 3.18/1.69 locali807230110ng_a_b @ r @ s ).
% 3.18/1.69
% 3.18/1.69 thf(fact_6_mult__submonoid__of__rng__axioms,axiom,
% 3.18/1.69 locali880295127ng_a_b @ r @ s ).
% 3.18/1.69
% 3.18/1.69 thf(fact_7_eq__obj__rng__of__frac_Orec__monoid__rng__of__frac_Ocong,axiom,
% 3.18/1.69 locali1944243560ac_a_b = locali1944243560ac_a_b ).
% 3.18/1.69
% 3.18/1.69 thf(fact_8_submonoid__axioms,axiom,
% 3.18/1.69 locali425460998xt_a_b @ r @ s ).
% 3.18/1.69
% 3.18/1.69 thf(fact_9_m__closed,axiom,
% 3.18/1.69 ! [X: a,Y: a] :
% 3.18/1.69 ( ( member_a @ X @ s )
% 3.18/1.69 => ( ( member_a @ Y @ s )
% 3.18/1.69 => ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ s ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_10_monoid__rng__of__frac,axiom,
% 3.18/1.69 monoid1242292471t_unit @ ( locali1944243560ac_a_b @ r @ s ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_11_local_Osemiring__axioms,axiom,
% 3.18/1.69 semiring_a_b @ r ).
% 3.18/1.69
% 3.18/1.69 thf(fact_12_one__closed,axiom,
% 3.18/1.69 member_a @ ( one_a_ring_ext_a_b @ r ) @ s ).
% 3.18/1.69
% 3.18/1.69 thf(fact_13_equiv__obj__rng__of__frac,axiom,
% 3.18/1.69 equiva2136214035t_unit @ rel ).
% 3.18/1.69
% 3.18/1.69 thf(fact_14_assms_I2_J,axiom,
% 3.18/1.69 member449909584od_a_a @ ( product_Pair_a_a @ r3 @ s3 ) @ ( partia206007992t_unit @ rel ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_15_assms_I1_J,axiom,
% 3.18/1.69 member449909584od_a_a @ ( product_Pair_a_a @ r2 @ s2 ) @ ( partia206007992t_unit @ rel ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_16_assoc__mult__rng__of__frac,axiom,
% 3.18/1.69 ! [R: a,S: a,T: a,U: a,V: a,W: a] :
% 3.18/1.69 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
% 3.18/1.69 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ T @ U ) @ ( partia206007992t_unit @ rel ) )
% 3.18/1.69 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ V @ W ) @ ( partia206007992t_unit @ rel ) )
% 3.18/1.69 => ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ R @ S ) @ ( locali1170246543t_unit @ rel @ T @ U ) ) @ ( locali1170246543t_unit @ rel @ V @ W ) )
% 3.18/1.69 = ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ R @ S ) @ ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ T @ U ) @ ( locali1170246543t_unit @ rel @ V @ W ) ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_17_abelian__monoid__axioms,axiom,
% 3.18/1.69 abelian_monoid_a_b @ r ).
% 3.18/1.69
% 3.18/1.69 thf(fact_18_mult__rng__of__frac__fundamental__lemma,axiom,
% 3.18/1.69 ! [R: a,S: a,R2: a,S2: a] :
% 3.18/1.69 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
% 3.18/1.69 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ R2 @ S2 ) @ ( partia206007992t_unit @ rel ) )
% 3.18/1.69 => ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ R @ S ) @ ( locali1170246543t_unit @ rel @ R2 @ S2 ) )
% 3.18/1.69 = ( locali1170246543t_unit @ rel @ ( mult_a_ring_ext_a_b @ r @ R @ R2 ) @ ( mult_a_ring_ext_a_b @ r @ S @ S2 ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_19_local_Oring__axioms,axiom,
% 3.18/1.69 ring_a_b @ r ).
% 3.18/1.69
% 3.18/1.69 thf(fact_20_closed__mult__rng__of__frac,axiom,
% 3.18/1.69 ! [R: a,S: a,T: a,U: a] :
% 3.18/1.69 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
% 3.18/1.69 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ T @ U ) @ ( partia206007992t_unit @ rel ) )
% 3.18/1.69 => ( member1838126896od_a_a @ ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ R @ S ) @ ( locali1170246543t_unit @ rel @ T @ U ) ) @ ( locali1538946157t_unit @ rel ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_21_submonoid_Oaxioms_I1_J,axiom,
% 3.18/1.69 ! [M: partia539144763t_unit,S3: set_se1596668135od_a_a] :
% 3.18/1.69 ( ( locali1488796916t_unit @ M @ S3 )
% 3.18/1.69 => ( monoid1242292471t_unit @ M ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_22_submonoid_Oaxioms_I1_J,axiom,
% 3.18/1.69 ! [M: partia1833973666xt_a_b,S3: set_a] :
% 3.18/1.69 ( ( locali425460998xt_a_b @ M @ S3 )
% 3.18/1.69 => ( monoid1022105667xt_a_b @ M ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_23_eq__obj__rng__of__frac_Oaxioms_I2_J,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b,S3: set_a] :
% 3.18/1.69 ( ( locali1648887798ac_a_b @ R3 @ S3 )
% 3.18/1.69 => ( locali807230110ng_a_b @ R3 @ S3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_24_mult__submonoid__of__rng_Oaxioms_I2_J,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b,S3: set_a] :
% 3.18/1.69 ( ( locali880295127ng_a_b @ R3 @ S3 )
% 3.18/1.69 => ( locali425460998xt_a_b @ R3 @ S3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_25_mult__submonoid__of__rng_Oaxioms_I1_J,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b,S3: set_a] :
% 3.18/1.69 ( ( locali880295127ng_a_b @ R3 @ S3 )
% 3.18/1.69 => ( ring_a_b @ R3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_26_mult__submonoid__of__crng_Oaxioms_I2_J,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b,S3: set_a] :
% 3.18/1.69 ( ( locali807230110ng_a_b @ R3 @ S3 )
% 3.18/1.69 => ( locali880295127ng_a_b @ R3 @ S3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_27_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Oone__closed,axiom,
% 3.18/1.69 ! [M: partia539144763t_unit,S3: set_se1596668135od_a_a] :
% 3.18/1.69 ( ( locali1488796916t_unit @ M @ S3 )
% 3.18/1.69 => ( member1838126896od_a_a @ ( one_se1488596747t_unit @ M ) @ S3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_28_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Oone__closed,axiom,
% 3.18/1.69 ! [M: partia1833973666xt_a_b,S3: set_a] :
% 3.18/1.69 ( ( locali425460998xt_a_b @ M @ S3 )
% 3.18/1.69 => ( member_a @ ( one_a_ring_ext_a_b @ M ) @ S3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_29_mult__submonoid__of__rng__def,axiom,
% 3.18/1.69 ( locali880295127ng_a_b
% 3.18/1.69 = ( ^ [R4: partia1833973666xt_a_b,S4: set_a] :
% 3.18/1.69 ( ( ring_a_b @ R4 )
% 3.18/1.69 & ( locali425460998xt_a_b @ R4 @ S4 ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_30_mult__submonoid__of__rng_Ointro,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b,S3: set_a] :
% 3.18/1.69 ( ( ring_a_b @ R3 )
% 3.18/1.69 => ( ( locali425460998xt_a_b @ R3 @ S3 )
% 3.18/1.69 => ( locali880295127ng_a_b @ R3 @ S3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_31_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Ois__submonoid,axiom,
% 3.18/1.69 ! [M: partia1833973666xt_a_b,S3: set_a] :
% 3.18/1.69 ( ( locali425460998xt_a_b @ M @ S3 )
% 3.18/1.69 => ( locali425460998xt_a_b @ M @ S3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_32_eq__obj__rng__of__frac_Omonoid__rng__of__frac,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b,S3: set_a] :
% 3.18/1.69 ( ( locali1648887798ac_a_b @ R3 @ S3 )
% 3.18/1.69 => ( monoid1242292471t_unit @ ( locali1944243560ac_a_b @ R3 @ S3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_33_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Om__closed,axiom,
% 3.18/1.69 ! [M: partia539144763t_unit,S3: set_se1596668135od_a_a,X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
% 3.18/1.69 ( ( locali1488796916t_unit @ M @ S3 )
% 3.18/1.69 => ( ( member1838126896od_a_a @ X @ S3 )
% 3.18/1.69 => ( ( member1838126896od_a_a @ Y @ S3 )
% 3.18/1.69 => ( member1838126896od_a_a @ ( mult_s977248631t_unit @ M @ X @ Y ) @ S3 ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_34_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Om__closed,axiom,
% 3.18/1.69 ! [M: partia1833973666xt_a_b,S3: set_a,X: a,Y: a] :
% 3.18/1.69 ( ( locali425460998xt_a_b @ M @ S3 )
% 3.18/1.69 => ( ( member_a @ X @ S3 )
% 3.18/1.69 => ( ( member_a @ Y @ S3 )
% 3.18/1.69 => ( member_a @ ( mult_a_ring_ext_a_b @ M @ X @ Y ) @ S3 ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_35_right__unit__mult__rng__of__frac,axiom,
% 3.18/1.69 ! [R: a,S: a] :
% 3.18/1.69 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
% 3.18/1.69 => ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ R @ S ) @ ( one_se1488596747t_unit @ ( locali1944243560ac_a_b @ r @ s ) ) )
% 3.18/1.69 = ( locali1170246543t_unit @ rel @ R @ S ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_36_left__unit__mult__rng__of__frac,axiom,
% 3.18/1.69 ! [R: a,S: a] :
% 3.18/1.69 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
% 3.18/1.69 => ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( one_se1488596747t_unit @ ( locali1944243560ac_a_b @ r @ s ) ) @ ( locali1170246543t_unit @ rel @ R @ S ) )
% 3.18/1.69 = ( locali1170246543t_unit @ rel @ R @ S ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_37_zero__in__mult__submonoid,axiom,
% 3.18/1.69 ! [R: a,S: a,R2: a,S2: a] :
% 3.18/1.69 ( ( member_a @ ( zero_a_b @ r ) @ s )
% 3.18/1.69 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
% 3.18/1.69 => ( ( member449909584od_a_a @ ( product_Pair_a_a @ R2 @ S2 ) @ ( partia206007992t_unit @ rel ) )
% 3.18/1.69 => ( ( locali1170246543t_unit @ rel @ R @ S )
% 3.18/1.69 = ( locali1170246543t_unit @ rel @ R2 @ S2 ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_38_non__empty__class,axiom,
% 3.18/1.69 ! [R: a,S: a] :
% 3.18/1.69 ( ( member449909584od_a_a @ ( product_Pair_a_a @ R @ S ) @ ( partia206007992t_unit @ rel ) )
% 3.18/1.69 => ( ( locali1170246543t_unit @ rel @ R @ S )
% 3.18/1.69 != bot_bo2131659635od_a_a ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_39_one__unique,axiom,
% 3.18/1.69 ! [U: a] :
% 3.18/1.69 ( ( member_a @ U @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ! [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ r @ U @ X2 )
% 3.18/1.69 = X2 ) )
% 3.18/1.69 => ( U
% 3.18/1.69 = ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_40_inv__unique,axiom,
% 3.18/1.69 ! [Y: a,X: a,Y2: a] :
% 3.18/1.69 ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
% 3.18/1.69 = ( one_a_ring_ext_a_b @ r ) )
% 3.18/1.69 => ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
% 3.18/1.69 = ( one_a_ring_ext_a_b @ r ) )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( member_a @ Y2 @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( Y = Y2 ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_41_Units__inv__comm,axiom,
% 3.18/1.69 ! [X: a,Y: a] :
% 3.18/1.69 ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
% 3.18/1.69 = ( one_a_ring_ext_a_b @ r ) )
% 3.18/1.69 => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 3.18/1.69 => ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
% 3.18/1.69 = ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_42_prod_Oinject,axiom,
% 3.18/1.69 ! [X1: a,X22: a,Y1: a,Y22: a] :
% 3.18/1.69 ( ( ( product_Pair_a_a @ X1 @ X22 )
% 3.18/1.69 = ( product_Pair_a_a @ Y1 @ Y22 ) )
% 3.18/1.69 = ( ( X1 = Y1 )
% 3.18/1.69 & ( X22 = Y22 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_43_old_Oprod_Oinject,axiom,
% 3.18/1.69 ! [A: a,B: a,A2: a,B2: a] :
% 3.18/1.69 ( ( ( product_Pair_a_a @ A @ B )
% 3.18/1.69 = ( product_Pair_a_a @ A2 @ B2 ) )
% 3.18/1.69 = ( ( A = A2 )
% 3.18/1.69 & ( B = B2 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_44_semiring_Oaxioms_I1_J,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( semiring_a_b @ R3 )
% 3.18/1.69 => ( abelian_monoid_a_b @ R3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_45_monoid__axioms,axiom,
% 3.18/1.69 monoid1022105667xt_a_b @ r ).
% 3.18/1.69
% 3.18/1.69 thf(fact_46_m__assoc,axiom,
% 3.18/1.69 ! [X: a,Y: a,Z: a] :
% 3.18/1.69 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ Z )
% 3.18/1.69 = ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_47_m__comm,axiom,
% 3.18/1.69 ! [X: a,Y: a] :
% 3.18/1.69 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
% 3.18/1.69 = ( mult_a_ring_ext_a_b @ r @ Y @ X ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_48_mem__Collect__eq,axiom,
% 3.18/1.69 ! [A: a,P: a > $o] :
% 3.18/1.69 ( ( member_a @ A @ ( collect_a @ P ) )
% 3.18/1.69 = ( P @ A ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_49_mem__Collect__eq,axiom,
% 3.18/1.69 ! [A: product_prod_a_a,P: product_prod_a_a > $o] :
% 3.18/1.69 ( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ P ) )
% 3.18/1.69 = ( P @ A ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_50_mem__Collect__eq,axiom,
% 3.18/1.69 ! [A: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
% 3.18/1.69 ( ( member1838126896od_a_a @ A @ ( collec183727474od_a_a @ P ) )
% 3.18/1.69 = ( P @ A ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_51_mem__Collect__eq,axiom,
% 3.18/1.69 ! [A: set_a,P: set_a > $o] :
% 3.18/1.69 ( ( member_set_a @ A @ ( collect_set_a @ P ) )
% 3.18/1.69 = ( P @ A ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_52_Collect__mem__eq,axiom,
% 3.18/1.69 ! [A3: set_a] :
% 3.18/1.69 ( ( collect_a
% 3.18/1.69 @ ^ [X3: a] : ( member_a @ X3 @ A3 ) )
% 3.18/1.69 = A3 ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_53_Collect__mem__eq,axiom,
% 3.18/1.69 ! [A3: set_Product_prod_a_a] :
% 3.18/1.69 ( ( collec645855634od_a_a
% 3.18/1.69 @ ^ [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A3 ) )
% 3.18/1.69 = A3 ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_54_Collect__mem__eq,axiom,
% 3.18/1.69 ! [A3: set_se1596668135od_a_a] :
% 3.18/1.69 ( ( collec183727474od_a_a
% 3.18/1.69 @ ^ [X3: set_Product_prod_a_a] : ( member1838126896od_a_a @ X3 @ A3 ) )
% 3.18/1.69 = A3 ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_55_Collect__mem__eq,axiom,
% 3.18/1.69 ! [A3: set_set_a] :
% 3.18/1.69 ( ( collect_set_a
% 3.18/1.69 @ ^ [X3: set_a] : ( member_set_a @ X3 @ A3 ) )
% 3.18/1.69 = A3 ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_56_m__lcomm,axiom,
% 3.18/1.69 ! [X: a,Y: a,Z: a] :
% 3.18/1.69 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
% 3.18/1.69 = ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_57_Units__closed,axiom,
% 3.18/1.69 ! [X: a] :
% 3.18/1.69 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 3.18/1.69 => ( member_a @ X @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_58_Units__l__inv__ex,axiom,
% 3.18/1.69 ! [X: a] :
% 3.18/1.69 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 3.18/1.69 => ? [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 & ( ( mult_a_ring_ext_a_b @ r @ X2 @ X )
% 3.18/1.69 = ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_59_Units__r__inv__ex,axiom,
% 3.18/1.69 ! [X: a] :
% 3.18/1.69 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 3.18/1.69 => ? [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 & ( ( mult_a_ring_ext_a_b @ r @ X @ X2 )
% 3.18/1.69 = ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_60_zero__closed,axiom,
% 3.18/1.69 member_a @ ( zero_a_b @ r ) @ ( partia1066395285xt_a_b @ r ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_61_semiring__simprules_I3_J,axiom,
% 3.18/1.69 ! [X: a,Y: a] :
% 3.18/1.69 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_62_semiring__simprules_I4_J,axiom,
% 3.18/1.69 member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia1066395285xt_a_b @ r ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_63_Units__m__closed,axiom,
% 3.18/1.69 ! [X: a,Y: a] :
% 3.18/1.69 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 3.18/1.69 => ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
% 3.18/1.69 => ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_64_Units__one__closed,axiom,
% 3.18/1.69 member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_65_l__null,axiom,
% 3.18/1.69 ! [X: a] :
% 3.18/1.69 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X )
% 3.18/1.69 = ( zero_a_b @ r ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_66_r__null,axiom,
% 3.18/1.69 ! [X: a] :
% 3.18/1.69 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( zero_a_b @ r ) )
% 3.18/1.69 = ( zero_a_b @ r ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_67_l__one,axiom,
% 3.18/1.69 ! [X: a] :
% 3.18/1.69 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X )
% 3.18/1.69 = X ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_68_r__one,axiom,
% 3.18/1.69 ! [X: a] :
% 3.18/1.69 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( one_a_ring_ext_a_b @ r ) )
% 3.18/1.69 = X ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_69_Units__l__cancel,axiom,
% 3.18/1.69 ! [X: a,Y: a,Z: a] :
% 3.18/1.69 ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
% 3.18/1.69 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
% 3.18/1.69 = ( mult_a_ring_ext_a_b @ r @ X @ Z ) )
% 3.18/1.69 = ( Y = Z ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_70_abelian__monoidE_I2_J,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( abelian_monoid_a_b @ R3 )
% 3.18/1.69 => ( member_a @ ( zero_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_71_ring_Oring__simprules_I2_J,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( ring_a_b @ R3 )
% 3.18/1.69 => ( member_a @ ( zero_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_72_semiring_Osemiring__simprules_I2_J,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( semiring_a_b @ R3 )
% 3.18/1.69 => ( member_a @ ( zero_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_73_abelian__monoid_Ozero__closed,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( abelian_monoid_a_b @ G )
% 3.18/1.69 => ( member_a @ ( zero_a_b @ G ) @ ( partia1066395285xt_a_b @ G ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_74_ring_Ois__monoid,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( ring_a_b @ R3 )
% 3.18/1.69 => ( monoid1022105667xt_a_b @ R3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_75_semiring_Oaxioms_I2_J,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( semiring_a_b @ R3 )
% 3.18/1.69 => ( monoid1022105667xt_a_b @ R3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_76_ring_Oring__simprules_I24_J,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b,X: a] :
% 3.18/1.69 ( ( ring_a_b @ R3 )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ R3 @ ( zero_a_b @ R3 ) @ X )
% 3.18/1.69 = ( zero_a_b @ R3 ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_77_ring_Oring__simprules_I25_J,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b,X: a] :
% 3.18/1.69 ( ( ring_a_b @ R3 )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ R3 @ X @ ( zero_a_b @ R3 ) )
% 3.18/1.69 = ( zero_a_b @ R3 ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_78_semiring_Or__null,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b,X: a] :
% 3.18/1.69 ( ( semiring_a_b @ R3 )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ R3 @ X @ ( zero_a_b @ R3 ) )
% 3.18/1.69 = ( zero_a_b @ R3 ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_79_semiring_Ol__null,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b,X: a] :
% 3.18/1.69 ( ( semiring_a_b @ R3 )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ R3 @ ( zero_a_b @ R3 ) @ X )
% 3.18/1.69 = ( zero_a_b @ R3 ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_80_ring_Oring__simprules_I5_J,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b,X: a,Y: a] :
% 3.18/1.69 ( ( ring_a_b @ R3 )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( member_a @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_81_ring_Oring__simprules_I11_J,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 3.18/1.69 ( ( ring_a_b @ R3 )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ R3 @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) @ Z )
% 3.18/1.69 = ( mult_a_ring_ext_a_b @ R3 @ X @ ( mult_a_ring_ext_a_b @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_82_semiring_Osemiring__simprules_I3_J,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b,X: a,Y: a] :
% 3.18/1.69 ( ( semiring_a_b @ R3 )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( member_a @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_83_semiring_Osemiring__simprules_I8_J,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 3.18/1.69 ( ( semiring_a_b @ R3 )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ R3 @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) @ Z )
% 3.18/1.69 = ( mult_a_ring_ext_a_b @ R3 @ X @ ( mult_a_ring_ext_a_b @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_84_ring_Oring__simprules_I6_J,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( ring_a_b @ R3 )
% 3.18/1.69 => ( member_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_85_semiring_Osemiring__simprules_I4_J,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( semiring_a_b @ R3 )
% 3.18/1.69 => ( member_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( partia1066395285xt_a_b @ R3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_86_ring_Oring__simprules_I12_J,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b,X: a] :
% 3.18/1.69 ( ( ring_a_b @ R3 )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ R3 @ ( one_a_ring_ext_a_b @ R3 ) @ X )
% 3.18/1.69 = X ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_87_semiring_Osemiring__simprules_I9_J,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b,X: a] :
% 3.18/1.69 ( ( semiring_a_b @ R3 )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ R3 @ ( one_a_ring_ext_a_b @ R3 ) @ X )
% 3.18/1.69 = X ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_88_old_Oprod_Oinducts,axiom,
% 3.18/1.69 ! [P: product_prod_a_a > $o,Prod: product_prod_a_a] :
% 3.18/1.69 ( ! [A4: a,B3: a] : ( P @ ( product_Pair_a_a @ A4 @ B3 ) )
% 3.18/1.69 => ( P @ Prod ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_89_old_Oprod_Oexhaust,axiom,
% 3.18/1.69 ! [Y: product_prod_a_a] :
% 3.18/1.69 ~ ! [A4: a,B3: a] :
% 3.18/1.69 ( Y
% 3.18/1.69 != ( product_Pair_a_a @ A4 @ B3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_90_Pair__inject,axiom,
% 3.18/1.69 ! [A: a,B: a,A2: a,B2: a] :
% 3.18/1.69 ( ( ( product_Pair_a_a @ A @ B )
% 3.18/1.69 = ( product_Pair_a_a @ A2 @ B2 ) )
% 3.18/1.69 => ~ ( ( A = A2 )
% 3.18/1.69 => ( B != B2 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_91_prod__cases,axiom,
% 3.18/1.69 ! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
% 3.18/1.69 ( ! [A4: a,B3: a] : ( P @ ( product_Pair_a_a @ A4 @ B3 ) )
% 3.18/1.69 => ( P @ P2 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_92_surj__pair,axiom,
% 3.18/1.69 ! [P2: product_prod_a_a] :
% 3.18/1.69 ? [X2: a,Y3: a] :
% 3.18/1.69 ( P2
% 3.18/1.69 = ( product_Pair_a_a @ X2 @ Y3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_93_ring_Ois__ring,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( ring_a_b @ R3 )
% 3.18/1.69 => ( ring_a_b @ R3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_94_cring__fieldI2,axiom,
% 3.18/1.69 ( ( ( zero_a_b @ r )
% 3.18/1.69 != ( one_a_ring_ext_a_b @ r ) )
% 3.18/1.69 => ( ! [A4: a] :
% 3.18/1.69 ( ( member_a @ A4 @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( A4
% 3.18/1.69 != ( zero_a_b @ r ) )
% 3.18/1.69 => ? [X4: a] :
% 3.18/1.69 ( ( member_a @ X4 @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 & ( ( mult_a_ring_ext_a_b @ r @ A4 @ X4 )
% 3.18/1.69 = ( one_a_ring_ext_a_b @ r ) ) ) ) )
% 3.18/1.69 => ( field_a_b @ r ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_95_monoid_Or__one,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit,X: set_Product_prod_a_a] :
% 3.18/1.69 ( ( monoid1242292471t_unit @ G )
% 3.18/1.69 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( mult_s977248631t_unit @ G @ X @ ( one_se1488596747t_unit @ G ) )
% 3.18/1.69 = X ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_96_monoid_Or__one,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b,X: a] :
% 3.18/1.69 ( ( monoid1022105667xt_a_b @ G )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ G @ X @ ( one_a_ring_ext_a_b @ G ) )
% 3.18/1.69 = X ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_97_monoid_Ol__one,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit,X: set_Product_prod_a_a] :
% 3.18/1.69 ( ( monoid1242292471t_unit @ G )
% 3.18/1.69 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( mult_s977248631t_unit @ G @ ( one_se1488596747t_unit @ G ) @ X )
% 3.18/1.69 = X ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_98_monoid_Ol__one,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b,X: a] :
% 3.18/1.69 ( ( monoid1022105667xt_a_b @ G )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X )
% 3.18/1.69 = X ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_99_monoid_Oone__closed,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit] :
% 3.18/1.69 ( ( monoid1242292471t_unit @ G )
% 3.18/1.69 => ( member1838126896od_a_a @ ( one_se1488596747t_unit @ G ) @ ( partia616129906t_unit @ G ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_100_monoid_Oone__closed,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( monoid1022105667xt_a_b @ G )
% 3.18/1.69 => ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia1066395285xt_a_b @ G ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_101_monoid_OUnits__r__inv__ex,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit,X: set_Product_prod_a_a] :
% 3.18/1.69 ( ( monoid1242292471t_unit @ G )
% 3.18/1.69 => ( ( member1838126896od_a_a @ X @ ( units_1734235042t_unit @ G ) )
% 3.18/1.69 => ? [X2: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 & ( ( mult_s977248631t_unit @ G @ X @ X2 )
% 3.18/1.69 = ( one_se1488596747t_unit @ G ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_102_monoid_OUnits__r__inv__ex,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b,X: a] :
% 3.18/1.69 ( ( monoid1022105667xt_a_b @ G )
% 3.18/1.69 => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
% 3.18/1.69 => ? [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 & ( ( mult_a_ring_ext_a_b @ G @ X @ X2 )
% 3.18/1.69 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_103_monoid_OUnits__l__inv__ex,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit,X: set_Product_prod_a_a] :
% 3.18/1.69 ( ( monoid1242292471t_unit @ G )
% 3.18/1.69 => ( ( member1838126896od_a_a @ X @ ( units_1734235042t_unit @ G ) )
% 3.18/1.69 => ? [X2: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 & ( ( mult_s977248631t_unit @ G @ X2 @ X )
% 3.18/1.69 = ( one_se1488596747t_unit @ G ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_104_monoid_OUnits__l__inv__ex,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b,X: a] :
% 3.18/1.69 ( ( monoid1022105667xt_a_b @ G )
% 3.18/1.69 => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
% 3.18/1.69 => ? [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 & ( ( mult_a_ring_ext_a_b @ G @ X2 @ X )
% 3.18/1.69 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_105_monoid_OUnits__inv__comm,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit,X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
% 3.18/1.69 ( ( monoid1242292471t_unit @ G )
% 3.18/1.69 => ( ( ( mult_s977248631t_unit @ G @ X @ Y )
% 3.18/1.69 = ( one_se1488596747t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ X @ ( units_1734235042t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ Y @ ( units_1734235042t_unit @ G ) )
% 3.18/1.69 => ( ( mult_s977248631t_unit @ G @ Y @ X )
% 3.18/1.69 = ( one_se1488596747t_unit @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_106_monoid_OUnits__inv__comm,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 3.18/1.69 ( ( monoid1022105667xt_a_b @ G )
% 3.18/1.69 => ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
% 3.18/1.69 = ( one_a_ring_ext_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ G ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
% 3.18/1.69 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_107_add__pow__ldistr__int,axiom,
% 3.18/1.69 ! [A: a,B: a,K: int] :
% 3.18/1.69 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_int @ r @ K @ A ) @ B )
% 3.18/1.69 = ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_108_add__pow__rdistr__int,axiom,
% 3.18/1.69 ! [A: a,B: a,K: int] :
% 3.18/1.69 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_int @ r @ K @ B ) )
% 3.18/1.69 = ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_109_add_Oint__pow__closed,axiom,
% 3.18/1.69 ! [X: a,I: int] :
% 3.18/1.69 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( member_a @ ( add_pow_a_b_int @ r @ I @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_110_add_Oint__pow__one,axiom,
% 3.18/1.69 ! [Z: int] :
% 3.18/1.69 ( ( add_pow_a_b_int @ r @ Z @ ( zero_a_b @ r ) )
% 3.18/1.69 = ( zero_a_b @ r ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_111_field_Ois__ring,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( field_a_b @ R3 )
% 3.18/1.69 => ( ring_a_b @ R3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_112_one__not__zero,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( field_a_b @ R3 )
% 3.18/1.69 => ( ( one_a_ring_ext_a_b @ R3 )
% 3.18/1.69 != ( zero_a_b @ R3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_113_integral,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b,A: a,B: a] :
% 3.18/1.69 ( ( field_a_b @ R3 )
% 3.18/1.69 => ( ( ( mult_a_ring_ext_a_b @ R3 @ A @ B )
% 3.18/1.69 = ( zero_a_b @ R3 ) )
% 3.18/1.69 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( A
% 3.18/1.69 = ( zero_a_b @ R3 ) )
% 3.18/1.69 | ( B
% 3.18/1.69 = ( zero_a_b @ R3 ) ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_114_ring_Oadd__pow__ldistr__int,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b,A: a,B: a,K: int] :
% 3.18/1.69 ( ( ring_a_b @ R3 )
% 3.18/1.69 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ R3 @ ( add_pow_a_b_int @ R3 @ K @ A ) @ B )
% 3.18/1.69 = ( add_pow_a_b_int @ R3 @ K @ ( mult_a_ring_ext_a_b @ R3 @ A @ B ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_115_ring_Oadd__pow__rdistr__int,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b,A: a,B: a,K: int] :
% 3.18/1.69 ( ( ring_a_b @ R3 )
% 3.18/1.69 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ R3 @ A @ ( add_pow_a_b_int @ R3 @ K @ B ) )
% 3.18/1.69 = ( add_pow_a_b_int @ R3 @ K @ ( mult_a_ring_ext_a_b @ R3 @ A @ B ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_116_monoid_Ocarrier__not__empty,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit] :
% 3.18/1.69 ( ( monoid1242292471t_unit @ G )
% 3.18/1.69 => ( ( partia616129906t_unit @ G )
% 3.18/1.69 != bot_bo1424703315od_a_a ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_117_monoid_Ocarrier__not__empty,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( monoid1022105667xt_a_b @ G )
% 3.18/1.69 => ( ( partia1066395285xt_a_b @ G )
% 3.18/1.69 != bot_bot_set_a ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_118_monoid_Om__assoc,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit,X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
% 3.18/1.69 ( ( monoid1242292471t_unit @ G )
% 3.18/1.69 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ Y @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ Z @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( mult_s977248631t_unit @ G @ ( mult_s977248631t_unit @ G @ X @ Y ) @ Z )
% 3.18/1.69 = ( mult_s977248631t_unit @ G @ X @ ( mult_s977248631t_unit @ G @ Y @ Z ) ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_119_monoid_Om__assoc,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 3.18/1.69 ( ( monoid1022105667xt_a_b @ G )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ Z )
% 3.18/1.69 = ( mult_a_ring_ext_a_b @ G @ X @ ( mult_a_ring_ext_a_b @ G @ Y @ Z ) ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_120_monoid_Om__closed,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit,X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
% 3.18/1.69 ( ( monoid1242292471t_unit @ G )
% 3.18/1.69 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ Y @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( member1838126896od_a_a @ ( mult_s977248631t_unit @ G @ X @ Y ) @ ( partia616129906t_unit @ G ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_121_monoid_Om__closed,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 3.18/1.69 ( ( monoid1022105667xt_a_b @ G )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( member_a @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ ( partia1066395285xt_a_b @ G ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_122_monoid_OUnits__closed,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit,X: set_Product_prod_a_a] :
% 3.18/1.69 ( ( monoid1242292471t_unit @ G )
% 3.18/1.69 => ( ( member1838126896od_a_a @ X @ ( units_1734235042t_unit @ G ) )
% 3.18/1.69 => ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_123_monoid_OUnits__closed,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b,X: a] :
% 3.18/1.69 ( ( monoid1022105667xt_a_b @ G )
% 3.18/1.69 => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
% 3.18/1.69 => ( member_a @ X @ ( partia1066395285xt_a_b @ G ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_124_monoid_OUnits__m__closed,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit,X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
% 3.18/1.69 ( ( monoid1242292471t_unit @ G )
% 3.18/1.69 => ( ( member1838126896od_a_a @ X @ ( units_1734235042t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ Y @ ( units_1734235042t_unit @ G ) )
% 3.18/1.69 => ( member1838126896od_a_a @ ( mult_s977248631t_unit @ G @ X @ Y ) @ ( units_1734235042t_unit @ G ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_125_monoid_OUnits__m__closed,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 3.18/1.69 ( ( monoid1022105667xt_a_b @ G )
% 3.18/1.69 => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ G ) )
% 3.18/1.69 => ( member_a @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_126_monoid_OUnits__one__closed,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( monoid1022105667xt_a_b @ G )
% 3.18/1.69 => ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( units_a_ring_ext_a_b @ G ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_127_monoid_OUnits__one__closed,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit] :
% 3.18/1.69 ( ( monoid1242292471t_unit @ G )
% 3.18/1.69 => ( member1838126896od_a_a @ ( one_se1488596747t_unit @ G ) @ ( units_1734235042t_unit @ G ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_128_monoidI,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit] :
% 3.18/1.69 ( ! [X2: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ Y3 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( member1838126896od_a_a @ ( mult_s977248631t_unit @ G @ X2 @ Y3 ) @ ( partia616129906t_unit @ G ) ) ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ ( one_se1488596747t_unit @ G ) @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ! [X2: set_Product_prod_a_a,Y3: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ Y3 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ Z2 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( mult_s977248631t_unit @ G @ ( mult_s977248631t_unit @ G @ X2 @ Y3 ) @ Z2 )
% 3.18/1.69 = ( mult_s977248631t_unit @ G @ X2 @ ( mult_s977248631t_unit @ G @ Y3 @ Z2 ) ) ) ) ) )
% 3.18/1.69 => ( ! [X2: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( mult_s977248631t_unit @ G @ ( one_se1488596747t_unit @ G ) @ X2 )
% 3.18/1.69 = X2 ) )
% 3.18/1.69 => ( ! [X2: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( mult_s977248631t_unit @ G @ X2 @ ( one_se1488596747t_unit @ G ) )
% 3.18/1.69 = X2 ) )
% 3.18/1.69 => ( monoid1242292471t_unit @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_129_monoidI,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b] :
% 3.18/1.69 ( ! [X2: a,Y3: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( member_a @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ ( partia1066395285xt_a_b @ G ) ) ) )
% 3.18/1.69 => ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ! [X2: a,Y3: a,Z2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ Z2 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ Z2 )
% 3.18/1.69 = ( mult_a_ring_ext_a_b @ G @ X2 @ ( mult_a_ring_ext_a_b @ G @ Y3 @ Z2 ) ) ) ) ) )
% 3.18/1.69 => ( ! [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X2 )
% 3.18/1.69 = X2 ) )
% 3.18/1.69 => ( ! [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ G @ X2 @ ( one_a_ring_ext_a_b @ G ) )
% 3.18/1.69 = X2 ) )
% 3.18/1.69 => ( monoid1022105667xt_a_b @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_130_Group_Omonoid__def,axiom,
% 3.18/1.69 ( monoid1242292471t_unit
% 3.18/1.69 = ( ^ [G2: partia539144763t_unit] :
% 3.18/1.69 ( ! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X3 @ ( partia616129906t_unit @ G2 ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ Y4 @ ( partia616129906t_unit @ G2 ) )
% 3.18/1.69 => ( member1838126896od_a_a @ ( mult_s977248631t_unit @ G2 @ X3 @ Y4 ) @ ( partia616129906t_unit @ G2 ) ) ) )
% 3.18/1.69 & ! [X3: set_Product_prod_a_a,Y4: set_Product_prod_a_a,Z3: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X3 @ ( partia616129906t_unit @ G2 ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ Y4 @ ( partia616129906t_unit @ G2 ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ Z3 @ ( partia616129906t_unit @ G2 ) )
% 3.18/1.69 => ( ( mult_s977248631t_unit @ G2 @ ( mult_s977248631t_unit @ G2 @ X3 @ Y4 ) @ Z3 )
% 3.18/1.69 = ( mult_s977248631t_unit @ G2 @ X3 @ ( mult_s977248631t_unit @ G2 @ Y4 @ Z3 ) ) ) ) ) )
% 3.18/1.69 & ( member1838126896od_a_a @ ( one_se1488596747t_unit @ G2 ) @ ( partia616129906t_unit @ G2 ) )
% 3.18/1.69 & ! [X3: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X3 @ ( partia616129906t_unit @ G2 ) )
% 3.18/1.69 => ( ( mult_s977248631t_unit @ G2 @ ( one_se1488596747t_unit @ G2 ) @ X3 )
% 3.18/1.69 = X3 ) )
% 3.18/1.69 & ! [X3: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X3 @ ( partia616129906t_unit @ G2 ) )
% 3.18/1.69 => ( ( mult_s977248631t_unit @ G2 @ X3 @ ( one_se1488596747t_unit @ G2 ) )
% 3.18/1.69 = X3 ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_131_Group_Omonoid__def,axiom,
% 3.18/1.69 ( monoid1022105667xt_a_b
% 3.18/1.69 = ( ^ [G2: partia1833973666xt_a_b] :
% 3.18/1.69 ( ! [X3: a,Y4: a] :
% 3.18/1.69 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G2 ) )
% 3.18/1.69 => ( ( member_a @ Y4 @ ( partia1066395285xt_a_b @ G2 ) )
% 3.18/1.69 => ( member_a @ ( mult_a_ring_ext_a_b @ G2 @ X3 @ Y4 ) @ ( partia1066395285xt_a_b @ G2 ) ) ) )
% 3.18/1.69 & ! [X3: a,Y4: a,Z3: a] :
% 3.18/1.69 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G2 ) )
% 3.18/1.69 => ( ( member_a @ Y4 @ ( partia1066395285xt_a_b @ G2 ) )
% 3.18/1.69 => ( ( member_a @ Z3 @ ( partia1066395285xt_a_b @ G2 ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ G2 @ ( mult_a_ring_ext_a_b @ G2 @ X3 @ Y4 ) @ Z3 )
% 3.18/1.69 = ( mult_a_ring_ext_a_b @ G2 @ X3 @ ( mult_a_ring_ext_a_b @ G2 @ Y4 @ Z3 ) ) ) ) ) )
% 3.18/1.69 & ( member_a @ ( one_a_ring_ext_a_b @ G2 ) @ ( partia1066395285xt_a_b @ G2 ) )
% 3.18/1.69 & ! [X3: a] :
% 3.18/1.69 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G2 ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ G2 @ ( one_a_ring_ext_a_b @ G2 ) @ X3 )
% 3.18/1.69 = X3 ) )
% 3.18/1.69 & ! [X3: a] :
% 3.18/1.69 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ G2 ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ G2 @ X3 @ ( one_a_ring_ext_a_b @ G2 ) )
% 3.18/1.69 = X3 ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_132_Group_Omonoid_Ointro,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit] :
% 3.18/1.69 ( ! [X2: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ Y3 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( member1838126896od_a_a @ ( mult_s977248631t_unit @ G @ X2 @ Y3 ) @ ( partia616129906t_unit @ G ) ) ) )
% 3.18/1.69 => ( ! [X2: set_Product_prod_a_a,Y3: set_Product_prod_a_a,Z2: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ Y3 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ Z2 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( mult_s977248631t_unit @ G @ ( mult_s977248631t_unit @ G @ X2 @ Y3 ) @ Z2 )
% 3.18/1.69 = ( mult_s977248631t_unit @ G @ X2 @ ( mult_s977248631t_unit @ G @ Y3 @ Z2 ) ) ) ) ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ ( one_se1488596747t_unit @ G ) @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ! [X2: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( mult_s977248631t_unit @ G @ ( one_se1488596747t_unit @ G ) @ X2 )
% 3.18/1.69 = X2 ) )
% 3.18/1.69 => ( ! [X2: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( mult_s977248631t_unit @ G @ X2 @ ( one_se1488596747t_unit @ G ) )
% 3.18/1.69 = X2 ) )
% 3.18/1.69 => ( monoid1242292471t_unit @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_133_Group_Omonoid_Ointro,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b] :
% 3.18/1.69 ( ! [X2: a,Y3: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( member_a @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ ( partia1066395285xt_a_b @ G ) ) ) )
% 3.18/1.69 => ( ! [X2: a,Y3: a,Z2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ Z2 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ Z2 )
% 3.18/1.69 = ( mult_a_ring_ext_a_b @ G @ X2 @ ( mult_a_ring_ext_a_b @ G @ Y3 @ Z2 ) ) ) ) ) )
% 3.18/1.69 => ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ! [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X2 )
% 3.18/1.69 = X2 ) )
% 3.18/1.69 => ( ! [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ G @ X2 @ ( one_a_ring_ext_a_b @ G ) )
% 3.18/1.69 = X2 ) )
% 3.18/1.69 => ( monoid1022105667xt_a_b @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_134_monoid_Oinv__unique,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit,Y: set_Product_prod_a_a,X: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
% 3.18/1.69 ( ( monoid1242292471t_unit @ G )
% 3.18/1.69 => ( ( ( mult_s977248631t_unit @ G @ Y @ X )
% 3.18/1.69 = ( one_se1488596747t_unit @ G ) )
% 3.18/1.69 => ( ( ( mult_s977248631t_unit @ G @ X @ Y2 )
% 3.18/1.69 = ( one_se1488596747t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ Y @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ Y2 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( Y = Y2 ) ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_135_monoid_Oinv__unique,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b,Y: a,X: a,Y2: a] :
% 3.18/1.69 ( ( monoid1022105667xt_a_b @ G )
% 3.18/1.69 => ( ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
% 3.18/1.69 = ( one_a_ring_ext_a_b @ G ) )
% 3.18/1.69 => ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y2 )
% 3.18/1.69 = ( one_a_ring_ext_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ Y2 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( Y = Y2 ) ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_136_monoid_Oone__unique,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit,U: set_Product_prod_a_a] :
% 3.18/1.69 ( ( monoid1242292471t_unit @ G )
% 3.18/1.69 => ( ( member1838126896od_a_a @ U @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ! [X2: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( mult_s977248631t_unit @ G @ U @ X2 )
% 3.18/1.69 = X2 ) )
% 3.18/1.69 => ( U
% 3.18/1.69 = ( one_se1488596747t_unit @ G ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_137_monoid_Oone__unique,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b,U: a] :
% 3.18/1.69 ( ( monoid1022105667xt_a_b @ G )
% 3.18/1.69 => ( ( member_a @ U @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ! [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ G @ U @ X2 )
% 3.18/1.69 = X2 ) )
% 3.18/1.69 => ( U
% 3.18/1.69 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_138_monoid_OUnits__l__cancel,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit,X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
% 3.18/1.69 ( ( monoid1242292471t_unit @ G )
% 3.18/1.69 => ( ( member1838126896od_a_a @ X @ ( units_1734235042t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ Y @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ Z @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( ( mult_s977248631t_unit @ G @ X @ Y )
% 3.18/1.69 = ( mult_s977248631t_unit @ G @ X @ Z ) )
% 3.18/1.69 = ( Y = Z ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_139_monoid_OUnits__l__cancel,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 3.18/1.69 ( ( monoid1022105667xt_a_b @ G )
% 3.18/1.69 => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
% 3.18/1.69 = ( mult_a_ring_ext_a_b @ G @ X @ Z ) )
% 3.18/1.69 = ( Y = Z ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_140_subset,axiom,
% 3.18/1.69 ord_less_eq_set_a @ s @ ( partia1066395285xt_a_b @ r ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_141_add__pow__rdistr,axiom,
% 3.18/1.69 ! [A: a,B: a,K: nat] :
% 3.18/1.69 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_nat @ r @ K @ B ) )
% 3.18/1.69 = ( add_pow_a_b_nat @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_142_add__pow__ldistr,axiom,
% 3.18/1.69 ! [A: a,B: a,K: nat] :
% 3.18/1.69 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_nat @ r @ K @ A ) @ B )
% 3.18/1.69 = ( add_pow_a_b_nat @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_143_group__l__invI,axiom,
% 3.18/1.69 ( ! [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ? [Xa: a] :
% 3.18/1.69 ( ( member_a @ Xa @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 & ( ( mult_a_ring_ext_a_b @ r @ Xa @ X2 )
% 3.18/1.69 = ( one_a_ring_ext_a_b @ r ) ) ) )
% 3.18/1.69 => ( group_a_ring_ext_a_b @ r ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_144_carrier__not__empty,axiom,
% 3.18/1.69 ( ( partia1066395285xt_a_b @ r )
% 3.18/1.69 != bot_bot_set_a ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_145_add_Onat__pow__closed,axiom,
% 3.18/1.69 ! [X: a,N: nat] :
% 3.18/1.69 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( member_a @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_146_add_Onat__pow__one,axiom,
% 3.18/1.69 ! [N: nat] :
% 3.18/1.69 ( ( add_pow_a_b_nat @ r @ N @ ( zero_a_b @ r ) )
% 3.18/1.69 = ( zero_a_b @ r ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_147_group_Ois__group,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( group_a_ring_ext_a_b @ G )
% 3.18/1.69 => ( group_a_ring_ext_a_b @ G ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_148_group_Ois__group,axiom,
% 3.18/1.69 ! [G: partia96731725t_unit] :
% 3.18/1.69 ( ( group_a_Product_unit @ G )
% 3.18/1.69 => ( group_a_Product_unit @ G ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_149_group_OUnits,axiom,
% 3.18/1.69 ! [G: partia96731725t_unit] :
% 3.18/1.69 ( ( group_a_Product_unit @ G )
% 3.18/1.69 => ( ord_less_eq_set_a @ ( partia1955795460t_unit @ G ) @ ( units_a_Product_unit @ G ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_150_group_OUnits,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( group_a_ring_ext_a_b @ G )
% 3.18/1.69 => ( ord_less_eq_set_a @ ( partia1066395285xt_a_b @ G ) @ ( units_a_ring_ext_a_b @ G ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_151_group_Ois__monoid,axiom,
% 3.18/1.69 ! [G: partia96731725t_unit] :
% 3.18/1.69 ( ( group_a_Product_unit @ G )
% 3.18/1.69 => ( monoid1447254976t_unit @ G ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_152_group_Ois__monoid,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit] :
% 3.18/1.69 ( ( group_52730354t_unit @ G )
% 3.18/1.69 => ( monoid1242292471t_unit @ G ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_153_group_Ois__monoid,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( group_a_ring_ext_a_b @ G )
% 3.18/1.69 => ( monoid1022105667xt_a_b @ G ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_154_Group_Ogroup_Oright__cancel,axiom,
% 3.18/1.69 ! [G: partia96731725t_unit,X: a,Y: a,Z: a] :
% 3.18/1.69 ( ( group_a_Product_unit @ G )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ( ( member_a @ Y @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ( ( member_a @ Z @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ( ( ( mult_a_Product_unit @ G @ Y @ X )
% 3.18/1.69 = ( mult_a_Product_unit @ G @ Z @ X ) )
% 3.18/1.69 = ( Y = Z ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_155_Group_Ogroup_Oright__cancel,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit,X: set_Product_prod_a_a,Y: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
% 3.18/1.69 ( ( group_52730354t_unit @ G )
% 3.18/1.69 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ Y @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ Z @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( ( mult_s977248631t_unit @ G @ Y @ X )
% 3.18/1.69 = ( mult_s977248631t_unit @ G @ Z @ X ) )
% 3.18/1.69 = ( Y = Z ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_156_Group_Ogroup_Oright__cancel,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 3.18/1.69 ( ( group_a_ring_ext_a_b @ G )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
% 3.18/1.69 = ( mult_a_ring_ext_a_b @ G @ Z @ X ) )
% 3.18/1.69 = ( Y = Z ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_157_group_OUnits__eq,axiom,
% 3.18/1.69 ! [G: partia96731725t_unit] :
% 3.18/1.69 ( ( group_a_Product_unit @ G )
% 3.18/1.69 => ( ( units_a_Product_unit @ G )
% 3.18/1.69 = ( partia1955795460t_unit @ G ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_158_group_OUnits__eq,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( group_a_ring_ext_a_b @ G )
% 3.18/1.69 => ( ( units_a_ring_ext_a_b @ G )
% 3.18/1.69 = ( partia1066395285xt_a_b @ G ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_159_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Osubset,axiom,
% 3.18/1.69 ! [M: partia1833973666xt_a_b,S3: set_a] :
% 3.18/1.69 ( ( locali425460998xt_a_b @ M @ S3 )
% 3.18/1.69 => ( ord_less_eq_set_a @ S3 @ ( partia1066395285xt_a_b @ M ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_160_group_Or__cancel__one_H,axiom,
% 3.18/1.69 ! [G: partia96731725t_unit,X: a,A: a] :
% 3.18/1.69 ( ( group_a_Product_unit @ G )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ( ( X
% 3.18/1.69 = ( mult_a_Product_unit @ G @ A @ X ) )
% 3.18/1.69 = ( A
% 3.18/1.69 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_161_group_Or__cancel__one_H,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit,X: set_Product_prod_a_a,A: set_Product_prod_a_a] :
% 3.18/1.69 ( ( group_52730354t_unit @ G )
% 3.18/1.69 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ A @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( X
% 3.18/1.69 = ( mult_s977248631t_unit @ G @ A @ X ) )
% 3.18/1.69 = ( A
% 3.18/1.69 = ( one_se1488596747t_unit @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_162_group_Or__cancel__one_H,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b,X: a,A: a] :
% 3.18/1.69 ( ( group_a_ring_ext_a_b @ G )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( X
% 3.18/1.69 = ( mult_a_ring_ext_a_b @ G @ A @ X ) )
% 3.18/1.69 = ( A
% 3.18/1.69 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_163_group_Ol__cancel__one_H,axiom,
% 3.18/1.69 ! [G: partia96731725t_unit,X: a,A: a] :
% 3.18/1.69 ( ( group_a_Product_unit @ G )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ( ( X
% 3.18/1.69 = ( mult_a_Product_unit @ G @ X @ A ) )
% 3.18/1.69 = ( A
% 3.18/1.69 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_164_group_Ol__cancel__one_H,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit,X: set_Product_prod_a_a,A: set_Product_prod_a_a] :
% 3.18/1.69 ( ( group_52730354t_unit @ G )
% 3.18/1.69 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ A @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( X
% 3.18/1.69 = ( mult_s977248631t_unit @ G @ X @ A ) )
% 3.18/1.69 = ( A
% 3.18/1.69 = ( one_se1488596747t_unit @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_165_group_Ol__cancel__one_H,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b,X: a,A: a] :
% 3.18/1.69 ( ( group_a_ring_ext_a_b @ G )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( X
% 3.18/1.69 = ( mult_a_ring_ext_a_b @ G @ X @ A ) )
% 3.18/1.69 = ( A
% 3.18/1.69 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_166_group_Or__cancel__one,axiom,
% 3.18/1.69 ! [G: partia96731725t_unit,X: a,A: a] :
% 3.18/1.69 ( ( group_a_Product_unit @ G )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ( ( ( mult_a_Product_unit @ G @ A @ X )
% 3.18/1.69 = X )
% 3.18/1.69 = ( A
% 3.18/1.69 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_167_group_Or__cancel__one,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit,X: set_Product_prod_a_a,A: set_Product_prod_a_a] :
% 3.18/1.69 ( ( group_52730354t_unit @ G )
% 3.18/1.69 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ A @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( ( mult_s977248631t_unit @ G @ A @ X )
% 3.18/1.69 = X )
% 3.18/1.69 = ( A
% 3.18/1.69 = ( one_se1488596747t_unit @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_168_group_Or__cancel__one,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b,X: a,A: a] :
% 3.18/1.69 ( ( group_a_ring_ext_a_b @ G )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( ( mult_a_ring_ext_a_b @ G @ A @ X )
% 3.18/1.69 = X )
% 3.18/1.69 = ( A
% 3.18/1.69 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_169_group_Ol__cancel__one,axiom,
% 3.18/1.69 ! [G: partia96731725t_unit,X: a,A: a] :
% 3.18/1.69 ( ( group_a_Product_unit @ G )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ( ( member_a @ A @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ( ( ( mult_a_Product_unit @ G @ X @ A )
% 3.18/1.69 = X )
% 3.18/1.69 = ( A
% 3.18/1.69 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_170_group_Ol__cancel__one,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit,X: set_Product_prod_a_a,A: set_Product_prod_a_a] :
% 3.18/1.69 ( ( group_52730354t_unit @ G )
% 3.18/1.69 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ A @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( ( mult_s977248631t_unit @ G @ X @ A )
% 3.18/1.69 = X )
% 3.18/1.69 = ( A
% 3.18/1.69 = ( one_se1488596747t_unit @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_171_group_Ol__cancel__one,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b,X: a,A: a] :
% 3.18/1.69 ( ( group_a_ring_ext_a_b @ G )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( ( mult_a_ring_ext_a_b @ G @ X @ A )
% 3.18/1.69 = X )
% 3.18/1.69 = ( A
% 3.18/1.69 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_172_group_Or__inv__ex,axiom,
% 3.18/1.69 ! [G: partia96731725t_unit,X: a] :
% 3.18/1.69 ( ( group_a_Product_unit @ G )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ? [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 & ( ( mult_a_Product_unit @ G @ X @ X2 )
% 3.18/1.69 = ( one_a_Product_unit @ G ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_173_group_Or__inv__ex,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit,X: set_Product_prod_a_a] :
% 3.18/1.69 ( ( group_52730354t_unit @ G )
% 3.18/1.69 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ? [X2: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 & ( ( mult_s977248631t_unit @ G @ X @ X2 )
% 3.18/1.69 = ( one_se1488596747t_unit @ G ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_174_group_Or__inv__ex,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b,X: a] :
% 3.18/1.69 ( ( group_a_ring_ext_a_b @ G )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ? [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 & ( ( mult_a_ring_ext_a_b @ G @ X @ X2 )
% 3.18/1.69 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_175_group_Ol__inv__ex,axiom,
% 3.18/1.69 ! [G: partia96731725t_unit,X: a] :
% 3.18/1.69 ( ( group_a_Product_unit @ G )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ? [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 & ( ( mult_a_Product_unit @ G @ X2 @ X )
% 3.18/1.69 = ( one_a_Product_unit @ G ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_176_group_Ol__inv__ex,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit,X: set_Product_prod_a_a] :
% 3.18/1.69 ( ( group_52730354t_unit @ G )
% 3.18/1.69 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ? [X2: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 & ( ( mult_s977248631t_unit @ G @ X2 @ X )
% 3.18/1.69 = ( one_se1488596747t_unit @ G ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_177_group_Ol__inv__ex,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b,X: a] :
% 3.18/1.69 ( ( group_a_ring_ext_a_b @ G )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ? [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 & ( ( mult_a_ring_ext_a_b @ G @ X2 @ X )
% 3.18/1.69 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_178_group_Oinv__comm,axiom,
% 3.18/1.69 ! [G: partia96731725t_unit,X: a,Y: a] :
% 3.18/1.69 ( ( group_a_Product_unit @ G )
% 3.18/1.69 => ( ( ( mult_a_Product_unit @ G @ X @ Y )
% 3.18/1.69 = ( one_a_Product_unit @ G ) )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ( ( member_a @ Y @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ( ( mult_a_Product_unit @ G @ Y @ X )
% 3.18/1.69 = ( one_a_Product_unit @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_179_group_Oinv__comm,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit,X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
% 3.18/1.69 ( ( group_52730354t_unit @ G )
% 3.18/1.69 => ( ( ( mult_s977248631t_unit @ G @ X @ Y )
% 3.18/1.69 = ( one_se1488596747t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ X @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ Y @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( mult_s977248631t_unit @ G @ Y @ X )
% 3.18/1.69 = ( one_se1488596747t_unit @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_180_group_Oinv__comm,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b,X: a,Y: a] :
% 3.18/1.69 ( ( group_a_ring_ext_a_b @ G )
% 3.18/1.69 => ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
% 3.18/1.69 = ( one_a_ring_ext_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
% 3.18/1.69 = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_181_groupI,axiom,
% 3.18/1.69 ! [G: partia96731725t_unit] :
% 3.18/1.69 ( ! [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ! [Y3: a] :
% 3.18/1.69 ( ( member_a @ Y3 @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ( member_a @ ( mult_a_Product_unit @ G @ X2 @ Y3 ) @ ( partia1955795460t_unit @ G ) ) ) )
% 3.18/1.69 => ( ( member_a @ ( one_a_Product_unit @ G ) @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ( ! [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ! [Y3: a] :
% 3.18/1.69 ( ( member_a @ Y3 @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ! [Z2: a] :
% 3.18/1.69 ( ( member_a @ Z2 @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ( ( mult_a_Product_unit @ G @ ( mult_a_Product_unit @ G @ X2 @ Y3 ) @ Z2 )
% 3.18/1.69 = ( mult_a_Product_unit @ G @ X2 @ ( mult_a_Product_unit @ G @ Y3 @ Z2 ) ) ) ) ) )
% 3.18/1.69 => ( ! [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ( ( mult_a_Product_unit @ G @ ( one_a_Product_unit @ G ) @ X2 )
% 3.18/1.69 = X2 ) )
% 3.18/1.69 => ( ! [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ? [Xa: a] :
% 3.18/1.69 ( ( member_a @ Xa @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 & ( ( mult_a_Product_unit @ G @ Xa @ X2 )
% 3.18/1.69 = ( one_a_Product_unit @ G ) ) ) )
% 3.18/1.69 => ( group_a_Product_unit @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_182_groupI,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit] :
% 3.18/1.69 ( ! [X2: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ! [Y3: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ Y3 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( member1838126896od_a_a @ ( mult_s977248631t_unit @ G @ X2 @ Y3 ) @ ( partia616129906t_unit @ G ) ) ) )
% 3.18/1.69 => ( ( member1838126896od_a_a @ ( one_se1488596747t_unit @ G ) @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ! [X2: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ! [Y3: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ Y3 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ! [Z2: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ Z2 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( mult_s977248631t_unit @ G @ ( mult_s977248631t_unit @ G @ X2 @ Y3 ) @ Z2 )
% 3.18/1.69 = ( mult_s977248631t_unit @ G @ X2 @ ( mult_s977248631t_unit @ G @ Y3 @ Z2 ) ) ) ) ) )
% 3.18/1.69 => ( ! [X2: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ( ( mult_s977248631t_unit @ G @ ( one_se1488596747t_unit @ G ) @ X2 )
% 3.18/1.69 = X2 ) )
% 3.18/1.69 => ( ! [X2: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ? [Xa: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ Xa @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 & ( ( mult_s977248631t_unit @ G @ Xa @ X2 )
% 3.18/1.69 = ( one_se1488596747t_unit @ G ) ) ) )
% 3.18/1.69 => ( group_52730354t_unit @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_183_groupI,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b] :
% 3.18/1.69 ( ! [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ! [Y3: a] :
% 3.18/1.69 ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( member_a @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ ( partia1066395285xt_a_b @ G ) ) ) )
% 3.18/1.69 => ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ! [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ! [Y3: a] :
% 3.18/1.69 ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ! [Z2: a] :
% 3.18/1.69 ( ( member_a @ Z2 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ Z2 )
% 3.18/1.69 = ( mult_a_ring_ext_a_b @ G @ X2 @ ( mult_a_ring_ext_a_b @ G @ Y3 @ Z2 ) ) ) ) ) )
% 3.18/1.69 => ( ! [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X2 )
% 3.18/1.69 = X2 ) )
% 3.18/1.69 => ( ! [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ? [Xa: a] :
% 3.18/1.69 ( ( member_a @ Xa @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 & ( ( mult_a_ring_ext_a_b @ G @ Xa @ X2 )
% 3.18/1.69 = ( one_a_ring_ext_a_b @ G ) ) ) )
% 3.18/1.69 => ( group_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_184_monoid_Ogroup__l__invI,axiom,
% 3.18/1.69 ! [G: partia96731725t_unit] :
% 3.18/1.69 ( ( monoid1447254976t_unit @ G )
% 3.18/1.69 => ( ! [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 => ? [Xa: a] :
% 3.18/1.69 ( ( member_a @ Xa @ ( partia1955795460t_unit @ G ) )
% 3.18/1.69 & ( ( mult_a_Product_unit @ G @ Xa @ X2 )
% 3.18/1.69 = ( one_a_Product_unit @ G ) ) ) )
% 3.18/1.69 => ( group_a_Product_unit @ G ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_185_monoid_Ogroup__l__invI,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit] :
% 3.18/1.69 ( ( monoid1242292471t_unit @ G )
% 3.18/1.69 => ( ! [X2: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X2 @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 => ? [Xa: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ Xa @ ( partia616129906t_unit @ G ) )
% 3.18/1.69 & ( ( mult_s977248631t_unit @ G @ Xa @ X2 )
% 3.18/1.69 = ( one_se1488596747t_unit @ G ) ) ) )
% 3.18/1.69 => ( group_52730354t_unit @ G ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_186_monoid_Ogroup__l__invI,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( monoid1022105667xt_a_b @ G )
% 3.18/1.69 => ( ! [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 => ? [Xa: a] :
% 3.18/1.69 ( ( member_a @ Xa @ ( partia1066395285xt_a_b @ G ) )
% 3.18/1.69 & ( ( mult_a_ring_ext_a_b @ G @ Xa @ X2 )
% 3.18/1.69 = ( one_a_ring_ext_a_b @ G ) ) ) )
% 3.18/1.69 => ( group_a_ring_ext_a_b @ G ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_187_semiring_Oadd__pow__rdistr,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b,A: a,B: a,K: nat] :
% 3.18/1.69 ( ( semiring_a_b @ R3 )
% 3.18/1.69 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ R3 @ A @ ( add_pow_a_b_nat @ R3 @ K @ B ) )
% 3.18/1.69 = ( add_pow_a_b_nat @ R3 @ K @ ( mult_a_ring_ext_a_b @ R3 @ A @ B ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_188_semiring_Oadd__pow__ldistr,axiom,
% 3.18/1.69 ! [R3: partia1833973666xt_a_b,A: a,B: a,K: nat] :
% 3.18/1.69 ( ( semiring_a_b @ R3 )
% 3.18/1.69 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ R3 ) )
% 3.18/1.69 => ( ( mult_a_ring_ext_a_b @ R3 @ ( add_pow_a_b_nat @ R3 @ K @ A ) @ B )
% 3.18/1.69 = ( add_pow_a_b_nat @ R3 @ K @ ( mult_a_ring_ext_a_b @ R3 @ A @ B ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_189_a__lcos__mult__one,axiom,
% 3.18/1.69 ! [M: set_a] :
% 3.18/1.69 ( ( ord_less_eq_set_a @ M @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M )
% 3.18/1.69 = M ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_190_set__add__closed,axiom,
% 3.18/1.69 ! [A3: set_a,B4: set_a] :
% 3.18/1.69 ( ( ord_less_eq_set_a @ A3 @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( ord_less_eq_set_a @ B4 @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A3 @ B4 ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_191_setadd__subset__G,axiom,
% 3.18/1.69 ! [H: set_a,K2: set_a] :
% 3.18/1.69 ( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( ord_less_eq_set_a @ K2 @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ord_less_eq_set_a @ ( set_add_a_b @ r @ H @ K2 ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_192_subset__empty,axiom,
% 3.18/1.69 ! [A3: set_Product_prod_a_a] :
% 3.18/1.69 ( ( ord_le1824328871od_a_a @ A3 @ bot_bo2131659635od_a_a )
% 3.18/1.69 = ( A3 = bot_bo2131659635od_a_a ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_193_subset__empty,axiom,
% 3.18/1.69 ! [A3: set_a] :
% 3.18/1.69 ( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
% 3.18/1.69 = ( A3 = bot_bot_set_a ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_194_subset__empty,axiom,
% 3.18/1.69 ! [A3: set_set_a] :
% 3.18/1.69 ( ( ord_le318720350_set_a @ A3 @ bot_bot_set_set_a )
% 3.18/1.69 = ( A3 = bot_bot_set_set_a ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_195_empty__iff,axiom,
% 3.18/1.69 ! [C: set_Product_prod_a_a] :
% 3.18/1.69 ~ ( member1838126896od_a_a @ C @ bot_bo1424703315od_a_a ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_196_empty__iff,axiom,
% 3.18/1.69 ! [C: set_a] :
% 3.18/1.69 ~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_197_empty__iff,axiom,
% 3.18/1.69 ! [C: product_prod_a_a] :
% 3.18/1.69 ~ ( member449909584od_a_a @ C @ bot_bo2131659635od_a_a ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_198_empty__iff,axiom,
% 3.18/1.69 ! [C: a] :
% 3.18/1.69 ~ ( member_a @ C @ bot_bot_set_a ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_199_all__not__in__conv,axiom,
% 3.18/1.69 ! [A3: set_se1596668135od_a_a] :
% 3.18/1.69 ( ( ! [X3: set_Product_prod_a_a] :
% 3.18/1.69 ~ ( member1838126896od_a_a @ X3 @ A3 ) )
% 3.18/1.69 = ( A3 = bot_bo1424703315od_a_a ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_200_all__not__in__conv,axiom,
% 3.18/1.69 ! [A3: set_set_a] :
% 3.18/1.69 ( ( ! [X3: set_a] :
% 3.18/1.69 ~ ( member_set_a @ X3 @ A3 ) )
% 3.18/1.69 = ( A3 = bot_bot_set_set_a ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_201_all__not__in__conv,axiom,
% 3.18/1.69 ! [A3: set_Product_prod_a_a] :
% 3.18/1.69 ( ( ! [X3: product_prod_a_a] :
% 3.18/1.69 ~ ( member449909584od_a_a @ X3 @ A3 ) )
% 3.18/1.69 = ( A3 = bot_bo2131659635od_a_a ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_202_all__not__in__conv,axiom,
% 3.18/1.69 ! [A3: set_a] :
% 3.18/1.69 ( ( ! [X3: a] :
% 3.18/1.69 ~ ( member_a @ X3 @ A3 ) )
% 3.18/1.69 = ( A3 = bot_bot_set_a ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_203_Collect__empty__eq,axiom,
% 3.18/1.69 ! [P: product_prod_a_a > $o] :
% 3.18/1.69 ( ( ( collec645855634od_a_a @ P )
% 3.18/1.69 = bot_bo2131659635od_a_a )
% 3.18/1.69 = ( ! [X3: product_prod_a_a] :
% 3.18/1.69 ~ ( P @ X3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_204_Collect__empty__eq,axiom,
% 3.18/1.69 ! [P: a > $o] :
% 3.18/1.69 ( ( ( collect_a @ P )
% 3.18/1.69 = bot_bot_set_a )
% 3.18/1.69 = ( ! [X3: a] :
% 3.18/1.69 ~ ( P @ X3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_205_empty__Collect__eq,axiom,
% 3.18/1.69 ! [P: product_prod_a_a > $o] :
% 3.18/1.69 ( ( bot_bo2131659635od_a_a
% 3.18/1.69 = ( collec645855634od_a_a @ P ) )
% 3.18/1.69 = ( ! [X3: product_prod_a_a] :
% 3.18/1.69 ~ ( P @ X3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_206_empty__Collect__eq,axiom,
% 3.18/1.69 ! [P: a > $o] :
% 3.18/1.69 ( ( bot_bot_set_a
% 3.18/1.69 = ( collect_a @ P ) )
% 3.18/1.69 = ( ! [X3: a] :
% 3.18/1.69 ~ ( P @ X3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_207_subsetI,axiom,
% 3.18/1.69 ! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
% 3.18/1.69 ( ! [X2: product_prod_a_a] :
% 3.18/1.69 ( ( member449909584od_a_a @ X2 @ A3 )
% 3.18/1.69 => ( member449909584od_a_a @ X2 @ B4 ) )
% 3.18/1.69 => ( ord_le1824328871od_a_a @ A3 @ B4 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_208_subsetI,axiom,
% 3.18/1.69 ! [A3: set_se1596668135od_a_a,B4: set_se1596668135od_a_a] :
% 3.18/1.69 ( ! [X2: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X2 @ A3 )
% 3.18/1.69 => ( member1838126896od_a_a @ X2 @ B4 ) )
% 3.18/1.69 => ( ord_le391835783od_a_a @ A3 @ B4 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_209_subsetI,axiom,
% 3.18/1.69 ! [A3: set_a,B4: set_a] :
% 3.18/1.69 ( ! [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ A3 )
% 3.18/1.69 => ( member_a @ X2 @ B4 ) )
% 3.18/1.69 => ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_210_subsetI,axiom,
% 3.18/1.69 ! [A3: set_set_a,B4: set_set_a] :
% 3.18/1.69 ( ! [X2: set_a] :
% 3.18/1.69 ( ( member_set_a @ X2 @ A3 )
% 3.18/1.69 => ( member_set_a @ X2 @ B4 ) )
% 3.18/1.69 => ( ord_le318720350_set_a @ A3 @ B4 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_211_subset__antisym,axiom,
% 3.18/1.69 ! [A3: set_a,B4: set_a] :
% 3.18/1.69 ( ( ord_less_eq_set_a @ A3 @ B4 )
% 3.18/1.69 => ( ( ord_less_eq_set_a @ B4 @ A3 )
% 3.18/1.69 => ( A3 = B4 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_212_subset__antisym,axiom,
% 3.18/1.69 ! [A3: set_set_a,B4: set_set_a] :
% 3.18/1.69 ( ( ord_le318720350_set_a @ A3 @ B4 )
% 3.18/1.69 => ( ( ord_le318720350_set_a @ B4 @ A3 )
% 3.18/1.69 => ( A3 = B4 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_213_a__l__coset__subset__G,axiom,
% 3.18/1.69 ! [H: set_a,X: a] :
% 3.18/1.69 ( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X @ H ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_214_empty__subsetI,axiom,
% 3.18/1.69 ! [A3: set_Product_prod_a_a] : ( ord_le1824328871od_a_a @ bot_bo2131659635od_a_a @ A3 ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_215_empty__subsetI,axiom,
% 3.18/1.69 ! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_216_empty__subsetI,axiom,
% 3.18/1.69 ! [A3: set_set_a] : ( ord_le318720350_set_a @ bot_bot_set_set_a @ A3 ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_217_emptyE,axiom,
% 3.18/1.69 ! [A: set_Product_prod_a_a] :
% 3.18/1.69 ~ ( member1838126896od_a_a @ A @ bot_bo1424703315od_a_a ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_218_emptyE,axiom,
% 3.18/1.69 ! [A: set_a] :
% 3.18/1.69 ~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_219_emptyE,axiom,
% 3.18/1.69 ! [A: product_prod_a_a] :
% 3.18/1.69 ~ ( member449909584od_a_a @ A @ bot_bo2131659635od_a_a ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_220_emptyE,axiom,
% 3.18/1.69 ! [A: a] :
% 3.18/1.69 ~ ( member_a @ A @ bot_bot_set_a ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_221_equals0D,axiom,
% 3.18/1.69 ! [A3: set_se1596668135od_a_a,A: set_Product_prod_a_a] :
% 3.18/1.69 ( ( A3 = bot_bo1424703315od_a_a )
% 3.18/1.69 => ~ ( member1838126896od_a_a @ A @ A3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_222_equals0D,axiom,
% 3.18/1.69 ! [A3: set_set_a,A: set_a] :
% 3.18/1.69 ( ( A3 = bot_bot_set_set_a )
% 3.18/1.69 => ~ ( member_set_a @ A @ A3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_223_equals0D,axiom,
% 3.18/1.69 ! [A3: set_Product_prod_a_a,A: product_prod_a_a] :
% 3.18/1.69 ( ( A3 = bot_bo2131659635od_a_a )
% 3.18/1.69 => ~ ( member449909584od_a_a @ A @ A3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_224_equals0D,axiom,
% 3.18/1.69 ! [A3: set_a,A: a] :
% 3.18/1.69 ( ( A3 = bot_bot_set_a )
% 3.18/1.69 => ~ ( member_a @ A @ A3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_225_equals0I,axiom,
% 3.18/1.69 ! [A3: set_se1596668135od_a_a] :
% 3.18/1.69 ( ! [Y3: set_Product_prod_a_a] :
% 3.18/1.69 ~ ( member1838126896od_a_a @ Y3 @ A3 )
% 3.18/1.69 => ( A3 = bot_bo1424703315od_a_a ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_226_equals0I,axiom,
% 3.18/1.69 ! [A3: set_set_a] :
% 3.18/1.69 ( ! [Y3: set_a] :
% 3.18/1.69 ~ ( member_set_a @ Y3 @ A3 )
% 3.18/1.69 => ( A3 = bot_bot_set_set_a ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_227_equals0I,axiom,
% 3.18/1.69 ! [A3: set_Product_prod_a_a] :
% 3.18/1.69 ( ! [Y3: product_prod_a_a] :
% 3.18/1.69 ~ ( member449909584od_a_a @ Y3 @ A3 )
% 3.18/1.69 => ( A3 = bot_bo2131659635od_a_a ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_228_equals0I,axiom,
% 3.18/1.69 ! [A3: set_a] :
% 3.18/1.69 ( ! [Y3: a] :
% 3.18/1.69 ~ ( member_a @ Y3 @ A3 )
% 3.18/1.69 => ( A3 = bot_bot_set_a ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_229_ex__in__conv,axiom,
% 3.18/1.69 ! [A3: set_se1596668135od_a_a] :
% 3.18/1.69 ( ( ? [X3: set_Product_prod_a_a] : ( member1838126896od_a_a @ X3 @ A3 ) )
% 3.18/1.69 = ( A3 != bot_bo1424703315od_a_a ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_230_ex__in__conv,axiom,
% 3.18/1.69 ! [A3: set_set_a] :
% 3.18/1.69 ( ( ? [X3: set_a] : ( member_set_a @ X3 @ A3 ) )
% 3.18/1.69 = ( A3 != bot_bot_set_set_a ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_231_ex__in__conv,axiom,
% 3.18/1.69 ! [A3: set_Product_prod_a_a] :
% 3.18/1.69 ( ( ? [X3: product_prod_a_a] : ( member449909584od_a_a @ X3 @ A3 ) )
% 3.18/1.69 = ( A3 != bot_bo2131659635od_a_a ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_232_ex__in__conv,axiom,
% 3.18/1.69 ! [A3: set_a] :
% 3.18/1.69 ( ( ? [X3: a] : ( member_a @ X3 @ A3 ) )
% 3.18/1.69 = ( A3 != bot_bot_set_a ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_233_in__mono,axiom,
% 3.18/1.69 ! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,X: product_prod_a_a] :
% 3.18/1.69 ( ( ord_le1824328871od_a_a @ A3 @ B4 )
% 3.18/1.69 => ( ( member449909584od_a_a @ X @ A3 )
% 3.18/1.69 => ( member449909584od_a_a @ X @ B4 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_234_in__mono,axiom,
% 3.18/1.69 ! [A3: set_se1596668135od_a_a,B4: set_se1596668135od_a_a,X: set_Product_prod_a_a] :
% 3.18/1.69 ( ( ord_le391835783od_a_a @ A3 @ B4 )
% 3.18/1.69 => ( ( member1838126896od_a_a @ X @ A3 )
% 3.18/1.69 => ( member1838126896od_a_a @ X @ B4 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_235_in__mono,axiom,
% 3.18/1.69 ! [A3: set_a,B4: set_a,X: a] :
% 3.18/1.69 ( ( ord_less_eq_set_a @ A3 @ B4 )
% 3.18/1.69 => ( ( member_a @ X @ A3 )
% 3.18/1.69 => ( member_a @ X @ B4 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_236_in__mono,axiom,
% 3.18/1.69 ! [A3: set_set_a,B4: set_set_a,X: set_a] :
% 3.18/1.69 ( ( ord_le318720350_set_a @ A3 @ B4 )
% 3.18/1.69 => ( ( member_set_a @ X @ A3 )
% 3.18/1.69 => ( member_set_a @ X @ B4 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_237_subsetD,axiom,
% 3.18/1.69 ! [A3: set_Product_prod_a_a,B4: set_Product_prod_a_a,C: product_prod_a_a] :
% 3.18/1.69 ( ( ord_le1824328871od_a_a @ A3 @ B4 )
% 3.18/1.69 => ( ( member449909584od_a_a @ C @ A3 )
% 3.18/1.69 => ( member449909584od_a_a @ C @ B4 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_238_subsetD,axiom,
% 3.18/1.69 ! [A3: set_se1596668135od_a_a,B4: set_se1596668135od_a_a,C: set_Product_prod_a_a] :
% 3.18/1.69 ( ( ord_le391835783od_a_a @ A3 @ B4 )
% 3.18/1.69 => ( ( member1838126896od_a_a @ C @ A3 )
% 3.18/1.69 => ( member1838126896od_a_a @ C @ B4 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_239_subsetD,axiom,
% 3.18/1.69 ! [A3: set_a,B4: set_a,C: a] :
% 3.18/1.69 ( ( ord_less_eq_set_a @ A3 @ B4 )
% 3.18/1.69 => ( ( member_a @ C @ A3 )
% 3.18/1.69 => ( member_a @ C @ B4 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_240_subsetD,axiom,
% 3.18/1.69 ! [A3: set_set_a,B4: set_set_a,C: set_a] :
% 3.18/1.69 ( ( ord_le318720350_set_a @ A3 @ B4 )
% 3.18/1.69 => ( ( member_set_a @ C @ A3 )
% 3.18/1.69 => ( member_set_a @ C @ B4 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_241_equalityE,axiom,
% 3.18/1.69 ! [A3: set_a,B4: set_a] :
% 3.18/1.69 ( ( A3 = B4 )
% 3.18/1.69 => ~ ( ( ord_less_eq_set_a @ A3 @ B4 )
% 3.18/1.69 => ~ ( ord_less_eq_set_a @ B4 @ A3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_242_equalityE,axiom,
% 3.18/1.69 ! [A3: set_set_a,B4: set_set_a] :
% 3.18/1.69 ( ( A3 = B4 )
% 3.18/1.69 => ~ ( ( ord_le318720350_set_a @ A3 @ B4 )
% 3.18/1.69 => ~ ( ord_le318720350_set_a @ B4 @ A3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_243_subset__eq,axiom,
% 3.18/1.69 ( ord_le1824328871od_a_a
% 3.18/1.69 = ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
% 3.18/1.69 ! [X3: product_prod_a_a] :
% 3.18/1.69 ( ( member449909584od_a_a @ X3 @ A5 )
% 3.18/1.69 => ( member449909584od_a_a @ X3 @ B5 ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_244_subset__eq,axiom,
% 3.18/1.69 ( ord_le391835783od_a_a
% 3.18/1.69 = ( ^ [A5: set_se1596668135od_a_a,B5: set_se1596668135od_a_a] :
% 3.18/1.69 ! [X3: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X3 @ A5 )
% 3.18/1.69 => ( member1838126896od_a_a @ X3 @ B5 ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_245_subset__eq,axiom,
% 3.18/1.69 ( ord_less_eq_set_a
% 3.18/1.69 = ( ^ [A5: set_a,B5: set_a] :
% 3.18/1.69 ! [X3: a] :
% 3.18/1.69 ( ( member_a @ X3 @ A5 )
% 3.18/1.69 => ( member_a @ X3 @ B5 ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_246_subset__eq,axiom,
% 3.18/1.69 ( ord_le318720350_set_a
% 3.18/1.69 = ( ^ [A5: set_set_a,B5: set_set_a] :
% 3.18/1.69 ! [X3: set_a] :
% 3.18/1.69 ( ( member_set_a @ X3 @ A5 )
% 3.18/1.69 => ( member_set_a @ X3 @ B5 ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_247_equalityD1,axiom,
% 3.18/1.69 ! [A3: set_a,B4: set_a] :
% 3.18/1.69 ( ( A3 = B4 )
% 3.18/1.69 => ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_248_equalityD1,axiom,
% 3.18/1.69 ! [A3: set_set_a,B4: set_set_a] :
% 3.18/1.69 ( ( A3 = B4 )
% 3.18/1.69 => ( ord_le318720350_set_a @ A3 @ B4 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_249_equalityD2,axiom,
% 3.18/1.69 ! [A3: set_a,B4: set_a] :
% 3.18/1.69 ( ( A3 = B4 )
% 3.18/1.69 => ( ord_less_eq_set_a @ B4 @ A3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_250_equalityD2,axiom,
% 3.18/1.69 ! [A3: set_set_a,B4: set_set_a] :
% 3.18/1.69 ( ( A3 = B4 )
% 3.18/1.69 => ( ord_le318720350_set_a @ B4 @ A3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_251_subset__iff,axiom,
% 3.18/1.69 ( ord_le1824328871od_a_a
% 3.18/1.69 = ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
% 3.18/1.69 ! [T2: product_prod_a_a] :
% 3.18/1.69 ( ( member449909584od_a_a @ T2 @ A5 )
% 3.18/1.69 => ( member449909584od_a_a @ T2 @ B5 ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_252_subset__iff,axiom,
% 3.18/1.69 ( ord_le391835783od_a_a
% 3.18/1.69 = ( ^ [A5: set_se1596668135od_a_a,B5: set_se1596668135od_a_a] :
% 3.18/1.69 ! [T2: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ T2 @ A5 )
% 3.18/1.69 => ( member1838126896od_a_a @ T2 @ B5 ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_253_subset__iff,axiom,
% 3.18/1.69 ( ord_less_eq_set_a
% 3.18/1.69 = ( ^ [A5: set_a,B5: set_a] :
% 3.18/1.69 ! [T2: a] :
% 3.18/1.69 ( ( member_a @ T2 @ A5 )
% 3.18/1.69 => ( member_a @ T2 @ B5 ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_254_subset__iff,axiom,
% 3.18/1.69 ( ord_le318720350_set_a
% 3.18/1.69 = ( ^ [A5: set_set_a,B5: set_set_a] :
% 3.18/1.69 ! [T2: set_a] :
% 3.18/1.69 ( ( member_set_a @ T2 @ A5 )
% 3.18/1.69 => ( member_set_a @ T2 @ B5 ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_255_subset__refl,axiom,
% 3.18/1.69 ! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_256_subset__refl,axiom,
% 3.18/1.69 ! [A3: set_set_a] : ( ord_le318720350_set_a @ A3 @ A3 ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_257_Collect__mono,axiom,
% 3.18/1.69 ! [P: a > $o,Q: a > $o] :
% 3.18/1.69 ( ! [X2: a] :
% 3.18/1.69 ( ( P @ X2 )
% 3.18/1.69 => ( Q @ X2 ) )
% 3.18/1.69 => ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_258_Collect__mono,axiom,
% 3.18/1.69 ! [P: set_a > $o,Q: set_a > $o] :
% 3.18/1.69 ( ! [X2: set_a] :
% 3.18/1.69 ( ( P @ X2 )
% 3.18/1.69 => ( Q @ X2 ) )
% 3.18/1.69 => ( ord_le318720350_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_259_subset__trans,axiom,
% 3.18/1.69 ! [A3: set_a,B4: set_a,C2: set_a] :
% 3.18/1.69 ( ( ord_less_eq_set_a @ A3 @ B4 )
% 3.18/1.69 => ( ( ord_less_eq_set_a @ B4 @ C2 )
% 3.18/1.69 => ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_260_subset__trans,axiom,
% 3.18/1.69 ! [A3: set_set_a,B4: set_set_a,C2: set_set_a] :
% 3.18/1.69 ( ( ord_le318720350_set_a @ A3 @ B4 )
% 3.18/1.69 => ( ( ord_le318720350_set_a @ B4 @ C2 )
% 3.18/1.69 => ( ord_le318720350_set_a @ A3 @ C2 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_261_set__eq__subset,axiom,
% 3.18/1.69 ( ( ^ [Y5: set_a,Z4: set_a] : ( Y5 = Z4 ) )
% 3.18/1.69 = ( ^ [A5: set_a,B5: set_a] :
% 3.18/1.69 ( ( ord_less_eq_set_a @ A5 @ B5 )
% 3.18/1.69 & ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_262_set__eq__subset,axiom,
% 3.18/1.69 ( ( ^ [Y5: set_set_a,Z4: set_set_a] : ( Y5 = Z4 ) )
% 3.18/1.69 = ( ^ [A5: set_set_a,B5: set_set_a] :
% 3.18/1.69 ( ( ord_le318720350_set_a @ A5 @ B5 )
% 3.18/1.69 & ( ord_le318720350_set_a @ B5 @ A5 ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_263_Collect__mono__iff,axiom,
% 3.18/1.69 ! [P: a > $o,Q: a > $o] :
% 3.18/1.69 ( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
% 3.18/1.69 = ( ! [X3: a] :
% 3.18/1.69 ( ( P @ X3 )
% 3.18/1.69 => ( Q @ X3 ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_264_Collect__mono__iff,axiom,
% 3.18/1.69 ! [P: set_a > $o,Q: set_a > $o] :
% 3.18/1.69 ( ( ord_le318720350_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
% 3.18/1.69 = ( ! [X3: set_a] :
% 3.18/1.69 ( ( P @ X3 )
% 3.18/1.69 => ( Q @ X3 ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_265_units__group,axiom,
% 3.18/1.69 group_a_Product_unit @ ( units_1411277569xt_a_b @ r ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_266_add__additive__subgroups,axiom,
% 3.18/1.69 ! [H: set_a,K2: set_a] :
% 3.18/1.69 ( ( additi2104487374up_a_b @ H @ r )
% 3.18/1.69 => ( ( additi2104487374up_a_b @ K2 @ r )
% 3.18/1.69 => ( additi2104487374up_a_b @ ( set_add_a_b @ r @ H @ K2 ) @ r ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_267_a__rcos__assoc__lcos,axiom,
% 3.18/1.69 ! [H: set_a,K2: set_a,X: a] :
% 3.18/1.69 ( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( ord_less_eq_set_a @ K2 @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( set_add_a_b @ r @ ( a_r_coset_a_b @ r @ H @ X ) @ K2 )
% 3.18/1.69 = ( set_add_a_b @ r @ H @ ( a_l_coset_a_b @ r @ X @ K2 ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_268_a__r__coset__subset__G,axiom,
% 3.18/1.69 ! [H: set_a,X: a] :
% 3.18/1.69 ( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ord_less_eq_set_a @ ( a_r_coset_a_b @ r @ H @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_269_a__setmult__rcos__assoc,axiom,
% 3.18/1.69 ! [H: set_a,K2: set_a,X: a] :
% 3.18/1.69 ( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( ord_less_eq_set_a @ K2 @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( set_add_a_b @ r @ H @ ( a_r_coset_a_b @ r @ K2 @ X ) )
% 3.18/1.69 = ( a_r_coset_a_b @ r @ ( set_add_a_b @ r @ H @ K2 ) @ X ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_270_units__of__units,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( units_a_Product_unit @ ( units_1411277569xt_a_b @ G ) )
% 3.18/1.69 = ( units_a_ring_ext_a_b @ G ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_271_a__coset__add__zero,axiom,
% 3.18/1.69 ! [M: set_a] :
% 3.18/1.69 ( ( ord_less_eq_set_a @ M @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( a_r_coset_a_b @ r @ M @ ( zero_a_b @ r ) )
% 3.18/1.69 = M ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_272_units__of__carrier,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( partia1955795460t_unit @ ( units_1411277569xt_a_b @ G ) )
% 3.18/1.69 = ( units_a_ring_ext_a_b @ G ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_273_monoid_Ounits__group,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit] :
% 3.18/1.69 ( ( monoid1242292471t_unit @ G )
% 3.18/1.69 => ( group_52730354t_unit @ ( units_390215097t_unit @ G ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_274_monoid_Ounits__group,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( monoid1022105667xt_a_b @ G )
% 3.18/1.69 => ( group_a_Product_unit @ ( units_1411277569xt_a_b @ G ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_275_units__of__mult,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( mult_a_Product_unit @ ( units_1411277569xt_a_b @ G ) )
% 3.18/1.69 = ( mult_a_ring_ext_a_b @ G ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_276_units__of__mult,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit] :
% 3.18/1.69 ( ( mult_s977248631t_unit @ ( units_390215097t_unit @ G ) )
% 3.18/1.69 = ( mult_s977248631t_unit @ G ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_277_units__of__one,axiom,
% 3.18/1.69 ! [G: partia1833973666xt_a_b] :
% 3.18/1.69 ( ( one_a_Product_unit @ ( units_1411277569xt_a_b @ G ) )
% 3.18/1.69 = ( one_a_ring_ext_a_b @ G ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_278_units__of__one,axiom,
% 3.18/1.69 ! [G: partia539144763t_unit] :
% 3.18/1.69 ( ( one_se1488596747t_unit @ ( units_390215097t_unit @ G ) )
% 3.18/1.69 = ( one_se1488596747t_unit @ G ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_279_a__rcosetsI,axiom,
% 3.18/1.69 ! [H: set_a,X: a] :
% 3.18/1.69 ( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 3.18/1.69 => ( member_set_a @ ( a_r_coset_a_b @ r @ H @ X ) @ ( a_RCOSETS_a_b @ r @ H ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_280_one__zeroI,axiom,
% 3.18/1.69 ( ( ( partia1066395285xt_a_b @ r )
% 3.18/1.69 = ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
% 3.18/1.69 => ( ( one_a_ring_ext_a_b @ r )
% 3.18/1.69 = ( zero_a_b @ r ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_281_one__zeroD,axiom,
% 3.18/1.69 ( ( ( one_a_ring_ext_a_b @ r )
% 3.18/1.69 = ( zero_a_b @ r ) )
% 3.18/1.69 => ( ( partia1066395285xt_a_b @ r )
% 3.18/1.69 = ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_282_insert__absorb2,axiom,
% 3.18/1.69 ! [X: a,A3: set_a] :
% 3.18/1.69 ( ( insert_a @ X @ ( insert_a @ X @ A3 ) )
% 3.18/1.69 = ( insert_a @ X @ A3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_283_insert__iff,axiom,
% 3.18/1.69 ! [A: a,B: a,A3: set_a] :
% 3.18/1.69 ( ( member_a @ A @ ( insert_a @ B @ A3 ) )
% 3.18/1.69 = ( ( A = B )
% 3.18/1.69 | ( member_a @ A @ A3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_284_insert__iff,axiom,
% 3.18/1.69 ! [A: product_prod_a_a,B: product_prod_a_a,A3: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ A3 ) )
% 3.18/1.69 = ( ( A = B )
% 3.18/1.69 | ( member449909584od_a_a @ A @ A3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_285_insert__iff,axiom,
% 3.18/1.69 ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,A3: set_se1596668135od_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ A @ ( insert1738756823od_a_a @ B @ A3 ) )
% 3.18/1.69 = ( ( A = B )
% 3.18/1.69 | ( member1838126896od_a_a @ A @ A3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_286_insert__iff,axiom,
% 3.18/1.69 ! [A: set_a,B: set_a,A3: set_set_a] :
% 3.18/1.69 ( ( member_set_a @ A @ ( insert_set_a @ B @ A3 ) )
% 3.18/1.69 = ( ( A = B )
% 3.18/1.69 | ( member_set_a @ A @ A3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_287_insertCI,axiom,
% 3.18/1.69 ! [A: a,B4: set_a,B: a] :
% 3.18/1.69 ( ( ~ ( member_a @ A @ B4 )
% 3.18/1.69 => ( A = B ) )
% 3.18/1.69 => ( member_a @ A @ ( insert_a @ B @ B4 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_288_insertCI,axiom,
% 3.18/1.69 ! [A: product_prod_a_a,B4: set_Product_prod_a_a,B: product_prod_a_a] :
% 3.18/1.69 ( ( ~ ( member449909584od_a_a @ A @ B4 )
% 3.18/1.69 => ( A = B ) )
% 3.18/1.69 => ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ B4 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_289_insertCI,axiom,
% 3.18/1.69 ! [A: set_Product_prod_a_a,B4: set_se1596668135od_a_a,B: set_Product_prod_a_a] :
% 3.18/1.69 ( ( ~ ( member1838126896od_a_a @ A @ B4 )
% 3.18/1.69 => ( A = B ) )
% 3.18/1.69 => ( member1838126896od_a_a @ A @ ( insert1738756823od_a_a @ B @ B4 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_290_insertCI,axiom,
% 3.18/1.69 ! [A: set_a,B4: set_set_a,B: set_a] :
% 3.18/1.69 ( ( ~ ( member_set_a @ A @ B4 )
% 3.18/1.69 => ( A = B ) )
% 3.18/1.69 => ( member_set_a @ A @ ( insert_set_a @ B @ B4 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_291_carrier__one__not__zero,axiom,
% 3.18/1.69 ( ( ( partia1066395285xt_a_b @ r )
% 3.18/1.69 != ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
% 3.18/1.69 = ( ( one_a_ring_ext_a_b @ r )
% 3.18/1.69 != ( zero_a_b @ r ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_292_carrier__one__zero,axiom,
% 3.18/1.69 ( ( ( partia1066395285xt_a_b @ r )
% 3.18/1.69 = ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
% 3.18/1.69 = ( ( one_a_ring_ext_a_b @ r )
% 3.18/1.69 = ( zero_a_b @ r ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_293_singletonI,axiom,
% 3.18/1.69 ! [A: set_Product_prod_a_a] : ( member1838126896od_a_a @ A @ ( insert1738756823od_a_a @ A @ bot_bo1424703315od_a_a ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_294_singletonI,axiom,
% 3.18/1.69 ! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_295_singletonI,axiom,
% 3.18/1.69 ! [A: product_prod_a_a] : ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_296_singletonI,axiom,
% 3.18/1.69 ! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_297_insert__subset,axiom,
% 3.18/1.69 ! [X: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
% 3.18/1.69 ( ( ord_le1824328871od_a_a @ ( insert1116662519od_a_a @ X @ A3 ) @ B4 )
% 3.18/1.69 = ( ( member449909584od_a_a @ X @ B4 )
% 3.18/1.69 & ( ord_le1824328871od_a_a @ A3 @ B4 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_298_insert__subset,axiom,
% 3.18/1.69 ! [X: set_Product_prod_a_a,A3: set_se1596668135od_a_a,B4: set_se1596668135od_a_a] :
% 3.18/1.69 ( ( ord_le391835783od_a_a @ ( insert1738756823od_a_a @ X @ A3 ) @ B4 )
% 3.18/1.69 = ( ( member1838126896od_a_a @ X @ B4 )
% 3.18/1.69 & ( ord_le391835783od_a_a @ A3 @ B4 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_299_insert__subset,axiom,
% 3.18/1.69 ! [X: a,A3: set_a,B4: set_a] :
% 3.18/1.69 ( ( ord_less_eq_set_a @ ( insert_a @ X @ A3 ) @ B4 )
% 3.18/1.69 = ( ( member_a @ X @ B4 )
% 3.18/1.69 & ( ord_less_eq_set_a @ A3 @ B4 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_300_insert__subset,axiom,
% 3.18/1.69 ! [X: set_a,A3: set_set_a,B4: set_set_a] :
% 3.18/1.69 ( ( ord_le318720350_set_a @ ( insert_set_a @ X @ A3 ) @ B4 )
% 3.18/1.69 = ( ( member_set_a @ X @ B4 )
% 3.18/1.69 & ( ord_le318720350_set_a @ A3 @ B4 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_301_singleton__insert__inj__eq_H,axiom,
% 3.18/1.69 ! [A: product_prod_a_a,A3: set_Product_prod_a_a,B: product_prod_a_a] :
% 3.18/1.69 ( ( ( insert1116662519od_a_a @ A @ A3 )
% 3.18/1.69 = ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a ) )
% 3.18/1.69 = ( ( A = B )
% 3.18/1.69 & ( ord_le1824328871od_a_a @ A3 @ ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_302_singleton__insert__inj__eq_H,axiom,
% 3.18/1.69 ! [A: a,A3: set_a,B: a] :
% 3.18/1.69 ( ( ( insert_a @ A @ A3 )
% 3.18/1.69 = ( insert_a @ B @ bot_bot_set_a ) )
% 3.18/1.69 = ( ( A = B )
% 3.18/1.69 & ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_303_singleton__insert__inj__eq_H,axiom,
% 3.18/1.69 ! [A: set_a,A3: set_set_a,B: set_a] :
% 3.18/1.69 ( ( ( insert_set_a @ A @ A3 )
% 3.18/1.69 = ( insert_set_a @ B @ bot_bot_set_set_a ) )
% 3.18/1.69 = ( ( A = B )
% 3.18/1.69 & ( ord_le318720350_set_a @ A3 @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_304_singleton__insert__inj__eq,axiom,
% 3.18/1.69 ! [B: product_prod_a_a,A: product_prod_a_a,A3: set_Product_prod_a_a] :
% 3.18/1.69 ( ( ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a )
% 3.18/1.69 = ( insert1116662519od_a_a @ A @ A3 ) )
% 3.18/1.69 = ( ( A = B )
% 3.18/1.69 & ( ord_le1824328871od_a_a @ A3 @ ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_305_singleton__insert__inj__eq,axiom,
% 3.18/1.69 ! [B: a,A: a,A3: set_a] :
% 3.18/1.69 ( ( ( insert_a @ B @ bot_bot_set_a )
% 3.18/1.69 = ( insert_a @ A @ A3 ) )
% 3.18/1.69 = ( ( A = B )
% 3.18/1.69 & ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_306_singleton__insert__inj__eq,axiom,
% 3.18/1.69 ! [B: set_a,A: set_a,A3: set_set_a] :
% 3.18/1.69 ( ( ( insert_set_a @ B @ bot_bot_set_set_a )
% 3.18/1.69 = ( insert_set_a @ A @ A3 ) )
% 3.18/1.69 = ( ( A = B )
% 3.18/1.69 & ( ord_le318720350_set_a @ A3 @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_307_subset__singleton__iff,axiom,
% 3.18/1.69 ! [X5: set_Product_prod_a_a,A: product_prod_a_a] :
% 3.18/1.69 ( ( ord_le1824328871od_a_a @ X5 @ ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a ) )
% 3.18/1.69 = ( ( X5 = bot_bo2131659635od_a_a )
% 3.18/1.69 | ( X5
% 3.18/1.69 = ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_308_subset__singleton__iff,axiom,
% 3.18/1.69 ! [X5: set_a,A: a] :
% 3.18/1.69 ( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A @ bot_bot_set_a ) )
% 3.18/1.69 = ( ( X5 = bot_bot_set_a )
% 3.18/1.69 | ( X5
% 3.18/1.69 = ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_309_subset__singleton__iff,axiom,
% 3.18/1.69 ! [X5: set_set_a,A: set_a] :
% 3.18/1.69 ( ( ord_le318720350_set_a @ X5 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
% 3.18/1.69 = ( ( X5 = bot_bot_set_set_a )
% 3.18/1.69 | ( X5
% 3.18/1.69 = ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_310_subset__singletonD,axiom,
% 3.18/1.69 ! [A3: set_Product_prod_a_a,X: product_prod_a_a] :
% 3.18/1.69 ( ( ord_le1824328871od_a_a @ A3 @ ( insert1116662519od_a_a @ X @ bot_bo2131659635od_a_a ) )
% 3.18/1.69 => ( ( A3 = bot_bo2131659635od_a_a )
% 3.18/1.69 | ( A3
% 3.18/1.69 = ( insert1116662519od_a_a @ X @ bot_bo2131659635od_a_a ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_311_subset__singletonD,axiom,
% 3.18/1.69 ! [A3: set_a,X: a] :
% 3.18/1.69 ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) )
% 3.18/1.69 => ( ( A3 = bot_bot_set_a )
% 3.18/1.69 | ( A3
% 3.18/1.69 = ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_312_subset__singletonD,axiom,
% 3.18/1.69 ! [A3: set_set_a,X: set_a] :
% 3.18/1.69 ( ( ord_le318720350_set_a @ A3 @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
% 3.18/1.69 => ( ( A3 = bot_bot_set_set_a )
% 3.18/1.69 | ( A3
% 3.18/1.69 = ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_313_mk__disjoint__insert,axiom,
% 3.18/1.69 ! [A: a,A3: set_a] :
% 3.18/1.69 ( ( member_a @ A @ A3 )
% 3.18/1.69 => ? [B6: set_a] :
% 3.18/1.69 ( ( A3
% 3.18/1.69 = ( insert_a @ A @ B6 ) )
% 3.18/1.69 & ~ ( member_a @ A @ B6 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_314_mk__disjoint__insert,axiom,
% 3.18/1.69 ! [A: product_prod_a_a,A3: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member449909584od_a_a @ A @ A3 )
% 3.18/1.69 => ? [B6: set_Product_prod_a_a] :
% 3.18/1.69 ( ( A3
% 3.18/1.69 = ( insert1116662519od_a_a @ A @ B6 ) )
% 3.18/1.69 & ~ ( member449909584od_a_a @ A @ B6 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_315_mk__disjoint__insert,axiom,
% 3.18/1.69 ! [A: set_Product_prod_a_a,A3: set_se1596668135od_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ A @ A3 )
% 3.18/1.69 => ? [B6: set_se1596668135od_a_a] :
% 3.18/1.69 ( ( A3
% 3.18/1.69 = ( insert1738756823od_a_a @ A @ B6 ) )
% 3.18/1.69 & ~ ( member1838126896od_a_a @ A @ B6 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_316_mk__disjoint__insert,axiom,
% 3.18/1.69 ! [A: set_a,A3: set_set_a] :
% 3.18/1.69 ( ( member_set_a @ A @ A3 )
% 3.18/1.69 => ? [B6: set_set_a] :
% 3.18/1.69 ( ( A3
% 3.18/1.69 = ( insert_set_a @ A @ B6 ) )
% 3.18/1.69 & ~ ( member_set_a @ A @ B6 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_317_insert__commute,axiom,
% 3.18/1.69 ! [X: a,Y: a,A3: set_a] :
% 3.18/1.69 ( ( insert_a @ X @ ( insert_a @ Y @ A3 ) )
% 3.18/1.69 = ( insert_a @ Y @ ( insert_a @ X @ A3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_318_insert__eq__iff,axiom,
% 3.18/1.69 ! [A: a,A3: set_a,B: a,B4: set_a] :
% 3.18/1.69 ( ~ ( member_a @ A @ A3 )
% 3.18/1.69 => ( ~ ( member_a @ B @ B4 )
% 3.18/1.69 => ( ( ( insert_a @ A @ A3 )
% 3.18/1.69 = ( insert_a @ B @ B4 ) )
% 3.18/1.69 = ( ( ( A = B )
% 3.18/1.69 => ( A3 = B4 ) )
% 3.18/1.69 & ( ( A != B )
% 3.18/1.69 => ? [C3: set_a] :
% 3.18/1.69 ( ( A3
% 3.18/1.69 = ( insert_a @ B @ C3 ) )
% 3.18/1.69 & ~ ( member_a @ B @ C3 )
% 3.18/1.69 & ( B4
% 3.18/1.69 = ( insert_a @ A @ C3 ) )
% 3.18/1.69 & ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_319_insert__eq__iff,axiom,
% 3.18/1.69 ! [A: product_prod_a_a,A3: set_Product_prod_a_a,B: product_prod_a_a,B4: set_Product_prod_a_a] :
% 3.18/1.69 ( ~ ( member449909584od_a_a @ A @ A3 )
% 3.18/1.69 => ( ~ ( member449909584od_a_a @ B @ B4 )
% 3.18/1.69 => ( ( ( insert1116662519od_a_a @ A @ A3 )
% 3.18/1.69 = ( insert1116662519od_a_a @ B @ B4 ) )
% 3.18/1.69 = ( ( ( A = B )
% 3.18/1.69 => ( A3 = B4 ) )
% 3.18/1.69 & ( ( A != B )
% 3.18/1.69 => ? [C3: set_Product_prod_a_a] :
% 3.18/1.69 ( ( A3
% 3.18/1.69 = ( insert1116662519od_a_a @ B @ C3 ) )
% 3.18/1.69 & ~ ( member449909584od_a_a @ B @ C3 )
% 3.18/1.69 & ( B4
% 3.18/1.69 = ( insert1116662519od_a_a @ A @ C3 ) )
% 3.18/1.69 & ~ ( member449909584od_a_a @ A @ C3 ) ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_320_insert__eq__iff,axiom,
% 3.18/1.69 ! [A: set_Product_prod_a_a,A3: set_se1596668135od_a_a,B: set_Product_prod_a_a,B4: set_se1596668135od_a_a] :
% 3.18/1.69 ( ~ ( member1838126896od_a_a @ A @ A3 )
% 3.18/1.69 => ( ~ ( member1838126896od_a_a @ B @ B4 )
% 3.18/1.69 => ( ( ( insert1738756823od_a_a @ A @ A3 )
% 3.18/1.69 = ( insert1738756823od_a_a @ B @ B4 ) )
% 3.18/1.69 = ( ( ( A = B )
% 3.18/1.69 => ( A3 = B4 ) )
% 3.18/1.69 & ( ( A != B )
% 3.18/1.69 => ? [C3: set_se1596668135od_a_a] :
% 3.18/1.69 ( ( A3
% 3.18/1.69 = ( insert1738756823od_a_a @ B @ C3 ) )
% 3.18/1.69 & ~ ( member1838126896od_a_a @ B @ C3 )
% 3.18/1.69 & ( B4
% 3.18/1.69 = ( insert1738756823od_a_a @ A @ C3 ) )
% 3.18/1.69 & ~ ( member1838126896od_a_a @ A @ C3 ) ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_321_insert__eq__iff,axiom,
% 3.18/1.69 ! [A: set_a,A3: set_set_a,B: set_a,B4: set_set_a] :
% 3.18/1.69 ( ~ ( member_set_a @ A @ A3 )
% 3.18/1.69 => ( ~ ( member_set_a @ B @ B4 )
% 3.18/1.69 => ( ( ( insert_set_a @ A @ A3 )
% 3.18/1.69 = ( insert_set_a @ B @ B4 ) )
% 3.18/1.69 = ( ( ( A = B )
% 3.18/1.69 => ( A3 = B4 ) )
% 3.18/1.69 & ( ( A != B )
% 3.18/1.69 => ? [C3: set_set_a] :
% 3.18/1.69 ( ( A3
% 3.18/1.69 = ( insert_set_a @ B @ C3 ) )
% 3.18/1.69 & ~ ( member_set_a @ B @ C3 )
% 3.18/1.69 & ( B4
% 3.18/1.69 = ( insert_set_a @ A @ C3 ) )
% 3.18/1.69 & ~ ( member_set_a @ A @ C3 ) ) ) ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_322_insert__absorb,axiom,
% 3.18/1.69 ! [A: a,A3: set_a] :
% 3.18/1.69 ( ( member_a @ A @ A3 )
% 3.18/1.69 => ( ( insert_a @ A @ A3 )
% 3.18/1.69 = A3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_323_insert__absorb,axiom,
% 3.18/1.69 ! [A: product_prod_a_a,A3: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member449909584od_a_a @ A @ A3 )
% 3.18/1.69 => ( ( insert1116662519od_a_a @ A @ A3 )
% 3.18/1.69 = A3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_324_insert__absorb,axiom,
% 3.18/1.69 ! [A: set_Product_prod_a_a,A3: set_se1596668135od_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ A @ A3 )
% 3.18/1.69 => ( ( insert1738756823od_a_a @ A @ A3 )
% 3.18/1.69 = A3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_325_insert__absorb,axiom,
% 3.18/1.69 ! [A: set_a,A3: set_set_a] :
% 3.18/1.69 ( ( member_set_a @ A @ A3 )
% 3.18/1.69 => ( ( insert_set_a @ A @ A3 )
% 3.18/1.69 = A3 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_326_insert__ident,axiom,
% 3.18/1.69 ! [X: a,A3: set_a,B4: set_a] :
% 3.18/1.69 ( ~ ( member_a @ X @ A3 )
% 3.18/1.69 => ( ~ ( member_a @ X @ B4 )
% 3.18/1.69 => ( ( ( insert_a @ X @ A3 )
% 3.18/1.69 = ( insert_a @ X @ B4 ) )
% 3.18/1.69 = ( A3 = B4 ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_327_insert__ident,axiom,
% 3.18/1.69 ! [X: product_prod_a_a,A3: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
% 3.18/1.69 ( ~ ( member449909584od_a_a @ X @ A3 )
% 3.18/1.69 => ( ~ ( member449909584od_a_a @ X @ B4 )
% 3.18/1.69 => ( ( ( insert1116662519od_a_a @ X @ A3 )
% 3.18/1.69 = ( insert1116662519od_a_a @ X @ B4 ) )
% 3.18/1.69 = ( A3 = B4 ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_328_insert__ident,axiom,
% 3.18/1.69 ! [X: set_Product_prod_a_a,A3: set_se1596668135od_a_a,B4: set_se1596668135od_a_a] :
% 3.18/1.69 ( ~ ( member1838126896od_a_a @ X @ A3 )
% 3.18/1.69 => ( ~ ( member1838126896od_a_a @ X @ B4 )
% 3.18/1.69 => ( ( ( insert1738756823od_a_a @ X @ A3 )
% 3.18/1.69 = ( insert1738756823od_a_a @ X @ B4 ) )
% 3.18/1.69 = ( A3 = B4 ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_329_insert__ident,axiom,
% 3.18/1.69 ! [X: set_a,A3: set_set_a,B4: set_set_a] :
% 3.18/1.69 ( ~ ( member_set_a @ X @ A3 )
% 3.18/1.69 => ( ~ ( member_set_a @ X @ B4 )
% 3.18/1.69 => ( ( ( insert_set_a @ X @ A3 )
% 3.18/1.69 = ( insert_set_a @ X @ B4 ) )
% 3.18/1.69 = ( A3 = B4 ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_330_Set_Oset__insert,axiom,
% 3.18/1.69 ! [X: a,A3: set_a] :
% 3.18/1.69 ( ( member_a @ X @ A3 )
% 3.18/1.69 => ~ ! [B6: set_a] :
% 3.18/1.69 ( ( A3
% 3.18/1.69 = ( insert_a @ X @ B6 ) )
% 3.18/1.69 => ( member_a @ X @ B6 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_331_Set_Oset__insert,axiom,
% 3.18/1.69 ! [X: product_prod_a_a,A3: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member449909584od_a_a @ X @ A3 )
% 3.18/1.69 => ~ ! [B6: set_Product_prod_a_a] :
% 3.18/1.69 ( ( A3
% 3.18/1.69 = ( insert1116662519od_a_a @ X @ B6 ) )
% 3.18/1.69 => ( member449909584od_a_a @ X @ B6 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_332_Set_Oset__insert,axiom,
% 3.18/1.69 ! [X: set_Product_prod_a_a,A3: set_se1596668135od_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ X @ A3 )
% 3.18/1.69 => ~ ! [B6: set_se1596668135od_a_a] :
% 3.18/1.69 ( ( A3
% 3.18/1.69 = ( insert1738756823od_a_a @ X @ B6 ) )
% 3.18/1.69 => ( member1838126896od_a_a @ X @ B6 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_333_Set_Oset__insert,axiom,
% 3.18/1.69 ! [X: set_a,A3: set_set_a] :
% 3.18/1.69 ( ( member_set_a @ X @ A3 )
% 3.18/1.69 => ~ ! [B6: set_set_a] :
% 3.18/1.69 ( ( A3
% 3.18/1.69 = ( insert_set_a @ X @ B6 ) )
% 3.18/1.69 => ( member_set_a @ X @ B6 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_334_insertI2,axiom,
% 3.18/1.69 ! [A: a,B4: set_a,B: a] :
% 3.18/1.69 ( ( member_a @ A @ B4 )
% 3.18/1.69 => ( member_a @ A @ ( insert_a @ B @ B4 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_335_insertI2,axiom,
% 3.18/1.69 ! [A: product_prod_a_a,B4: set_Product_prod_a_a,B: product_prod_a_a] :
% 3.18/1.69 ( ( member449909584od_a_a @ A @ B4 )
% 3.18/1.69 => ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ B4 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_336_insertI2,axiom,
% 3.18/1.69 ! [A: set_Product_prod_a_a,B4: set_se1596668135od_a_a,B: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ A @ B4 )
% 3.18/1.69 => ( member1838126896od_a_a @ A @ ( insert1738756823od_a_a @ B @ B4 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_337_insertI2,axiom,
% 3.18/1.69 ! [A: set_a,B4: set_set_a,B: set_a] :
% 3.18/1.69 ( ( member_set_a @ A @ B4 )
% 3.18/1.69 => ( member_set_a @ A @ ( insert_set_a @ B @ B4 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_338_insertI1,axiom,
% 3.18/1.69 ! [A: a,B4: set_a] : ( member_a @ A @ ( insert_a @ A @ B4 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_339_insertI1,axiom,
% 3.18/1.69 ! [A: product_prod_a_a,B4: set_Product_prod_a_a] : ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ A @ B4 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_340_insertI1,axiom,
% 3.18/1.69 ! [A: set_Product_prod_a_a,B4: set_se1596668135od_a_a] : ( member1838126896od_a_a @ A @ ( insert1738756823od_a_a @ A @ B4 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_341_insertI1,axiom,
% 3.18/1.69 ! [A: set_a,B4: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B4 ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_342_insertE,axiom,
% 3.18/1.69 ! [A: a,B: a,A3: set_a] :
% 3.18/1.69 ( ( member_a @ A @ ( insert_a @ B @ A3 ) )
% 3.18/1.69 => ( ( A != B )
% 3.18/1.69 => ( member_a @ A @ A3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_343_insertE,axiom,
% 3.18/1.69 ! [A: product_prod_a_a,B: product_prod_a_a,A3: set_Product_prod_a_a] :
% 3.18/1.69 ( ( member449909584od_a_a @ A @ ( insert1116662519od_a_a @ B @ A3 ) )
% 3.18/1.69 => ( ( A != B )
% 3.18/1.69 => ( member449909584od_a_a @ A @ A3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_344_insertE,axiom,
% 3.18/1.69 ! [A: set_Product_prod_a_a,B: set_Product_prod_a_a,A3: set_se1596668135od_a_a] :
% 3.18/1.69 ( ( member1838126896od_a_a @ A @ ( insert1738756823od_a_a @ B @ A3 ) )
% 3.18/1.69 => ( ( A != B )
% 3.18/1.69 => ( member1838126896od_a_a @ A @ A3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_345_insertE,axiom,
% 3.18/1.69 ! [A: set_a,B: set_a,A3: set_set_a] :
% 3.18/1.69 ( ( member_set_a @ A @ ( insert_set_a @ B @ A3 ) )
% 3.18/1.69 => ( ( A != B )
% 3.18/1.69 => ( member_set_a @ A @ A3 ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_346_singleton__inject,axiom,
% 3.18/1.69 ! [A: product_prod_a_a,B: product_prod_a_a] :
% 3.18/1.69 ( ( ( insert1116662519od_a_a @ A @ bot_bo2131659635od_a_a )
% 3.18/1.69 = ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a ) )
% 3.18/1.69 => ( A = B ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_347_singleton__inject,axiom,
% 3.18/1.69 ! [A: a,B: a] :
% 3.18/1.69 ( ( ( insert_a @ A @ bot_bot_set_a )
% 3.18/1.69 = ( insert_a @ B @ bot_bot_set_a ) )
% 3.18/1.69 => ( A = B ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_348_insert__not__empty,axiom,
% 3.18/1.69 ! [A: product_prod_a_a,A3: set_Product_prod_a_a] :
% 3.18/1.69 ( ( insert1116662519od_a_a @ A @ A3 )
% 3.18/1.69 != bot_bo2131659635od_a_a ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_349_insert__not__empty,axiom,
% 3.18/1.69 ! [A: a,A3: set_a] :
% 3.18/1.69 ( ( insert_a @ A @ A3 )
% 3.18/1.69 != bot_bot_set_a ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_350_doubleton__eq__iff,axiom,
% 3.18/1.69 ! [A: product_prod_a_a,B: product_prod_a_a,C: product_prod_a_a,D: product_prod_a_a] :
% 3.18/1.69 ( ( ( insert1116662519od_a_a @ A @ ( insert1116662519od_a_a @ B @ bot_bo2131659635od_a_a ) )
% 3.18/1.69 = ( insert1116662519od_a_a @ C @ ( insert1116662519od_a_a @ D @ bot_bo2131659635od_a_a ) ) )
% 3.18/1.69 = ( ( ( A = C )
% 3.18/1.69 & ( B = D ) )
% 3.18/1.69 | ( ( A = D )
% 3.18/1.69 & ( B = C ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_351_doubleton__eq__iff,axiom,
% 3.18/1.69 ! [A: a,B: a,C: a,D: a] :
% 3.18/1.69 ( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
% 3.18/1.69 = ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
% 3.18/1.69 = ( ( ( A = C )
% 3.18/1.69 & ( B = D ) )
% 3.18/1.69 | ( ( A = D )
% 3.18/1.69 & ( B = C ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_352_field__intro2,axiom,
% 3.18/1.69 ( ( ( zero_a_b @ r )
% 3.18/1.69 != ( one_a_ring_ext_a_b @ r ) )
% 3.18/1.69 => ( ! [X2: a] :
% 3.18/1.69 ( ( member_a @ X2 @ ( minus_minus_set_a @ ( partia1066395285xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
% 3.18/1.69 => ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) ) )
% 3.18/1.69 => ( field_a_b @ r ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_353_cring__fieldI,axiom,
% 3.18/1.69 ( ( ( units_a_ring_ext_a_b @ r )
% 3.18/1.69 = ( minus_minus_set_a @ ( partia1066395285xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
% 3.18/1.69 => ( field_a_b @ r ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(fact_354_rcosets__subset__PowG,axiom,
% 3.18/1.69 ! [H: set_a] :
% 3.18/1.69 ( ( additi2104487374up_a_b @ H @ r )
% 3.18/1.69 => ( ord_le318720350_set_a @ ( a_RCOSETS_a_b @ r @ H ) @ ( pow_a @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 3.18/1.69
% 3.18/1.69 thf(conj_0,conjecture,
% 3.18/1.69 ( ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) )
% 3.18/1.69 = ( mult_s977248631t_unit @ ( locali1944243560ac_a_b @ r @ s ) @ ( locali1170246543t_unit @ rel @ r3 @ s3 ) @ ( locali1170246543t_unit @ rel @ r2 @ s2 ) ) ) ).
% 3.18/1.69 ------------------------
% 3.31/1.72 % (6517)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_6318 for (2999ds/27Mi)
% 3.31/1.72 % (6518)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_6318 for (2999ds/2Mi)
% 3.31/1.72 % (6521)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_6318 for (2999ds/18Mi)
% 3.31/1.72 % (6516)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_6318 for (2999ds/4Mi)
% 3.31/1.72 % (6520)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_6318 for (2999ds/275Mi)
% 3.31/1.72 % (6518)Instruction limit reached!
% 3.31/1.72 % (6518)------------------------------
% 3.31/1.72 % (6518)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.31/1.72 % (6518)Termination reason: Unknown
% 3.31/1.72 % (6518)Termination phase: shuffling
% 3.31/1.72
% 3.31/1.72 % (6518)Memory used [KB]: 1535
% 3.31/1.72 % (6518)Time elapsed: 0.003 s
% 3.31/1.72 % (6518)Instructions burned: 4 (million)
% 3.31/1.72 % (6518)------------------------------
% 3.31/1.72 % (6518)------------------------------
% 3.31/1.72 % (6516)Instruction limit reached!
% 3.31/1.72 % (6516)------------------------------
% 3.31/1.72 % (6516)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.31/1.72 % (6516)Termination reason: Unknown
% 3.31/1.72 % (6516)Termination phase: shuffling
% 3.31/1.72
% 3.31/1.72 % (6516)Memory used [KB]: 1535
% 3.31/1.72 % (6516)Time elapsed: 0.003 s
% 3.31/1.72 % (6516)Instructions burned: 6 (million)
% 3.31/1.72 % (6516)------------------------------
% 3.31/1.72 % (6516)------------------------------
% 3.31/1.72 % (6521)Instruction limit reached!
% 3.31/1.72 % (6521)------------------------------
% 3.31/1.72 % (6521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.31/1.72 % (6521)Termination reason: Unknown
% 3.31/1.72 % (6521)Termination phase: shuffling
% 3.31/1.72
% 3.31/1.72 % (6521)Memory used [KB]: 1791
% 3.31/1.72 % (6521)Time elapsed: 0.008 s
% 3.31/1.72 % (6521)Instructions burned: 21 (million)
% 3.31/1.72 % (6521)------------------------------
% 3.31/1.72 % (6521)------------------------------
% 3.31/1.72 % (6515)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_6318 for (2999ds/183Mi)
% 3.31/1.72 % (6519)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_6318 for (2999ds/2Mi)
% 3.31/1.72 % (6517)Instruction limit reached!
% 3.31/1.72 % (6517)------------------------------
% 3.31/1.72 % (6517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.31/1.72 % (6517)Termination reason: Unknown
% 3.31/1.72 % (6517)Termination phase: shuffling
% 3.31/1.72
% 3.31/1.72 % (6517)Memory used [KB]: 1918
% 3.31/1.72 % (6517)Time elapsed: 0.010 s
% 3.31/1.72 % (6517)Instructions burned: 27 (million)
% 3.31/1.72 % (6517)------------------------------
% 3.31/1.72 % (6517)------------------------------
% 3.31/1.73 % (6519)Instruction limit reached!
% 3.31/1.73 % (6519)------------------------------
% 3.31/1.73 % (6519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.31/1.73 % (6519)Termination reason: Unknown
% 3.31/1.73 % (6519)Termination phase: shuffling
% 3.31/1.73
% 3.31/1.73 % (6519)Memory used [KB]: 1535
% 3.31/1.73 % (6519)Time elapsed: 0.003 s
% 3.31/1.73 % (6519)Instructions burned: 5 (million)
% 3.31/1.73 % (6519)------------------------------
% 3.31/1.73 % (6519)------------------------------
% 3.31/1.73 % (6523)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_6318 for (2999ds/3Mi)
% 3.31/1.73 % (6523)Instruction limit reached!
% 3.31/1.73 % (6523)------------------------------
% 3.31/1.73 % (6523)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.31/1.73 % (6523)Termination reason: Unknown
% 3.31/1.73 % (6523)Termination phase: shuffling
% 3.31/1.73
% 3.31/1.73 % (6523)Memory used [KB]: 1407
% 3.31/1.73 % (6523)Time elapsed: 0.002 s
% 3.31/1.73 % (6523)Instructions burned: 3 (million)
% 3.31/1.73 % (6523)------------------------------
% 3.31/1.73 % (6523)------------------------------
% 3.31/1.73 % (6524)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_6318 for (2999ds/37Mi)
% 3.31/1.73 % (6526)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_6318 for (2999ds/15Mi)
% 3.31/1.73 % (6527)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_6318 for (2999ds/3Mi)
% 3.31/1.73 % (6527)Instruction limit reached!
% 3.31/1.73 % (6527)------------------------------
% 3.31/1.73 % (6527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.31/1.73 % (6527)Termination reason: Unknown
% 3.31/1.73 % (6527)Termination phase: shuffling
% 3.31/1.73
% 3.31/1.73 % (6527)Memory used [KB]: 1407
% 3.31/1.73 % (6527)Time elapsed: 0.002 s
% 3.31/1.74 % (6527)Instructions burned: 3 (million)
% 3.31/1.74 % (6527)------------------------------
% 3.31/1.74 % (6527)------------------------------
% 3.31/1.74 % (6528)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_6318 for (2999ds/1041Mi)
% 3.31/1.74 % (6529)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_6318 for (2999ds/7Mi)
% 3.31/1.74 % (6526)Instruction limit reached!
% 3.31/1.74 % (6526)------------------------------
% 3.31/1.74 % (6526)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.31/1.74 % (6526)Termination reason: Unknown
% 3.31/1.74 % (6526)Termination phase: shuffling
% 3.31/1.74
% 3.31/1.74 % (6526)Memory used [KB]: 1663
% 3.31/1.74 % (6526)Time elapsed: 0.007 s
% 3.31/1.74 % (6526)Instructions burned: 16 (million)
% 3.31/1.74 % (6526)------------------------------
% 3.31/1.74 % (6526)------------------------------
% 3.31/1.74 % (6529)Instruction limit reached!
% 3.31/1.74 % (6529)------------------------------
% 3.31/1.74 % (6529)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.31/1.74 % (6529)Termination reason: Unknown
% 3.31/1.74 % (6529)Termination phase: shuffling
% 3.31/1.74
% 3.31/1.74 % (6529)Memory used [KB]: 1535
% 3.31/1.74 % (6529)Time elapsed: 0.003 s
% 3.31/1.74 % (6529)Instructions burned: 7 (million)
% 3.31/1.74 % (6529)------------------------------
% 3.31/1.74 % (6529)------------------------------
% 3.31/1.74 % (6520)First to succeed.
% 3.31/1.74 % (6524)Instruction limit reached!
% 3.31/1.74 % (6524)------------------------------
% 3.31/1.74 % (6524)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.31/1.74 % (6524)Termination reason: Unknown
% 3.31/1.74 % (6524)Termination phase: shuffling
% 3.31/1.74
% 3.31/1.74 % (6524)Memory used [KB]: 2174
% 3.31/1.74 % (6524)Time elapsed: 0.013 s
% 3.31/1.74 % (6524)Instructions burned: 39 (million)
% 3.31/1.74 % (6524)------------------------------
% 3.31/1.74 % (6524)------------------------------
% 3.31/1.74 % (6520)Refutation found. Thanks to Tanya!
% 3.31/1.74 % SZS status Theorem for DTF2THF_6318
% 3.31/1.74 % SZS output start Proof for DTF2THF_6318
% 3.31/1.74 thf(type_def_5, type, partia1833973666xt_a_b: $tType).
% 3.31/1.74 thf(type_def_6, type, set_a: $tType).
% 3.31/1.74 thf(type_def_7, type, set_set_a: $tType).
% 3.31/1.74 thf(type_def_9, type, a: $tType).
% 3.31/1.74 thf(type_def_10, type, partia1993116613t_unit: $tType).
% 3.31/1.74 thf(type_def_11, type, set_Product_prod_a_a: $tType).
% 3.31/1.74 thf(type_def_12, type, partia539144763t_unit: $tType).
% 3.31/1.74 thf(type_def_13, type, set_se1596668135od_a_a: $tType).
% 3.31/1.74 thf(type_def_14, type, partia96731725t_unit: $tType).
% 3.31/1.74 thf(type_def_15, type, product_prod_a_a: $tType).
% 3.31/1.74 thf(type_def_16, type, int: $tType).
% 3.31/1.74 thf(type_def_17, type, nat: $tType).
% 3.31/1.74 thf(func_def_0, type, partia539144763t_unit: $tType).
% 3.31/1.74 thf(func_def_1, type, partia1993116613t_unit: $tType).
% 3.31/1.74 thf(func_def_2, type, partia1833973666xt_a_b: $tType).
% 3.31/1.74 thf(func_def_3, type, partia96731725t_unit: $tType).
% 3.31/1.74 thf(func_def_4, type, set_se1596668135od_a_a: $tType).
% 3.31/1.74 thf(func_def_5, type, set_Product_prod_a_a: $tType).
% 3.31/1.74 thf(func_def_6, type, product_prod_a_a: $tType).
% 3.31/1.74 thf(func_def_7, type, set_set_a: $tType).
% 3.31/1.74 thf(func_def_8, type, set_a: $tType).
% 3.31/1.74 thf(func_def_9, type, nat: $tType).
% 3.31/1.74 thf(func_def_10, type, int: $tType).
% 3.31/1.74 thf(func_def_11, type, a: $tType).
% 3.31/1.74 thf(func_def_12, type, a_RCOSETS_a_b: partia1833973666xt_a_b > set_a > set_set_a).
% 3.31/1.74 thf(func_def_13, type, a_l_coset_a_b: partia1833973666xt_a_b > a > set_a > set_a).
% 3.31/1.74 thf(func_def_14, type, a_r_coset_a_b: partia1833973666xt_a_b > set_a > a > set_a).
% 3.31/1.74 thf(func_def_15, type, additi2104487374up_a_b: set_a > partia1833973666xt_a_b > $o).
% 3.31/1.74 thf(func_def_16, type, set_add_a_b: partia1833973666xt_a_b > set_a > set_a > set_a).
% 3.31/1.74 thf(func_def_17, type, equiva2136214035t_unit: partia1993116613t_unit > $o).
% 3.31/1.74 thf(func_def_18, type, partia206007992t_unit: partia1993116613t_unit > set_Product_prod_a_a).
% 3.31/1.74 thf(func_def_19, type, partia616129906t_unit: partia539144763t_unit > set_se1596668135od_a_a).
% 3.31/1.74 thf(func_def_20, type, partia1955795460t_unit: partia96731725t_unit > set_a).
% 3.31/1.74 thf(func_def_21, type, partia1066395285xt_a_b: partia1833973666xt_a_b > set_a).
% 3.31/1.74 thf(func_def_22, type, units_1734235042t_unit: partia539144763t_unit > set_se1596668135od_a_a).
% 3.31/1.74 thf(func_def_23, type, units_a_Product_unit: partia96731725t_unit > set_a).
% 3.31/1.74 thf(func_def_24, type, units_a_ring_ext_a_b: partia1833973666xt_a_b > set_a).
% 3.31/1.74 thf(func_def_25, type, group_52730354t_unit: partia539144763t_unit > $o).
% 3.31/1.74 thf(func_def_26, type, group_a_Product_unit: partia96731725t_unit > $o).
% 3.31/1.74 thf(func_def_27, type, group_a_ring_ext_a_b: partia1833973666xt_a_b > $o).
% 3.31/1.74 thf(func_def_28, type, monoid1242292471t_unit: partia539144763t_unit > $o).
% 3.31/1.74 thf(func_def_29, type, monoid1447254976t_unit: partia96731725t_unit > $o).
% 3.31/1.74 thf(func_def_30, type, monoid1022105667xt_a_b: partia1833973666xt_a_b > $o).
% 3.31/1.74 thf(func_def_31, type, mult_s977248631t_unit: partia539144763t_unit > set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a).
% 3.31/1.74 thf(func_def_32, type, mult_a_Product_unit: partia96731725t_unit > a > a > a).
% 3.31/1.74 thf(func_def_33, type, mult_a_ring_ext_a_b: partia1833973666xt_a_b > a > a > a).
% 3.31/1.74 thf(func_def_34, type, one_se1488596747t_unit: partia539144763t_unit > set_Product_prod_a_a).
% 3.31/1.74 thf(func_def_35, type, one_a_Product_unit: partia96731725t_unit > a).
% 3.31/1.74 thf(func_def_36, type, one_a_ring_ext_a_b: partia1833973666xt_a_b > a).
% 3.31/1.74 thf(func_def_37, type, units_390215097t_unit: partia539144763t_unit > partia539144763t_unit).
% 3.31/1.74 thf(func_def_38, type, units_1411277569xt_a_b: partia1833973666xt_a_b > partia96731725t_unit).
% 3.31/1.74 thf(func_def_39, type, minus_minus_set_a: set_a > set_a > set_a).
% 3.31/1.74 thf(func_def_40, type, locali1170246543t_unit: partia1993116613t_unit > a > a > set_Product_prod_a_a).
% 3.31/1.74 thf(func_def_41, type, locali1648887798ac_a_b: partia1833973666xt_a_b > set_a > $o).
% 3.31/1.74 thf(func_def_42, type, locali1944243560ac_a_b: partia1833973666xt_a_b > set_a > partia539144763t_unit).
% 3.31/1.74 thf(func_def_43, type, locali807230110ng_a_b: partia1833973666xt_a_b > set_a > $o).
% 3.31/1.74 thf(func_def_44, type, locali880295127ng_a_b: partia1833973666xt_a_b > set_a > $o).
% 3.31/1.74 thf(func_def_45, type, locali1538946157t_unit: partia1993116613t_unit > set_se1596668135od_a_a).
% 3.31/1.74 thf(func_def_46, type, locali1488796916t_unit: partia539144763t_unit > set_se1596668135od_a_a > $o).
% 3.31/1.74 thf(func_def_47, type, locali425460998xt_a_b: partia1833973666xt_a_b > set_a > $o).
% 3.31/1.74 thf(func_def_48, type, bot_bo2131659635od_a_a: set_Product_prod_a_a).
% 3.31/1.74 thf(func_def_49, type, bot_bo1424703315od_a_a: set_se1596668135od_a_a).
% 3.31/1.74 thf(func_def_50, type, bot_bot_set_set_a: set_set_a).
% 3.31/1.74 thf(func_def_51, type, bot_bot_set_a: set_a).
% 3.31/1.74 thf(func_def_52, type, ord_le1824328871od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o).
% 3.31/1.74 thf(func_def_53, type, ord_le391835783od_a_a: set_se1596668135od_a_a > set_se1596668135od_a_a > $o).
% 3.31/1.74 thf(func_def_54, type, ord_le318720350_set_a: set_set_a > set_set_a > $o).
% 3.31/1.74 thf(func_def_55, type, ord_less_eq_set_a: set_a > set_a > $o).
% 3.31/1.74 thf(func_def_56, type, product_Pair_a_a: a > a > product_prod_a_a).
% 3.31/1.74 thf(func_def_57, type, abelian_monoid_a_b: partia1833973666xt_a_b > $o).
% 3.31/1.74 thf(func_def_58, type, add_pow_a_b_int: partia1833973666xt_a_b > int > a > a).
% 3.31/1.74 thf(func_def_59, type, add_pow_a_b_nat: partia1833973666xt_a_b > nat > a > a).
% 3.31/1.74 thf(func_def_60, type, field_a_b: partia1833973666xt_a_b > $o).
% 3.31/1.74 thf(func_def_61, type, ring_a_b: partia1833973666xt_a_b > $o).
% 3.31/1.74 thf(func_def_62, type, zero_a_b: partia1833973666xt_a_b > a).
% 3.31/1.74 thf(func_def_63, type, semiring_a_b: partia1833973666xt_a_b > $o).
% 3.31/1.74 thf(func_def_64, type, collec645855634od_a_a: (product_prod_a_a > $o) > set_Product_prod_a_a).
% 3.31/1.74 thf(func_def_65, type, collec183727474od_a_a: (set_Product_prod_a_a > $o) > set_se1596668135od_a_a).
% 3.31/1.74 thf(func_def_66, type, collect_set_a: (set_a > $o) > set_set_a).
% 3.31/1.74 thf(func_def_67, type, collect_a: (a > $o) > set_a).
% 3.31/1.74 thf(func_def_68, type, pow_a: set_a > set_set_a).
% 3.31/1.74 thf(func_def_69, type, insert1116662519od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a).
% 3.31/1.74 thf(func_def_70, type, insert1738756823od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a > set_se1596668135od_a_a).
% 3.31/1.74 thf(func_def_71, type, insert_set_a: set_a > set_set_a > set_set_a).
% 3.31/1.74 thf(func_def_72, type, insert_a: a > set_a > set_a).
% 3.31/1.74 thf(func_def_73, type, member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o).
% 3.31/1.74 thf(func_def_74, type, member1838126896od_a_a: set_Product_prod_a_a > set_se1596668135od_a_a > $o).
% 3.31/1.74 thf(func_def_75, type, member_set_a: set_a > set_set_a > $o).
% 3.31/1.74 thf(func_def_76, type, member_a: a > set_a > $o).
% 3.31/1.74 thf(func_def_77, type, r: partia1833973666xt_a_b).
% 3.31/1.74 thf(func_def_78, type, s: set_a).
% 3.31/1.74 thf(func_def_79, type, r2: a).
% 3.31/1.74 thf(func_def_80, type, r3: a).
% 3.31/1.74 thf(func_def_81, type, rel: partia1993116613t_unit).
% 3.31/1.74 thf(func_def_82, type, s2: a).
% 3.31/1.74 thf(func_def_83, type, s3: a).
% 3.31/1.74 thf(func_def_96, type, sP0: partia539144763t_unit > $o).
% 3.31/1.74 thf(func_def_97, type, sP1: partia539144763t_unit > $o).
% 3.31/1.74 thf(func_def_98, type, sP2: partia539144763t_unit > $o).
% 3.31/1.74 thf(func_def_99, type, sP3: partia539144763t_unit > $o).
% 3.31/1.74 thf(func_def_100, type, sK4: product_prod_a_a > a).
% 3.31/1.74 thf(func_def_101, type, sK5: product_prod_a_a > a).
% 3.31/1.74 thf(func_def_102, type, sK6: set_a > set_a > a).
% 3.31/1.74 thf(func_def_103, type, sK7: partia539144763t_unit > set_Product_prod_a_a > set_Product_prod_a_a).
% 3.31/1.74 thf(func_def_104, type, sK8: (product_prod_a_a > $o) > a).
% 3.31/1.74 thf(func_def_105, type, sK9: (product_prod_a_a > $o) > a).
% 3.31/1.74 thf(func_def_106, type, sK10: partia539144763t_unit > set_Product_prod_a_a).
% 3.31/1.74 thf(func_def_107, type, sK11: partia539144763t_unit > set_Product_prod_a_a).
% 3.31/1.74 thf(func_def_108, type, sK12: partia539144763t_unit > set_Product_prod_a_a).
% 3.31/1.74 thf(func_def_109, type, sK13: partia539144763t_unit > set_Product_prod_a_a).
% 3.31/1.74 thf(func_def_110, type, sK14: partia539144763t_unit > set_Product_prod_a_a).
% 3.31/1.74 thf(func_def_111, type, sK15: partia539144763t_unit > set_Product_prod_a_a).
% 3.31/1.74 thf(func_def_112, type, sK16: partia539144763t_unit > set_Product_prod_a_a).
% 3.31/1.74 thf(func_def_113, type, sK17: (product_prod_a_a > $o) > a).
% 3.31/1.74 thf(func_def_114, type, sK18: (product_prod_a_a > $o) > a).
% 3.31/1.74 thf(func_def_115, type, sK19: product_prod_a_a > a).
% 3.31/1.74 thf(func_def_116, type, sK20: product_prod_a_a > a).
% 3.31/1.74 thf(func_def_117, type, sK21: partia539144763t_unit > set_Product_prod_a_a).
% 3.31/1.74 thf(func_def_118, type, sK22: partia539144763t_unit > set_Product_prod_a_a).
% 3.31/1.74 thf(func_def_119, type, sK23: partia539144763t_unit > set_Product_prod_a_a).
% 3.31/1.74 thf(func_def_120, type, sK24: partia539144763t_unit > set_Product_prod_a_a).
% 3.31/1.74 thf(func_def_121, type, sK25: partia539144763t_unit > set_Product_prod_a_a).
% 3.31/1.74 thf(func_def_122, type, sK26: partia539144763t_unit > set_Product_prod_a_a).
% 3.31/1.74 thf(func_def_123, type, sK27: partia539144763t_unit > set_Product_prod_a_a).
% 3.31/1.74 thf(func_def_124, type, sK28: a > a).
% 3.31/1.74 thf(func_def_126, type, ph30: !>[X0: $tType]:(X0)).
% 3.31/1.74 thf(f1210,plain,(
% 3.31/1.74 $false),
% 3.31/1.74 inference(subsumption_resolution,[status(thm)],[f1209,f1147])).
% 3.31/1.74 thf(f1147,plain,(
% 3.31/1.74 ((mult_s977248631t_unit @ (locali1944243560ac_a_b @ r @ s) @ (locali1170246543t_unit @ rel @ r2 @ s2) @ (locali1170246543t_unit @ rel @ r3 @ s3)) = (locali1170246543t_unit @ rel @ (mult_a_ring_ext_a_b @ r @ r2 @ r3) @ (mult_a_ring_ext_a_b @ r @ s2 @ s3)))),
% 3.31/1.74 inference(cnf_transformation,[status(thm)],[f4])).
% 3.31/1.74 thf(f4,axiom,(
% 3.31/1.74 ((mult_s977248631t_unit @ (locali1944243560ac_a_b @ r @ s) @ (locali1170246543t_unit @ rel @ r2 @ s2) @ (locali1170246543t_unit @ rel @ r3 @ s3)) = (locali1170246543t_unit @ rel @ (mult_a_ring_ext_a_b @ r @ r2 @ r3) @ (mult_a_ring_ext_a_b @ r @ s2 @ s3)))),
% 3.31/1.74 file('/export/starexec/sandbox/tmp/tmp.7ORhJzyl8w/DTF2THF_6318.p',fact_3_f1)).
% 3.31/1.74 thf(f1209,plain,(
% 3.31/1.74 ((mult_s977248631t_unit @ (locali1944243560ac_a_b @ r @ s) @ (locali1170246543t_unit @ rel @ r2 @ s2) @ (locali1170246543t_unit @ rel @ r3 @ s3)) != (locali1170246543t_unit @ rel @ (mult_a_ring_ext_a_b @ r @ r2 @ r3) @ (mult_a_ring_ext_a_b @ r @ s2 @ s3)))),
% 3.31/1.74 inference(forward_demodulation,[status(thm)],[f1208,f1152])).
% 3.31/1.74 thf(f1152,plain,(
% 3.31/1.74 ((mult_a_ring_ext_a_b @ r @ s2 @ s3) = (mult_a_ring_ext_a_b @ r @ s3 @ s2))),
% 3.31/1.74 inference(cnf_transformation,[status(thm)],[f2])).
% 3.31/1.74 thf(f2,axiom,(
% 3.31/1.74 ((mult_a_ring_ext_a_b @ r @ s2 @ s3) = (mult_a_ring_ext_a_b @ r @ s3 @ s2))),
% 3.31/1.74 file('/export/starexec/sandbox/tmp/tmp.7ORhJzyl8w/DTF2THF_6318.p',fact_1_f4)).
% 3.31/1.74 thf(f1208,plain,(
% 3.31/1.74 ((mult_s977248631t_unit @ (locali1944243560ac_a_b @ r @ s) @ (locali1170246543t_unit @ rel @ r2 @ s2) @ (locali1170246543t_unit @ rel @ r3 @ s3)) != (locali1170246543t_unit @ rel @ (mult_a_ring_ext_a_b @ r @ r2 @ r3) @ (mult_a_ring_ext_a_b @ r @ s3 @ s2)))),
% 3.31/1.74 inference(forward_demodulation,[status(thm)],[f1207,f1158])).
% 3.31/1.74 thf(f1158,plain,(
% 3.31/1.74 ((mult_a_ring_ext_a_b @ r @ r2 @ r3) = (mult_a_ring_ext_a_b @ r @ r3 @ r2))),
% 3.31/1.74 inference(cnf_transformation,[status(thm)],[f1])).
% 3.31/1.74 thf(f1,axiom,(
% 3.31/1.74 ((mult_a_ring_ext_a_b @ r @ r2 @ r3) = (mult_a_ring_ext_a_b @ r @ r3 @ r2))),
% 3.31/1.74 file('/export/starexec/sandbox/tmp/tmp.7ORhJzyl8w/DTF2THF_6318.p',fact_0_f3)).
% 3.31/1.74 thf(f1207,plain,(
% 3.31/1.74 ((mult_s977248631t_unit @ (locali1944243560ac_a_b @ r @ s) @ (locali1170246543t_unit @ rel @ r2 @ s2) @ (locali1170246543t_unit @ rel @ r3 @ s3)) != (locali1170246543t_unit @ rel @ (mult_a_ring_ext_a_b @ r @ r3 @ r2) @ (mult_a_ring_ext_a_b @ r @ s3 @ s2)))),
% 3.31/1.74 inference(forward_demodulation,[status(thm)],[f1182,f1184])).
% 3.31/1.74 thf(f1184,plain,(
% 3.31/1.74 ((mult_s977248631t_unit @ (locali1944243560ac_a_b @ r @ s) @ (locali1170246543t_unit @ rel @ r3 @ s3) @ (locali1170246543t_unit @ rel @ r2 @ s2)) = (locali1170246543t_unit @ rel @ (mult_a_ring_ext_a_b @ r @ r3 @ r2) @ (mult_a_ring_ext_a_b @ r @ s3 @ s2)))),
% 3.31/1.74 inference(cnf_transformation,[status(thm)],[f5])).
% 3.31/1.74 thf(f5,axiom,(
% 3.31/1.74 ((mult_s977248631t_unit @ (locali1944243560ac_a_b @ r @ s) @ (locali1170246543t_unit @ rel @ r3 @ s3) @ (locali1170246543t_unit @ rel @ r2 @ s2)) = (locali1170246543t_unit @ rel @ (mult_a_ring_ext_a_b @ r @ r3 @ r2) @ (mult_a_ring_ext_a_b @ r @ s3 @ s2)))),
% 3.31/1.74 file('/export/starexec/sandbox/tmp/tmp.7ORhJzyl8w/DTF2THF_6318.p',fact_4_f2)).
% 3.31/1.74 thf(f1182,plain,(
% 3.31/1.74 ((mult_s977248631t_unit @ (locali1944243560ac_a_b @ r @ s) @ (locali1170246543t_unit @ rel @ r2 @ s2) @ (locali1170246543t_unit @ rel @ r3 @ s3)) != (mult_s977248631t_unit @ (locali1944243560ac_a_b @ r @ s) @ (locali1170246543t_unit @ rel @ r3 @ s3) @ (locali1170246543t_unit @ rel @ r2 @ s2)))),
% 3.31/1.74 inference(cnf_transformation,[status(thm)],[f1009])).
% 3.31/1.74 thf(f1009,plain,(
% 3.31/1.74 ((mult_s977248631t_unit @ (locali1944243560ac_a_b @ r @ s) @ (locali1170246543t_unit @ rel @ r2 @ s2) @ (locali1170246543t_unit @ rel @ r3 @ s3)) != (mult_s977248631t_unit @ (locali1944243560ac_a_b @ r @ s) @ (locali1170246543t_unit @ rel @ r3 @ s3) @ (locali1170246543t_unit @ rel @ r2 @ s2)))),
% 3.31/1.74 inference(flattening,[status(thm)],[f357])).
% 3.31/1.74 thf(f357,negated_conjecture,(
% 3.31/1.74 ~((mult_s977248631t_unit @ (locali1944243560ac_a_b @ r @ s) @ (locali1170246543t_unit @ rel @ r2 @ s2) @ (locali1170246543t_unit @ rel @ r3 @ s3)) = (mult_s977248631t_unit @ (locali1944243560ac_a_b @ r @ s) @ (locali1170246543t_unit @ rel @ r3 @ s3) @ (locali1170246543t_unit @ rel @ r2 @ s2)))),
% 3.31/1.74 inference(negated_conjecture,[status(cth)],[f356])).
% 3.31/1.74 thf(f356,conjecture,(
% 3.31/1.74 ((mult_s977248631t_unit @ (locali1944243560ac_a_b @ r @ s) @ (locali1170246543t_unit @ rel @ r2 @ s2) @ (locali1170246543t_unit @ rel @ r3 @ s3)) = (mult_s977248631t_unit @ (locali1944243560ac_a_b @ r @ s) @ (locali1170246543t_unit @ rel @ r3 @ s3) @ (locali1170246543t_unit @ rel @ r2 @ s2)))),
% 3.31/1.74 file('/export/starexec/sandbox/tmp/tmp.7ORhJzyl8w/DTF2THF_6318.p',conj_0)).
% 3.31/1.74 % SZS output end Proof for DTF2THF_6318
% 3.31/1.74 % (6520)------------------------------
% 3.31/1.74 % (6520)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.31/1.74 % (6520)Termination reason: Refutation
% 3.31/1.74
% 3.31/1.74 % (6520)Memory used [KB]: 6780
% 3.31/1.74 % (6520)Time elapsed: 0.024 s
% 3.31/1.74 % (6520)Instructions burned: 75 (million)
% 3.31/1.74 % (6520)------------------------------
% 3.31/1.74 % (6520)------------------------------
% 3.31/1.74 % (6514)Success in time 0.037 s
%------------------------------------------------------------------------------