TPTP Problem File: SLH0909^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Khovanskii_Theorem/0004_FiniteProduct/prob_00161_005089__13314950_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1497 ( 431 unt; 223 typ; 0 def)
% Number of atoms : 4326 (1282 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 14975 ( 426 ~; 48 |; 273 &;11884 @)
% ( 0 <=>;2344 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Number of types : 22 ( 21 usr)
% Number of type conns : 3377 (3377 >; 0 *; 0 +; 0 <<)
% Number of symbols : 204 ( 202 usr; 14 con; 0-6 aty)
% Number of variables : 4402 ( 403 ^;3939 !; 60 ?;4402 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:13:30.540
%------------------------------------------------------------------------------
% Could-be-implicit typings (21)
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_J_J,type,
set_nat_nat_a_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_Mt__Nat__Onat_J_J,type,
set_nat_a_a_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_Mtf__a_J_J,type,
set_nat_a_a_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__a_J_J_J,type,
set_nat_nat_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mt__Nat__Onat_J_J,type,
set_nat_a_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_It__Nat__Onat_Mtf__a_J_J_J,type,
set_a_nat_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_Itf__a_Mtf__a_J_J_J,type,
set_nat_a_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_Mtf__a_J_Mt__Nat__Onat_J_J,type,
set_a_a_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_J,type,
set_nat_a_a2: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_M_062_Itf__a_Mtf__a_J_J_J,type,
set_a_a_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
set_a_a_a2: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Nat__Onat_J_J,type,
set_a_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
set_nat_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
set_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (202)
thf(sy_c_FiniteProduct_Ocommutative__monoid_OM__ify_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
commut4120000896247601890at_a_a: set_nat_a_a2 > ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_OM__ify_001_062_It__Nat__Onat_Mtf__a_J,type,
commut2316704705022288065_nat_a: set_nat_a > ( nat > a ) > ( nat > a ) > nat > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_OM__ify_001_062_Itf__a_Mtf__a_J,type,
commut6112553959220001673fy_a_a: set_a_a > ( a > a ) > ( a > a ) > a > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_OM__ify_001t__Nat__Onat,type,
commut810702690453168372fy_nat: set_nat > nat > nat > nat ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_OM__ify_001tf__a,type,
commutative_M_ify_a: set_a > a > a > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001_062_It__Nat__Onat_Mtf__a_J_001t__Nat__Onat,type,
commut6753747983606973455_a_nat: set_nat_a > ( ( nat > a ) > ( nat > a ) > nat > a ) > ( nat > a ) > ( nat > nat > a ) > set_nat > nat > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001_062_It__Nat__Onat_Mtf__a_J_001tf__a,type,
commut1274061894236046463at_a_a: set_nat_a > ( ( nat > a ) > ( nat > a ) > nat > a ) > ( nat > a ) > ( a > nat > a ) > set_a > nat > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001_062_Itf__a_Mtf__a_J_001t__Nat__Onat,type,
commut6621034724473204317_a_nat: set_a_a > ( ( a > a ) > ( a > a ) > a > a ) > ( a > a ) > ( nat > a > a ) > set_nat > a > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001_062_Itf__a_Mtf__a_J_001tf__a,type,
commut5480430193892889009_a_a_a: set_a_a > ( ( a > a ) > ( a > a ) > a > a ) > ( a > a ) > ( a > a > a ) > set_a > a > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001t__Nat__Onat_001_062_It__Nat__Onat_Mtf__a_J,type,
commut5709917066755550957_nat_a: set_nat > ( nat > nat > nat ) > nat > ( ( nat > a ) > nat ) > set_nat_a > nat ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001t__Nat__Onat_001_062_Itf__a_Mtf__a_J,type,
commut5797115372127264861at_a_a: set_nat > ( nat > nat > nat ) > nat > ( ( a > a ) > nat ) > set_a_a > nat ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001t__Nat__Onat_001t__Nat__Onat,type,
commut1028764413824576968at_nat: set_nat > ( nat > nat > nat ) > nat > ( nat > nat ) > set_nat > nat ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001t__Nat__Onat_001tf__a,type,
commut1549887680474846982_nat_a: set_nat > ( nat > nat > nat ) > nat > ( a > nat ) > set_a > nat ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001tf__a_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
commut8121142741902956950at_a_a: set_a > ( a > a > a ) > a > ( ( ( nat > a ) > a ) > a ) > set_nat_a_a2 > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
commut5242989786243415821_nat_a: set_a > ( a > a > a ) > a > ( ( nat > a ) > a ) > set_nat_a > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001tf__a_001_062_Itf__a_Mtf__a_J,type,
commut6344323929752164413_a_a_a: set_a > ( a > a > a ) > a > ( ( a > a ) > a ) > set_a_a > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001tf__a_001t__Nat__Onat,type,
commut6741328216151336360_a_nat: set_a > ( a > a > a ) > a > ( nat > a ) > set_nat > a ).
thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001tf__a_001tf__a,type,
commut5005951359559292710mp_a_a: set_a > ( a > a > a ) > a > ( a > a ) > set_a > a ).
thf(sy_c_Finite__Set_Ocomp__fun__commute__on_001_062_It__Nat__Onat_Mtf__a_J_001tf__a,type,
finite1127406183625600809at_a_a: set_nat_a > ( ( nat > a ) > a > a ) > $o ).
thf(sy_c_Finite__Set_Ocomp__fun__commute__on_001t__Nat__Onat_001tf__a,type,
finite1071566134745755356_nat_a: set_nat > ( nat > a > a ) > $o ).
thf(sy_c_Finite__Set_Ocomp__fun__commute__on_001tf__a_001tf__a,type,
finite2737277698481670352on_a_a: set_a > ( a > a > a ) > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
finite7239108116303828181at_a_a: set_nat_a_a2 > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mtf__a_J,type,
finite_finite_nat_a: set_nat_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_Itf__a_Mtf__a_J,type,
finite_finite_a_a: set_a_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Finite__Set_Ofold_001_062_It__Nat__Onat_Mtf__a_J_001t__Nat__Onat,type,
finite7774500027257897325_a_nat: ( ( nat > a ) > nat > nat ) > nat > set_nat_a > nat ).
thf(sy_c_Finite__Set_Ofold_001_062_It__Nat__Onat_Mtf__a_J_001tf__a,type,
finite_fold_nat_a_a: ( ( nat > a ) > a > a ) > a > set_nat_a > a ).
thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001_062_It__Nat__Onat_Mtf__a_J,type,
finite6730669110406474827_nat_a: ( nat > ( nat > a ) > nat > a ) > ( nat > a ) > set_nat > nat > a ).
thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001_062_Itf__a_Mtf__a_J,type,
finite_fold_nat_a_a2: ( nat > ( a > a ) > a > a ) > ( a > a ) > set_nat > a > a ).
thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
finite_fold_nat_nat: ( nat > nat > nat ) > nat > set_nat > nat ).
thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001tf__a,type,
finite_fold_nat_a: ( nat > a > a ) > a > set_nat > a ).
thf(sy_c_Finite__Set_Ofold_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
finite_fold_a_nat_a: ( a > ( nat > a ) > nat > a ) > ( nat > a ) > set_a > nat > a ).
thf(sy_c_Finite__Set_Ofold_001tf__a_001_062_Itf__a_Mtf__a_J,type,
finite_fold_a_a_a: ( a > ( a > a ) > a > a ) > ( a > a ) > set_a > a > a ).
thf(sy_c_Finite__Set_Ofold_001tf__a_001t__Nat__Onat,type,
finite_fold_a_nat: ( a > nat > nat ) > nat > set_a > nat ).
thf(sy_c_Finite__Set_Ofold_001tf__a_001tf__a,type,
finite_fold_a_a: ( a > a > a ) > a > set_a > a ).
thf(sy_c_FuncSet_OPi_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_001t__Nat__Onat,type,
pi_nat_a_a_nat: set_nat_a_a2 > ( ( ( nat > a ) > a ) > set_nat ) > set_nat_a_a_nat ).
thf(sy_c_FuncSet_OPi_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_001tf__a,type,
pi_nat_a_a_a: set_nat_a_a2 > ( ( ( nat > a ) > a ) > set_a ) > set_nat_a_a_a ).
thf(sy_c_FuncSet_OPi_001_062_It__Nat__Onat_Mtf__a_J_001t__Nat__Onat,type,
pi_nat_a_nat: set_nat_a > ( ( nat > a ) > set_nat ) > set_nat_a_nat ).
thf(sy_c_FuncSet_OPi_001_062_It__Nat__Onat_Mtf__a_J_001tf__a,type,
pi_nat_a_a: set_nat_a > ( ( nat > a ) > set_a ) > set_nat_a_a2 ).
thf(sy_c_FuncSet_OPi_001_062_Itf__a_Mtf__a_J_001t__Nat__Onat,type,
pi_a_a_nat: set_a_a > ( ( a > a ) > set_nat ) > set_a_a_nat ).
thf(sy_c_FuncSet_OPi_001_062_Itf__a_Mtf__a_J_001tf__a,type,
pi_a_a_a: set_a_a > ( ( a > a ) > set_a ) > set_a_a_a2 ).
thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
pi_nat_nat_a_a: set_nat > ( nat > set_nat_a_a2 ) > set_nat_nat_a_a ).
thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001_062_It__Nat__Onat_Mtf__a_J,type,
pi_nat_nat_a: set_nat > ( nat > set_nat_a ) > set_nat_nat_a ).
thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001_062_Itf__a_Mtf__a_J,type,
pi_nat_a_a2: set_nat > ( nat > set_a_a ) > set_nat_a_a ).
thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001t__Nat__Onat,type,
pi_nat_nat: set_nat > ( nat > set_nat ) > set_nat_nat ).
thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001tf__a,type,
pi_nat_a: set_nat > ( nat > set_a ) > set_nat_a ).
thf(sy_c_FuncSet_OPi_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
pi_a_nat_a: set_a > ( a > set_nat_a ) > set_a_nat_a ).
thf(sy_c_FuncSet_OPi_001tf__a_001_062_Itf__a_Mtf__a_J,type,
pi_a_a_a2: set_a > ( a > set_a_a ) > set_a_a_a ).
thf(sy_c_FuncSet_OPi_001tf__a_001t__Nat__Onat,type,
pi_a_nat: set_a > ( a > set_nat ) > set_a_nat ).
thf(sy_c_FuncSet_OPi_001tf__a_001tf__a,type,
pi_a_a: set_a > ( a > set_a ) > set_a_a ).
thf(sy_c_FuncSet_Orestrict_001tf__a_001tf__a,type,
restrict_a_a: ( a > a ) > set_a > a > a ).
thf(sy_c_Group__Theory_Ocommutative__monoid_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
group_1237985806048136895at_a_a: set_nat_a_a2 > ( ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a ) > ( ( nat > a ) > a ) > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid_001_062_It__Nat__Onat_Mtf__a_J,type,
group_3093379471365697572_nat_a: set_nat_a > ( ( nat > a ) > ( nat > a ) > nat > a ) > ( nat > a ) > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid_001_062_Itf__a_Mtf__a_J,type,
group_6976245611985207014id_a_a: set_a_a > ( ( a > a ) > ( a > a ) > a > a ) > ( a > a ) > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid_001t__Nat__Onat,type,
group_6791354081887936081id_nat: set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid_001tf__a,type,
group_4866109990395492029noid_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid__axioms_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
group_3498422669771298786at_a_a: set_nat_a_a2 > ( ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a ) > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid__axioms_001_062_It__Nat__Onat_Mtf__a_J,type,
group_5745313098993787841_nat_a: set_nat_a > ( ( nat > a ) > ( nat > a ) > nat > a ) > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid__axioms_001_062_Itf__a_Mtf__a_J,type,
group_7053950466465069193ms_a_a: set_a_a > ( ( a > a ) > ( a > a ) > a > a ) > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid__axioms_001t__Nat__Onat,type,
group_5685275631618022900ms_nat: set_nat > ( nat > nat > nat ) > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid__axioms_001tf__a,type,
group_2081300317213596122ioms_a: set_a > ( a > a > a ) > $o ).
thf(sy_c_Group__Theory_Ogroup_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
group_group_nat_a_a: set_nat_a_a2 > ( ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a ) > ( ( nat > a ) > a ) > $o ).
thf(sy_c_Group__Theory_Ogroup_001_062_It__Nat__Onat_Mtf__a_J,type,
group_group_nat_a: set_nat_a > ( ( nat > a ) > ( nat > a ) > nat > a ) > ( nat > a ) > $o ).
thf(sy_c_Group__Theory_Ogroup_001_062_Itf__a_Mtf__a_J,type,
group_group_a_a: set_a_a > ( ( a > a ) > ( a > a ) > a > a ) > ( a > a ) > $o ).
thf(sy_c_Group__Theory_Ogroup_001t__Nat__Onat,type,
group_group_nat: set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Group__Theory_Ogroup_001tf__a,type,
group_group_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Omonoid_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
group_monoid_nat_a_a: set_nat_a_a2 > ( ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a ) > ( ( nat > a ) > a ) > $o ).
thf(sy_c_Group__Theory_Omonoid_001_062_It__Nat__Onat_Mtf__a_J,type,
group_monoid_nat_a: set_nat_a > ( ( nat > a ) > ( nat > a ) > nat > a ) > ( nat > a ) > $o ).
thf(sy_c_Group__Theory_Omonoid_001_062_Itf__a_Mtf__a_J,type,
group_monoid_a_a: set_a_a > ( ( a > a ) > ( a > a ) > a > a ) > ( a > a ) > $o ).
thf(sy_c_Group__Theory_Omonoid_001t__Nat__Onat,type,
group_monoid_nat: set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Group__Theory_Omonoid_001tf__a,type,
group_monoid_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Omonoid_OUnits_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
group_Units_nat_a_a: set_nat_a_a2 > ( ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a ) > ( ( nat > a ) > a ) > set_nat_a_a2 ).
thf(sy_c_Group__Theory_Omonoid_OUnits_001_062_It__Nat__Onat_Mtf__a_J,type,
group_Units_nat_a: set_nat_a > ( ( nat > a ) > ( nat > a ) > nat > a ) > ( nat > a ) > set_nat_a ).
thf(sy_c_Group__Theory_Omonoid_OUnits_001_062_Itf__a_Mtf__a_J,type,
group_Units_a_a: set_a_a > ( ( a > a ) > ( a > a ) > a > a ) > ( a > a ) > set_a_a ).
thf(sy_c_Group__Theory_Omonoid_OUnits_001t__Nat__Onat,type,
group_Units_nat: set_nat > ( nat > nat > nat ) > nat > set_nat ).
thf(sy_c_Group__Theory_Omonoid_OUnits_001tf__a,type,
group_Units_a: set_a > ( a > a > a ) > a > set_a ).
thf(sy_c_Group__Theory_Omonoid_Oinverse_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
group_6240212554837956873at_a_a: set_nat_a_a2 > ( ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a ).
thf(sy_c_Group__Theory_Omonoid_Oinverse_001_062_It__Nat__Onat_Mtf__a_J,type,
group_inverse_nat_a: set_nat_a > ( ( nat > a ) > ( nat > a ) > nat > a ) > ( nat > a ) > ( nat > a ) > nat > a ).
thf(sy_c_Group__Theory_Omonoid_Oinverse_001_062_Itf__a_Mtf__a_J,type,
group_inverse_a_a: set_a_a > ( ( a > a ) > ( a > a ) > a > a ) > ( a > a ) > ( a > a ) > a > a ).
thf(sy_c_Group__Theory_Omonoid_Oinverse_001t__Nat__Onat,type,
group_inverse_nat: set_nat > ( nat > nat > nat ) > nat > nat > nat ).
thf(sy_c_Group__Theory_Omonoid_Oinverse_001tf__a,type,
group_inverse_a: set_a > ( a > a > a ) > a > a > a ).
thf(sy_c_Group__Theory_Omonoid_Oinvertible_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
group_1438879830998425741at_a_a: set_nat_a_a2 > ( ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > $o ).
thf(sy_c_Group__Theory_Omonoid_Oinvertible_001_062_It__Nat__Onat_Mtf__a_J,type,
group_645299334525884886_nat_a: set_nat_a > ( ( nat > a ) > ( nat > a ) > nat > a ) > ( nat > a ) > ( nat > a ) > $o ).
thf(sy_c_Group__Theory_Omonoid_Oinvertible_001_062_Itf__a_Mtf__a_J,type,
group_invertible_a_a: set_a_a > ( ( a > a ) > ( a > a ) > a > a ) > ( a > a ) > ( a > a ) > $o ).
thf(sy_c_Group__Theory_Omonoid_Oinvertible_001t__Nat__Onat,type,
group_invertible_nat: set_nat > ( nat > nat > nat ) > nat > nat > $o ).
thf(sy_c_Group__Theory_Omonoid_Oinvertible_001tf__a,type,
group_invertible_a: set_a > ( a > a > a ) > a > a > $o ).
thf(sy_c_Group__Theory_Osubgroup_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
group_5398023533210767094at_a_a: set_nat_a_a2 > set_nat_a_a2 > ( ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a ) > ( ( nat > a ) > a ) > $o ).
thf(sy_c_Group__Theory_Osubgroup_001_062_It__Nat__Onat_Mtf__a_J,type,
group_subgroup_nat_a: set_nat_a > set_nat_a > ( ( nat > a ) > ( nat > a ) > nat > a ) > ( nat > a ) > $o ).
thf(sy_c_Group__Theory_Osubgroup_001_062_Itf__a_Mtf__a_J,type,
group_subgroup_a_a: set_a_a > set_a_a > ( ( a > a ) > ( a > a ) > a > a ) > ( a > a ) > $o ).
thf(sy_c_Group__Theory_Osubgroup_001t__Nat__Onat,type,
group_subgroup_nat: set_nat > set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Group__Theory_Osubgroup_001tf__a,type,
group_subgroup_a: set_a > set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_J,type,
minus_1482667089342205261at_a_a: set_nat_a_a2 > set_nat_a_a2 > set_nat_a_a2 ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
minus_490503922182417452_nat_a: set_nat_a > set_nat_a > set_nat_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
minus_minus_set_a_a: set_a_a > set_a_a > set_a_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_HOL_OThe_001tf__a,type,
the_a: ( a > $o ) > a ).
thf(sy_c_HOL_Oundefined_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
undefined_nat_a_a: ( nat > a ) > a ).
thf(sy_c_HOL_Oundefined_001_062_It__Nat__Onat_Mtf__a_J,type,
undefined_nat_a: nat > a ).
thf(sy_c_HOL_Oundefined_001_062_Itf__a_Mtf__a_J,type,
undefined_a_a: a > a ).
thf(sy_c_HOL_Oundefined_001t__Nat__Onat,type,
undefined_nat: nat ).
thf(sy_c_HOL_Oundefined_001tf__a,type,
undefined_a: a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_J,type,
inf_inf_set_nat_a_a: set_nat_a_a2 > set_nat_a_a2 > set_nat_a_a2 ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
inf_inf_set_nat_a: set_nat_a > set_nat_a > set_nat_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
inf_inf_set_a_a: set_a_a > set_a_a > set_a_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_J,type,
sup_sup_set_nat_a_a: set_nat_a_a2 > set_nat_a_a2 > set_nat_a_a2 ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
sup_sup_set_nat_a: set_nat_a > set_nat_a > set_nat_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
sup_sup_set_a_a: set_a_a > set_a_a > set_a_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_001t__Nat__Onat,type,
lattic7650876701696636585_a_nat: ( ( ( nat > a ) > a ) > nat ) > set_nat_a_a2 > ( nat > a ) > a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_062_It__Nat__Onat_Mtf__a_J_001t__Nat__Onat,type,
lattic6419734799033661276_a_nat: ( ( nat > a ) > nat ) > set_nat_a > nat > a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_062_Itf__a_Mtf__a_J_001t__Nat__Onat,type,
lattic5687691984537318480_a_nat: ( ( a > a ) > nat ) > set_a_a > a > a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001t__Nat__Onat,type,
lattic6340287419671400565_a_nat: ( a > nat ) > set_a > a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_J,type,
bot_bot_set_nat_a_a: set_nat_a_a2 ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
bot_bot_set_nat_a: set_nat_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
bot_bot_set_a_a: set_a_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_M_Eo_J,type,
ord_le3623034401944517937_a_a_o: ( ( ( nat > a ) > a ) > $o ) > ( ( ( nat > a ) > a ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mtf__a_J_M_Eo_J,type,
ord_less_eq_nat_a_o: ( ( nat > a ) > $o ) > ( ( nat > a ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_Itf__a_Mtf__a_J_M_Eo_J,type,
ord_less_eq_a_a_o: ( ( a > a ) > $o ) > ( ( a > a ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_Mt__Nat__Onat_J_J,type,
ord_le6513183344911049539_a_nat: set_nat_a_a_nat > set_nat_a_a_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_Mtf__a_J_J,type,
ord_le7944041390409729655_a_a_a: set_nat_a_a_a > set_nat_a_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mt__Nat__Onat_J_J,type,
ord_le8014916076312755750_a_nat: set_nat_a_nat > set_nat_a_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_J,type,
ord_le3509452538356653652at_a_a: set_nat_a_a2 > set_nat_a_a2 > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_Itf__a_Mtf__a_J_Mt__Nat__Onat_J_J,type,
ord_le4090943279780567402_a_nat: set_a_a_nat > set_a_a_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
ord_le7181591058469194768_a_a_a: set_a_a_a2 > set_a_a_a2 > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le9059583361652607317at_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
ord_le871467723717165285_nat_a: set_nat_a > set_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__a_Mt__Nat__Onat_J_J,type,
ord_le1612561287239139007_a_nat: set_a_nat > set_a_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
ord_less_eq_set_a_a: set_a_a > set_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
collect_nat_a_a: ( ( ( nat > a ) > a ) > $o ) > set_nat_a_a2 ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mtf__a_J,type,
collect_nat_a: ( ( nat > a ) > $o ) > set_nat_a ).
thf(sy_c_Set_OCollect_001_062_Itf__a_Mtf__a_J,type,
collect_a_a: ( ( a > a ) > $o ) > set_a_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
insert_nat_a_a: ( ( nat > a ) > a ) > set_nat_a_a2 > set_nat_a_a2 ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mtf__a_J,type,
insert_nat_a: ( nat > a ) > set_nat_a > set_nat_a ).
thf(sy_c_Set_Oinsert_001_062_Itf__a_Mtf__a_J,type,
insert_a_a: ( a > a ) > set_a_a > set_a_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__singleton_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
is_singleton_nat_a_a: set_nat_a_a2 > $o ).
thf(sy_c_Set_Ois__singleton_001_062_It__Nat__Onat_Mtf__a_J,type,
is_singleton_nat_a: set_nat_a > $o ).
thf(sy_c_Set_Ois__singleton_001_062_Itf__a_Mtf__a_J,type,
is_singleton_a_a: set_a_a > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_Itf__a_J,type,
set_or6288561110385358355_set_a: set_a > set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_Itf__a_J,type,
set_ord_atMost_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(sy_c_member_001_062_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_Mtf__a_J,type,
member_nat_a_a_a: ( ( ( nat > a ) > a ) > a ) > set_nat_a_a_a > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mt__Nat__Onat_J,type,
member_nat_a_nat: ( ( nat > a ) > nat ) > set_nat_a_nat > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
member_nat_a_a: ( ( nat > a ) > a ) > set_nat_a_a2 > $o ).
thf(sy_c_member_001_062_I_062_Itf__a_Mtf__a_J_Mt__Nat__Onat_J,type,
member_a_a_nat: ( ( a > a ) > nat ) > set_a_a_nat > $o ).
thf(sy_c_member_001_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J,type,
member_a_a_a: ( ( a > a ) > a ) > set_a_a_a2 > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_J,type,
member_nat_nat_a_a: ( nat > ( nat > a ) > a ) > set_nat_nat_a_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mtf__a_J_J,type,
member_nat_nat_a: ( nat > nat > a ) > set_nat_nat_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_062_Itf__a_Mtf__a_J_J,type,
member_nat_a_a2: ( nat > a > a ) > set_nat_a_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(sy_c_member_001_062_Itf__a_M_062_It__Nat__Onat_Mtf__a_J_J,type,
member_a_nat_a: ( a > nat > a ) > set_a_nat_a > $o ).
thf(sy_c_member_001_062_Itf__a_M_062_Itf__a_Mtf__a_J_J,type,
member_a_a_a2: ( a > a > a ) > set_a_a_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Nat__Onat_J,type,
member_a_nat: ( a > nat ) > set_a_nat > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_M,type,
m: set_a ).
thf(sy_v_composition,type,
composition: a > a > a ).
thf(sy_v_f,type,
f: nat > a ).
thf(sy_v_unit,type,
unit: a ).
% Relevant facts (1273)
thf(fact_0_commutative,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ m )
=> ( ( member_a @ Y @ m )
=> ( ( composition @ X @ Y )
= ( composition @ Y @ X ) ) ) ) ).
% commutative
thf(fact_1_left__commute,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ m )
=> ( ( member_a @ B @ m )
=> ( ( member_a @ C @ m )
=> ( ( composition @ B @ ( composition @ A @ C ) )
= ( composition @ A @ ( composition @ B @ C ) ) ) ) ) ) ).
% left_commute
thf(fact_2__C0_C,axiom,
( member_nat_a @ f
@ ( pi_nat_a @ ( set_ord_atMost_nat @ ( suc @ zero_zero_nat ) )
@ ^ [Uu: nat] : m ) ) ).
% "0"
thf(fact_3_local_Oinverse__unique,axiom,
! [U: a,V: a,V2: a] :
( ( ( composition @ U @ V )
= unit )
=> ( ( ( composition @ V2 @ U )
= unit )
=> ( ( member_a @ U @ m )
=> ( ( member_a @ V2 @ m )
=> ( ( member_a @ V @ m )
=> ( V2 = V ) ) ) ) ) ) ).
% local.inverse_unique
thf(fact_4_fincomp__unit__eqI,axiom,
! [A2: set_nat_a_a2,F: ( ( nat > a ) > a ) > a] :
( ! [X2: ( nat > a ) > a] :
( ( member_nat_a_a @ X2 @ A2 )
=> ( ( F @ X2 )
= unit ) )
=> ( ( commut8121142741902956950at_a_a @ m @ composition @ unit @ F @ A2 )
= unit ) ) ).
% fincomp_unit_eqI
thf(fact_5_fincomp__unit__eqI,axiom,
! [A2: set_a_a,F: ( a > a ) > a] :
( ! [X2: a > a] :
( ( member_a_a @ X2 @ A2 )
=> ( ( F @ X2 )
= unit ) )
=> ( ( commut6344323929752164413_a_a_a @ m @ composition @ unit @ F @ A2 )
= unit ) ) ).
% fincomp_unit_eqI
thf(fact_6_fincomp__unit__eqI,axiom,
! [A2: set_a,F: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( F @ X2 )
= unit ) )
=> ( ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ F @ A2 )
= unit ) ) ).
% fincomp_unit_eqI
thf(fact_7_fincomp__unit__eqI,axiom,
! [A2: set_nat_a,F: ( nat > a ) > a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( ( F @ X2 )
= unit ) )
=> ( ( commut5242989786243415821_nat_a @ m @ composition @ unit @ F @ A2 )
= unit ) ) ).
% fincomp_unit_eqI
thf(fact_8_fincomp__unit__eqI,axiom,
! [A2: set_nat,F: nat > a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( F @ X2 )
= unit ) )
=> ( ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ A2 )
= unit ) ) ).
% fincomp_unit_eqI
thf(fact_9_commutative__monoid__axioms,axiom,
group_4866109990395492029noid_a @ m @ composition @ unit ).
% commutative_monoid_axioms
thf(fact_10_fincomp__closed,axiom,
! [F: ( nat > a ) > a,F2: set_nat_a] :
( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ F2
@ ^ [Uu: nat > a] : m ) )
=> ( member_a @ ( commut5242989786243415821_nat_a @ m @ composition @ unit @ F @ F2 ) @ m ) ) ).
% fincomp_closed
thf(fact_11_fincomp__closed,axiom,
! [F: a > a,F2: set_a] :
( ( member_a_a @ F
@ ( pi_a_a @ F2
@ ^ [Uu: a] : m ) )
=> ( member_a @ ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ F @ F2 ) @ m ) ) ).
% fincomp_closed
thf(fact_12_fincomp__closed,axiom,
! [F: nat > a,F2: set_nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ F2
@ ^ [Uu: nat] : m ) )
=> ( member_a @ ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ F2 ) @ m ) ) ).
% fincomp_closed
thf(fact_13_fincomp__comp,axiom,
! [F: ( nat > a ) > a,A2: set_nat_a,G: ( nat > a ) > a] :
( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : m ) )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : m ) )
=> ( ( commut5242989786243415821_nat_a @ m @ composition @ unit
@ ^ [X3: nat > a] : ( composition @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( composition @ ( commut5242989786243415821_nat_a @ m @ composition @ unit @ F @ A2 ) @ ( commut5242989786243415821_nat_a @ m @ composition @ unit @ G @ A2 ) ) ) ) ) ).
% fincomp_comp
thf(fact_14_fincomp__comp,axiom,
! [F: a > a,A2: set_a,G: a > a] :
( ( member_a_a @ F
@ ( pi_a_a @ A2
@ ^ [Uu: a] : m ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A2
@ ^ [Uu: a] : m ) )
=> ( ( commut5005951359559292710mp_a_a @ m @ composition @ unit
@ ^ [X3: a] : ( composition @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( composition @ ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ F @ A2 ) @ ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ G @ A2 ) ) ) ) ) ).
% fincomp_comp
thf(fact_15_fincomp__comp,axiom,
! [F: nat > a,A2: set_nat,G: nat > a] :
( ( member_nat_a @ F
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : m ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : m ) )
=> ( ( commut6741328216151336360_a_nat @ m @ composition @ unit
@ ^ [X3: nat] : ( composition @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( composition @ ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ A2 ) @ ( commut6741328216151336360_a_nat @ m @ composition @ unit @ G @ A2 ) ) ) ) ) ).
% fincomp_comp
thf(fact_16_fincomp__cong_H,axiom,
! [A2: set_nat_a_a2,B2: set_nat_a_a2,G: ( ( nat > a ) > a ) > a,F: ( ( nat > a ) > a ) > a] :
( ( A2 = B2 )
=> ( ( member_nat_a_a_a @ G
@ ( pi_nat_a_a_a @ B2
@ ^ [Uu: ( nat > a ) > a] : m ) )
=> ( ! [I: ( nat > a ) > a] :
( ( member_nat_a_a @ I @ B2 )
=> ( ( F @ I )
= ( G @ I ) ) )
=> ( ( commut8121142741902956950at_a_a @ m @ composition @ unit @ F @ A2 )
= ( commut8121142741902956950at_a_a @ m @ composition @ unit @ G @ B2 ) ) ) ) ) ).
% fincomp_cong'
thf(fact_17_fincomp__cong_H,axiom,
! [A2: set_a_a,B2: set_a_a,G: ( a > a ) > a,F: ( a > a ) > a] :
( ( A2 = B2 )
=> ( ( member_a_a_a @ G
@ ( pi_a_a_a @ B2
@ ^ [Uu: a > a] : m ) )
=> ( ! [I: a > a] :
( ( member_a_a @ I @ B2 )
=> ( ( F @ I )
= ( G @ I ) ) )
=> ( ( commut6344323929752164413_a_a_a @ m @ composition @ unit @ F @ A2 )
= ( commut6344323929752164413_a_a_a @ m @ composition @ unit @ G @ B2 ) ) ) ) ) ).
% fincomp_cong'
thf(fact_18_fincomp__cong_H,axiom,
! [A2: set_a,B2: set_a,G: a > a,F: a > a] :
( ( A2 = B2 )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : m ) )
=> ( ! [I: a] :
( ( member_a @ I @ B2 )
=> ( ( F @ I )
= ( G @ I ) ) )
=> ( ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ F @ A2 )
= ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ G @ B2 ) ) ) ) ) ).
% fincomp_cong'
thf(fact_19_fincomp__cong_H,axiom,
! [A2: set_nat_a,B2: set_nat_a,G: ( nat > a ) > a,F: ( nat > a ) > a] :
( ( A2 = B2 )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ B2
@ ^ [Uu: nat > a] : m ) )
=> ( ! [I: nat > a] :
( ( member_nat_a @ I @ B2 )
=> ( ( F @ I )
= ( G @ I ) ) )
=> ( ( commut5242989786243415821_nat_a @ m @ composition @ unit @ F @ A2 )
= ( commut5242989786243415821_nat_a @ m @ composition @ unit @ G @ B2 ) ) ) ) ) ).
% fincomp_cong'
thf(fact_20_fincomp__cong_H,axiom,
! [A2: set_nat,B2: set_nat,G: nat > a,F: nat > a] :
( ( A2 = B2 )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : m ) )
=> ( ! [I: nat] :
( ( member_nat @ I @ B2 )
=> ( ( F @ I )
= ( G @ I ) ) )
=> ( ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ A2 )
= ( commut6741328216151336360_a_nat @ m @ composition @ unit @ G @ B2 ) ) ) ) ) ).
% fincomp_cong'
thf(fact_21_associative,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ m )
=> ( ( member_a @ B @ m )
=> ( ( member_a @ C @ m )
=> ( ( composition @ ( composition @ A @ B ) @ C )
= ( composition @ A @ ( composition @ B @ C ) ) ) ) ) ) ).
% associative
thf(fact_22_composition__closed,axiom,
! [A: a,B: a] :
( ( member_a @ A @ m )
=> ( ( member_a @ B @ m )
=> ( member_a @ ( composition @ A @ B ) @ m ) ) ) ).
% composition_closed
thf(fact_23_unit__closed,axiom,
member_a @ unit @ m ).
% unit_closed
thf(fact_24_left__unit,axiom,
! [A: a] :
( ( member_a @ A @ m )
=> ( ( composition @ unit @ A )
= A ) ) ).
% left_unit
thf(fact_25_right__unit,axiom,
! [A: a] :
( ( member_a @ A @ m )
=> ( ( composition @ A @ unit )
= A ) ) ).
% right_unit
thf(fact_26_fincomp__unit,axiom,
! [A2: set_nat] :
( ( commut6741328216151336360_a_nat @ m @ composition @ unit
@ ^ [I2: nat] : unit
@ A2 )
= unit ) ).
% fincomp_unit
thf(fact_27_fincomp__unit,axiom,
! [A2: set_nat_a] :
( ( commut5242989786243415821_nat_a @ m @ composition @ unit
@ ^ [I2: nat > a] : unit
@ A2 )
= unit ) ).
% fincomp_unit
thf(fact_28_fincomp__unit,axiom,
! [A2: set_a] :
( ( commut5005951359559292710mp_a_a @ m @ composition @ unit
@ ^ [I2: a] : unit
@ A2 )
= unit ) ).
% fincomp_unit
thf(fact_29_M__ify__def,axiom,
! [X: a] :
( ( ( member_a @ X @ m )
=> ( ( commutative_M_ify_a @ m @ unit @ X )
= X ) )
& ( ~ ( member_a @ X @ m )
=> ( ( commutative_M_ify_a @ m @ unit @ X )
= unit ) ) ) ).
% M_ify_def
thf(fact_30_commutative__monoid_Ofincomp_Ocong,axiom,
commut6741328216151336360_a_nat = commut6741328216151336360_a_nat ).
% commutative_monoid.fincomp.cong
thf(fact_31_commutative__monoid_Ofincomp_Ocong,axiom,
commut5242989786243415821_nat_a = commut5242989786243415821_nat_a ).
% commutative_monoid.fincomp.cong
thf(fact_32_commutative__monoid_Ofincomp_Ocong,axiom,
commut5005951359559292710mp_a_a = commut5005951359559292710mp_a_a ).
% commutative_monoid.fincomp.cong
thf(fact_33_fincomp__Suc,axiom,
! [F: nat > a,N: nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ ( suc @ N ) )
@ ^ [Uu: nat] : m ) )
=> ( ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( composition @ ( F @ ( suc @ N ) ) @ ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ ( set_ord_atMost_nat @ N ) ) ) ) ) ).
% fincomp_Suc
thf(fact_34_fincomp__0_H,axiom,
! [F: nat > a,N: nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ N )
@ ^ [Uu: nat] : m ) )
=> ( ( composition @ ( F @ zero_zero_nat ) @ ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% fincomp_0'
thf(fact_35_inverse__equality,axiom,
! [U: a,V2: a] :
( ( ( composition @ U @ V2 )
= unit )
=> ( ( ( composition @ V2 @ U )
= unit )
=> ( ( member_a @ U @ m )
=> ( ( member_a @ V2 @ m )
=> ( ( group_inverse_a @ m @ composition @ unit @ U )
= V2 ) ) ) ) ) ).
% inverse_equality
thf(fact_36_invertibleE,axiom,
! [U: a] :
( ( group_invertible_a @ m @ composition @ unit @ U )
=> ( ! [V3: a] :
( ( ( ( composition @ U @ V3 )
= unit )
& ( ( composition @ V3 @ U )
= unit ) )
=> ~ ( member_a @ V3 @ m ) )
=> ~ ( member_a @ U @ m ) ) ) ).
% invertibleE
thf(fact_37_invertible__def,axiom,
! [U: a] :
( ( member_a @ U @ m )
=> ( ( group_invertible_a @ m @ composition @ unit @ U )
= ( ? [X3: a] :
( ( member_a @ X3 @ m )
& ( ( composition @ U @ X3 )
= unit )
& ( ( composition @ X3 @ U )
= unit ) ) ) ) ) ).
% invertible_def
thf(fact_38_unit__invertible,axiom,
group_invertible_a @ m @ composition @ unit @ unit ).
% unit_invertible
thf(fact_39_monoid__axioms,axiom,
group_monoid_a @ m @ composition @ unit ).
% monoid_axioms
thf(fact_40_atMost__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X )
= ( set_ord_atMost_nat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_41_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_42_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_43_invertible__right__inverse2,axiom,
! [U: a,V2: a] :
( ( group_invertible_a @ m @ composition @ unit @ U )
=> ( ( member_a @ U @ m )
=> ( ( member_a @ V2 @ m )
=> ( ( composition @ U @ ( composition @ ( group_inverse_a @ m @ composition @ unit @ U ) @ V2 ) )
= V2 ) ) ) ) ).
% invertible_right_inverse2
thf(fact_44_invertible__left__inverse2,axiom,
! [U: a,V2: a] :
( ( group_invertible_a @ m @ composition @ unit @ U )
=> ( ( member_a @ U @ m )
=> ( ( member_a @ V2 @ m )
=> ( ( composition @ ( group_inverse_a @ m @ composition @ unit @ U ) @ ( composition @ U @ V2 ) )
= V2 ) ) ) ) ).
% invertible_left_inverse2
thf(fact_45_inverse__composition__commute,axiom,
! [X: a,Y: a] :
( ( group_invertible_a @ m @ composition @ unit @ X )
=> ( ( group_invertible_a @ m @ composition @ unit @ Y )
=> ( ( member_a @ X @ m )
=> ( ( member_a @ Y @ m )
=> ( ( group_inverse_a @ m @ composition @ unit @ ( composition @ X @ Y ) )
= ( composition @ ( group_inverse_a @ m @ composition @ unit @ Y ) @ ( group_inverse_a @ m @ composition @ unit @ X ) ) ) ) ) ) ) ).
% inverse_composition_commute
thf(fact_46_invertible__right__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( group_invertible_a @ m @ composition @ unit @ X )
=> ( ( member_a @ X @ m )
=> ( ( member_a @ Y @ m )
=> ( ( member_a @ Z @ m )
=> ( ( ( composition @ Y @ X )
= ( composition @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ).
% invertible_right_cancel
thf(fact_47_invertible__left__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( group_invertible_a @ m @ composition @ unit @ X )
=> ( ( member_a @ X @ m )
=> ( ( member_a @ Y @ m )
=> ( ( member_a @ Z @ m )
=> ( ( ( composition @ X @ Y )
= ( composition @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ).
% invertible_left_cancel
thf(fact_48_invertibleI,axiom,
! [U: a,V2: a] :
( ( ( composition @ U @ V2 )
= unit )
=> ( ( ( composition @ V2 @ U )
= unit )
=> ( ( member_a @ U @ m )
=> ( ( member_a @ V2 @ m )
=> ( group_invertible_a @ m @ composition @ unit @ U ) ) ) ) ) ).
% invertibleI
thf(fact_49_composition__invertible,axiom,
! [X: a,Y: a] :
( ( group_invertible_a @ m @ composition @ unit @ X )
=> ( ( group_invertible_a @ m @ composition @ unit @ Y )
=> ( ( member_a @ X @ m )
=> ( ( member_a @ Y @ m )
=> ( group_invertible_a @ m @ composition @ unit @ ( composition @ X @ Y ) ) ) ) ) ) ).
% composition_invertible
thf(fact_50_inverse__unit,axiom,
( ( group_inverse_a @ m @ composition @ unit @ unit )
= unit ) ).
% inverse_unit
thf(fact_51_invertible__right__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ m @ composition @ unit @ U )
=> ( ( member_a @ U @ m )
=> ( ( composition @ U @ ( group_inverse_a @ m @ composition @ unit @ U ) )
= unit ) ) ) ).
% invertible_right_inverse
thf(fact_52_invertible__left__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ m @ composition @ unit @ U )
=> ( ( member_a @ U @ m )
=> ( ( composition @ ( group_inverse_a @ m @ composition @ unit @ U ) @ U )
= unit ) ) ) ).
% invertible_left_inverse
thf(fact_53_invertible__inverse__invertible,axiom,
! [U: a] :
( ( group_invertible_a @ m @ composition @ unit @ U )
=> ( ( member_a @ U @ m )
=> ( group_invertible_a @ m @ composition @ unit @ ( group_inverse_a @ m @ composition @ unit @ U ) ) ) ) ).
% invertible_inverse_invertible
thf(fact_54_invertible__inverse__inverse,axiom,
! [U: a] :
( ( group_invertible_a @ m @ composition @ unit @ U )
=> ( ( member_a @ U @ m )
=> ( ( group_inverse_a @ m @ composition @ unit @ ( group_inverse_a @ m @ composition @ unit @ U ) )
= U ) ) ) ).
% invertible_inverse_inverse
thf(fact_55_invertible__inverse__closed,axiom,
! [U: a] :
( ( group_invertible_a @ m @ composition @ unit @ U )
=> ( ( member_a @ U @ m )
=> ( member_a @ ( group_inverse_a @ m @ composition @ unit @ U ) @ m ) ) ) ).
% invertible_inverse_closed
thf(fact_56_commutative__monoid_OM__ify__def,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,X: nat > a] :
( ( group_3093379471365697572_nat_a @ M @ Composition @ Unit )
=> ( ( ( member_nat_a @ X @ M )
=> ( ( commut2316704705022288065_nat_a @ M @ Unit @ X )
= X ) )
& ( ~ ( member_nat_a @ X @ M )
=> ( ( commut2316704705022288065_nat_a @ M @ Unit @ X )
= Unit ) ) ) ) ).
% commutative_monoid.M_ify_def
thf(fact_57_commutative__monoid_OM__ify__def,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat] :
( ( group_6791354081887936081id_nat @ M @ Composition @ Unit )
=> ( ( ( member_nat @ X @ M )
=> ( ( commut810702690453168372fy_nat @ M @ Unit @ X )
= X ) )
& ( ~ ( member_nat @ X @ M )
=> ( ( commut810702690453168372fy_nat @ M @ Unit @ X )
= Unit ) ) ) ) ).
% commutative_monoid.M_ify_def
thf(fact_58_commutative__monoid_OM__ify__def,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,X: ( nat > a ) > a] :
( ( group_1237985806048136895at_a_a @ M @ Composition @ Unit )
=> ( ( ( member_nat_a_a @ X @ M )
=> ( ( commut4120000896247601890at_a_a @ M @ Unit @ X )
= X ) )
& ( ~ ( member_nat_a_a @ X @ M )
=> ( ( commut4120000896247601890at_a_a @ M @ Unit @ X )
= Unit ) ) ) ) ).
% commutative_monoid.M_ify_def
thf(fact_59_commutative__monoid_OM__ify__def,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,X: a > a] :
( ( group_6976245611985207014id_a_a @ M @ Composition @ Unit )
=> ( ( ( member_a_a @ X @ M )
=> ( ( commut6112553959220001673fy_a_a @ M @ Unit @ X )
= X ) )
& ( ~ ( member_a_a @ X @ M )
=> ( ( commut6112553959220001673fy_a_a @ M @ Unit @ X )
= Unit ) ) ) ) ).
% commutative_monoid.M_ify_def
thf(fact_60_commutative__monoid_OM__ify__def,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( ( member_a @ X @ M )
=> ( ( commutative_M_ify_a @ M @ Unit @ X )
= X ) )
& ( ~ ( member_a @ X @ M )
=> ( ( commutative_M_ify_a @ M @ Unit @ X )
= Unit ) ) ) ) ).
% commutative_monoid.M_ify_def
thf(fact_61_commutative__monoid_Oleft__commute,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,A: nat > a,B: nat > a,C: nat > a] :
( ( group_3093379471365697572_nat_a @ M @ Composition @ Unit )
=> ( ( member_nat_a @ A @ M )
=> ( ( member_nat_a @ B @ M )
=> ( ( member_nat_a @ C @ M )
=> ( ( Composition @ B @ ( Composition @ A @ C ) )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% commutative_monoid.left_commute
thf(fact_62_commutative__monoid_Oleft__commute,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,A: nat,B: nat,C: nat] :
( ( group_6791354081887936081id_nat @ M @ Composition @ Unit )
=> ( ( member_nat @ A @ M )
=> ( ( member_nat @ B @ M )
=> ( ( member_nat @ C @ M )
=> ( ( Composition @ B @ ( Composition @ A @ C ) )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% commutative_monoid.left_commute
thf(fact_63_commutative__monoid_Oleft__commute,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,A: ( nat > a ) > a,B: ( nat > a ) > a,C: ( nat > a ) > a] :
( ( group_1237985806048136895at_a_a @ M @ Composition @ Unit )
=> ( ( member_nat_a_a @ A @ M )
=> ( ( member_nat_a_a @ B @ M )
=> ( ( member_nat_a_a @ C @ M )
=> ( ( Composition @ B @ ( Composition @ A @ C ) )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% commutative_monoid.left_commute
thf(fact_64_commutative__monoid_Oleft__commute,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,A: a > a,B: a > a,C: a > a] :
( ( group_6976245611985207014id_a_a @ M @ Composition @ Unit )
=> ( ( member_a_a @ A @ M )
=> ( ( member_a_a @ B @ M )
=> ( ( member_a_a @ C @ M )
=> ( ( Composition @ B @ ( Composition @ A @ C ) )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% commutative_monoid.left_commute
thf(fact_65_commutative__monoid_Oleft__commute,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A: a,B: a,C: a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_a @ A @ M )
=> ( ( member_a @ B @ M )
=> ( ( member_a @ C @ M )
=> ( ( Composition @ B @ ( Composition @ A @ C ) )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% commutative_monoid.left_commute
thf(fact_66_mem__Collect__eq,axiom,
! [A: nat > a,P: ( nat > a ) > $o] :
( ( member_nat_a @ A @ ( collect_nat_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_67_mem__Collect__eq,axiom,
! [A: ( nat > a ) > a,P: ( ( nat > a ) > a ) > $o] :
( ( member_nat_a_a @ A @ ( collect_nat_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_68_mem__Collect__eq,axiom,
! [A: a > a,P: ( a > a ) > $o] :
( ( member_a_a @ A @ ( collect_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_69_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_70_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A2: set_nat_a] :
( ( collect_nat_a
@ ^ [X3: nat > a] : ( member_nat_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A2: set_nat_a_a2] :
( ( collect_nat_a_a
@ ^ [X3: ( nat > a ) > a] : ( member_nat_a_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
! [A2: set_a_a] :
( ( collect_a_a
@ ^ [X3: a > a] : ( member_a_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_74_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_75_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_76_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_77_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_78_commutative__monoid_OM__ify_Ocong,axiom,
commutative_M_ify_a = commutative_M_ify_a ).
% commutative_monoid.M_ify.cong
thf(fact_79_atMost__atLeast0,axiom,
( set_ord_atMost_nat
= ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% atMost_atLeast0
thf(fact_80_commutative__monoid_Ofincomp__closed,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: nat > a,F2: set_nat] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ F2
@ ^ [Uu: nat] : M ) )
=> ( member_a @ ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ F2 ) @ M ) ) ) ).
% commutative_monoid.fincomp_closed
thf(fact_81_commutative__monoid_Ofincomp__closed,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: ( nat > a ) > a,F2: set_nat_a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ F2
@ ^ [Uu: nat > a] : M ) )
=> ( member_a @ ( commut5242989786243415821_nat_a @ M @ Composition @ Unit @ F @ F2 ) @ M ) ) ) ).
% commutative_monoid.fincomp_closed
thf(fact_82_commutative__monoid_Ofincomp__closed,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: a > a,F2: set_a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_a_a @ F
@ ( pi_a_a @ F2
@ ^ [Uu: a] : M ) )
=> ( member_a @ ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ F @ F2 ) @ M ) ) ) ).
% commutative_monoid.fincomp_closed
thf(fact_83_commutative__monoid_Ofincomp__cong_H,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_nat_a_a2,B2: set_nat_a_a2,G: ( ( nat > a ) > a ) > a,F: ( ( nat > a ) > a ) > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( A2 = B2 )
=> ( ( member_nat_a_a_a @ G
@ ( pi_nat_a_a_a @ B2
@ ^ [Uu: ( nat > a ) > a] : M ) )
=> ( ! [I: ( nat > a ) > a] :
( ( member_nat_a_a @ I @ B2 )
=> ( ( F @ I )
= ( G @ I ) ) )
=> ( ( commut8121142741902956950at_a_a @ M @ Composition @ Unit @ F @ A2 )
= ( commut8121142741902956950at_a_a @ M @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_cong'
thf(fact_84_commutative__monoid_Ofincomp__cong_H,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_a_a,B2: set_a_a,G: ( a > a ) > a,F: ( a > a ) > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( A2 = B2 )
=> ( ( member_a_a_a @ G
@ ( pi_a_a_a @ B2
@ ^ [Uu: a > a] : M ) )
=> ( ! [I: a > a] :
( ( member_a_a @ I @ B2 )
=> ( ( F @ I )
= ( G @ I ) ) )
=> ( ( commut6344323929752164413_a_a_a @ M @ Composition @ Unit @ F @ A2 )
= ( commut6344323929752164413_a_a_a @ M @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_cong'
thf(fact_85_commutative__monoid_Ofincomp__cong_H,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_nat,B2: set_nat,G: nat > a,F: nat > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( A2 = B2 )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : M ) )
=> ( ! [I: nat] :
( ( member_nat @ I @ B2 )
=> ( ( F @ I )
= ( G @ I ) ) )
=> ( ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ A2 )
= ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_cong'
thf(fact_86_commutative__monoid_Ofincomp__cong_H,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_nat_a,B2: set_nat_a,G: ( nat > a ) > a,F: ( nat > a ) > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( A2 = B2 )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ B2
@ ^ [Uu: nat > a] : M ) )
=> ( ! [I: nat > a] :
( ( member_nat_a @ I @ B2 )
=> ( ( F @ I )
= ( G @ I ) ) )
=> ( ( commut5242989786243415821_nat_a @ M @ Composition @ Unit @ F @ A2 )
= ( commut5242989786243415821_nat_a @ M @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_cong'
thf(fact_87_commutative__monoid_Ofincomp__cong_H,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_a,B2: set_a,G: a > a,F: a > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( A2 = B2 )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : M ) )
=> ( ! [I: a] :
( ( member_a @ I @ B2 )
=> ( ( F @ I )
= ( G @ I ) ) )
=> ( ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ F @ A2 )
= ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ G @ B2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_cong'
thf(fact_88_commutative__monoid_Ofincomp__comp,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: nat > a,A2: set_nat,G: nat > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : M ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : M ) )
=> ( ( commut6741328216151336360_a_nat @ M @ Composition @ Unit
@ ^ [X3: nat] : ( Composition @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( Composition @ ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ A2 ) @ ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ G @ A2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_comp
thf(fact_89_commutative__monoid_Ofincomp__comp,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: ( nat > a ) > a,A2: set_nat_a,G: ( nat > a ) > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : M ) )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : M ) )
=> ( ( commut5242989786243415821_nat_a @ M @ Composition @ Unit
@ ^ [X3: nat > a] : ( Composition @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( Composition @ ( commut5242989786243415821_nat_a @ M @ Composition @ Unit @ F @ A2 ) @ ( commut5242989786243415821_nat_a @ M @ Composition @ Unit @ G @ A2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_comp
thf(fact_90_commutative__monoid_Ofincomp__comp,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: a > a,A2: set_a,G: a > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_a_a @ F
@ ( pi_a_a @ A2
@ ^ [Uu: a] : M ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A2
@ ^ [Uu: a] : M ) )
=> ( ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit
@ ^ [X3: a] : ( Composition @ ( F @ X3 ) @ ( G @ X3 ) )
@ A2 )
= ( Composition @ ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ F @ A2 ) @ ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ G @ A2 ) ) ) ) ) ) ).
% commutative_monoid.fincomp_comp
thf(fact_91_commutative__monoid_Ofincomp__unit__eqI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_nat_a_a2,F: ( ( nat > a ) > a ) > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ! [X2: ( nat > a ) > a] :
( ( member_nat_a_a @ X2 @ A2 )
=> ( ( F @ X2 )
= Unit ) )
=> ( ( commut8121142741902956950at_a_a @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ).
% commutative_monoid.fincomp_unit_eqI
thf(fact_92_commutative__monoid_Ofincomp__unit__eqI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_a_a,F: ( a > a ) > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ! [X2: a > a] :
( ( member_a_a @ X2 @ A2 )
=> ( ( F @ X2 )
= Unit ) )
=> ( ( commut6344323929752164413_a_a_a @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ).
% commutative_monoid.fincomp_unit_eqI
thf(fact_93_commutative__monoid_Ofincomp__unit__eqI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_nat,F: nat > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( F @ X2 )
= Unit ) )
=> ( ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ).
% commutative_monoid.fincomp_unit_eqI
thf(fact_94_commutative__monoid_Ofincomp__unit__eqI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_nat_a,F: ( nat > a ) > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( ( F @ X2 )
= Unit ) )
=> ( ( commut5242989786243415821_nat_a @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ).
% commutative_monoid.fincomp_unit_eqI
thf(fact_95_commutative__monoid_Ofincomp__unit__eqI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_a,F: a > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( F @ X2 )
= Unit ) )
=> ( ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ).
% commutative_monoid.fincomp_unit_eqI
thf(fact_96_commutative__monoid_Ofincomp__unit,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_nat] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( commut6741328216151336360_a_nat @ M @ Composition @ Unit
@ ^ [I2: nat] : Unit
@ A2 )
= Unit ) ) ).
% commutative_monoid.fincomp_unit
thf(fact_97_commutative__monoid_Ofincomp__unit,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_nat_a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( commut5242989786243415821_nat_a @ M @ Composition @ Unit
@ ^ [I2: nat > a] : Unit
@ A2 )
= Unit ) ) ).
% commutative_monoid.fincomp_unit
thf(fact_98_commutative__monoid_Ofincomp__unit,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit
@ ^ [I2: a] : Unit
@ A2 )
= Unit ) ) ).
% commutative_monoid.fincomp_unit
thf(fact_99_commutative__monoid_Ofincomp__0_H,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: nat > a,N: nat] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ N )
@ ^ [Uu: nat] : M ) )
=> ( ( Composition @ ( F @ zero_zero_nat ) @ ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ ( set_ord_atMost_nat @ N ) ) ) ) ) ).
% commutative_monoid.fincomp_0'
thf(fact_100_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_101_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_102_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_103_commutative__monoid_Ofincomp__Suc,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: nat > a,N: nat] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ ( set_ord_atMost_nat @ ( suc @ N ) )
@ ^ [Uu: nat] : M ) )
=> ( ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( Composition @ ( F @ ( suc @ N ) ) @ ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ ( set_ord_atMost_nat @ N ) ) ) ) ) ) ).
% commutative_monoid.fincomp_Suc
thf(fact_104_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_105_Zero__not__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_not_Suc
thf(fact_106_Zero__neq__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_neq_Suc
thf(fact_107_Suc__neq__Zero,axiom,
! [M3: nat] :
( ( suc @ M3 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_108_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_109_diff__induct,axiom,
! [P: nat > nat > $o,M3: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X2: nat,Y3: nat] :
( ( P @ X2 @ Y3 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y3 ) ) )
=> ( P @ M3 @ N ) ) ) ) ).
% diff_induct
thf(fact_110_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_111_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_112_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_113_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_114_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_115_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_116_Units__def,axiom,
( ( group_Units_a @ m @ composition @ unit )
= ( collect_a
@ ^ [U2: a] :
( ( member_a @ U2 @ m )
& ( group_invertible_a @ m @ composition @ unit @ U2 ) ) ) ) ).
% Units_def
thf(fact_117_mem__UnitsI,axiom,
! [U: a] :
( ( group_invertible_a @ m @ composition @ unit @ U )
=> ( ( member_a @ U @ m )
=> ( member_a @ U @ ( group_Units_a @ m @ composition @ unit ) ) ) ) ).
% mem_UnitsI
thf(fact_118_mem__UnitsD,axiom,
! [U: a] :
( ( member_a @ U @ ( group_Units_a @ m @ composition @ unit ) )
=> ( ( group_invertible_a @ m @ composition @ unit @ U )
& ( member_a @ U @ m ) ) ) ).
% mem_UnitsD
thf(fact_119_comp__fun__commute__onI,axiom,
! [F: nat > a,F2: set_nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ F2
@ ^ [Uu: nat] : m ) )
=> ( finite1071566134745755356_nat_a @ F2
@ ^ [X3: nat,Y4: a] : ( composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ m @ unit @ Y4 ) ) ) ) ).
% comp_fun_commute_onI
thf(fact_120_comp__fun__commute__onI,axiom,
! [F: ( nat > a ) > a,F2: set_nat_a] :
( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ F2
@ ^ [Uu: nat > a] : m ) )
=> ( finite1127406183625600809at_a_a @ F2
@ ^ [X3: nat > a,Y4: a] : ( composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ m @ unit @ Y4 ) ) ) ) ).
% comp_fun_commute_onI
thf(fact_121_comp__fun__commute__onI,axiom,
! [F: a > a,F2: set_a] :
( ( member_a_a @ F
@ ( pi_a_a @ F2
@ ^ [Uu: a] : m ) )
=> ( finite2737277698481670352on_a_a @ F2
@ ^ [X3: a,Y4: a] : ( composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ m @ unit @ Y4 ) ) ) ) ).
% comp_fun_commute_onI
thf(fact_122_monoid_Oinvertible__left__inverse,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,U: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( group_645299334525884886_nat_a @ M @ Composition @ Unit @ U )
=> ( ( member_nat_a @ U @ M )
=> ( ( Composition @ ( group_inverse_nat_a @ M @ Composition @ Unit @ U ) @ U )
= Unit ) ) ) ) ).
% monoid.invertible_left_inverse
thf(fact_123_monoid_Oinvertible__left__inverse,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( ( Composition @ ( group_inverse_nat @ M @ Composition @ Unit @ U ) @ U )
= Unit ) ) ) ) ).
% monoid.invertible_left_inverse
thf(fact_124_monoid_Oinvertible__left__inverse,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,U: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( group_1438879830998425741at_a_a @ M @ Composition @ Unit @ U )
=> ( ( member_nat_a_a @ U @ M )
=> ( ( Composition @ ( group_6240212554837956873at_a_a @ M @ Composition @ Unit @ U ) @ U )
= Unit ) ) ) ) ).
% monoid.invertible_left_inverse
thf(fact_125_monoid_Oinvertible__left__inverse,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,U: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a_a @ M @ Composition @ Unit @ U )
=> ( ( member_a_a @ U @ M )
=> ( ( Composition @ ( group_inverse_a_a @ M @ Composition @ Unit @ U ) @ U )
= Unit ) ) ) ) ).
% monoid.invertible_left_inverse
thf(fact_126_monoid_Oinvertible__left__inverse,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( ( Composition @ ( group_inverse_a @ M @ Composition @ Unit @ U ) @ U )
= Unit ) ) ) ) ).
% monoid.invertible_left_inverse
thf(fact_127_monoid_Oinvertible__left__inverse2,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,U: nat > a,V2: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( group_645299334525884886_nat_a @ M @ Composition @ Unit @ U )
=> ( ( member_nat_a @ U @ M )
=> ( ( member_nat_a @ V2 @ M )
=> ( ( Composition @ ( group_inverse_nat_a @ M @ Composition @ Unit @ U ) @ ( Composition @ U @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_left_inverse2
thf(fact_128_monoid_Oinvertible__left__inverse2,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat,V2: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( ( member_nat @ V2 @ M )
=> ( ( Composition @ ( group_inverse_nat @ M @ Composition @ Unit @ U ) @ ( Composition @ U @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_left_inverse2
thf(fact_129_monoid_Oinvertible__left__inverse2,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,U: ( nat > a ) > a,V2: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( group_1438879830998425741at_a_a @ M @ Composition @ Unit @ U )
=> ( ( member_nat_a_a @ U @ M )
=> ( ( member_nat_a_a @ V2 @ M )
=> ( ( Composition @ ( group_6240212554837956873at_a_a @ M @ Composition @ Unit @ U ) @ ( Composition @ U @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_left_inverse2
thf(fact_130_monoid_Oinvertible__left__inverse2,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,U: a > a,V2: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a_a @ M @ Composition @ Unit @ U )
=> ( ( member_a_a @ U @ M )
=> ( ( member_a_a @ V2 @ M )
=> ( ( Composition @ ( group_inverse_a_a @ M @ Composition @ Unit @ U ) @ ( Composition @ U @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_left_inverse2
thf(fact_131_monoid_Oinvertible__left__inverse2,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a,V2: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( ( Composition @ ( group_inverse_a @ M @ Composition @ Unit @ U ) @ ( Composition @ U @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_left_inverse2
thf(fact_132_monoid_Oinvertible__right__inverse,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,U: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( group_645299334525884886_nat_a @ M @ Composition @ Unit @ U )
=> ( ( member_nat_a @ U @ M )
=> ( ( Composition @ U @ ( group_inverse_nat_a @ M @ Composition @ Unit @ U ) )
= Unit ) ) ) ) ).
% monoid.invertible_right_inverse
thf(fact_133_monoid_Oinvertible__right__inverse,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( ( Composition @ U @ ( group_inverse_nat @ M @ Composition @ Unit @ U ) )
= Unit ) ) ) ) ).
% monoid.invertible_right_inverse
thf(fact_134_monoid_Oinvertible__right__inverse,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,U: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( group_1438879830998425741at_a_a @ M @ Composition @ Unit @ U )
=> ( ( member_nat_a_a @ U @ M )
=> ( ( Composition @ U @ ( group_6240212554837956873at_a_a @ M @ Composition @ Unit @ U ) )
= Unit ) ) ) ) ).
% monoid.invertible_right_inverse
thf(fact_135_monoid_Oinvertible__right__inverse,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,U: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a_a @ M @ Composition @ Unit @ U )
=> ( ( member_a_a @ U @ M )
=> ( ( Composition @ U @ ( group_inverse_a_a @ M @ Composition @ Unit @ U ) )
= Unit ) ) ) ) ).
% monoid.invertible_right_inverse
thf(fact_136_monoid_Oinvertible__right__inverse,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( ( Composition @ U @ ( group_inverse_a @ M @ Composition @ Unit @ U ) )
= Unit ) ) ) ) ).
% monoid.invertible_right_inverse
thf(fact_137_monoid_Oinvertible__inverse__closed,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,U: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( group_645299334525884886_nat_a @ M @ Composition @ Unit @ U )
=> ( ( member_nat_a @ U @ M )
=> ( member_nat_a @ ( group_inverse_nat_a @ M @ Composition @ Unit @ U ) @ M ) ) ) ) ).
% monoid.invertible_inverse_closed
thf(fact_138_monoid_Oinvertible__inverse__closed,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( member_nat @ ( group_inverse_nat @ M @ Composition @ Unit @ U ) @ M ) ) ) ) ).
% monoid.invertible_inverse_closed
thf(fact_139_monoid_Oinvertible__inverse__closed,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,U: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( group_1438879830998425741at_a_a @ M @ Composition @ Unit @ U )
=> ( ( member_nat_a_a @ U @ M )
=> ( member_nat_a_a @ ( group_6240212554837956873at_a_a @ M @ Composition @ Unit @ U ) @ M ) ) ) ) ).
% monoid.invertible_inverse_closed
thf(fact_140_monoid_Oinvertible__inverse__closed,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,U: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a_a @ M @ Composition @ Unit @ U )
=> ( ( member_a_a @ U @ M )
=> ( member_a_a @ ( group_inverse_a_a @ M @ Composition @ Unit @ U ) @ M ) ) ) ) ).
% monoid.invertible_inverse_closed
thf(fact_141_monoid_Oinvertible__inverse__closed,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( member_a @ ( group_inverse_a @ M @ Composition @ Unit @ U ) @ M ) ) ) ) ).
% monoid.invertible_inverse_closed
thf(fact_142_monoid_Oinvertible__right__inverse2,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,U: nat > a,V2: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( group_645299334525884886_nat_a @ M @ Composition @ Unit @ U )
=> ( ( member_nat_a @ U @ M )
=> ( ( member_nat_a @ V2 @ M )
=> ( ( Composition @ U @ ( Composition @ ( group_inverse_nat_a @ M @ Composition @ Unit @ U ) @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_right_inverse2
thf(fact_143_monoid_Oinvertible__right__inverse2,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat,V2: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( ( member_nat @ V2 @ M )
=> ( ( Composition @ U @ ( Composition @ ( group_inverse_nat @ M @ Composition @ Unit @ U ) @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_right_inverse2
thf(fact_144_monoid_Oinvertible__right__inverse2,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,U: ( nat > a ) > a,V2: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( group_1438879830998425741at_a_a @ M @ Composition @ Unit @ U )
=> ( ( member_nat_a_a @ U @ M )
=> ( ( member_nat_a_a @ V2 @ M )
=> ( ( Composition @ U @ ( Composition @ ( group_6240212554837956873at_a_a @ M @ Composition @ Unit @ U ) @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_right_inverse2
thf(fact_145_monoid_Oinvertible__right__inverse2,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,U: a > a,V2: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a_a @ M @ Composition @ Unit @ U )
=> ( ( member_a_a @ U @ M )
=> ( ( member_a_a @ V2 @ M )
=> ( ( Composition @ U @ ( Composition @ ( group_inverse_a_a @ M @ Composition @ Unit @ U ) @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_right_inverse2
thf(fact_146_monoid_Oinvertible__right__inverse2,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a,V2: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( ( Composition @ U @ ( Composition @ ( group_inverse_a @ M @ Composition @ Unit @ U ) @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_right_inverse2
thf(fact_147_monoid_Oinvertible__inverse__inverse,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,U: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( group_645299334525884886_nat_a @ M @ Composition @ Unit @ U )
=> ( ( member_nat_a @ U @ M )
=> ( ( group_inverse_nat_a @ M @ Composition @ Unit @ ( group_inverse_nat_a @ M @ Composition @ Unit @ U ) )
= U ) ) ) ) ).
% monoid.invertible_inverse_inverse
thf(fact_148_monoid_Oinvertible__inverse__inverse,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( ( group_inverse_nat @ M @ Composition @ Unit @ ( group_inverse_nat @ M @ Composition @ Unit @ U ) )
= U ) ) ) ) ).
% monoid.invertible_inverse_inverse
thf(fact_149_monoid_Oinvertible__inverse__inverse,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,U: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( group_1438879830998425741at_a_a @ M @ Composition @ Unit @ U )
=> ( ( member_nat_a_a @ U @ M )
=> ( ( group_6240212554837956873at_a_a @ M @ Composition @ Unit @ ( group_6240212554837956873at_a_a @ M @ Composition @ Unit @ U ) )
= U ) ) ) ) ).
% monoid.invertible_inverse_inverse
thf(fact_150_monoid_Oinvertible__inverse__inverse,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,U: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a_a @ M @ Composition @ Unit @ U )
=> ( ( member_a_a @ U @ M )
=> ( ( group_inverse_a_a @ M @ Composition @ Unit @ ( group_inverse_a_a @ M @ Composition @ Unit @ U ) )
= U ) ) ) ) ).
% monoid.invertible_inverse_inverse
thf(fact_151_monoid_Oinvertible__inverse__inverse,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( ( group_inverse_a @ M @ Composition @ Unit @ ( group_inverse_a @ M @ Composition @ Unit @ U ) )
= U ) ) ) ) ).
% monoid.invertible_inverse_inverse
thf(fact_152_monoid_Oinverse__composition__commute,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,X: nat > a,Y: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( group_645299334525884886_nat_a @ M @ Composition @ Unit @ X )
=> ( ( group_645299334525884886_nat_a @ M @ Composition @ Unit @ Y )
=> ( ( member_nat_a @ X @ M )
=> ( ( member_nat_a @ Y @ M )
=> ( ( group_inverse_nat_a @ M @ Composition @ Unit @ ( Composition @ X @ Y ) )
= ( Composition @ ( group_inverse_nat_a @ M @ Composition @ Unit @ Y ) @ ( group_inverse_nat_a @ M @ Composition @ Unit @ X ) ) ) ) ) ) ) ) ).
% monoid.inverse_composition_commute
thf(fact_153_monoid_Oinverse__composition__commute,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat,Y: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ Y )
=> ( ( member_nat @ X @ M )
=> ( ( member_nat @ Y @ M )
=> ( ( group_inverse_nat @ M @ Composition @ Unit @ ( Composition @ X @ Y ) )
= ( Composition @ ( group_inverse_nat @ M @ Composition @ Unit @ Y ) @ ( group_inverse_nat @ M @ Composition @ Unit @ X ) ) ) ) ) ) ) ) ).
% monoid.inverse_composition_commute
thf(fact_154_monoid_Oinverse__composition__commute,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,X: ( nat > a ) > a,Y: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( group_1438879830998425741at_a_a @ M @ Composition @ Unit @ X )
=> ( ( group_1438879830998425741at_a_a @ M @ Composition @ Unit @ Y )
=> ( ( member_nat_a_a @ X @ M )
=> ( ( member_nat_a_a @ Y @ M )
=> ( ( group_6240212554837956873at_a_a @ M @ Composition @ Unit @ ( Composition @ X @ Y ) )
= ( Composition @ ( group_6240212554837956873at_a_a @ M @ Composition @ Unit @ Y ) @ ( group_6240212554837956873at_a_a @ M @ Composition @ Unit @ X ) ) ) ) ) ) ) ) ).
% monoid.inverse_composition_commute
thf(fact_155_monoid_Oinverse__composition__commute,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,X: a > a,Y: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a_a @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_a_a @ M @ Composition @ Unit @ Y )
=> ( ( member_a_a @ X @ M )
=> ( ( member_a_a @ Y @ M )
=> ( ( group_inverse_a_a @ M @ Composition @ Unit @ ( Composition @ X @ Y ) )
= ( Composition @ ( group_inverse_a_a @ M @ Composition @ Unit @ Y ) @ ( group_inverse_a_a @ M @ Composition @ Unit @ X ) ) ) ) ) ) ) ) ).
% monoid.inverse_composition_commute
thf(fact_156_monoid_Oinverse__composition__commute,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a,Y: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ Y )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( ( group_inverse_a @ M @ Composition @ Unit @ ( Composition @ X @ Y ) )
= ( Composition @ ( group_inverse_a @ M @ Composition @ Unit @ Y ) @ ( group_inverse_a @ M @ Composition @ Unit @ X ) ) ) ) ) ) ) ) ).
% monoid.inverse_composition_commute
thf(fact_157_monoid_OUnits_Ocong,axiom,
group_Units_a = group_Units_a ).
% monoid.Units.cong
thf(fact_158_monoid_Omem__UnitsI,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,U: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( group_645299334525884886_nat_a @ M @ Composition @ Unit @ U )
=> ( ( member_nat_a @ U @ M )
=> ( member_nat_a @ U @ ( group_Units_nat_a @ M @ Composition @ Unit ) ) ) ) ) ).
% monoid.mem_UnitsI
thf(fact_159_monoid_Omem__UnitsI,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( member_nat @ U @ ( group_Units_nat @ M @ Composition @ Unit ) ) ) ) ) ).
% monoid.mem_UnitsI
thf(fact_160_monoid_Omem__UnitsI,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,U: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( group_1438879830998425741at_a_a @ M @ Composition @ Unit @ U )
=> ( ( member_nat_a_a @ U @ M )
=> ( member_nat_a_a @ U @ ( group_Units_nat_a_a @ M @ Composition @ Unit ) ) ) ) ) ).
% monoid.mem_UnitsI
thf(fact_161_monoid_Omem__UnitsI,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,U: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a_a @ M @ Composition @ Unit @ U )
=> ( ( member_a_a @ U @ M )
=> ( member_a_a @ U @ ( group_Units_a_a @ M @ Composition @ Unit ) ) ) ) ) ).
% monoid.mem_UnitsI
thf(fact_162_monoid_Omem__UnitsI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( member_a @ U @ ( group_Units_a @ M @ Composition @ Unit ) ) ) ) ) ).
% monoid.mem_UnitsI
thf(fact_163_monoid_Omem__UnitsD,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,U: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( member_nat_a @ U @ ( group_Units_nat_a @ M @ Composition @ Unit ) )
=> ( ( group_645299334525884886_nat_a @ M @ Composition @ Unit @ U )
& ( member_nat_a @ U @ M ) ) ) ) ).
% monoid.mem_UnitsD
thf(fact_164_monoid_Omem__UnitsD,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( member_nat @ U @ ( group_Units_nat @ M @ Composition @ Unit ) )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
& ( member_nat @ U @ M ) ) ) ) ).
% monoid.mem_UnitsD
thf(fact_165_monoid_Omem__UnitsD,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,U: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( member_nat_a_a @ U @ ( group_Units_nat_a_a @ M @ Composition @ Unit ) )
=> ( ( group_1438879830998425741at_a_a @ M @ Composition @ Unit @ U )
& ( member_nat_a_a @ U @ M ) ) ) ) ).
% monoid.mem_UnitsD
thf(fact_166_monoid_Omem__UnitsD,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,U: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( member_a_a @ U @ ( group_Units_a_a @ M @ Composition @ Unit ) )
=> ( ( group_invertible_a_a @ M @ Composition @ Unit @ U )
& ( member_a_a @ U @ M ) ) ) ) ).
% monoid.mem_UnitsD
thf(fact_167_monoid_Omem__UnitsD,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( member_a @ U @ ( group_Units_a @ M @ Composition @ Unit ) )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
& ( member_a @ U @ M ) ) ) ) ).
% monoid.mem_UnitsD
thf(fact_168_monoid_OUnits__def,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( group_Units_nat_a @ M @ Composition @ Unit )
= ( collect_nat_a
@ ^ [U2: nat > a] :
( ( member_nat_a @ U2 @ M )
& ( group_645299334525884886_nat_a @ M @ Composition @ Unit @ U2 ) ) ) ) ) ).
% monoid.Units_def
thf(fact_169_monoid_OUnits__def,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( group_Units_nat_a_a @ M @ Composition @ Unit )
= ( collect_nat_a_a
@ ^ [U2: ( nat > a ) > a] :
( ( member_nat_a_a @ U2 @ M )
& ( group_1438879830998425741at_a_a @ M @ Composition @ Unit @ U2 ) ) ) ) ) ).
% monoid.Units_def
thf(fact_170_monoid_OUnits__def,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( group_Units_a_a @ M @ Composition @ Unit )
= ( collect_a_a
@ ^ [U2: a > a] :
( ( member_a_a @ U2 @ M )
& ( group_invertible_a_a @ M @ Composition @ Unit @ U2 ) ) ) ) ) ).
% monoid.Units_def
thf(fact_171_monoid_OUnits__def,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_Units_nat @ M @ Composition @ Unit )
= ( collect_nat
@ ^ [U2: nat] :
( ( member_nat @ U2 @ M )
& ( group_invertible_nat @ M @ Composition @ Unit @ U2 ) ) ) ) ) ).
% monoid.Units_def
thf(fact_172_monoid_OUnits__def,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_Units_a @ M @ Composition @ Unit )
= ( collect_a
@ ^ [U2: a] :
( ( member_a @ U2 @ M )
& ( group_invertible_a @ M @ Composition @ Unit @ U2 ) ) ) ) ) ).
% monoid.Units_def
thf(fact_173_Group__Theory_Omonoid__def,axiom,
( group_monoid_nat_a
= ( ^ [M4: set_nat_a,Composition2: ( nat > a ) > ( nat > a ) > nat > a,Unit2: nat > a] :
( ! [A3: nat > a,B3: nat > a] :
( ( member_nat_a @ A3 @ M4 )
=> ( ( member_nat_a @ B3 @ M4 )
=> ( member_nat_a @ ( Composition2 @ A3 @ B3 ) @ M4 ) ) )
& ( member_nat_a @ Unit2 @ M4 )
& ! [A3: nat > a,B3: nat > a,C2: nat > a] :
( ( member_nat_a @ A3 @ M4 )
=> ( ( member_nat_a @ B3 @ M4 )
=> ( ( member_nat_a @ C2 @ M4 )
=> ( ( Composition2 @ ( Composition2 @ A3 @ B3 ) @ C2 )
= ( Composition2 @ A3 @ ( Composition2 @ B3 @ C2 ) ) ) ) ) )
& ! [A3: nat > a] :
( ( member_nat_a @ A3 @ M4 )
=> ( ( Composition2 @ Unit2 @ A3 )
= A3 ) )
& ! [A3: nat > a] :
( ( member_nat_a @ A3 @ M4 )
=> ( ( Composition2 @ A3 @ Unit2 )
= A3 ) ) ) ) ) ).
% Group_Theory.monoid_def
thf(fact_174_Group__Theory_Omonoid__def,axiom,
( group_monoid_nat
= ( ^ [M4: set_nat,Composition2: nat > nat > nat,Unit2: nat] :
( ! [A3: nat,B3: nat] :
( ( member_nat @ A3 @ M4 )
=> ( ( member_nat @ B3 @ M4 )
=> ( member_nat @ ( Composition2 @ A3 @ B3 ) @ M4 ) ) )
& ( member_nat @ Unit2 @ M4 )
& ! [A3: nat,B3: nat,C2: nat] :
( ( member_nat @ A3 @ M4 )
=> ( ( member_nat @ B3 @ M4 )
=> ( ( member_nat @ C2 @ M4 )
=> ( ( Composition2 @ ( Composition2 @ A3 @ B3 ) @ C2 )
= ( Composition2 @ A3 @ ( Composition2 @ B3 @ C2 ) ) ) ) ) )
& ! [A3: nat] :
( ( member_nat @ A3 @ M4 )
=> ( ( Composition2 @ Unit2 @ A3 )
= A3 ) )
& ! [A3: nat] :
( ( member_nat @ A3 @ M4 )
=> ( ( Composition2 @ A3 @ Unit2 )
= A3 ) ) ) ) ) ).
% Group_Theory.monoid_def
thf(fact_175_Group__Theory_Omonoid__def,axiom,
( group_monoid_nat_a_a
= ( ^ [M4: set_nat_a_a2,Composition2: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit2: ( nat > a ) > a] :
( ! [A3: ( nat > a ) > a,B3: ( nat > a ) > a] :
( ( member_nat_a_a @ A3 @ M4 )
=> ( ( member_nat_a_a @ B3 @ M4 )
=> ( member_nat_a_a @ ( Composition2 @ A3 @ B3 ) @ M4 ) ) )
& ( member_nat_a_a @ Unit2 @ M4 )
& ! [A3: ( nat > a ) > a,B3: ( nat > a ) > a,C2: ( nat > a ) > a] :
( ( member_nat_a_a @ A3 @ M4 )
=> ( ( member_nat_a_a @ B3 @ M4 )
=> ( ( member_nat_a_a @ C2 @ M4 )
=> ( ( Composition2 @ ( Composition2 @ A3 @ B3 ) @ C2 )
= ( Composition2 @ A3 @ ( Composition2 @ B3 @ C2 ) ) ) ) ) )
& ! [A3: ( nat > a ) > a] :
( ( member_nat_a_a @ A3 @ M4 )
=> ( ( Composition2 @ Unit2 @ A3 )
= A3 ) )
& ! [A3: ( nat > a ) > a] :
( ( member_nat_a_a @ A3 @ M4 )
=> ( ( Composition2 @ A3 @ Unit2 )
= A3 ) ) ) ) ) ).
% Group_Theory.monoid_def
thf(fact_176_Group__Theory_Omonoid__def,axiom,
( group_monoid_a_a
= ( ^ [M4: set_a_a,Composition2: ( a > a ) > ( a > a ) > a > a,Unit2: a > a] :
( ! [A3: a > a,B3: a > a] :
( ( member_a_a @ A3 @ M4 )
=> ( ( member_a_a @ B3 @ M4 )
=> ( member_a_a @ ( Composition2 @ A3 @ B3 ) @ M4 ) ) )
& ( member_a_a @ Unit2 @ M4 )
& ! [A3: a > a,B3: a > a,C2: a > a] :
( ( member_a_a @ A3 @ M4 )
=> ( ( member_a_a @ B3 @ M4 )
=> ( ( member_a_a @ C2 @ M4 )
=> ( ( Composition2 @ ( Composition2 @ A3 @ B3 ) @ C2 )
= ( Composition2 @ A3 @ ( Composition2 @ B3 @ C2 ) ) ) ) ) )
& ! [A3: a > a] :
( ( member_a_a @ A3 @ M4 )
=> ( ( Composition2 @ Unit2 @ A3 )
= A3 ) )
& ! [A3: a > a] :
( ( member_a_a @ A3 @ M4 )
=> ( ( Composition2 @ A3 @ Unit2 )
= A3 ) ) ) ) ) ).
% Group_Theory.monoid_def
thf(fact_177_Group__Theory_Omonoid__def,axiom,
( group_monoid_a
= ( ^ [M4: set_a,Composition2: a > a > a,Unit2: a] :
( ! [A3: a,B3: a] :
( ( member_a @ A3 @ M4 )
=> ( ( member_a @ B3 @ M4 )
=> ( member_a @ ( Composition2 @ A3 @ B3 ) @ M4 ) ) )
& ( member_a @ Unit2 @ M4 )
& ! [A3: a,B3: a,C2: a] :
( ( member_a @ A3 @ M4 )
=> ( ( member_a @ B3 @ M4 )
=> ( ( member_a @ C2 @ M4 )
=> ( ( Composition2 @ ( Composition2 @ A3 @ B3 ) @ C2 )
= ( Composition2 @ A3 @ ( Composition2 @ B3 @ C2 ) ) ) ) ) )
& ! [A3: a] :
( ( member_a @ A3 @ M4 )
=> ( ( Composition2 @ Unit2 @ A3 )
= A3 ) )
& ! [A3: a] :
( ( member_a @ A3 @ M4 )
=> ( ( Composition2 @ A3 @ Unit2 )
= A3 ) ) ) ) ) ).
% Group_Theory.monoid_def
thf(fact_178_monoid_Ocomposition__closed,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,A: nat > a,B: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( member_nat_a @ A @ M )
=> ( ( member_nat_a @ B @ M )
=> ( member_nat_a @ ( Composition @ A @ B ) @ M ) ) ) ) ).
% monoid.composition_closed
thf(fact_179_monoid_Ocomposition__closed,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,A: nat,B: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( member_nat @ A @ M )
=> ( ( member_nat @ B @ M )
=> ( member_nat @ ( Composition @ A @ B ) @ M ) ) ) ) ).
% monoid.composition_closed
thf(fact_180_monoid_Ocomposition__closed,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,A: ( nat > a ) > a,B: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( member_nat_a_a @ A @ M )
=> ( ( member_nat_a_a @ B @ M )
=> ( member_nat_a_a @ ( Composition @ A @ B ) @ M ) ) ) ) ).
% monoid.composition_closed
thf(fact_181_monoid_Ocomposition__closed,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,A: a > a,B: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( member_a_a @ A @ M )
=> ( ( member_a_a @ B @ M )
=> ( member_a_a @ ( Composition @ A @ B ) @ M ) ) ) ) ).
% monoid.composition_closed
thf(fact_182_monoid_Ocomposition__closed,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A: a,B: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( member_a @ A @ M )
=> ( ( member_a @ B @ M )
=> ( member_a @ ( Composition @ A @ B ) @ M ) ) ) ) ).
% monoid.composition_closed
thf(fact_183_monoid_Oinverse__unique,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,U: nat > a,V: nat > a,V2: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_nat_a @ U @ M )
=> ( ( member_nat_a @ V2 @ M )
=> ( ( member_nat_a @ V @ M )
=> ( V2 = V ) ) ) ) ) ) ) ).
% monoid.inverse_unique
thf(fact_184_monoid_Oinverse__unique,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat,V: nat,V2: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_nat @ U @ M )
=> ( ( member_nat @ V2 @ M )
=> ( ( member_nat @ V @ M )
=> ( V2 = V ) ) ) ) ) ) ) ).
% monoid.inverse_unique
thf(fact_185_monoid_Oinverse__unique,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,U: ( nat > a ) > a,V: ( nat > a ) > a,V2: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_nat_a_a @ U @ M )
=> ( ( member_nat_a_a @ V2 @ M )
=> ( ( member_nat_a_a @ V @ M )
=> ( V2 = V ) ) ) ) ) ) ) ).
% monoid.inverse_unique
thf(fact_186_monoid_Oinverse__unique,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,U: a > a,V: a > a,V2: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_a_a @ U @ M )
=> ( ( member_a_a @ V2 @ M )
=> ( ( member_a_a @ V @ M )
=> ( V2 = V ) ) ) ) ) ) ) ).
% monoid.inverse_unique
thf(fact_187_monoid_Oinverse__unique,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a,V: a,V2: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( ( member_a @ V @ M )
=> ( V2 = V ) ) ) ) ) ) ) ).
% monoid.inverse_unique
thf(fact_188_monoid_Ounit__closed,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( member_nat_a @ Unit @ M ) ) ).
% monoid.unit_closed
thf(fact_189_monoid_Ounit__closed,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( member_nat @ Unit @ M ) ) ).
% monoid.unit_closed
thf(fact_190_monoid_Ounit__closed,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( member_nat_a_a @ Unit @ M ) ) ).
% monoid.unit_closed
thf(fact_191_monoid_Ounit__closed,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( member_a_a @ Unit @ M ) ) ).
% monoid.unit_closed
thf(fact_192_monoid_Ounit__closed,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( member_a @ Unit @ M ) ) ).
% monoid.unit_closed
thf(fact_193_monoid_Oassociative,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,A: nat > a,B: nat > a,C: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( member_nat_a @ A @ M )
=> ( ( member_nat_a @ B @ M )
=> ( ( member_nat_a @ C @ M )
=> ( ( Composition @ ( Composition @ A @ B ) @ C )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% monoid.associative
thf(fact_194_monoid_Oassociative,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,A: nat,B: nat,C: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( member_nat @ A @ M )
=> ( ( member_nat @ B @ M )
=> ( ( member_nat @ C @ M )
=> ( ( Composition @ ( Composition @ A @ B ) @ C )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% monoid.associative
thf(fact_195_monoid_Oassociative,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,A: ( nat > a ) > a,B: ( nat > a ) > a,C: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( member_nat_a_a @ A @ M )
=> ( ( member_nat_a_a @ B @ M )
=> ( ( member_nat_a_a @ C @ M )
=> ( ( Composition @ ( Composition @ A @ B ) @ C )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% monoid.associative
thf(fact_196_monoid_Oassociative,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,A: a > a,B: a > a,C: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( member_a_a @ A @ M )
=> ( ( member_a_a @ B @ M )
=> ( ( member_a_a @ C @ M )
=> ( ( Composition @ ( Composition @ A @ B ) @ C )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% monoid.associative
thf(fact_197_monoid_Oassociative,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A: a,B: a,C: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( member_a @ A @ M )
=> ( ( member_a @ B @ M )
=> ( ( member_a @ C @ M )
=> ( ( Composition @ ( Composition @ A @ B ) @ C )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% monoid.associative
thf(fact_198_monoid_Oright__unit,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,A: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( member_nat_a @ A @ M )
=> ( ( Composition @ A @ Unit )
= A ) ) ) ).
% monoid.right_unit
thf(fact_199_monoid_Oright__unit,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,A: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( member_nat @ A @ M )
=> ( ( Composition @ A @ Unit )
= A ) ) ) ).
% monoid.right_unit
thf(fact_200_monoid_Oright__unit,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,A: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( member_nat_a_a @ A @ M )
=> ( ( Composition @ A @ Unit )
= A ) ) ) ).
% monoid.right_unit
thf(fact_201_monoid_Oright__unit,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,A: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( member_a_a @ A @ M )
=> ( ( Composition @ A @ Unit )
= A ) ) ) ).
% monoid.right_unit
thf(fact_202_monoid_Oright__unit,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( member_a @ A @ M )
=> ( ( Composition @ A @ Unit )
= A ) ) ) ).
% monoid.right_unit
thf(fact_203_monoid_Oleft__unit,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,A: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( member_nat_a @ A @ M )
=> ( ( Composition @ Unit @ A )
= A ) ) ) ).
% monoid.left_unit
thf(fact_204_monoid_Oleft__unit,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,A: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( member_nat @ A @ M )
=> ( ( Composition @ Unit @ A )
= A ) ) ) ).
% monoid.left_unit
thf(fact_205_monoid_Oleft__unit,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,A: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( member_nat_a_a @ A @ M )
=> ( ( Composition @ Unit @ A )
= A ) ) ) ).
% monoid.left_unit
thf(fact_206_monoid_Oleft__unit,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,A: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( member_a_a @ A @ M )
=> ( ( Composition @ Unit @ A )
= A ) ) ) ).
% monoid.left_unit
thf(fact_207_monoid_Oleft__unit,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( member_a @ A @ M )
=> ( ( Composition @ Unit @ A )
= A ) ) ) ).
% monoid.left_unit
thf(fact_208_Group__Theory_Omonoid_Ointro,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a] :
( ! [A4: nat > a,B4: nat > a] :
( ( member_nat_a @ A4 @ M )
=> ( ( member_nat_a @ B4 @ M )
=> ( member_nat_a @ ( Composition @ A4 @ B4 ) @ M ) ) )
=> ( ( member_nat_a @ Unit @ M )
=> ( ! [A4: nat > a,B4: nat > a,C3: nat > a] :
( ( member_nat_a @ A4 @ M )
=> ( ( member_nat_a @ B4 @ M )
=> ( ( member_nat_a @ C3 @ M )
=> ( ( Composition @ ( Composition @ A4 @ B4 ) @ C3 )
= ( Composition @ A4 @ ( Composition @ B4 @ C3 ) ) ) ) ) )
=> ( ! [A4: nat > a] :
( ( member_nat_a @ A4 @ M )
=> ( ( Composition @ Unit @ A4 )
= A4 ) )
=> ( ! [A4: nat > a] :
( ( member_nat_a @ A4 @ M )
=> ( ( Composition @ A4 @ Unit )
= A4 ) )
=> ( group_monoid_nat_a @ M @ Composition @ Unit ) ) ) ) ) ) ).
% Group_Theory.monoid.intro
thf(fact_209_Group__Theory_Omonoid_Ointro,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat] :
( ! [A4: nat,B4: nat] :
( ( member_nat @ A4 @ M )
=> ( ( member_nat @ B4 @ M )
=> ( member_nat @ ( Composition @ A4 @ B4 ) @ M ) ) )
=> ( ( member_nat @ Unit @ M )
=> ( ! [A4: nat,B4: nat,C3: nat] :
( ( member_nat @ A4 @ M )
=> ( ( member_nat @ B4 @ M )
=> ( ( member_nat @ C3 @ M )
=> ( ( Composition @ ( Composition @ A4 @ B4 ) @ C3 )
= ( Composition @ A4 @ ( Composition @ B4 @ C3 ) ) ) ) ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ M )
=> ( ( Composition @ Unit @ A4 )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ M )
=> ( ( Composition @ A4 @ Unit )
= A4 ) )
=> ( group_monoid_nat @ M @ Composition @ Unit ) ) ) ) ) ) ).
% Group_Theory.monoid.intro
thf(fact_210_Group__Theory_Omonoid_Ointro,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a] :
( ! [A4: ( nat > a ) > a,B4: ( nat > a ) > a] :
( ( member_nat_a_a @ A4 @ M )
=> ( ( member_nat_a_a @ B4 @ M )
=> ( member_nat_a_a @ ( Composition @ A4 @ B4 ) @ M ) ) )
=> ( ( member_nat_a_a @ Unit @ M )
=> ( ! [A4: ( nat > a ) > a,B4: ( nat > a ) > a,C3: ( nat > a ) > a] :
( ( member_nat_a_a @ A4 @ M )
=> ( ( member_nat_a_a @ B4 @ M )
=> ( ( member_nat_a_a @ C3 @ M )
=> ( ( Composition @ ( Composition @ A4 @ B4 ) @ C3 )
= ( Composition @ A4 @ ( Composition @ B4 @ C3 ) ) ) ) ) )
=> ( ! [A4: ( nat > a ) > a] :
( ( member_nat_a_a @ A4 @ M )
=> ( ( Composition @ Unit @ A4 )
= A4 ) )
=> ( ! [A4: ( nat > a ) > a] :
( ( member_nat_a_a @ A4 @ M )
=> ( ( Composition @ A4 @ Unit )
= A4 ) )
=> ( group_monoid_nat_a_a @ M @ Composition @ Unit ) ) ) ) ) ) ).
% Group_Theory.monoid.intro
thf(fact_211_Group__Theory_Omonoid_Ointro,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a] :
( ! [A4: a > a,B4: a > a] :
( ( member_a_a @ A4 @ M )
=> ( ( member_a_a @ B4 @ M )
=> ( member_a_a @ ( Composition @ A4 @ B4 ) @ M ) ) )
=> ( ( member_a_a @ Unit @ M )
=> ( ! [A4: a > a,B4: a > a,C3: a > a] :
( ( member_a_a @ A4 @ M )
=> ( ( member_a_a @ B4 @ M )
=> ( ( member_a_a @ C3 @ M )
=> ( ( Composition @ ( Composition @ A4 @ B4 ) @ C3 )
= ( Composition @ A4 @ ( Composition @ B4 @ C3 ) ) ) ) ) )
=> ( ! [A4: a > a] :
( ( member_a_a @ A4 @ M )
=> ( ( Composition @ Unit @ A4 )
= A4 ) )
=> ( ! [A4: a > a] :
( ( member_a_a @ A4 @ M )
=> ( ( Composition @ A4 @ Unit )
= A4 ) )
=> ( group_monoid_a_a @ M @ Composition @ Unit ) ) ) ) ) ) ).
% Group_Theory.monoid.intro
thf(fact_212_Group__Theory_Omonoid_Ointro,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ! [A4: a,B4: a] :
( ( member_a @ A4 @ M )
=> ( ( member_a @ B4 @ M )
=> ( member_a @ ( Composition @ A4 @ B4 ) @ M ) ) )
=> ( ( member_a @ Unit @ M )
=> ( ! [A4: a,B4: a,C3: a] :
( ( member_a @ A4 @ M )
=> ( ( member_a @ B4 @ M )
=> ( ( member_a @ C3 @ M )
=> ( ( Composition @ ( Composition @ A4 @ B4 ) @ C3 )
= ( Composition @ A4 @ ( Composition @ B4 @ C3 ) ) ) ) ) )
=> ( ! [A4: a] :
( ( member_a @ A4 @ M )
=> ( ( Composition @ Unit @ A4 )
= A4 ) )
=> ( ! [A4: a] :
( ( member_a @ A4 @ M )
=> ( ( Composition @ A4 @ Unit )
= A4 ) )
=> ( group_monoid_a @ M @ Composition @ Unit ) ) ) ) ) ) ).
% Group_Theory.monoid.intro
thf(fact_213_monoid_Oinvertible_Ocong,axiom,
group_invertible_a = group_invertible_a ).
% monoid.invertible.cong
thf(fact_214_monoid_Oinverse_Ocong,axiom,
group_inverse_a = group_inverse_a ).
% monoid.inverse.cong
thf(fact_215_commutative__monoid_Ocommutative,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,X: nat > a,Y: nat > a] :
( ( group_3093379471365697572_nat_a @ M @ Composition @ Unit )
=> ( ( member_nat_a @ X @ M )
=> ( ( member_nat_a @ Y @ M )
=> ( ( Composition @ X @ Y )
= ( Composition @ Y @ X ) ) ) ) ) ).
% commutative_monoid.commutative
thf(fact_216_commutative__monoid_Ocommutative,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat,Y: nat] :
( ( group_6791354081887936081id_nat @ M @ Composition @ Unit )
=> ( ( member_nat @ X @ M )
=> ( ( member_nat @ Y @ M )
=> ( ( Composition @ X @ Y )
= ( Composition @ Y @ X ) ) ) ) ) ).
% commutative_monoid.commutative
thf(fact_217_commutative__monoid_Ocommutative,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,X: ( nat > a ) > a,Y: ( nat > a ) > a] :
( ( group_1237985806048136895at_a_a @ M @ Composition @ Unit )
=> ( ( member_nat_a_a @ X @ M )
=> ( ( member_nat_a_a @ Y @ M )
=> ( ( Composition @ X @ Y )
= ( Composition @ Y @ X ) ) ) ) ) ).
% commutative_monoid.commutative
thf(fact_218_commutative__monoid_Ocommutative,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,X: a > a,Y: a > a] :
( ( group_6976245611985207014id_a_a @ M @ Composition @ Unit )
=> ( ( member_a_a @ X @ M )
=> ( ( member_a_a @ Y @ M )
=> ( ( Composition @ X @ Y )
= ( Composition @ Y @ X ) ) ) ) ) ).
% commutative_monoid.commutative
thf(fact_219_commutative__monoid_Ocommutative,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a,Y: a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( ( Composition @ X @ Y )
= ( Composition @ Y @ X ) ) ) ) ) ).
% commutative_monoid.commutative
thf(fact_220_commutative__monoid_Ocomp__fun__commute__onI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: nat > a,F2: set_nat] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ F2
@ ^ [Uu: nat] : M ) )
=> ( finite1071566134745755356_nat_a @ F2
@ ^ [X3: nat,Y4: a] : ( Composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ M @ Unit @ Y4 ) ) ) ) ) ).
% commutative_monoid.comp_fun_commute_onI
thf(fact_221_commutative__monoid_Ocomp__fun__commute__onI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: ( nat > a ) > a,F2: set_nat_a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ F2
@ ^ [Uu: nat > a] : M ) )
=> ( finite1127406183625600809at_a_a @ F2
@ ^ [X3: nat > a,Y4: a] : ( Composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ M @ Unit @ Y4 ) ) ) ) ) ).
% commutative_monoid.comp_fun_commute_onI
thf(fact_222_commutative__monoid_Ocomp__fun__commute__onI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: a > a,F2: set_a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_a_a @ F
@ ( pi_a_a @ F2
@ ^ [Uu: a] : M ) )
=> ( finite2737277698481670352on_a_a @ F2
@ ^ [X3: a,Y4: a] : ( Composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ M @ Unit @ Y4 ) ) ) ) ) ).
% commutative_monoid.comp_fun_commute_onI
thf(fact_223_monoid_Oinvertible__right__cancel,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,X: nat > a,Y: nat > a,Z: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( group_645299334525884886_nat_a @ M @ Composition @ Unit @ X )
=> ( ( member_nat_a @ X @ M )
=> ( ( member_nat_a @ Y @ M )
=> ( ( member_nat_a @ Z @ M )
=> ( ( ( Composition @ Y @ X )
= ( Composition @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_right_cancel
thf(fact_224_monoid_Oinvertible__right__cancel,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat,Y: nat,Z: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ X )
=> ( ( member_nat @ X @ M )
=> ( ( member_nat @ Y @ M )
=> ( ( member_nat @ Z @ M )
=> ( ( ( Composition @ Y @ X )
= ( Composition @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_right_cancel
thf(fact_225_monoid_Oinvertible__right__cancel,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,X: ( nat > a ) > a,Y: ( nat > a ) > a,Z: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( group_1438879830998425741at_a_a @ M @ Composition @ Unit @ X )
=> ( ( member_nat_a_a @ X @ M )
=> ( ( member_nat_a_a @ Y @ M )
=> ( ( member_nat_a_a @ Z @ M )
=> ( ( ( Composition @ Y @ X )
= ( Composition @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_right_cancel
thf(fact_226_monoid_Oinvertible__right__cancel,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,X: a > a,Y: a > a,Z: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a_a @ M @ Composition @ Unit @ X )
=> ( ( member_a_a @ X @ M )
=> ( ( member_a_a @ Y @ M )
=> ( ( member_a_a @ Z @ M )
=> ( ( ( Composition @ Y @ X )
= ( Composition @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_right_cancel
thf(fact_227_monoid_Oinvertible__right__cancel,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a,Y: a,Z: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ X )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( ( member_a @ Z @ M )
=> ( ( ( Composition @ Y @ X )
= ( Composition @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_right_cancel
thf(fact_228_monoid_Oinvertible__left__cancel,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,X: nat > a,Y: nat > a,Z: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( group_645299334525884886_nat_a @ M @ Composition @ Unit @ X )
=> ( ( member_nat_a @ X @ M )
=> ( ( member_nat_a @ Y @ M )
=> ( ( member_nat_a @ Z @ M )
=> ( ( ( Composition @ X @ Y )
= ( Composition @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_left_cancel
thf(fact_229_monoid_Oinvertible__left__cancel,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat,Y: nat,Z: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ X )
=> ( ( member_nat @ X @ M )
=> ( ( member_nat @ Y @ M )
=> ( ( member_nat @ Z @ M )
=> ( ( ( Composition @ X @ Y )
= ( Composition @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_left_cancel
thf(fact_230_monoid_Oinvertible__left__cancel,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,X: ( nat > a ) > a,Y: ( nat > a ) > a,Z: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( group_1438879830998425741at_a_a @ M @ Composition @ Unit @ X )
=> ( ( member_nat_a_a @ X @ M )
=> ( ( member_nat_a_a @ Y @ M )
=> ( ( member_nat_a_a @ Z @ M )
=> ( ( ( Composition @ X @ Y )
= ( Composition @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_left_cancel
thf(fact_231_monoid_Oinvertible__left__cancel,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,X: a > a,Y: a > a,Z: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a_a @ M @ Composition @ Unit @ X )
=> ( ( member_a_a @ X @ M )
=> ( ( member_a_a @ Y @ M )
=> ( ( member_a_a @ Z @ M )
=> ( ( ( Composition @ X @ Y )
= ( Composition @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_left_cancel
thf(fact_232_monoid_Oinvertible__left__cancel,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a,Y: a,Z: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ X )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( ( member_a @ Z @ M )
=> ( ( ( Composition @ X @ Y )
= ( Composition @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ) ).
% monoid.invertible_left_cancel
thf(fact_233_monoid_Ocomposition__invertible,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,X: nat > a,Y: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( group_645299334525884886_nat_a @ M @ Composition @ Unit @ X )
=> ( ( group_645299334525884886_nat_a @ M @ Composition @ Unit @ Y )
=> ( ( member_nat_a @ X @ M )
=> ( ( member_nat_a @ Y @ M )
=> ( group_645299334525884886_nat_a @ M @ Composition @ Unit @ ( Composition @ X @ Y ) ) ) ) ) ) ) ).
% monoid.composition_invertible
thf(fact_234_monoid_Ocomposition__invertible,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat,Y: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ Y )
=> ( ( member_nat @ X @ M )
=> ( ( member_nat @ Y @ M )
=> ( group_invertible_nat @ M @ Composition @ Unit @ ( Composition @ X @ Y ) ) ) ) ) ) ) ).
% monoid.composition_invertible
thf(fact_235_monoid_Ocomposition__invertible,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,X: ( nat > a ) > a,Y: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( group_1438879830998425741at_a_a @ M @ Composition @ Unit @ X )
=> ( ( group_1438879830998425741at_a_a @ M @ Composition @ Unit @ Y )
=> ( ( member_nat_a_a @ X @ M )
=> ( ( member_nat_a_a @ Y @ M )
=> ( group_1438879830998425741at_a_a @ M @ Composition @ Unit @ ( Composition @ X @ Y ) ) ) ) ) ) ) ).
% monoid.composition_invertible
thf(fact_236_monoid_Ocomposition__invertible,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,X: a > a,Y: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a_a @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_a_a @ M @ Composition @ Unit @ Y )
=> ( ( member_a_a @ X @ M )
=> ( ( member_a_a @ Y @ M )
=> ( group_invertible_a_a @ M @ Composition @ Unit @ ( Composition @ X @ Y ) ) ) ) ) ) ) ).
% monoid.composition_invertible
thf(fact_237_monoid_Ocomposition__invertible,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a,Y: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ Y )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( group_invertible_a @ M @ Composition @ Unit @ ( Composition @ X @ Y ) ) ) ) ) ) ) ).
% monoid.composition_invertible
thf(fact_238_monoid_Ounit__invertible,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( group_invertible_a @ M @ Composition @ Unit @ Unit ) ) ).
% monoid.unit_invertible
thf(fact_239_monoid_Oinvertible__def,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,U: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( member_nat_a @ U @ M )
=> ( ( group_645299334525884886_nat_a @ M @ Composition @ Unit @ U )
= ( ? [X3: nat > a] :
( ( member_nat_a @ X3 @ M )
& ( ( Composition @ U @ X3 )
= Unit )
& ( ( Composition @ X3 @ U )
= Unit ) ) ) ) ) ) ).
% monoid.invertible_def
thf(fact_240_monoid_Oinvertible__def,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( member_nat @ U @ M )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ M )
& ( ( Composition @ U @ X3 )
= Unit )
& ( ( Composition @ X3 @ U )
= Unit ) ) ) ) ) ) ).
% monoid.invertible_def
thf(fact_241_monoid_Oinvertible__def,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,U: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( member_nat_a_a @ U @ M )
=> ( ( group_1438879830998425741at_a_a @ M @ Composition @ Unit @ U )
= ( ? [X3: ( nat > a ) > a] :
( ( member_nat_a_a @ X3 @ M )
& ( ( Composition @ U @ X3 )
= Unit )
& ( ( Composition @ X3 @ U )
= Unit ) ) ) ) ) ) ).
% monoid.invertible_def
thf(fact_242_monoid_Oinvertible__def,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,U: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( member_a_a @ U @ M )
=> ( ( group_invertible_a_a @ M @ Composition @ Unit @ U )
= ( ? [X3: a > a] :
( ( member_a_a @ X3 @ M )
& ( ( Composition @ U @ X3 )
= Unit )
& ( ( Composition @ X3 @ U )
= Unit ) ) ) ) ) ) ).
% monoid.invertible_def
thf(fact_243_monoid_Oinvertible__def,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( member_a @ U @ M )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
= ( ? [X3: a] :
( ( member_a @ X3 @ M )
& ( ( Composition @ U @ X3 )
= Unit )
& ( ( Composition @ X3 @ U )
= Unit ) ) ) ) ) ) ).
% monoid.invertible_def
thf(fact_244_monoid_OinvertibleI,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,U: nat > a,V2: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_nat_a @ U @ M )
=> ( ( member_nat_a @ V2 @ M )
=> ( group_645299334525884886_nat_a @ M @ Composition @ Unit @ U ) ) ) ) ) ) ).
% monoid.invertibleI
thf(fact_245_monoid_OinvertibleI,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat,V2: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_nat @ U @ M )
=> ( ( member_nat @ V2 @ M )
=> ( group_invertible_nat @ M @ Composition @ Unit @ U ) ) ) ) ) ) ).
% monoid.invertibleI
thf(fact_246_monoid_OinvertibleI,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,U: ( nat > a ) > a,V2: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_nat_a_a @ U @ M )
=> ( ( member_nat_a_a @ V2 @ M )
=> ( group_1438879830998425741at_a_a @ M @ Composition @ Unit @ U ) ) ) ) ) ) ).
% monoid.invertibleI
thf(fact_247_monoid_OinvertibleI,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,U: a > a,V2: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_a_a @ U @ M )
=> ( ( member_a_a @ V2 @ M )
=> ( group_invertible_a_a @ M @ Composition @ Unit @ U ) ) ) ) ) ) ).
% monoid.invertibleI
thf(fact_248_monoid_OinvertibleI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a,V2: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( group_invertible_a @ M @ Composition @ Unit @ U ) ) ) ) ) ) ).
% monoid.invertibleI
thf(fact_249_monoid_OinvertibleE,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,U: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( group_645299334525884886_nat_a @ M @ Composition @ Unit @ U )
=> ( ! [V3: nat > a] :
( ( ( ( Composition @ U @ V3 )
= Unit )
& ( ( Composition @ V3 @ U )
= Unit ) )
=> ~ ( member_nat_a @ V3 @ M ) )
=> ~ ( member_nat_a @ U @ M ) ) ) ) ).
% monoid.invertibleE
thf(fact_250_monoid_OinvertibleE,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ! [V3: nat] :
( ( ( ( Composition @ U @ V3 )
= Unit )
& ( ( Composition @ V3 @ U )
= Unit ) )
=> ~ ( member_nat @ V3 @ M ) )
=> ~ ( member_nat @ U @ M ) ) ) ) ).
% monoid.invertibleE
thf(fact_251_monoid_OinvertibleE,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,U: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( group_1438879830998425741at_a_a @ M @ Composition @ Unit @ U )
=> ( ! [V3: ( nat > a ) > a] :
( ( ( ( Composition @ U @ V3 )
= Unit )
& ( ( Composition @ V3 @ U )
= Unit ) )
=> ~ ( member_nat_a_a @ V3 @ M ) )
=> ~ ( member_nat_a_a @ U @ M ) ) ) ) ).
% monoid.invertibleE
thf(fact_252_monoid_OinvertibleE,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,U: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a_a @ M @ Composition @ Unit @ U )
=> ( ! [V3: a > a] :
( ( ( ( Composition @ U @ V3 )
= Unit )
& ( ( Composition @ V3 @ U )
= Unit ) )
=> ~ ( member_a_a @ V3 @ M ) )
=> ~ ( member_a_a @ U @ M ) ) ) ) ).
% monoid.invertibleE
thf(fact_253_monoid_OinvertibleE,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ! [V3: a] :
( ( ( ( Composition @ U @ V3 )
= Unit )
& ( ( Composition @ V3 @ U )
= Unit ) )
=> ~ ( member_a @ V3 @ M ) )
=> ~ ( member_a @ U @ M ) ) ) ) ).
% monoid.invertibleE
thf(fact_254_monoid_Oinverse__equality,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,U: nat > a,V2: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_nat_a @ U @ M )
=> ( ( member_nat_a @ V2 @ M )
=> ( ( group_inverse_nat_a @ M @ Composition @ Unit @ U )
= V2 ) ) ) ) ) ) ).
% monoid.inverse_equality
thf(fact_255_monoid_Oinverse__equality,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat,V2: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_nat @ U @ M )
=> ( ( member_nat @ V2 @ M )
=> ( ( group_inverse_nat @ M @ Composition @ Unit @ U )
= V2 ) ) ) ) ) ) ).
% monoid.inverse_equality
thf(fact_256_monoid_Oinverse__equality,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,U: ( nat > a ) > a,V2: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_nat_a_a @ U @ M )
=> ( ( member_nat_a_a @ V2 @ M )
=> ( ( group_6240212554837956873at_a_a @ M @ Composition @ Unit @ U )
= V2 ) ) ) ) ) ) ).
% monoid.inverse_equality
thf(fact_257_monoid_Oinverse__equality,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,U: a > a,V2: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_a_a @ U @ M )
=> ( ( member_a_a @ V2 @ M )
=> ( ( group_inverse_a_a @ M @ Composition @ Unit @ U )
= V2 ) ) ) ) ) ) ).
% monoid.inverse_equality
thf(fact_258_monoid_Oinverse__equality,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a,V2: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_a @ U @ M )
=> ( ( member_a @ V2 @ M )
=> ( ( group_inverse_a @ M @ Composition @ Unit @ U )
= V2 ) ) ) ) ) ) ).
% monoid.inverse_equality
thf(fact_259_monoid_Oinverse__unit,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_inverse_a @ M @ Composition @ Unit @ Unit )
= Unit ) ) ).
% monoid.inverse_unit
thf(fact_260_commutative__monoid_Oaxioms_I1_J,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( group_monoid_a @ M @ Composition @ Unit ) ) ).
% commutative_monoid.axioms(1)
thf(fact_261_monoid_Oinvertible__inverse__invertible,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,U: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( group_645299334525884886_nat_a @ M @ Composition @ Unit @ U )
=> ( ( member_nat_a @ U @ M )
=> ( group_645299334525884886_nat_a @ M @ Composition @ Unit @ ( group_inverse_nat_a @ M @ Composition @ Unit @ U ) ) ) ) ) ).
% monoid.invertible_inverse_invertible
thf(fact_262_monoid_Oinvertible__inverse__invertible,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ U )
=> ( ( member_nat @ U @ M )
=> ( group_invertible_nat @ M @ Composition @ Unit @ ( group_inverse_nat @ M @ Composition @ Unit @ U ) ) ) ) ) ).
% monoid.invertible_inverse_invertible
thf(fact_263_monoid_Oinvertible__inverse__invertible,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,U: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( group_1438879830998425741at_a_a @ M @ Composition @ Unit @ U )
=> ( ( member_nat_a_a @ U @ M )
=> ( group_1438879830998425741at_a_a @ M @ Composition @ Unit @ ( group_6240212554837956873at_a_a @ M @ Composition @ Unit @ U ) ) ) ) ) ).
% monoid.invertible_inverse_invertible
thf(fact_264_monoid_Oinvertible__inverse__invertible,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,U: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a_a @ M @ Composition @ Unit @ U )
=> ( ( member_a_a @ U @ M )
=> ( group_invertible_a_a @ M @ Composition @ Unit @ ( group_inverse_a_a @ M @ Composition @ Unit @ U ) ) ) ) ) ).
% monoid.invertible_inverse_invertible
thf(fact_265_monoid_Oinvertible__inverse__invertible,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ U )
=> ( ( member_a @ U @ M )
=> ( group_invertible_a @ M @ Composition @ Unit @ ( group_inverse_a @ M @ Composition @ Unit @ U ) ) ) ) ) ).
% monoid.invertible_inverse_invertible
thf(fact_266_Pi__I,axiom,
! [A2: set_a,F: a > a,B2: a > set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_a_a @ F @ ( pi_a_a @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_267_Pi__I,axiom,
! [A2: set_a,F: a > nat,B2: a > set_nat] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_a_nat @ F @ ( pi_a_nat @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_268_Pi__I,axiom,
! [A2: set_nat,F: nat > a,B2: nat > set_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_nat_a @ F @ ( pi_nat_a @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_269_Pi__I,axiom,
! [A2: set_nat,F: nat > nat,B2: nat > set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_nat_nat @ F @ ( pi_nat_nat @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_270_Pi__I,axiom,
! [A2: set_a,F: a > nat > a,B2: a > set_nat_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_nat_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_a_nat_a @ F @ ( pi_a_nat_a @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_271_Pi__I,axiom,
! [A2: set_a,F: a > a > a,B2: a > set_a_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_a_a_a2 @ F @ ( pi_a_a_a2 @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_272_Pi__I,axiom,
! [A2: set_nat_a,F: ( nat > a ) > a,B2: ( nat > a ) > set_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_nat_a_a @ F @ ( pi_nat_a_a @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_273_Pi__I,axiom,
! [A2: set_nat_a,F: ( nat > a ) > nat,B2: ( nat > a ) > set_nat] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_nat_a_nat @ F @ ( pi_nat_a_nat @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_274_Pi__I,axiom,
! [A2: set_nat,F: nat > nat > a,B2: nat > set_nat_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_nat_nat_a @ F @ ( pi_nat_nat_a @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_275_Pi__I,axiom,
! [A2: set_nat,F: nat > a > a,B2: nat > set_a_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_a_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_nat_a_a2 @ F @ ( pi_nat_a_a2 @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_276_group__of__Units,axiom,
group_group_a @ ( group_Units_a @ m @ composition @ unit ) @ composition @ unit ).
% group_of_Units
thf(fact_277_fincomp__empty,axiom,
! [F: nat > a] :
( ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ bot_bot_set_nat )
= unit ) ).
% fincomp_empty
thf(fact_278_fincomp__empty,axiom,
! [F: ( nat > a ) > a] :
( ( commut5242989786243415821_nat_a @ m @ composition @ unit @ F @ bot_bot_set_nat_a )
= unit ) ).
% fincomp_empty
thf(fact_279_fincomp__empty,axiom,
! [F: a > a] :
( ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ F @ bot_bot_set_a )
= unit ) ).
% fincomp_empty
thf(fact_280_inverse__undefined,axiom,
! [U: a] :
( ~ ( member_a @ U @ m )
=> ( ( group_inverse_a @ m @ composition @ unit @ U )
= undefined_a ) ) ).
% inverse_undefined
thf(fact_281_fincomp__infinite,axiom,
! [A2: set_nat,F: nat > a] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ A2 )
= unit ) ) ).
% fincomp_infinite
thf(fact_282_fincomp__infinite,axiom,
! [A2: set_nat_a,F: ( nat > a ) > a] :
( ~ ( finite_finite_nat_a @ A2 )
=> ( ( commut5242989786243415821_nat_a @ m @ composition @ unit @ F @ A2 )
= unit ) ) ).
% fincomp_infinite
thf(fact_283_fincomp__infinite,axiom,
! [A2: set_a,F: a > a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ F @ A2 )
= unit ) ) ).
% fincomp_infinite
thf(fact_284_subgroupI,axiom,
! [G2: set_a] :
( ( ord_less_eq_set_a @ G2 @ m )
=> ( ( member_a @ unit @ G2 )
=> ( ! [G3: a,H: a] :
( ( member_a @ G3 @ G2 )
=> ( ( member_a @ H @ G2 )
=> ( member_a @ ( composition @ G3 @ H ) @ G2 ) ) )
=> ( ! [G3: a] :
( ( member_a @ G3 @ G2 )
=> ( group_invertible_a @ m @ composition @ unit @ G3 ) )
=> ( ! [G3: a] :
( ( member_a @ G3 @ G2 )
=> ( member_a @ ( group_inverse_a @ m @ composition @ unit @ G3 ) @ G2 ) )
=> ( group_subgroup_a @ G2 @ m @ composition @ unit ) ) ) ) ) ) ).
% subgroupI
thf(fact_285_fincomp__def,axiom,
! [A2: set_nat,F: nat > a] :
( ( ( finite_finite_nat @ A2 )
=> ( ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ A2 )
= ( finite_fold_nat_a
@ ^ [X3: nat,Y4: a] : ( composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ m @ unit @ Y4 ) )
@ unit
@ A2 ) ) )
& ( ~ ( finite_finite_nat @ A2 )
=> ( ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ A2 )
= unit ) ) ) ).
% fincomp_def
thf(fact_286_fincomp__def,axiom,
! [A2: set_nat_a,F: ( nat > a ) > a] :
( ( ( finite_finite_nat_a @ A2 )
=> ( ( commut5242989786243415821_nat_a @ m @ composition @ unit @ F @ A2 )
= ( finite_fold_nat_a_a
@ ^ [X3: nat > a,Y4: a] : ( composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ m @ unit @ Y4 ) )
@ unit
@ A2 ) ) )
& ( ~ ( finite_finite_nat_a @ A2 )
=> ( ( commut5242989786243415821_nat_a @ m @ composition @ unit @ F @ A2 )
= unit ) ) ) ).
% fincomp_def
thf(fact_287_fincomp__def,axiom,
! [A2: set_a,F: a > a] :
( ( ( finite_finite_a @ A2 )
=> ( ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ F @ A2 )
= ( finite_fold_a_a
@ ^ [X3: a,Y4: a] : ( composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ m @ unit @ Y4 ) )
@ unit
@ A2 ) ) )
& ( ~ ( finite_finite_a @ A2 )
=> ( ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ F @ A2 )
= unit ) ) ) ).
% fincomp_def
thf(fact_288_fincomp__0,axiom,
! [F: nat > a] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat )
@ ^ [Uu: nat] : m ) )
=> ( ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ ( set_ord_atMost_nat @ zero_zero_nat ) )
= ( F @ zero_zero_nat ) ) ) ).
% fincomp_0
thf(fact_289_commutative__monoid__def,axiom,
( group_4866109990395492029noid_a
= ( ^ [M4: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_monoid_a @ M4 @ Composition2 @ Unit2 )
& ( group_2081300317213596122ioms_a @ M4 @ Composition2 ) ) ) ) ).
% commutative_monoid_def
thf(fact_290_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% finite_atLeastAtMost
thf(fact_291_finite__atMost,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% finite_atMost
thf(fact_292_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_293_Icc__eq__Icc,axiom,
! [L: set_a,H2: set_a,L2: set_a,H3: set_a] :
( ( ( set_or6288561110385358355_set_a @ L @ H2 )
= ( set_or6288561110385358355_set_a @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_set_a @ L @ H2 )
& ~ ( ord_less_eq_set_a @ L2 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_294_Icc__eq__Icc,axiom,
! [L: set_nat,H2: set_nat,L2: set_nat,H3: set_nat] :
( ( ( set_or4548717258645045905et_nat @ L @ H2 )
= ( set_or4548717258645045905et_nat @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_set_nat @ L @ H2 )
& ~ ( ord_less_eq_set_nat @ L2 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_295_Icc__eq__Icc,axiom,
! [L: nat,H2: nat,L2: nat,H3: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H2 )
= ( set_or1269000886237332187st_nat @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_nat @ L @ H2 )
& ~ ( ord_less_eq_nat @ L2 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_296_atLeastAtMost__iff,axiom,
! [I3: set_a,L: set_a,U: set_a] :
( ( member_set_a @ I3 @ ( set_or6288561110385358355_set_a @ L @ U ) )
= ( ( ord_less_eq_set_a @ L @ I3 )
& ( ord_less_eq_set_a @ I3 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_297_atLeastAtMost__iff,axiom,
! [I3: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I3 @ ( set_or4548717258645045905et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I3 )
& ( ord_less_eq_set_nat @ I3 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_298_atLeastAtMost__iff,axiom,
! [I3: nat,L: nat,U: nat] :
( ( member_nat @ I3 @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I3 )
& ( ord_less_eq_nat @ I3 @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_299_atMost__iff,axiom,
! [I3: set_a,K: set_a] :
( ( member_set_a @ I3 @ ( set_ord_atMost_set_a @ K ) )
= ( ord_less_eq_set_a @ I3 @ K ) ) ).
% atMost_iff
thf(fact_300_atMost__iff,axiom,
! [I3: set_nat,K: set_nat] :
( ( member_set_nat @ I3 @ ( set_or4236626031148496127et_nat @ K ) )
= ( ord_less_eq_set_nat @ I3 @ K ) ) ).
% atMost_iff
thf(fact_301_atMost__iff,axiom,
! [I3: nat,K: nat] :
( ( member_nat @ I3 @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I3 @ K ) ) ).
% atMost_iff
thf(fact_302_Pi__split__insert__domain,axiom,
! [X: nat > nat > a,I3: nat,I4: set_nat,X4: nat > set_nat_a] :
( ( member_nat_nat_a @ X @ ( pi_nat_nat_a @ ( insert_nat @ I3 @ I4 ) @ X4 ) )
= ( ( member_nat_nat_a @ X @ ( pi_nat_nat_a @ I4 @ X4 ) )
& ( member_nat_a @ ( X @ I3 ) @ ( X4 @ I3 ) ) ) ) ).
% Pi_split_insert_domain
thf(fact_303_Pi__split__insert__domain,axiom,
! [X: nat > nat,I3: nat,I4: set_nat,X4: nat > set_nat] :
( ( member_nat_nat @ X @ ( pi_nat_nat @ ( insert_nat @ I3 @ I4 ) @ X4 ) )
= ( ( member_nat_nat @ X @ ( pi_nat_nat @ I4 @ X4 ) )
& ( member_nat @ ( X @ I3 ) @ ( X4 @ I3 ) ) ) ) ).
% Pi_split_insert_domain
thf(fact_304_Pi__split__insert__domain,axiom,
! [X: nat > ( nat > a ) > a,I3: nat,I4: set_nat,X4: nat > set_nat_a_a2] :
( ( member_nat_nat_a_a @ X @ ( pi_nat_nat_a_a @ ( insert_nat @ I3 @ I4 ) @ X4 ) )
= ( ( member_nat_nat_a_a @ X @ ( pi_nat_nat_a_a @ I4 @ X4 ) )
& ( member_nat_a_a @ ( X @ I3 ) @ ( X4 @ I3 ) ) ) ) ).
% Pi_split_insert_domain
thf(fact_305_Pi__split__insert__domain,axiom,
! [X: nat > a > a,I3: nat,I4: set_nat,X4: nat > set_a_a] :
( ( member_nat_a_a2 @ X @ ( pi_nat_a_a2 @ ( insert_nat @ I3 @ I4 ) @ X4 ) )
= ( ( member_nat_a_a2 @ X @ ( pi_nat_a_a2 @ I4 @ X4 ) )
& ( member_a_a @ ( X @ I3 ) @ ( X4 @ I3 ) ) ) ) ).
% Pi_split_insert_domain
thf(fact_306_Pi__split__insert__domain,axiom,
! [X: nat > a,I3: nat,I4: set_nat,X4: nat > set_a] :
( ( member_nat_a @ X @ ( pi_nat_a @ ( insert_nat @ I3 @ I4 ) @ X4 ) )
= ( ( member_nat_a @ X @ ( pi_nat_a @ I4 @ X4 ) )
& ( member_a @ ( X @ I3 ) @ ( X4 @ I3 ) ) ) ) ).
% Pi_split_insert_domain
thf(fact_307_Pi__split__insert__domain,axiom,
! [X: ( nat > a ) > a,I3: nat > a,I4: set_nat_a,X4: ( nat > a ) > set_a] :
( ( member_nat_a_a @ X @ ( pi_nat_a_a @ ( insert_nat_a @ I3 @ I4 ) @ X4 ) )
= ( ( member_nat_a_a @ X @ ( pi_nat_a_a @ I4 @ X4 ) )
& ( member_a @ ( X @ I3 ) @ ( X4 @ I3 ) ) ) ) ).
% Pi_split_insert_domain
thf(fact_308_Pi__split__insert__domain,axiom,
! [X: a > a,I3: a,I4: set_a,X4: a > set_a] :
( ( member_a_a @ X @ ( pi_a_a @ ( insert_a @ I3 @ I4 ) @ X4 ) )
= ( ( member_a_a @ X @ ( pi_a_a @ I4 @ X4 ) )
& ( member_a @ ( X @ I3 ) @ ( X4 @ I3 ) ) ) ) ).
% Pi_split_insert_domain
thf(fact_309_Pi__eq__empty,axiom,
! [A2: set_nat,B2: nat > set_a] :
( ( ( pi_nat_a @ A2 @ B2 )
= bot_bot_set_nat_a )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ( B2 @ X3 )
= bot_bot_set_a ) ) ) ) ).
% Pi_eq_empty
thf(fact_310_Pi__eq__empty,axiom,
! [A2: set_nat_a,B2: ( nat > a ) > set_a] :
( ( ( pi_nat_a_a @ A2 @ B2 )
= bot_bot_set_nat_a_a )
= ( ? [X3: nat > a] :
( ( member_nat_a @ X3 @ A2 )
& ( ( B2 @ X3 )
= bot_bot_set_a ) ) ) ) ).
% Pi_eq_empty
thf(fact_311_Pi__eq__empty,axiom,
! [A2: set_a,B2: a > set_a] :
( ( ( pi_a_a @ A2 @ B2 )
= bot_bot_set_a_a )
= ( ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( ( B2 @ X3 )
= bot_bot_set_a ) ) ) ) ).
% Pi_eq_empty
thf(fact_312_atLeastatMost__empty__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( set_or6288561110385358355_set_a @ A @ B )
= bot_bot_set_set_a )
= ( ~ ( ord_less_eq_set_a @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_313_atLeastatMost__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( set_or4548717258645045905et_nat @ A @ B )
= bot_bot_set_set_nat )
= ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_314_atLeastatMost__empty__iff,axiom,
! [A: nat,B: nat] :
( ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_315_atLeastatMost__empty__iff2,axiom,
! [A: set_a,B: set_a] :
( ( bot_bot_set_set_a
= ( set_or6288561110385358355_set_a @ A @ B ) )
= ( ~ ( ord_less_eq_set_a @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_316_atLeastatMost__empty__iff2,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_set_nat
= ( set_or4548717258645045905et_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_317_atLeastatMost__empty__iff2,axiom,
! [A: nat,B: nat] :
( ( bot_bot_set_nat
= ( set_or1269000886237332187st_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_318_atLeastatMost__subset__iff,axiom,
! [A: set_a,B: set_a,C: set_a,D: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ A @ B ) @ ( set_or6288561110385358355_set_a @ C @ D ) )
= ( ~ ( ord_less_eq_set_a @ A @ B )
| ( ( ord_less_eq_set_a @ C @ A )
& ( ord_less_eq_set_a @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_319_atLeastatMost__subset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_eq_set_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_320_atLeastatMost__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_321_atLeastAtMost__singleton,axiom,
! [A: nat] :
( ( set_or1269000886237332187st_nat @ A @ A )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% atLeastAtMost_singleton
thf(fact_322_atLeastAtMost__singleton__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( set_or1269000886237332187st_nat @ A @ B )
= ( insert_nat @ C @ bot_bot_set_nat ) )
= ( ( A = B )
& ( B = C ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_323_atMost__subset__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_ord_atMost_set_a @ X ) @ ( set_ord_atMost_set_a @ Y ) )
= ( ord_less_eq_set_a @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_324_atMost__subset__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X ) @ ( set_or4236626031148496127et_nat @ Y ) )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_325_atMost__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_326_Icc__subset__Iic__iff,axiom,
! [L: set_a,H2: set_a,H3: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_or6288561110385358355_set_a @ L @ H2 ) @ ( set_ord_atMost_set_a @ H3 ) )
= ( ~ ( ord_less_eq_set_a @ L @ H2 )
| ( ord_less_eq_set_a @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_327_Icc__subset__Iic__iff,axiom,
! [L: set_nat,H2: set_nat,H3: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ L @ H2 ) @ ( set_or4236626031148496127et_nat @ H3 ) )
= ( ~ ( ord_less_eq_set_nat @ L @ H2 )
| ( ord_less_eq_set_nat @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_328_Icc__subset__Iic__iff,axiom,
! [L: nat,H2: nat,H3: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H2 ) @ ( set_ord_atMost_nat @ H3 ) )
= ( ~ ( ord_less_eq_nat @ L @ H2 )
| ( ord_less_eq_nat @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_329_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_330_fincomp__insert,axiom,
! [F2: set_nat_a_a2,A: ( nat > a ) > a,F: ( ( nat > a ) > a ) > a] :
( ( finite7239108116303828181at_a_a @ F2 )
=> ( ~ ( member_nat_a_a @ A @ F2 )
=> ( ( member_nat_a_a_a @ F
@ ( pi_nat_a_a_a @ F2
@ ^ [Uu: ( nat > a ) > a] : m ) )
=> ( ( member_a @ ( F @ A ) @ m )
=> ( ( commut8121142741902956950at_a_a @ m @ composition @ unit @ F @ ( insert_nat_a_a @ A @ F2 ) )
= ( composition @ ( F @ A ) @ ( commut8121142741902956950at_a_a @ m @ composition @ unit @ F @ F2 ) ) ) ) ) ) ) ).
% fincomp_insert
thf(fact_331_fincomp__insert,axiom,
! [F2: set_a_a,A: a > a,F: ( a > a ) > a] :
( ( finite_finite_a_a @ F2 )
=> ( ~ ( member_a_a @ A @ F2 )
=> ( ( member_a_a_a @ F
@ ( pi_a_a_a @ F2
@ ^ [Uu: a > a] : m ) )
=> ( ( member_a @ ( F @ A ) @ m )
=> ( ( commut6344323929752164413_a_a_a @ m @ composition @ unit @ F @ ( insert_a_a @ A @ F2 ) )
= ( composition @ ( F @ A ) @ ( commut6344323929752164413_a_a_a @ m @ composition @ unit @ F @ F2 ) ) ) ) ) ) ) ).
% fincomp_insert
thf(fact_332_fincomp__insert,axiom,
! [F2: set_nat,A: nat,F: nat > a] :
( ( finite_finite_nat @ F2 )
=> ( ~ ( member_nat @ A @ F2 )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ F2
@ ^ [Uu: nat] : m ) )
=> ( ( member_a @ ( F @ A ) @ m )
=> ( ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ ( insert_nat @ A @ F2 ) )
= ( composition @ ( F @ A ) @ ( commut6741328216151336360_a_nat @ m @ composition @ unit @ F @ F2 ) ) ) ) ) ) ) ).
% fincomp_insert
thf(fact_333_fincomp__insert,axiom,
! [F2: set_nat_a,A: nat > a,F: ( nat > a ) > a] :
( ( finite_finite_nat_a @ F2 )
=> ( ~ ( member_nat_a @ A @ F2 )
=> ( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ F2
@ ^ [Uu: nat > a] : m ) )
=> ( ( member_a @ ( F @ A ) @ m )
=> ( ( commut5242989786243415821_nat_a @ m @ composition @ unit @ F @ ( insert_nat_a @ A @ F2 ) )
= ( composition @ ( F @ A ) @ ( commut5242989786243415821_nat_a @ m @ composition @ unit @ F @ F2 ) ) ) ) ) ) ) ).
% fincomp_insert
thf(fact_334_fincomp__insert,axiom,
! [F2: set_a,A: a,F: a > a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ A @ F2 )
=> ( ( member_a_a @ F
@ ( pi_a_a @ F2
@ ^ [Uu: a] : m ) )
=> ( ( member_a @ ( F @ A ) @ m )
=> ( ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ F @ ( insert_a @ A @ F2 ) )
= ( composition @ ( F @ A ) @ ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ F @ F2 ) ) ) ) ) ) ) ).
% fincomp_insert
thf(fact_335_Pi__mono,axiom,
! [A2: set_a,B2: a > set_a,C4: a > set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( B2 @ X2 ) @ ( C4 @ X2 ) ) )
=> ( ord_less_eq_set_a_a @ ( pi_a_a @ A2 @ B2 ) @ ( pi_a_a @ A2 @ C4 ) ) ) ).
% Pi_mono
thf(fact_336_Pi__mono,axiom,
! [A2: set_nat_a,B2: ( nat > a ) > set_a,C4: ( nat > a ) > set_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( B2 @ X2 ) @ ( C4 @ X2 ) ) )
=> ( ord_le3509452538356653652at_a_a @ ( pi_nat_a_a @ A2 @ B2 ) @ ( pi_nat_a_a @ A2 @ C4 ) ) ) ).
% Pi_mono
thf(fact_337_Pi__mono,axiom,
! [A2: set_nat,B2: nat > set_a,C4: nat > set_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( B2 @ X2 ) @ ( C4 @ X2 ) ) )
=> ( ord_le871467723717165285_nat_a @ ( pi_nat_a @ A2 @ B2 ) @ ( pi_nat_a @ A2 @ C4 ) ) ) ).
% Pi_mono
thf(fact_338_Pi__mono,axiom,
! [A2: set_nat_a_a2,B2: ( ( nat > a ) > a ) > set_a,C4: ( ( nat > a ) > a ) > set_a] :
( ! [X2: ( nat > a ) > a] :
( ( member_nat_a_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( B2 @ X2 ) @ ( C4 @ X2 ) ) )
=> ( ord_le7944041390409729655_a_a_a @ ( pi_nat_a_a_a @ A2 @ B2 ) @ ( pi_nat_a_a_a @ A2 @ C4 ) ) ) ).
% Pi_mono
thf(fact_339_Pi__mono,axiom,
! [A2: set_a_a,B2: ( a > a ) > set_a,C4: ( a > a ) > set_a] :
( ! [X2: a > a] :
( ( member_a_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( B2 @ X2 ) @ ( C4 @ X2 ) ) )
=> ( ord_le7181591058469194768_a_a_a @ ( pi_a_a_a @ A2 @ B2 ) @ ( pi_a_a_a @ A2 @ C4 ) ) ) ).
% Pi_mono
thf(fact_340_Pi__mono,axiom,
! [A2: set_a,B2: a > set_nat,C4: a > set_nat] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X2 ) @ ( C4 @ X2 ) ) )
=> ( ord_le1612561287239139007_a_nat @ ( pi_a_nat @ A2 @ B2 ) @ ( pi_a_nat @ A2 @ C4 ) ) ) ).
% Pi_mono
thf(fact_341_Pi__mono,axiom,
! [A2: set_nat_a,B2: ( nat > a ) > set_nat,C4: ( nat > a ) > set_nat] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X2 ) @ ( C4 @ X2 ) ) )
=> ( ord_le8014916076312755750_a_nat @ ( pi_nat_a_nat @ A2 @ B2 ) @ ( pi_nat_a_nat @ A2 @ C4 ) ) ) ).
% Pi_mono
thf(fact_342_Pi__mono,axiom,
! [A2: set_nat,B2: nat > set_nat,C4: nat > set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X2 ) @ ( C4 @ X2 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( pi_nat_nat @ A2 @ B2 ) @ ( pi_nat_nat @ A2 @ C4 ) ) ) ).
% Pi_mono
thf(fact_343_Pi__mono,axiom,
! [A2: set_nat_a_a2,B2: ( ( nat > a ) > a ) > set_nat,C4: ( ( nat > a ) > a ) > set_nat] :
( ! [X2: ( nat > a ) > a] :
( ( member_nat_a_a @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X2 ) @ ( C4 @ X2 ) ) )
=> ( ord_le6513183344911049539_a_nat @ ( pi_nat_a_a_nat @ A2 @ B2 ) @ ( pi_nat_a_a_nat @ A2 @ C4 ) ) ) ).
% Pi_mono
thf(fact_344_Pi__mono,axiom,
! [A2: set_a_a,B2: ( a > a ) > set_nat,C4: ( a > a ) > set_nat] :
( ! [X2: a > a] :
( ( member_a_a @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( B2 @ X2 ) @ ( C4 @ X2 ) ) )
=> ( ord_le4090943279780567402_a_nat @ ( pi_a_a_nat @ A2 @ B2 ) @ ( pi_a_a_nat @ A2 @ C4 ) ) ) ).
% Pi_mono
thf(fact_345_Pi__anti__mono,axiom,
! [A5: set_nat_a,A2: set_nat_a,B2: ( nat > a ) > set_a] :
( ( ord_le871467723717165285_nat_a @ A5 @ A2 )
=> ( ord_le3509452538356653652at_a_a @ ( pi_nat_a_a @ A2 @ B2 ) @ ( pi_nat_a_a @ A5 @ B2 ) ) ) ).
% Pi_anti_mono
thf(fact_346_Pi__anti__mono,axiom,
! [A5: set_a,A2: set_a,B2: a > set_a] :
( ( ord_less_eq_set_a @ A5 @ A2 )
=> ( ord_less_eq_set_a_a @ ( pi_a_a @ A2 @ B2 ) @ ( pi_a_a @ A5 @ B2 ) ) ) ).
% Pi_anti_mono
thf(fact_347_Pi__anti__mono,axiom,
! [A5: set_nat,A2: set_nat,B2: nat > set_a] :
( ( ord_less_eq_set_nat @ A5 @ A2 )
=> ( ord_le871467723717165285_nat_a @ ( pi_nat_a @ A2 @ B2 ) @ ( pi_nat_a @ A5 @ B2 ) ) ) ).
% Pi_anti_mono
thf(fact_348_subgroup__transitive,axiom,
! [K2: set_a,H4: set_a,Composition: a > a > a,Unit: a,G2: set_a] :
( ( group_subgroup_a @ K2 @ H4 @ Composition @ Unit )
=> ( ( group_subgroup_a @ H4 @ G2 @ Composition @ Unit )
=> ( group_subgroup_a @ K2 @ G2 @ Composition @ Unit ) ) ) ).
% subgroup_transitive
thf(fact_349_subgroup_Oaxioms_I2_J,axiom,
! [G2: set_a,M: set_a,Composition: a > a > a,Unit: a] :
( ( group_subgroup_a @ G2 @ M @ Composition @ Unit )
=> ( group_group_a @ G2 @ Composition @ Unit ) ) ).
% subgroup.axioms(2)
thf(fact_350_lift__Suc__mono__le,axiom,
! [F: nat > set_a,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_a @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_351_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_352_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_353_lift__Suc__antimono__le,axiom,
! [F: nat > set_a,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_a @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_354_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_355_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_356_atLeastAtMost__singleton_H,axiom,
! [A: nat,B: nat] :
( ( A = B )
=> ( ( set_or1269000886237332187st_nat @ A @ B )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_357_commutative__monoid__axioms_Ointro,axiom,
! [M: set_a,Composition: a > a > a] :
( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ M )
=> ( ( member_a @ Y3 @ M )
=> ( ( Composition @ X2 @ Y3 )
= ( Composition @ Y3 @ X2 ) ) ) )
=> ( group_2081300317213596122ioms_a @ M @ Composition ) ) ).
% commutative_monoid_axioms.intro
thf(fact_358_commutative__monoid__axioms_Ointro,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a] :
( ! [X2: nat > a,Y3: nat > a] :
( ( member_nat_a @ X2 @ M )
=> ( ( member_nat_a @ Y3 @ M )
=> ( ( Composition @ X2 @ Y3 )
= ( Composition @ Y3 @ X2 ) ) ) )
=> ( group_5745313098993787841_nat_a @ M @ Composition ) ) ).
% commutative_monoid_axioms.intro
thf(fact_359_commutative__monoid__axioms_Ointro,axiom,
! [M: set_nat,Composition: nat > nat > nat] :
( ! [X2: nat,Y3: nat] :
( ( member_nat @ X2 @ M )
=> ( ( member_nat @ Y3 @ M )
=> ( ( Composition @ X2 @ Y3 )
= ( Composition @ Y3 @ X2 ) ) ) )
=> ( group_5685275631618022900ms_nat @ M @ Composition ) ) ).
% commutative_monoid_axioms.intro
thf(fact_360_commutative__monoid__axioms_Ointro,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a] :
( ! [X2: ( nat > a ) > a,Y3: ( nat > a ) > a] :
( ( member_nat_a_a @ X2 @ M )
=> ( ( member_nat_a_a @ Y3 @ M )
=> ( ( Composition @ X2 @ Y3 )
= ( Composition @ Y3 @ X2 ) ) ) )
=> ( group_3498422669771298786at_a_a @ M @ Composition ) ) ).
% commutative_monoid_axioms.intro
thf(fact_361_commutative__monoid__axioms_Ointro,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a] :
( ! [X2: a > a,Y3: a > a] :
( ( member_a_a @ X2 @ M )
=> ( ( member_a_a @ Y3 @ M )
=> ( ( Composition @ X2 @ Y3 )
= ( Composition @ Y3 @ X2 ) ) ) )
=> ( group_7053950466465069193ms_a_a @ M @ Composition ) ) ).
% commutative_monoid_axioms.intro
thf(fact_362_commutative__monoid__axioms__def,axiom,
( group_2081300317213596122ioms_a
= ( ^ [M4: set_a,Composition2: a > a > a] :
! [X3: a,Y4: a] :
( ( member_a @ X3 @ M4 )
=> ( ( member_a @ Y4 @ M4 )
=> ( ( Composition2 @ X3 @ Y4 )
= ( Composition2 @ Y4 @ X3 ) ) ) ) ) ) ).
% commutative_monoid_axioms_def
thf(fact_363_commutative__monoid__axioms__def,axiom,
( group_5745313098993787841_nat_a
= ( ^ [M4: set_nat_a,Composition2: ( nat > a ) > ( nat > a ) > nat > a] :
! [X3: nat > a,Y4: nat > a] :
( ( member_nat_a @ X3 @ M4 )
=> ( ( member_nat_a @ Y4 @ M4 )
=> ( ( Composition2 @ X3 @ Y4 )
= ( Composition2 @ Y4 @ X3 ) ) ) ) ) ) ).
% commutative_monoid_axioms_def
thf(fact_364_commutative__monoid__axioms__def,axiom,
( group_5685275631618022900ms_nat
= ( ^ [M4: set_nat,Composition2: nat > nat > nat] :
! [X3: nat,Y4: nat] :
( ( member_nat @ X3 @ M4 )
=> ( ( member_nat @ Y4 @ M4 )
=> ( ( Composition2 @ X3 @ Y4 )
= ( Composition2 @ Y4 @ X3 ) ) ) ) ) ) ).
% commutative_monoid_axioms_def
thf(fact_365_commutative__monoid__axioms__def,axiom,
( group_3498422669771298786at_a_a
= ( ^ [M4: set_nat_a_a2,Composition2: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a] :
! [X3: ( nat > a ) > a,Y4: ( nat > a ) > a] :
( ( member_nat_a_a @ X3 @ M4 )
=> ( ( member_nat_a_a @ Y4 @ M4 )
=> ( ( Composition2 @ X3 @ Y4 )
= ( Composition2 @ Y4 @ X3 ) ) ) ) ) ) ).
% commutative_monoid_axioms_def
thf(fact_366_commutative__monoid__axioms__def,axiom,
( group_7053950466465069193ms_a_a
= ( ^ [M4: set_a_a,Composition2: ( a > a ) > ( a > a ) > a > a] :
! [X3: a > a,Y4: a > a] :
( ( member_a_a @ X3 @ M4 )
=> ( ( member_a_a @ Y4 @ M4 )
=> ( ( Composition2 @ X3 @ Y4 )
= ( Composition2 @ Y4 @ X3 ) ) ) ) ) ) ).
% commutative_monoid_axioms_def
thf(fact_367_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_368_not__empty__eq__Iic__eq__empty,axiom,
! [H2: nat] :
( bot_bot_set_nat
!= ( set_ord_atMost_nat @ H2 ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_369_Group__Theory_Ogroup_Oaxioms_I1_J,axiom,
! [G2: set_a,Composition: a > a > a,Unit: a] :
( ( group_group_a @ G2 @ Composition @ Unit )
=> ( group_monoid_a @ G2 @ Composition @ Unit ) ) ).
% Group_Theory.group.axioms(1)
thf(fact_370_group_Oinvertible,axiom,
! [G2: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,U: nat > a] :
( ( group_group_nat_a @ G2 @ Composition @ Unit )
=> ( ( member_nat_a @ U @ G2 )
=> ( group_645299334525884886_nat_a @ G2 @ Composition @ Unit @ U ) ) ) ).
% group.invertible
thf(fact_371_group_Oinvertible,axiom,
! [G2: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_group_nat @ G2 @ Composition @ Unit )
=> ( ( member_nat @ U @ G2 )
=> ( group_invertible_nat @ G2 @ Composition @ Unit @ U ) ) ) ).
% group.invertible
thf(fact_372_group_Oinvertible,axiom,
! [G2: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,U: ( nat > a ) > a] :
( ( group_group_nat_a_a @ G2 @ Composition @ Unit )
=> ( ( member_nat_a_a @ U @ G2 )
=> ( group_1438879830998425741at_a_a @ G2 @ Composition @ Unit @ U ) ) ) ).
% group.invertible
thf(fact_373_group_Oinvertible,axiom,
! [G2: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,U: a > a] :
( ( group_group_a_a @ G2 @ Composition @ Unit )
=> ( ( member_a_a @ U @ G2 )
=> ( group_invertible_a_a @ G2 @ Composition @ Unit @ U ) ) ) ).
% group.invertible
thf(fact_374_group_Oinvertible,axiom,
! [G2: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_group_a @ G2 @ Composition @ Unit )
=> ( ( member_a @ U @ G2 )
=> ( group_invertible_a @ G2 @ Composition @ Unit @ U ) ) ) ).
% group.invertible
thf(fact_375_atMost__def,axiom,
( set_ord_atMost_set_a
= ( ^ [U2: set_a] :
( collect_set_a
@ ^ [X3: set_a] : ( ord_less_eq_set_a @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_376_atMost__def,axiom,
( set_or4236626031148496127et_nat
= ( ^ [U2: set_nat] :
( collect_set_nat
@ ^ [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_377_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X3: nat] : ( ord_less_eq_nat @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_378_subgroup_Osubgroup__inverse__equality,axiom,
! [G2: set_nat_a,M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,U: nat > a] :
( ( group_subgroup_nat_a @ G2 @ M @ Composition @ Unit )
=> ( ( member_nat_a @ U @ G2 )
=> ( ( group_inverse_nat_a @ M @ Composition @ Unit @ U )
= ( group_inverse_nat_a @ G2 @ Composition @ Unit @ U ) ) ) ) ).
% subgroup.subgroup_inverse_equality
thf(fact_379_subgroup_Osubgroup__inverse__equality,axiom,
! [G2: set_nat,M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_subgroup_nat @ G2 @ M @ Composition @ Unit )
=> ( ( member_nat @ U @ G2 )
=> ( ( group_inverse_nat @ M @ Composition @ Unit @ U )
= ( group_inverse_nat @ G2 @ Composition @ Unit @ U ) ) ) ) ).
% subgroup.subgroup_inverse_equality
thf(fact_380_subgroup_Osubgroup__inverse__equality,axiom,
! [G2: set_nat_a_a2,M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,U: ( nat > a ) > a] :
( ( group_5398023533210767094at_a_a @ G2 @ M @ Composition @ Unit )
=> ( ( member_nat_a_a @ U @ G2 )
=> ( ( group_6240212554837956873at_a_a @ M @ Composition @ Unit @ U )
= ( group_6240212554837956873at_a_a @ G2 @ Composition @ Unit @ U ) ) ) ) ).
% subgroup.subgroup_inverse_equality
thf(fact_381_subgroup_Osubgroup__inverse__equality,axiom,
! [G2: set_a_a,M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,U: a > a] :
( ( group_subgroup_a_a @ G2 @ M @ Composition @ Unit )
=> ( ( member_a_a @ U @ G2 )
=> ( ( group_inverse_a_a @ M @ Composition @ Unit @ U )
= ( group_inverse_a_a @ G2 @ Composition @ Unit @ U ) ) ) ) ).
% subgroup.subgroup_inverse_equality
thf(fact_382_subgroup_Osubgroup__inverse__equality,axiom,
! [G2: set_a,M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_subgroup_a @ G2 @ M @ Composition @ Unit )
=> ( ( member_a @ U @ G2 )
=> ( ( group_inverse_a @ M @ Composition @ Unit @ U )
= ( group_inverse_a @ G2 @ Composition @ Unit @ U ) ) ) ) ).
% subgroup.subgroup_inverse_equality
thf(fact_383_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,F2: set_a,A: a,F: a > nat] :
( ( group_6791354081887936081id_nat @ M @ Composition @ Unit )
=> ( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ A @ F2 )
=> ( ( member_a_nat @ F
@ ( pi_a_nat @ F2
@ ^ [Uu: a] : M ) )
=> ( ( member_nat @ ( F @ A ) @ M )
=> ( ( commut1549887680474846982_nat_a @ M @ Composition @ Unit @ F @ ( insert_a @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut1549887680474846982_nat_a @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_384_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,F2: set_nat,A: nat,F: nat > nat] :
( ( group_6791354081887936081id_nat @ M @ Composition @ Unit )
=> ( ( finite_finite_nat @ F2 )
=> ( ~ ( member_nat @ A @ F2 )
=> ( ( member_nat_nat @ F
@ ( pi_nat_nat @ F2
@ ^ [Uu: nat] : M ) )
=> ( ( member_nat @ ( F @ A ) @ M )
=> ( ( commut1028764413824576968at_nat @ M @ Composition @ Unit @ F @ ( insert_nat @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut1028764413824576968at_nat @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_385_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F2: set_nat,A: nat,F: nat > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( finite_finite_nat @ F2 )
=> ( ~ ( member_nat @ A @ F2 )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ F2
@ ^ [Uu: nat] : M ) )
=> ( ( member_a @ ( F @ A ) @ M )
=> ( ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ ( insert_nat @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_386_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F2: set_a,A: a,F: a > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ A @ F2 )
=> ( ( member_a_a @ F
@ ( pi_a_a @ F2
@ ^ [Uu: a] : M ) )
=> ( ( member_a @ ( F @ A ) @ M )
=> ( ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ F @ ( insert_a @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_387_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,F2: set_a,A: a,F: a > nat > a] :
( ( group_3093379471365697572_nat_a @ M @ Composition @ Unit )
=> ( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ A @ F2 )
=> ( ( member_a_nat_a @ F
@ ( pi_a_nat_a @ F2
@ ^ [Uu: a] : M ) )
=> ( ( member_nat_a @ ( F @ A ) @ M )
=> ( ( commut1274061894236046463at_a_a @ M @ Composition @ Unit @ F @ ( insert_a @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut1274061894236046463at_a_a @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_388_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,F2: set_a,A: a,F: a > a > a] :
( ( group_6976245611985207014id_a_a @ M @ Composition @ Unit )
=> ( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ A @ F2 )
=> ( ( member_a_a_a2 @ F
@ ( pi_a_a_a2 @ F2
@ ^ [Uu: a] : M ) )
=> ( ( member_a_a @ ( F @ A ) @ M )
=> ( ( commut5480430193892889009_a_a_a @ M @ Composition @ Unit @ F @ ( insert_a @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut5480430193892889009_a_a_a @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_389_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,F2: set_nat_a,A: nat > a,F: ( nat > a ) > nat] :
( ( group_6791354081887936081id_nat @ M @ Composition @ Unit )
=> ( ( finite_finite_nat_a @ F2 )
=> ( ~ ( member_nat_a @ A @ F2 )
=> ( ( member_nat_a_nat @ F
@ ( pi_nat_a_nat @ F2
@ ^ [Uu: nat > a] : M ) )
=> ( ( member_nat @ ( F @ A ) @ M )
=> ( ( commut5709917066755550957_nat_a @ M @ Composition @ Unit @ F @ ( insert_nat_a @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut5709917066755550957_nat_a @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_390_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,F2: set_a_a,A: a > a,F: ( a > a ) > nat] :
( ( group_6791354081887936081id_nat @ M @ Composition @ Unit )
=> ( ( finite_finite_a_a @ F2 )
=> ( ~ ( member_a_a @ A @ F2 )
=> ( ( member_a_a_nat @ F
@ ( pi_a_a_nat @ F2
@ ^ [Uu: a > a] : M ) )
=> ( ( member_nat @ ( F @ A ) @ M )
=> ( ( commut5797115372127264861at_a_a @ M @ Composition @ Unit @ F @ ( insert_a_a @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut5797115372127264861at_a_a @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_391_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,F2: set_nat,A: nat,F: nat > nat > a] :
( ( group_3093379471365697572_nat_a @ M @ Composition @ Unit )
=> ( ( finite_finite_nat @ F2 )
=> ( ~ ( member_nat @ A @ F2 )
=> ( ( member_nat_nat_a @ F
@ ( pi_nat_nat_a @ F2
@ ^ [Uu: nat] : M ) )
=> ( ( member_nat_a @ ( F @ A ) @ M )
=> ( ( commut6753747983606973455_a_nat @ M @ Composition @ Unit @ F @ ( insert_nat @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut6753747983606973455_a_nat @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_392_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,F2: set_nat,A: nat,F: nat > a > a] :
( ( group_6976245611985207014id_a_a @ M @ Composition @ Unit )
=> ( ( finite_finite_nat @ F2 )
=> ( ~ ( member_nat @ A @ F2 )
=> ( ( member_nat_a_a2 @ F
@ ( pi_nat_a_a2 @ F2
@ ^ [Uu: nat] : M ) )
=> ( ( member_a_a @ ( F @ A ) @ M )
=> ( ( commut6621034724473204317_a_nat @ M @ Composition @ Unit @ F @ ( insert_nat @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut6621034724473204317_a_nat @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_393_monoid_OsubgroupI,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,G2: set_nat_a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ( ord_le871467723717165285_nat_a @ G2 @ M )
=> ( ( member_nat_a @ Unit @ G2 )
=> ( ! [G3: nat > a,H: nat > a] :
( ( member_nat_a @ G3 @ G2 )
=> ( ( member_nat_a @ H @ G2 )
=> ( member_nat_a @ ( Composition @ G3 @ H ) @ G2 ) ) )
=> ( ! [G3: nat > a] :
( ( member_nat_a @ G3 @ G2 )
=> ( group_645299334525884886_nat_a @ M @ Composition @ Unit @ G3 ) )
=> ( ! [G3: nat > a] :
( ( member_nat_a @ G3 @ G2 )
=> ( member_nat_a @ ( group_inverse_nat_a @ M @ Composition @ Unit @ G3 ) @ G2 ) )
=> ( group_subgroup_nat_a @ G2 @ M @ Composition @ Unit ) ) ) ) ) ) ) ).
% monoid.subgroupI
thf(fact_394_monoid_OsubgroupI,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,G2: set_nat_a_a2] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ( ord_le3509452538356653652at_a_a @ G2 @ M )
=> ( ( member_nat_a_a @ Unit @ G2 )
=> ( ! [G3: ( nat > a ) > a,H: ( nat > a ) > a] :
( ( member_nat_a_a @ G3 @ G2 )
=> ( ( member_nat_a_a @ H @ G2 )
=> ( member_nat_a_a @ ( Composition @ G3 @ H ) @ G2 ) ) )
=> ( ! [G3: ( nat > a ) > a] :
( ( member_nat_a_a @ G3 @ G2 )
=> ( group_1438879830998425741at_a_a @ M @ Composition @ Unit @ G3 ) )
=> ( ! [G3: ( nat > a ) > a] :
( ( member_nat_a_a @ G3 @ G2 )
=> ( member_nat_a_a @ ( group_6240212554837956873at_a_a @ M @ Composition @ Unit @ G3 ) @ G2 ) )
=> ( group_5398023533210767094at_a_a @ G2 @ M @ Composition @ Unit ) ) ) ) ) ) ) ).
% monoid.subgroupI
thf(fact_395_monoid_OsubgroupI,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,G2: set_a_a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ( ord_less_eq_set_a_a @ G2 @ M )
=> ( ( member_a_a @ Unit @ G2 )
=> ( ! [G3: a > a,H: a > a] :
( ( member_a_a @ G3 @ G2 )
=> ( ( member_a_a @ H @ G2 )
=> ( member_a_a @ ( Composition @ G3 @ H ) @ G2 ) ) )
=> ( ! [G3: a > a] :
( ( member_a_a @ G3 @ G2 )
=> ( group_invertible_a_a @ M @ Composition @ Unit @ G3 ) )
=> ( ! [G3: a > a] :
( ( member_a_a @ G3 @ G2 )
=> ( member_a_a @ ( group_inverse_a_a @ M @ Composition @ Unit @ G3 ) @ G2 ) )
=> ( group_subgroup_a_a @ G2 @ M @ Composition @ Unit ) ) ) ) ) ) ) ).
% monoid.subgroupI
thf(fact_396_monoid_OsubgroupI,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,G2: set_nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( ord_less_eq_set_nat @ G2 @ M )
=> ( ( member_nat @ Unit @ G2 )
=> ( ! [G3: nat,H: nat] :
( ( member_nat @ G3 @ G2 )
=> ( ( member_nat @ H @ G2 )
=> ( member_nat @ ( Composition @ G3 @ H ) @ G2 ) ) )
=> ( ! [G3: nat] :
( ( member_nat @ G3 @ G2 )
=> ( group_invertible_nat @ M @ Composition @ Unit @ G3 ) )
=> ( ! [G3: nat] :
( ( member_nat @ G3 @ G2 )
=> ( member_nat @ ( group_inverse_nat @ M @ Composition @ Unit @ G3 ) @ G2 ) )
=> ( group_subgroup_nat @ G2 @ M @ Composition @ Unit ) ) ) ) ) ) ) ).
% monoid.subgroupI
thf(fact_397_monoid_OsubgroupI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,G2: set_a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( ord_less_eq_set_a @ G2 @ M )
=> ( ( member_a @ Unit @ G2 )
=> ( ! [G3: a,H: a] :
( ( member_a @ G3 @ G2 )
=> ( ( member_a @ H @ G2 )
=> ( member_a @ ( Composition @ G3 @ H ) @ G2 ) ) )
=> ( ! [G3: a] :
( ( member_a @ G3 @ G2 )
=> ( group_invertible_a @ M @ Composition @ Unit @ G3 ) )
=> ( ! [G3: a] :
( ( member_a @ G3 @ G2 )
=> ( member_a @ ( group_inverse_a @ M @ Composition @ Unit @ G3 ) @ G2 ) )
=> ( group_subgroup_a @ G2 @ M @ Composition @ Unit ) ) ) ) ) ) ) ).
% monoid.subgroupI
thf(fact_398_commutative__monoid_Ofincomp__def,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_nat,F: nat > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( ( finite_finite_nat @ A2 )
=> ( ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ A2 )
= ( finite_fold_nat_a
@ ^ [X3: nat,Y4: a] : ( Composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ M @ Unit @ Y4 ) )
@ Unit
@ A2 ) ) )
& ( ~ ( finite_finite_nat @ A2 )
=> ( ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ) ).
% commutative_monoid.fincomp_def
thf(fact_399_commutative__monoid_Ofincomp__def,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_nat_a,F: ( nat > a ) > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( ( finite_finite_nat_a @ A2 )
=> ( ( commut5242989786243415821_nat_a @ M @ Composition @ Unit @ F @ A2 )
= ( finite_fold_nat_a_a
@ ^ [X3: nat > a,Y4: a] : ( Composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ M @ Unit @ Y4 ) )
@ Unit
@ A2 ) ) )
& ( ~ ( finite_finite_nat_a @ A2 )
=> ( ( commut5242989786243415821_nat_a @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ) ).
% commutative_monoid.fincomp_def
thf(fact_400_commutative__monoid_Ofincomp__def,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_a,F: a > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( ( finite_finite_a @ A2 )
=> ( ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ F @ A2 )
= ( finite_fold_a_a
@ ^ [X3: a,Y4: a] : ( Composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ M @ Unit @ Y4 ) )
@ Unit
@ A2 ) ) )
& ( ~ ( finite_finite_a @ A2 )
=> ( ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ) ).
% commutative_monoid.fincomp_def
thf(fact_401_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost_nat @ ( suc @ K ) )
= ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% atMost_Suc
thf(fact_402_commutative__monoid_Ofincomp__infinite,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_nat,F: nat > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ~ ( finite_finite_nat @ A2 )
=> ( ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ).
% commutative_monoid.fincomp_infinite
thf(fact_403_commutative__monoid_Ofincomp__infinite,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_nat_a,F: ( nat > a ) > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ~ ( finite_finite_nat_a @ A2 )
=> ( ( commut5242989786243415821_nat_a @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ).
% commutative_monoid.fincomp_infinite
thf(fact_404_commutative__monoid_Ofincomp__infinite,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_a,F: a > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ~ ( finite_finite_a @ A2 )
=> ( ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ).
% commutative_monoid.fincomp_infinite
thf(fact_405_monoid_Oinverse__undefined,axiom,
! [M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,U: nat > a] :
( ( group_monoid_nat_a @ M @ Composition @ Unit )
=> ( ~ ( member_nat_a @ U @ M )
=> ( ( group_inverse_nat_a @ M @ Composition @ Unit @ U )
= undefined_nat_a ) ) ) ).
% monoid.inverse_undefined
thf(fact_406_monoid_Oinverse__undefined,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,U: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ~ ( member_nat @ U @ M )
=> ( ( group_inverse_nat @ M @ Composition @ Unit @ U )
= undefined_nat ) ) ) ).
% monoid.inverse_undefined
thf(fact_407_monoid_Oinverse__undefined,axiom,
! [M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,U: ( nat > a ) > a] :
( ( group_monoid_nat_a_a @ M @ Composition @ Unit )
=> ( ~ ( member_nat_a_a @ U @ M )
=> ( ( group_6240212554837956873at_a_a @ M @ Composition @ Unit @ U )
= undefined_nat_a_a ) ) ) ).
% monoid.inverse_undefined
thf(fact_408_monoid_Oinverse__undefined,axiom,
! [M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,U: a > a] :
( ( group_monoid_a_a @ M @ Composition @ Unit )
=> ( ~ ( member_a_a @ U @ M )
=> ( ( group_inverse_a_a @ M @ Composition @ Unit @ U )
= undefined_a_a ) ) ) ).
% monoid.inverse_undefined
thf(fact_409_monoid_Oinverse__undefined,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,U: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ~ ( member_a @ U @ M )
=> ( ( group_inverse_a @ M @ Composition @ Unit @ U )
= undefined_a ) ) ) ).
% monoid.inverse_undefined
thf(fact_410_commutative__monoid_Ofincomp__empty,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: nat > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ bot_bot_set_nat )
= Unit ) ) ).
% commutative_monoid.fincomp_empty
thf(fact_411_commutative__monoid_Ofincomp__empty,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: ( nat > a ) > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( commut5242989786243415821_nat_a @ M @ Composition @ Unit @ F @ bot_bot_set_nat_a )
= Unit ) ) ).
% commutative_monoid.fincomp_empty
thf(fact_412_commutative__monoid_Ofincomp__empty,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: a > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( commut5005951359559292710mp_a_a @ M @ Composition @ Unit @ F @ bot_bot_set_a )
= Unit ) ) ).
% commutative_monoid.fincomp_empty
thf(fact_413_monoid_Ogroup__of__Units,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( group_group_a @ ( group_Units_a @ M @ Composition @ Unit ) @ Composition @ Unit ) ) ).
% monoid.group_of_Units
thf(fact_414_atLeast0__atMost__Suc,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_415_subgroup_Osubgroup__inverse__iff,axiom,
! [G2: set_nat_a,M: set_nat_a,Composition: ( nat > a ) > ( nat > a ) > nat > a,Unit: nat > a,X: nat > a] :
( ( group_subgroup_nat_a @ G2 @ M @ Composition @ Unit )
=> ( ( group_645299334525884886_nat_a @ M @ Composition @ Unit @ X )
=> ( ( member_nat_a @ X @ M )
=> ( ( member_nat_a @ ( group_inverse_nat_a @ M @ Composition @ Unit @ X ) @ G2 )
= ( member_nat_a @ X @ G2 ) ) ) ) ) ).
% subgroup.subgroup_inverse_iff
thf(fact_416_subgroup_Osubgroup__inverse__iff,axiom,
! [G2: set_nat,M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat] :
( ( group_subgroup_nat @ G2 @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ X )
=> ( ( member_nat @ X @ M )
=> ( ( member_nat @ ( group_inverse_nat @ M @ Composition @ Unit @ X ) @ G2 )
= ( member_nat @ X @ G2 ) ) ) ) ) ).
% subgroup.subgroup_inverse_iff
thf(fact_417_subgroup_Osubgroup__inverse__iff,axiom,
! [G2: set_nat_a_a2,M: set_nat_a_a2,Composition: ( ( nat > a ) > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a,Unit: ( nat > a ) > a,X: ( nat > a ) > a] :
( ( group_5398023533210767094at_a_a @ G2 @ M @ Composition @ Unit )
=> ( ( group_1438879830998425741at_a_a @ M @ Composition @ Unit @ X )
=> ( ( member_nat_a_a @ X @ M )
=> ( ( member_nat_a_a @ ( group_6240212554837956873at_a_a @ M @ Composition @ Unit @ X ) @ G2 )
= ( member_nat_a_a @ X @ G2 ) ) ) ) ) ).
% subgroup.subgroup_inverse_iff
thf(fact_418_subgroup_Osubgroup__inverse__iff,axiom,
! [G2: set_a_a,M: set_a_a,Composition: ( a > a ) > ( a > a ) > a > a,Unit: a > a,X: a > a] :
( ( group_subgroup_a_a @ G2 @ M @ Composition @ Unit )
=> ( ( group_invertible_a_a @ M @ Composition @ Unit @ X )
=> ( ( member_a_a @ X @ M )
=> ( ( member_a_a @ ( group_inverse_a_a @ M @ Composition @ Unit @ X ) @ G2 )
= ( member_a_a @ X @ G2 ) ) ) ) ) ).
% subgroup.subgroup_inverse_iff
thf(fact_419_subgroup_Osubgroup__inverse__iff,axiom,
! [G2: set_a,M: set_a,Composition: a > a > a,Unit: a,X: a] :
( ( group_subgroup_a @ G2 @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ X )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ ( group_inverse_a @ M @ Composition @ Unit @ X ) @ G2 )
= ( member_a @ X @ G2 ) ) ) ) ) ).
% subgroup.subgroup_inverse_iff
thf(fact_420_commutative__monoid_Oaxioms_I2_J,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( group_2081300317213596122ioms_a @ M @ Composition ) ) ).
% commutative_monoid.axioms(2)
thf(fact_421_commutative__monoid_Ofincomp__0,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: nat > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat )
@ ^ [Uu: nat] : M ) )
=> ( ( commut6741328216151336360_a_nat @ M @ Composition @ Unit @ F @ ( set_ord_atMost_nat @ zero_zero_nat ) )
= ( F @ zero_zero_nat ) ) ) ) ).
% commutative_monoid.fincomp_0
thf(fact_422_PiE,axiom,
! [F: a > nat,A2: set_a,B2: a > set_nat,X: a] :
( ( member_a_nat @ F @ ( pi_a_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_a @ X @ A2 ) ) ) ).
% PiE
thf(fact_423_PiE,axiom,
! [F: nat > nat,A2: set_nat,B2: nat > set_nat,X: nat] :
( ( member_nat_nat @ F @ ( pi_nat_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ).
% PiE
thf(fact_424_PiE,axiom,
! [F: nat > a,A2: set_nat,B2: nat > set_a,X: nat] :
( ( member_nat_a @ F @ ( pi_nat_a @ A2 @ B2 ) )
=> ( ~ ( member_a @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ).
% PiE
thf(fact_425_PiE,axiom,
! [F: a > a,A2: set_a,B2: a > set_a,X: a] :
( ( member_a_a @ F @ ( pi_a_a @ A2 @ B2 ) )
=> ( ~ ( member_a @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_a @ X @ A2 ) ) ) ).
% PiE
thf(fact_426_PiE,axiom,
! [F: ( a > a ) > a,A2: set_a_a,B2: ( a > a ) > set_a,X: a > a] :
( ( member_a_a_a @ F @ ( pi_a_a_a @ A2 @ B2 ) )
=> ( ~ ( member_a @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_a_a @ X @ A2 ) ) ) ).
% PiE
thf(fact_427_PiE,axiom,
! [F: a > nat > a,A2: set_a,B2: a > set_nat_a,X: a] :
( ( member_a_nat_a @ F @ ( pi_a_nat_a @ A2 @ B2 ) )
=> ( ~ ( member_nat_a @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_a @ X @ A2 ) ) ) ).
% PiE
thf(fact_428_PiE,axiom,
! [F: nat > nat > a,A2: set_nat,B2: nat > set_nat_a,X: nat] :
( ( member_nat_nat_a @ F @ ( pi_nat_nat_a @ A2 @ B2 ) )
=> ( ~ ( member_nat_a @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ).
% PiE
thf(fact_429_PiE,axiom,
! [F: ( nat > a ) > nat,A2: set_nat_a,B2: ( nat > a ) > set_nat,X: nat > a] :
( ( member_nat_a_nat @ F @ ( pi_nat_a_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_nat_a @ X @ A2 ) ) ) ).
% PiE
thf(fact_430_PiE,axiom,
! [F: ( a > a ) > nat,A2: set_a_a,B2: ( a > a ) > set_nat,X: a > a] :
( ( member_a_a_nat @ F @ ( pi_a_a_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_a_a @ X @ A2 ) ) ) ).
% PiE
thf(fact_431_PiE,axiom,
! [F: a > a > a,A2: set_a,B2: a > set_a_a,X: a] :
( ( member_a_a_a2 @ F @ ( pi_a_a_a2 @ A2 @ B2 ) )
=> ( ~ ( member_a_a @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_a @ X @ A2 ) ) ) ).
% PiE
thf(fact_432_Pi__I_H,axiom,
! [A2: set_a,F: a > a,B2: a > set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_a_a @ F @ ( pi_a_a @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_433_Pi__I_H,axiom,
! [A2: set_a,F: a > nat,B2: a > set_nat] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_a_nat @ F @ ( pi_a_nat @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_434_Pi__I_H,axiom,
! [A2: set_nat,F: nat > a,B2: nat > set_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_nat_a @ F @ ( pi_nat_a @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_435_Pi__I_H,axiom,
! [A2: set_nat,F: nat > nat,B2: nat > set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_nat_nat @ F @ ( pi_nat_nat @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_436_Pi__I_H,axiom,
! [A2: set_a,F: a > nat > a,B2: a > set_nat_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_nat_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_a_nat_a @ F @ ( pi_a_nat_a @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_437_Pi__I_H,axiom,
! [A2: set_a,F: a > a > a,B2: a > set_a_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_a_a_a2 @ F @ ( pi_a_a_a2 @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_438_Pi__I_H,axiom,
! [A2: set_nat_a,F: ( nat > a ) > a,B2: ( nat > a ) > set_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_nat_a_a @ F @ ( pi_nat_a_a @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_439_Pi__I_H,axiom,
! [A2: set_nat_a,F: ( nat > a ) > nat,B2: ( nat > a ) > set_nat] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_nat_a_nat @ F @ ( pi_nat_a_nat @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_440_Pi__I_H,axiom,
! [A2: set_nat,F: nat > nat > a,B2: nat > set_nat_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_nat_nat_a @ F @ ( pi_nat_nat_a @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_441_Pi__I_H,axiom,
! [A2: set_nat,F: nat > a > a,B2: nat > set_a_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_a_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_nat_a_a2 @ F @ ( pi_nat_a_a2 @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_442_Pi__iff,axiom,
! [F: nat > a,I4: set_nat,X4: nat > set_a] :
( ( member_nat_a @ F @ ( pi_nat_a @ I4 @ X4 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I4 )
=> ( member_a @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% Pi_iff
thf(fact_443_Pi__iff,axiom,
! [F: ( nat > a ) > a,I4: set_nat_a,X4: ( nat > a ) > set_a] :
( ( member_nat_a_a @ F @ ( pi_nat_a_a @ I4 @ X4 ) )
= ( ! [X3: nat > a] :
( ( member_nat_a @ X3 @ I4 )
=> ( member_a @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% Pi_iff
thf(fact_444_Pi__iff,axiom,
! [F: a > a,I4: set_a,X4: a > set_a] :
( ( member_a_a @ F @ ( pi_a_a @ I4 @ X4 ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ I4 )
=> ( member_a @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% Pi_iff
thf(fact_445_Pi__mem,axiom,
! [F: a > nat,A2: set_a,B2: a > set_nat,X: a] :
( ( member_a_nat @ F @ ( pi_a_nat @ A2 @ B2 ) )
=> ( ( member_a @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_446_Pi__mem,axiom,
! [F: nat > nat,A2: set_nat,B2: nat > set_nat,X: nat] :
( ( member_nat_nat @ F @ ( pi_nat_nat @ A2 @ B2 ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_447_Pi__mem,axiom,
! [F: nat > a,A2: set_nat,B2: nat > set_a,X: nat] :
( ( member_nat_a @ F @ ( pi_nat_a @ A2 @ B2 ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_448_Pi__mem,axiom,
! [F: a > a,A2: set_a,B2: a > set_a,X: a] :
( ( member_a_a @ F @ ( pi_a_a @ A2 @ B2 ) )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_449_Pi__mem,axiom,
! [F: a > nat > a,A2: set_a,B2: a > set_nat_a,X: a] :
( ( member_a_nat_a @ F @ ( pi_a_nat_a @ A2 @ B2 ) )
=> ( ( member_a @ X @ A2 )
=> ( member_nat_a @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_450_Pi__mem,axiom,
! [F: a > a > a,A2: set_a,B2: a > set_a_a,X: a] :
( ( member_a_a_a2 @ F @ ( pi_a_a_a2 @ A2 @ B2 ) )
=> ( ( member_a @ X @ A2 )
=> ( member_a_a @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_451_Pi__mem,axiom,
! [F: ( nat > a ) > nat,A2: set_nat_a,B2: ( nat > a ) > set_nat,X: nat > a] :
( ( member_nat_a_nat @ F @ ( pi_nat_a_nat @ A2 @ B2 ) )
=> ( ( member_nat_a @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_452_Pi__mem,axiom,
! [F: nat > nat > a,A2: set_nat,B2: nat > set_nat_a,X: nat] :
( ( member_nat_nat_a @ F @ ( pi_nat_nat_a @ A2 @ B2 ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat_a @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_453_Pi__mem,axiom,
! [F: nat > a > a,A2: set_nat,B2: nat > set_a_a,X: nat] :
( ( member_nat_a_a2 @ F @ ( pi_nat_a_a2 @ A2 @ B2 ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_a_a @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_454_Pi__mem,axiom,
! [F: ( a > a ) > a,A2: set_a_a,B2: ( a > a ) > set_a,X: a > a] :
( ( member_a_a_a @ F @ ( pi_a_a_a @ A2 @ B2 ) )
=> ( ( member_a_a @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_455_Pi__cong,axiom,
! [A2: set_a,F: a > a,G: a > a,B2: a > set_a] :
( ! [W: a] :
( ( member_a @ W @ A2 )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_a_a @ F @ ( pi_a_a @ A2 @ B2 ) )
= ( member_a_a @ G @ ( pi_a_a @ A2 @ B2 ) ) ) ) ).
% Pi_cong
thf(fact_456_Pi__cong,axiom,
! [A2: set_nat_a,F: ( nat > a ) > a,G: ( nat > a ) > a,B2: ( nat > a ) > set_a] :
( ! [W: nat > a] :
( ( member_nat_a @ W @ A2 )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_nat_a_a @ F @ ( pi_nat_a_a @ A2 @ B2 ) )
= ( member_nat_a_a @ G @ ( pi_nat_a_a @ A2 @ B2 ) ) ) ) ).
% Pi_cong
thf(fact_457_Pi__cong,axiom,
! [A2: set_nat,F: nat > a,G: nat > a,B2: nat > set_a] :
( ! [W: nat] :
( ( member_nat @ W @ A2 )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_nat_a @ F @ ( pi_nat_a @ A2 @ B2 ) )
= ( member_nat_a @ G @ ( pi_nat_a @ A2 @ B2 ) ) ) ) ).
% Pi_cong
thf(fact_458_funcset__mem,axiom,
! [F: a > nat,A2: set_a,B2: set_nat,X: a] :
( ( member_a_nat @ F
@ ( pi_a_nat @ A2
@ ^ [Uu: a] : B2 ) )
=> ( ( member_a @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_459_funcset__mem,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat,X: nat] :
( ( member_nat_nat @ F
@ ( pi_nat_nat @ A2
@ ^ [Uu: nat] : B2 ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_460_funcset__mem,axiom,
! [F: nat > a,A2: set_nat,B2: set_a,X: nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : B2 ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_a @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_461_funcset__mem,axiom,
! [F: a > a,A2: set_a,B2: set_a,X: a] :
( ( member_a_a @ F
@ ( pi_a_a @ A2
@ ^ [Uu: a] : B2 ) )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_462_funcset__mem,axiom,
! [F: a > nat > a,A2: set_a,B2: set_nat_a,X: a] :
( ( member_a_nat_a @ F
@ ( pi_a_nat_a @ A2
@ ^ [Uu: a] : B2 ) )
=> ( ( member_a @ X @ A2 )
=> ( member_nat_a @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_463_funcset__mem,axiom,
! [F: a > a > a,A2: set_a,B2: set_a_a,X: a] :
( ( member_a_a_a2 @ F
@ ( pi_a_a_a2 @ A2
@ ^ [Uu: a] : B2 ) )
=> ( ( member_a @ X @ A2 )
=> ( member_a_a @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_464_funcset__mem,axiom,
! [F: ( nat > a ) > nat,A2: set_nat_a,B2: set_nat,X: nat > a] :
( ( member_nat_a_nat @ F
@ ( pi_nat_a_nat @ A2
@ ^ [Uu: nat > a] : B2 ) )
=> ( ( member_nat_a @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_465_funcset__mem,axiom,
! [F: nat > nat > a,A2: set_nat,B2: set_nat_a,X: nat] :
( ( member_nat_nat_a @ F
@ ( pi_nat_nat_a @ A2
@ ^ [Uu: nat] : B2 ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat_a @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_466_funcset__mem,axiom,
! [F: nat > a > a,A2: set_nat,B2: set_a_a,X: nat] :
( ( member_nat_a_a2 @ F
@ ( pi_nat_a_a2 @ A2
@ ^ [Uu: nat] : B2 ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_a_a @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_467_funcset__mem,axiom,
! [F: ( a > a ) > a,A2: set_a_a,B2: set_a,X: a > a] :
( ( member_a_a_a @ F
@ ( pi_a_a_a @ A2
@ ^ [Uu: a > a] : B2 ) )
=> ( ( member_a_a @ X @ A2 )
=> ( member_a @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_468_funcset__id,axiom,
! [A2: set_a] :
( member_a_a
@ ^ [X3: a] : X3
@ ( pi_a_a @ A2
@ ^ [Uu: a] : A2 ) ) ).
% funcset_id
thf(fact_469_funcsetI,axiom,
! [A2: set_a,F: a > a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ B2 ) )
=> ( member_a_a @ F
@ ( pi_a_a @ A2
@ ^ [Uu: a] : B2 ) ) ) ).
% funcsetI
thf(fact_470_funcsetI,axiom,
! [A2: set_a,F: a > nat,B2: set_nat] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( member_a_nat @ F
@ ( pi_a_nat @ A2
@ ^ [Uu: a] : B2 ) ) ) ).
% funcsetI
thf(fact_471_funcsetI,axiom,
! [A2: set_nat,F: nat > a,B2: set_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ B2 ) )
=> ( member_nat_a @ F
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : B2 ) ) ) ).
% funcsetI
thf(fact_472_funcsetI,axiom,
! [A2: set_nat,F: nat > nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( member_nat_nat @ F
@ ( pi_nat_nat @ A2
@ ^ [Uu: nat] : B2 ) ) ) ).
% funcsetI
thf(fact_473_funcsetI,axiom,
! [A2: set_a,F: a > nat > a,B2: set_nat_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_nat_a @ ( F @ X2 ) @ B2 ) )
=> ( member_a_nat_a @ F
@ ( pi_a_nat_a @ A2
@ ^ [Uu: a] : B2 ) ) ) ).
% funcsetI
thf(fact_474_funcsetI,axiom,
! [A2: set_a,F: a > a > a,B2: set_a_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a_a @ ( F @ X2 ) @ B2 ) )
=> ( member_a_a_a2 @ F
@ ( pi_a_a_a2 @ A2
@ ^ [Uu: a] : B2 ) ) ) ).
% funcsetI
thf(fact_475_funcsetI,axiom,
! [A2: set_nat_a,F: ( nat > a ) > a,B2: set_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ B2 ) )
=> ( member_nat_a_a @ F
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : B2 ) ) ) ).
% funcsetI
thf(fact_476_funcsetI,axiom,
! [A2: set_nat_a,F: ( nat > a ) > nat,B2: set_nat] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( member_nat_a_nat @ F
@ ( pi_nat_a_nat @ A2
@ ^ [Uu: nat > a] : B2 ) ) ) ).
% funcsetI
thf(fact_477_funcsetI,axiom,
! [A2: set_nat,F: nat > nat > a,B2: set_nat_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat_a @ ( F @ X2 ) @ B2 ) )
=> ( member_nat_nat_a @ F
@ ( pi_nat_nat_a @ A2
@ ^ [Uu: nat] : B2 ) ) ) ).
% funcsetI
thf(fact_478_funcsetI,axiom,
! [A2: set_nat,F: nat > a > a,B2: set_a_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_a_a @ ( F @ X2 ) @ B2 ) )
=> ( member_nat_a_a2 @ F
@ ( pi_nat_a_a2 @ A2
@ ^ [Uu: nat] : B2 ) ) ) ).
% funcsetI
thf(fact_479_commutative__monoid_Ointro,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_2081300317213596122ioms_a @ M @ Composition )
=> ( group_4866109990395492029noid_a @ M @ Composition @ Unit ) ) ) ).
% commutative_monoid.intro
thf(fact_480_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A2: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_481_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A2: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_482_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_483_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B: nat] :
( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_484_singleton__conv,axiom,
! [A: a] :
( ( collect_a
@ ^ [X3: a] : ( X3 = A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_485_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X3: nat] : ( X3 = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_486_singleton__conv2,axiom,
! [A: a] :
( ( collect_a
@ ( ^ [Y5: a,Z2: a] : ( Y5 = Z2 )
@ A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_487_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_488_finite__Collect__subsets,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B5: set_a] : ( ord_less_eq_set_a @ B5 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_489_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B5: set_nat] : ( ord_less_eq_set_nat @ B5 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_490_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_491_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_492_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_493_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_494_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_495_all__not__in__conv,axiom,
! [A2: set_nat_a] :
( ( ! [X3: nat > a] :
~ ( member_nat_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat_a ) ) ).
% all_not_in_conv
thf(fact_496_all__not__in__conv,axiom,
! [A2: set_nat_a_a2] :
( ( ! [X3: ( nat > a ) > a] :
~ ( member_nat_a_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat_a_a ) ) ).
% all_not_in_conv
thf(fact_497_all__not__in__conv,axiom,
! [A2: set_a_a] :
( ( ! [X3: a > a] :
~ ( member_a_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a_a ) ) ).
% all_not_in_conv
thf(fact_498_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_499_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_500_empty__iff,axiom,
! [C: nat > a] :
~ ( member_nat_a @ C @ bot_bot_set_nat_a ) ).
% empty_iff
thf(fact_501_empty__iff,axiom,
! [C: ( nat > a ) > a] :
~ ( member_nat_a_a @ C @ bot_bot_set_nat_a_a ) ).
% empty_iff
thf(fact_502_empty__iff,axiom,
! [C: a > a] :
~ ( member_a_a @ C @ bot_bot_set_a_a ) ).
% empty_iff
thf(fact_503_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_504_subset__antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_505_subset__antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_506_subsetI,axiom,
! [A2: set_nat_a,B2: set_nat_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( member_nat_a @ X2 @ B2 ) )
=> ( ord_le871467723717165285_nat_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_507_subsetI,axiom,
! [A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ! [X2: ( nat > a ) > a] :
( ( member_nat_a_a @ X2 @ A2 )
=> ( member_nat_a_a @ X2 @ B2 ) )
=> ( ord_le3509452538356653652at_a_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_508_subsetI,axiom,
! [A2: set_a_a,B2: set_a_a] :
( ! [X2: a > a] :
( ( member_a_a @ X2 @ A2 )
=> ( member_a_a @ X2 @ B2 ) )
=> ( ord_less_eq_set_a_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_509_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_510_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_511_insert__absorb2,axiom,
! [X: nat,A2: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ X @ A2 ) )
= ( insert_nat @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_512_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_513_insert__iff,axiom,
! [A: nat > a,B: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ ( insert_nat_a @ B @ A2 ) )
= ( ( A = B )
| ( member_nat_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_514_insert__iff,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_515_insert__iff,axiom,
! [A: ( nat > a ) > a,B: ( nat > a ) > a,A2: set_nat_a_a2] :
( ( member_nat_a_a @ A @ ( insert_nat_a_a @ B @ A2 ) )
= ( ( A = B )
| ( member_nat_a_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_516_insert__iff,axiom,
! [A: a > a,B: a > a,A2: set_a_a] :
( ( member_a_a @ A @ ( insert_a_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_517_insertCI,axiom,
! [A: a,B2: set_a,B: a] :
( ( ~ ( member_a @ A @ B2 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_518_insertCI,axiom,
! [A: nat > a,B2: set_nat_a,B: nat > a] :
( ( ~ ( member_nat_a @ A @ B2 )
=> ( A = B ) )
=> ( member_nat_a @ A @ ( insert_nat_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_519_insertCI,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_520_insertCI,axiom,
! [A: ( nat > a ) > a,B2: set_nat_a_a2,B: ( nat > a ) > a] :
( ( ~ ( member_nat_a_a @ A @ B2 )
=> ( A = B ) )
=> ( member_nat_a_a @ A @ ( insert_nat_a_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_521_insertCI,axiom,
! [A: a > a,B2: set_a_a,B: a > a] :
( ( ~ ( member_a_a @ A @ B2 )
=> ( A = B ) )
=> ( member_a_a @ A @ ( insert_a_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_522_finite__Collect__disjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_523_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_524_finite__Collect__conjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P ) )
| ( finite_finite_a @ ( collect_a @ Q ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_525_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_526_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_527_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_528_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_529_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_530_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_531_singletonI,axiom,
! [A: nat > a] : ( member_nat_a @ A @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) ) ).
% singletonI
thf(fact_532_singletonI,axiom,
! [A: ( nat > a ) > a] : ( member_nat_a_a @ A @ ( insert_nat_a_a @ A @ bot_bot_set_nat_a_a ) ) ).
% singletonI
thf(fact_533_singletonI,axiom,
! [A: a > a] : ( member_a_a @ A @ ( insert_a_a @ A @ bot_bot_set_a_a ) ) ).
% singletonI
thf(fact_534_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_535_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_536_insert__subset,axiom,
! [X: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ ( insert_nat_a @ X @ A2 ) @ B2 )
= ( ( member_nat_a @ X @ B2 )
& ( ord_le871467723717165285_nat_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_537_insert__subset,axiom,
! [X: ( nat > a ) > a,A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ( ord_le3509452538356653652at_a_a @ ( insert_nat_a_a @ X @ A2 ) @ B2 )
= ( ( member_nat_a_a @ X @ B2 )
& ( ord_le3509452538356653652at_a_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_538_insert__subset,axiom,
! [X: a > a,A2: set_a_a,B2: set_a_a] :
( ( ord_less_eq_set_a_a @ ( insert_a_a @ X @ A2 ) @ B2 )
= ( ( member_a_a @ X @ B2 )
& ( ord_less_eq_set_a_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_539_insert__subset,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( ( member_a @ X @ B2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_540_insert__subset,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
= ( ( member_nat @ X @ B2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_541_Suc__le__mono,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M3 ) )
= ( ord_less_eq_nat @ N @ M3 ) ) ).
% Suc_le_mono
thf(fact_542_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_543_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_544_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_545_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M5: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_eq_nat @ X3 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_546_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M ) )
=> ~ ! [M2: nat] :
( ( P @ M2 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_547_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_548_nat__le__linear,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
| ( ord_less_eq_nat @ N @ M3 ) ) ).
% nat_le_linear
thf(fact_549_le__antisym,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( ord_less_eq_nat @ N @ M3 )
=> ( M3 = N ) ) ) ).
% le_antisym
thf(fact_550_eq__imp__le,axiom,
! [M3: nat,N: nat] :
( ( M3 = N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% eq_imp_le
thf(fact_551_le__trans,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_552_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_553_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_554_less__eq__set__def,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A6: set_nat_a,B5: set_nat_a] :
( ord_less_eq_nat_a_o
@ ^ [X3: nat > a] : ( member_nat_a @ X3 @ A6 )
@ ^ [X3: nat > a] : ( member_nat_a @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_555_less__eq__set__def,axiom,
( ord_le3509452538356653652at_a_a
= ( ^ [A6: set_nat_a_a2,B5: set_nat_a_a2] :
( ord_le3623034401944517937_a_a_o
@ ^ [X3: ( nat > a ) > a] : ( member_nat_a_a @ X3 @ A6 )
@ ^ [X3: ( nat > a ) > a] : ( member_nat_a_a @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_556_less__eq__set__def,axiom,
( ord_less_eq_set_a_a
= ( ^ [A6: set_a_a,B5: set_a_a] :
( ord_less_eq_a_a_o
@ ^ [X3: a > a] : ( member_a_a @ X3 @ A6 )
@ ^ [X3: a > a] : ( member_a_a @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_557_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
( ord_less_eq_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A6 )
@ ^ [X3: a] : ( member_a @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_558_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B5: set_nat] :
( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_559_Suc__leD,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% Suc_leD
thf(fact_560_le__SucE,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M3 @ N )
=> ( M3
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_561_le__SucI,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ M3 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_562_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M2: nat] :
( M6
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_563_le__Suc__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M3 @ N )
| ( M3
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_564_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_565_not__less__eq__eq,axiom,
! [M3: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M3 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M3 ) ) ).
% not_less_eq_eq
thf(fact_566_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M7: nat] :
( ( ord_less_eq_nat @ ( suc @ M7 ) @ N2 )
=> ( P @ M7 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_567_nat__induct__at__least,axiom,
! [M3: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( P @ M3 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_568_transitive__stepwise__le,axiom,
! [M3: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ! [X2: nat] : ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y3: nat,Z3: nat] :
( ( R @ X2 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X2 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M3 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_569_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_570_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_571_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_572_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_573_subset__eq__atLeast0__atMost__finite,axiom,
! [N6: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N6 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N6 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_574_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_575_ex__in__conv,axiom,
! [A2: set_nat_a] :
( ( ? [X3: nat > a] : ( member_nat_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat_a ) ) ).
% ex_in_conv
thf(fact_576_ex__in__conv,axiom,
! [A2: set_nat_a_a2] :
( ( ? [X3: ( nat > a ) > a] : ( member_nat_a_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat_a_a ) ) ).
% ex_in_conv
thf(fact_577_ex__in__conv,axiom,
! [A2: set_a_a] :
( ( ? [X3: a > a] : ( member_a_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a_a ) ) ).
% ex_in_conv
thf(fact_578_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_579_equals0I,axiom,
! [A2: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_580_equals0I,axiom,
! [A2: set_nat_a] :
( ! [Y3: nat > a] :
~ ( member_nat_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_nat_a ) ) ).
% equals0I
thf(fact_581_equals0I,axiom,
! [A2: set_nat_a_a2] :
( ! [Y3: ( nat > a ) > a] :
~ ( member_nat_a_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_nat_a_a ) ) ).
% equals0I
thf(fact_582_equals0I,axiom,
! [A2: set_a_a] :
( ! [Y3: a > a] :
~ ( member_a_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_a_a ) ) ).
% equals0I
thf(fact_583_equals0I,axiom,
! [A2: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_584_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_585_equals0D,axiom,
! [A2: set_nat_a,A: nat > a] :
( ( A2 = bot_bot_set_nat_a )
=> ~ ( member_nat_a @ A @ A2 ) ) ).
% equals0D
thf(fact_586_equals0D,axiom,
! [A2: set_nat_a_a2,A: ( nat > a ) > a] :
( ( A2 = bot_bot_set_nat_a_a )
=> ~ ( member_nat_a_a @ A @ A2 ) ) ).
% equals0D
thf(fact_587_equals0D,axiom,
! [A2: set_a_a,A: a > a] :
( ( A2 = bot_bot_set_a_a )
=> ~ ( member_a_a @ A @ A2 ) ) ).
% equals0D
thf(fact_588_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_589_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_590_emptyE,axiom,
! [A: nat > a] :
~ ( member_nat_a @ A @ bot_bot_set_nat_a ) ).
% emptyE
thf(fact_591_emptyE,axiom,
! [A: ( nat > a ) > a] :
~ ( member_nat_a_a @ A @ bot_bot_set_nat_a_a ) ).
% emptyE
thf(fact_592_emptyE,axiom,
! [A: a > a] :
~ ( member_a_a @ A @ bot_bot_set_a_a ) ).
% emptyE
thf(fact_593_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_594_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_595_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_596_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A6: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A6 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_597_set__eq__subset,axiom,
( ( ^ [Y5: set_nat,Z2: set_nat] : ( Y5 = Z2 ) )
= ( ^ [A6: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_598_subset__trans,axiom,
! [A2: set_a,B2: set_a,C4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C4 )
=> ( ord_less_eq_set_a @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_599_subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C4 )
=> ( ord_less_eq_set_nat @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_600_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_601_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_602_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_603_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_604_subset__iff,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A6: set_nat_a,B5: set_nat_a] :
! [T: nat > a] :
( ( member_nat_a @ T @ A6 )
=> ( member_nat_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_605_subset__iff,axiom,
( ord_le3509452538356653652at_a_a
= ( ^ [A6: set_nat_a_a2,B5: set_nat_a_a2] :
! [T: ( nat > a ) > a] :
( ( member_nat_a_a @ T @ A6 )
=> ( member_nat_a_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_606_subset__iff,axiom,
( ord_less_eq_set_a_a
= ( ^ [A6: set_a_a,B5: set_a_a] :
! [T: a > a] :
( ( member_a_a @ T @ A6 )
=> ( member_a_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_607_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
! [T: a] :
( ( member_a @ T @ A6 )
=> ( member_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_608_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B5: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A6 )
=> ( member_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_609_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_610_equalityD2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_611_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_612_equalityD1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_613_subset__eq,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A6: set_nat_a,B5: set_nat_a] :
! [X3: nat > a] :
( ( member_nat_a @ X3 @ A6 )
=> ( member_nat_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_614_subset__eq,axiom,
( ord_le3509452538356653652at_a_a
= ( ^ [A6: set_nat_a_a2,B5: set_nat_a_a2] :
! [X3: ( nat > a ) > a] :
( ( member_nat_a_a @ X3 @ A6 )
=> ( member_nat_a_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_615_subset__eq,axiom,
( ord_less_eq_set_a_a
= ( ^ [A6: set_a_a,B5: set_a_a] :
! [X3: a > a] :
( ( member_a_a @ X3 @ A6 )
=> ( member_a_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_616_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A6 )
=> ( member_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_617_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B5: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( member_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_618_equalityE,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_619_equalityE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_620_subsetD,axiom,
! [A2: set_nat_a,B2: set_nat_a,C: nat > a] :
( ( ord_le871467723717165285_nat_a @ A2 @ B2 )
=> ( ( member_nat_a @ C @ A2 )
=> ( member_nat_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_621_subsetD,axiom,
! [A2: set_nat_a_a2,B2: set_nat_a_a2,C: ( nat > a ) > a] :
( ( ord_le3509452538356653652at_a_a @ A2 @ B2 )
=> ( ( member_nat_a_a @ C @ A2 )
=> ( member_nat_a_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_622_subsetD,axiom,
! [A2: set_a_a,B2: set_a_a,C: a > a] :
( ( ord_less_eq_set_a_a @ A2 @ B2 )
=> ( ( member_a_a @ C @ A2 )
=> ( member_a_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_623_subsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_624_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_625_in__mono,axiom,
! [A2: set_nat_a,B2: set_nat_a,X: nat > a] :
( ( ord_le871467723717165285_nat_a @ A2 @ B2 )
=> ( ( member_nat_a @ X @ A2 )
=> ( member_nat_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_626_in__mono,axiom,
! [A2: set_nat_a_a2,B2: set_nat_a_a2,X: ( nat > a ) > a] :
( ( ord_le3509452538356653652at_a_a @ A2 @ B2 )
=> ( ( member_nat_a_a @ X @ A2 )
=> ( member_nat_a_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_627_in__mono,axiom,
! [A2: set_a_a,B2: set_a_a,X: a > a] :
( ( ord_less_eq_set_a_a @ A2 @ B2 )
=> ( ( member_a_a @ X @ A2 )
=> ( member_a_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_628_in__mono,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_629_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_630_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B6: set_a] :
( ( A2
= ( insert_a @ A @ B6 ) )
& ~ ( member_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_631_mk__disjoint__insert,axiom,
! [A: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ A2 )
=> ? [B6: set_nat_a] :
( ( A2
= ( insert_nat_a @ A @ B6 ) )
& ~ ( member_nat_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_632_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B6: set_nat] :
( ( A2
= ( insert_nat @ A @ B6 ) )
& ~ ( member_nat @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_633_mk__disjoint__insert,axiom,
! [A: ( nat > a ) > a,A2: set_nat_a_a2] :
( ( member_nat_a_a @ A @ A2 )
=> ? [B6: set_nat_a_a2] :
( ( A2
= ( insert_nat_a_a @ A @ B6 ) )
& ~ ( member_nat_a_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_634_mk__disjoint__insert,axiom,
! [A: a > a,A2: set_a_a] :
( ( member_a_a @ A @ A2 )
=> ? [B6: set_a_a] :
( ( A2
= ( insert_a_a @ A @ B6 ) )
& ~ ( member_a_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_635_insert__commute,axiom,
! [X: nat,Y: nat,A2: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ Y @ A2 ) )
= ( insert_nat @ Y @ ( insert_nat @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_636_insert__eq__iff,axiom,
! [A: a,A2: set_a,B: a,B2: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B @ B2 )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C5: set_a] :
( ( A2
= ( insert_a @ B @ C5 ) )
& ~ ( member_a @ B @ C5 )
& ( B2
= ( insert_a @ A @ C5 ) )
& ~ ( member_a @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_637_insert__eq__iff,axiom,
! [A: nat > a,A2: set_nat_a,B: nat > a,B2: set_nat_a] :
( ~ ( member_nat_a @ A @ A2 )
=> ( ~ ( member_nat_a @ B @ B2 )
=> ( ( ( insert_nat_a @ A @ A2 )
= ( insert_nat_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C5: set_nat_a] :
( ( A2
= ( insert_nat_a @ B @ C5 ) )
& ~ ( member_nat_a @ B @ C5 )
& ( B2
= ( insert_nat_a @ A @ C5 ) )
& ~ ( member_nat_a @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_638_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B @ B2 )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C5: set_nat] :
( ( A2
= ( insert_nat @ B @ C5 ) )
& ~ ( member_nat @ B @ C5 )
& ( B2
= ( insert_nat @ A @ C5 ) )
& ~ ( member_nat @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_639_insert__eq__iff,axiom,
! [A: ( nat > a ) > a,A2: set_nat_a_a2,B: ( nat > a ) > a,B2: set_nat_a_a2] :
( ~ ( member_nat_a_a @ A @ A2 )
=> ( ~ ( member_nat_a_a @ B @ B2 )
=> ( ( ( insert_nat_a_a @ A @ A2 )
= ( insert_nat_a_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C5: set_nat_a_a2] :
( ( A2
= ( insert_nat_a_a @ B @ C5 ) )
& ~ ( member_nat_a_a @ B @ C5 )
& ( B2
= ( insert_nat_a_a @ A @ C5 ) )
& ~ ( member_nat_a_a @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_640_insert__eq__iff,axiom,
! [A: a > a,A2: set_a_a,B: a > a,B2: set_a_a] :
( ~ ( member_a_a @ A @ A2 )
=> ( ~ ( member_a_a @ B @ B2 )
=> ( ( ( insert_a_a @ A @ A2 )
= ( insert_a_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C5: set_a_a] :
( ( A2
= ( insert_a_a @ B @ C5 ) )
& ~ ( member_a_a @ B @ C5 )
& ( B2
= ( insert_a_a @ A @ C5 ) )
& ~ ( member_a_a @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_641_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_642_insert__absorb,axiom,
! [A: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ A2 )
=> ( ( insert_nat_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_643_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_644_insert__absorb,axiom,
! [A: ( nat > a ) > a,A2: set_nat_a_a2] :
( ( member_nat_a_a @ A @ A2 )
=> ( ( insert_nat_a_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_645_insert__absorb,axiom,
! [A: a > a,A2: set_a_a] :
( ( member_a_a @ A @ A2 )
=> ( ( insert_a_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_646_insert__ident,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B2 )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_647_insert__ident,axiom,
! [X: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ~ ( member_nat_a @ X @ A2 )
=> ( ~ ( member_nat_a @ X @ B2 )
=> ( ( ( insert_nat_a @ X @ A2 )
= ( insert_nat_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_648_insert__ident,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ~ ( member_nat @ X @ B2 )
=> ( ( ( insert_nat @ X @ A2 )
= ( insert_nat @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_649_insert__ident,axiom,
! [X: ( nat > a ) > a,A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ~ ( member_nat_a_a @ X @ A2 )
=> ( ~ ( member_nat_a_a @ X @ B2 )
=> ( ( ( insert_nat_a_a @ X @ A2 )
= ( insert_nat_a_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_650_insert__ident,axiom,
! [X: a > a,A2: set_a_a,B2: set_a_a] :
( ~ ( member_a_a @ X @ A2 )
=> ( ~ ( member_a_a @ X @ B2 )
=> ( ( ( insert_a_a @ X @ A2 )
= ( insert_a_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_651_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B6: set_a] :
( ( A2
= ( insert_a @ X @ B6 ) )
=> ( member_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_652_Set_Oset__insert,axiom,
! [X: nat > a,A2: set_nat_a] :
( ( member_nat_a @ X @ A2 )
=> ~ ! [B6: set_nat_a] :
( ( A2
= ( insert_nat_a @ X @ B6 ) )
=> ( member_nat_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_653_Set_Oset__insert,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ~ ! [B6: set_nat] :
( ( A2
= ( insert_nat @ X @ B6 ) )
=> ( member_nat @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_654_Set_Oset__insert,axiom,
! [X: ( nat > a ) > a,A2: set_nat_a_a2] :
( ( member_nat_a_a @ X @ A2 )
=> ~ ! [B6: set_nat_a_a2] :
( ( A2
= ( insert_nat_a_a @ X @ B6 ) )
=> ( member_nat_a_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_655_Set_Oset__insert,axiom,
! [X: a > a,A2: set_a_a] :
( ( member_a_a @ X @ A2 )
=> ~ ! [B6: set_a_a] :
( ( A2
= ( insert_a_a @ X @ B6 ) )
=> ( member_a_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_656_insertI2,axiom,
! [A: a,B2: set_a,B: a] :
( ( member_a @ A @ B2 )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_657_insertI2,axiom,
! [A: nat > a,B2: set_nat_a,B: nat > a] :
( ( member_nat_a @ A @ B2 )
=> ( member_nat_a @ A @ ( insert_nat_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_658_insertI2,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( member_nat @ A @ B2 )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_659_insertI2,axiom,
! [A: ( nat > a ) > a,B2: set_nat_a_a2,B: ( nat > a ) > a] :
( ( member_nat_a_a @ A @ B2 )
=> ( member_nat_a_a @ A @ ( insert_nat_a_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_660_insertI2,axiom,
! [A: a > a,B2: set_a_a,B: a > a] :
( ( member_a_a @ A @ B2 )
=> ( member_a_a @ A @ ( insert_a_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_661_insertI1,axiom,
! [A: a,B2: set_a] : ( member_a @ A @ ( insert_a @ A @ B2 ) ) ).
% insertI1
thf(fact_662_insertI1,axiom,
! [A: nat > a,B2: set_nat_a] : ( member_nat_a @ A @ ( insert_nat_a @ A @ B2 ) ) ).
% insertI1
thf(fact_663_insertI1,axiom,
! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_664_insertI1,axiom,
! [A: ( nat > a ) > a,B2: set_nat_a_a2] : ( member_nat_a_a @ A @ ( insert_nat_a_a @ A @ B2 ) ) ).
% insertI1
thf(fact_665_insertI1,axiom,
! [A: a > a,B2: set_a_a] : ( member_a_a @ A @ ( insert_a_a @ A @ B2 ) ) ).
% insertI1
thf(fact_666_insertE,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_667_insertE,axiom,
! [A: nat > a,B: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ ( insert_nat_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_668_insertE,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_669_insertE,axiom,
! [A: ( nat > a ) > a,B: ( nat > a ) > a,A2: set_nat_a_a2] :
( ( member_nat_a_a @ A @ ( insert_nat_a_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat_a_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_670_insertE,axiom,
! [A: a > a,B: a > a,A2: set_a_a] :
( ( member_a_a @ A @ ( insert_a_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_a_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_671_atLeastAtMost__insertL,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( insert_nat @ M3 @ ( set_or1269000886237332187st_nat @ ( suc @ M3 ) @ N ) )
= ( set_or1269000886237332187st_nat @ M3 @ N ) ) ) ).
% atLeastAtMost_insertL
thf(fact_672_atLeastAtMostSuc__conv,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N ) )
=> ( ( set_or1269000886237332187st_nat @ M3 @ ( suc @ N ) )
= ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M3 @ N ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_673_Icc__eq__insert__lb__nat,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( set_or1269000886237332187st_nat @ M3 @ N )
= ( insert_nat @ M3 @ ( set_or1269000886237332187st_nat @ ( suc @ M3 ) @ N ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_674_fold__closed__eq,axiom,
! [A2: set_a,B2: set_a,F: a > a > a,G: a > a > a,Z: a] :
( ! [A4: a,B4: a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_a @ B4 @ B2 )
=> ( ( F @ A4 @ B4 )
= ( G @ A4 @ B4 ) ) ) )
=> ( ! [A4: a,B4: a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_a @ B4 @ B2 )
=> ( member_a @ ( G @ A4 @ B4 ) @ B2 ) ) )
=> ( ( member_a @ Z @ B2 )
=> ( ( finite_fold_a_a @ F @ Z @ A2 )
= ( finite_fold_a_a @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_675_fold__closed__eq,axiom,
! [A2: set_a,B2: set_nat,F: a > nat > nat,G: a > nat > nat,Z: nat] :
( ! [A4: a,B4: nat] :
( ( member_a @ A4 @ A2 )
=> ( ( member_nat @ B4 @ B2 )
=> ( ( F @ A4 @ B4 )
= ( G @ A4 @ B4 ) ) ) )
=> ( ! [A4: a,B4: nat] :
( ( member_a @ A4 @ A2 )
=> ( ( member_nat @ B4 @ B2 )
=> ( member_nat @ ( G @ A4 @ B4 ) @ B2 ) ) )
=> ( ( member_nat @ Z @ B2 )
=> ( ( finite_fold_a_nat @ F @ Z @ A2 )
= ( finite_fold_a_nat @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_676_fold__closed__eq,axiom,
! [A2: set_nat,B2: set_a,F: nat > a > a,G: nat > a > a,Z: a] :
( ! [A4: nat,B4: a] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_a @ B4 @ B2 )
=> ( ( F @ A4 @ B4 )
= ( G @ A4 @ B4 ) ) ) )
=> ( ! [A4: nat,B4: a] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_a @ B4 @ B2 )
=> ( member_a @ ( G @ A4 @ B4 ) @ B2 ) ) )
=> ( ( member_a @ Z @ B2 )
=> ( ( finite_fold_nat_a @ F @ Z @ A2 )
= ( finite_fold_nat_a @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_677_fold__closed__eq,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat > nat,G: nat > nat > nat,Z: nat] :
( ! [A4: nat,B4: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_nat @ B4 @ B2 )
=> ( ( F @ A4 @ B4 )
= ( G @ A4 @ B4 ) ) ) )
=> ( ! [A4: nat,B4: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_nat @ B4 @ B2 )
=> ( member_nat @ ( G @ A4 @ B4 ) @ B2 ) ) )
=> ( ( member_nat @ Z @ B2 )
=> ( ( finite_fold_nat_nat @ F @ Z @ A2 )
= ( finite_fold_nat_nat @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_678_fold__closed__eq,axiom,
! [A2: set_a,B2: set_nat_a,F: a > ( nat > a ) > nat > a,G: a > ( nat > a ) > nat > a,Z: nat > a] :
( ! [A4: a,B4: nat > a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_nat_a @ B4 @ B2 )
=> ( ( F @ A4 @ B4 )
= ( G @ A4 @ B4 ) ) ) )
=> ( ! [A4: a,B4: nat > a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_nat_a @ B4 @ B2 )
=> ( member_nat_a @ ( G @ A4 @ B4 ) @ B2 ) ) )
=> ( ( member_nat_a @ Z @ B2 )
=> ( ( finite_fold_a_nat_a @ F @ Z @ A2 )
= ( finite_fold_a_nat_a @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_679_fold__closed__eq,axiom,
! [A2: set_a,B2: set_a_a,F: a > ( a > a ) > a > a,G: a > ( a > a ) > a > a,Z: a > a] :
( ! [A4: a,B4: a > a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_a_a @ B4 @ B2 )
=> ( ( F @ A4 @ B4 )
= ( G @ A4 @ B4 ) ) ) )
=> ( ! [A4: a,B4: a > a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_a_a @ B4 @ B2 )
=> ( member_a_a @ ( G @ A4 @ B4 ) @ B2 ) ) )
=> ( ( member_a_a @ Z @ B2 )
=> ( ( finite_fold_a_a_a @ F @ Z @ A2 )
= ( finite_fold_a_a_a @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_680_fold__closed__eq,axiom,
! [A2: set_nat_a,B2: set_a,F: ( nat > a ) > a > a,G: ( nat > a ) > a > a,Z: a] :
( ! [A4: nat > a,B4: a] :
( ( member_nat_a @ A4 @ A2 )
=> ( ( member_a @ B4 @ B2 )
=> ( ( F @ A4 @ B4 )
= ( G @ A4 @ B4 ) ) ) )
=> ( ! [A4: nat > a,B4: a] :
( ( member_nat_a @ A4 @ A2 )
=> ( ( member_a @ B4 @ B2 )
=> ( member_a @ ( G @ A4 @ B4 ) @ B2 ) ) )
=> ( ( member_a @ Z @ B2 )
=> ( ( finite_fold_nat_a_a @ F @ Z @ A2 )
= ( finite_fold_nat_a_a @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_681_fold__closed__eq,axiom,
! [A2: set_nat_a,B2: set_nat,F: ( nat > a ) > nat > nat,G: ( nat > a ) > nat > nat,Z: nat] :
( ! [A4: nat > a,B4: nat] :
( ( member_nat_a @ A4 @ A2 )
=> ( ( member_nat @ B4 @ B2 )
=> ( ( F @ A4 @ B4 )
= ( G @ A4 @ B4 ) ) ) )
=> ( ! [A4: nat > a,B4: nat] :
( ( member_nat_a @ A4 @ A2 )
=> ( ( member_nat @ B4 @ B2 )
=> ( member_nat @ ( G @ A4 @ B4 ) @ B2 ) ) )
=> ( ( member_nat @ Z @ B2 )
=> ( ( finite7774500027257897325_a_nat @ F @ Z @ A2 )
= ( finite7774500027257897325_a_nat @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_682_fold__closed__eq,axiom,
! [A2: set_nat,B2: set_nat_a,F: nat > ( nat > a ) > nat > a,G: nat > ( nat > a ) > nat > a,Z: nat > a] :
( ! [A4: nat,B4: nat > a] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_nat_a @ B4 @ B2 )
=> ( ( F @ A4 @ B4 )
= ( G @ A4 @ B4 ) ) ) )
=> ( ! [A4: nat,B4: nat > a] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_nat_a @ B4 @ B2 )
=> ( member_nat_a @ ( G @ A4 @ B4 ) @ B2 ) ) )
=> ( ( member_nat_a @ Z @ B2 )
=> ( ( finite6730669110406474827_nat_a @ F @ Z @ A2 )
= ( finite6730669110406474827_nat_a @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_683_fold__closed__eq,axiom,
! [A2: set_nat,B2: set_a_a,F: nat > ( a > a ) > a > a,G: nat > ( a > a ) > a > a,Z: a > a] :
( ! [A4: nat,B4: a > a] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_a_a @ B4 @ B2 )
=> ( ( F @ A4 @ B4 )
= ( G @ A4 @ B4 ) ) ) )
=> ( ! [A4: nat,B4: a > a] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_a_a @ B4 @ B2 )
=> ( member_a_a @ ( G @ A4 @ B4 ) @ B2 ) ) )
=> ( ( member_a_a @ Z @ B2 )
=> ( ( finite_fold_nat_a_a2 @ F @ Z @ A2 )
= ( finite_fold_nat_a_a2 @ G @ Z @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_684_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X3: a] : $false ) ) ).
% empty_def
thf(fact_685_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X3: nat] : $false ) ) ).
% empty_def
thf(fact_686_pigeonhole__infinite__rel,axiom,
! [A2: set_nat_a,B2: set_nat,R: ( nat > a ) > nat > $o] :
( ~ ( finite_finite_nat_a @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite_finite_nat_a
@ ( collect_nat_a
@ ^ [A3: nat > a] :
( ( member_nat_a @ A3 @ A2 )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_687_pigeonhole__infinite__rel,axiom,
! [A2: set_nat_a_a2,B2: set_nat,R: ( ( nat > a ) > a ) > nat > $o] :
( ~ ( finite7239108116303828181at_a_a @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: ( nat > a ) > a] :
( ( member_nat_a_a @ X2 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite7239108116303828181at_a_a
@ ( collect_nat_a_a
@ ^ [A3: ( nat > a ) > a] :
( ( member_nat_a_a @ A3 @ A2 )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_688_pigeonhole__infinite__rel,axiom,
! [A2: set_a_a,B2: set_nat,R: ( a > a ) > nat > $o] :
( ~ ( finite_finite_a_a @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: a > a] :
( ( member_a_a @ X2 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite_finite_a_a
@ ( collect_a_a
@ ^ [A3: a > a] :
( ( member_a_a @ A3 @ A2 )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_689_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B2: set_nat,R: a > nat > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A3: a] :
( ( member_a @ A3 @ A2 )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_690_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_691_not__finite__existsD,axiom,
! [P: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P ) )
=> ? [X_1: a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_692_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_693_Collect__subset,axiom,
! [A2: set_nat_a,P: ( nat > a ) > $o] :
( ord_le871467723717165285_nat_a
@ ( collect_nat_a
@ ^ [X3: nat > a] :
( ( member_nat_a @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_694_Collect__subset,axiom,
! [A2: set_nat_a_a2,P: ( ( nat > a ) > a ) > $o] :
( ord_le3509452538356653652at_a_a
@ ( collect_nat_a_a
@ ^ [X3: ( nat > a ) > a] :
( ( member_nat_a_a @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_695_Collect__subset,axiom,
! [A2: set_a_a,P: ( a > a ) > $o] :
( ord_less_eq_set_a_a
@ ( collect_a_a
@ ^ [X3: a > a] :
( ( member_a_a @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_696_Collect__subset,axiom,
! [A2: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_697_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_698_insert__Collect,axiom,
! [A: a,P: a > $o] :
( ( insert_a @ A @ ( collect_a @ P ) )
= ( collect_a
@ ^ [U2: a] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_699_insert__Collect,axiom,
! [A: nat,P: nat > $o] :
( ( insert_nat @ A @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U2: nat] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_700_insert__compr,axiom,
( insert_nat_a
= ( ^ [A3: nat > a,B5: set_nat_a] :
( collect_nat_a
@ ^ [X3: nat > a] :
( ( X3 = A3 )
| ( member_nat_a @ X3 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_701_insert__compr,axiom,
( insert_nat_a_a
= ( ^ [A3: ( nat > a ) > a,B5: set_nat_a_a2] :
( collect_nat_a_a
@ ^ [X3: ( nat > a ) > a] :
( ( X3 = A3 )
| ( member_nat_a_a @ X3 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_702_insert__compr,axiom,
( insert_a_a
= ( ^ [A3: a > a,B5: set_a_a] :
( collect_a_a
@ ^ [X3: a > a] :
( ( X3 = A3 )
| ( member_a_a @ X3 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_703_insert__compr,axiom,
( insert_a
= ( ^ [A3: a,B5: set_a] :
( collect_a
@ ^ [X3: a] :
( ( X3 = A3 )
| ( member_a @ X3 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_704_insert__compr,axiom,
( insert_nat
= ( ^ [A3: nat,B5: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( X3 = A3 )
| ( member_nat @ X3 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_705_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( ord_less_eq_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_706_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_707_finite__has__minimal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ( ord_less_eq_set_nat @ X2 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_708_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( ord_less_eq_set_a @ A @ X2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_709_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ A @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_710_finite__has__maximal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ( ord_less_eq_set_nat @ A @ X2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_711_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_712_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_713_rev__finite__subset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_714_rev__finite__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_715_infinite__super,axiom,
! [S: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_716_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_717_finite__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_718_finite__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_719_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_720_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_721_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_722_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_723_singleton__iff,axiom,
! [B: nat > a,A: nat > a] :
( ( member_nat_a @ B @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_724_singleton__iff,axiom,
! [B: ( nat > a ) > a,A: ( nat > a ) > a] :
( ( member_nat_a_a @ B @ ( insert_nat_a_a @ A @ bot_bot_set_nat_a_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_725_singleton__iff,axiom,
! [B: a > a,A: a > a] :
( ( member_a_a @ B @ ( insert_a_a @ A @ bot_bot_set_a_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_726_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_727_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_728_singletonD,axiom,
! [B: nat > a,A: nat > a] :
( ( member_nat_a @ B @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_729_singletonD,axiom,
! [B: ( nat > a ) > a,A: ( nat > a ) > a] :
( ( member_nat_a_a @ B @ ( insert_nat_a_a @ A @ bot_bot_set_nat_a_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_730_singletonD,axiom,
! [B: a > a,A: a > a] :
( ( member_a_a @ B @ ( insert_a_a @ A @ bot_bot_set_a_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_731_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_732_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_733_subset__insertI2,axiom,
! [A2: set_a,B2: set_a,B: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_734_subset__insertI2,axiom,
! [A2: set_nat,B2: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_735_subset__insertI,axiom,
! [B2: set_a,A: a] : ( ord_less_eq_set_a @ B2 @ ( insert_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_736_subset__insertI,axiom,
! [B2: set_nat,A: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A @ B2 ) ) ).
% subset_insertI
thf(fact_737_subset__insert,axiom,
! [X: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ~ ( member_nat_a @ X @ A2 )
=> ( ( ord_le871467723717165285_nat_a @ A2 @ ( insert_nat_a @ X @ B2 ) )
= ( ord_le871467723717165285_nat_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_738_subset__insert,axiom,
! [X: ( nat > a ) > a,A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ~ ( member_nat_a_a @ X @ A2 )
=> ( ( ord_le3509452538356653652at_a_a @ A2 @ ( insert_nat_a_a @ X @ B2 ) )
= ( ord_le3509452538356653652at_a_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_739_subset__insert,axiom,
! [X: a > a,A2: set_a_a,B2: set_a_a] :
( ~ ( member_a_a @ X @ A2 )
=> ( ( ord_less_eq_set_a_a @ A2 @ ( insert_a_a @ X @ B2 ) )
= ( ord_less_eq_set_a_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_740_subset__insert,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_741_subset__insert,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_742_insert__mono,axiom,
! [C4: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C4 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C4 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_743_insert__mono,axiom,
! [C4: set_nat,D2: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C4 @ D2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C4 ) @ ( insert_nat @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_744_Collect__conv__if2,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X3: a] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X3: a] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_745_Collect__conv__if2,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_746_Collect__conv__if,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X3: a] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X3: a] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_747_Collect__conv__if,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_748_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_749_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_750_finite__has__minimal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_751_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_752_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_753_finite__has__maximal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_754_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A7: set_a] :
( ~ ( finite_finite_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_755_infinite__finite__induct,axiom,
! [P: set_nat_a > $o,A2: set_nat_a] :
( ! [A7: set_nat_a] :
( ~ ( finite_finite_nat_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_nat_a )
=> ( ! [X2: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ~ ( member_nat_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_756_infinite__finite__induct,axiom,
! [P: set_nat_a_a2 > $o,A2: set_nat_a_a2] :
( ! [A7: set_nat_a_a2] :
( ~ ( finite7239108116303828181at_a_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_nat_a_a )
=> ( ! [X2: ( nat > a ) > a,F3: set_nat_a_a2] :
( ( finite7239108116303828181at_a_a @ F3 )
=> ( ~ ( member_nat_a_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_757_infinite__finite__induct,axiom,
! [P: set_a_a > $o,A2: set_a_a] :
( ! [A7: set_a_a] :
( ~ ( finite_finite_a_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_a_a )
=> ( ! [X2: a > a,F3: set_a_a] :
( ( finite_finite_a_a @ F3 )
=> ( ~ ( member_a_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_758_infinite__finite__induct,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ! [A7: set_nat] :
( ~ ( finite_finite_nat @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_759_finite__ne__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X2: a] : ( P @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_760_finite__ne__induct,axiom,
! [F2: set_nat_a,P: set_nat_a > $o] :
( ( finite_finite_nat_a @ F2 )
=> ( ( F2 != bot_bot_set_nat_a )
=> ( ! [X2: nat > a] : ( P @ ( insert_nat_a @ X2 @ bot_bot_set_nat_a ) )
=> ( ! [X2: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ( F3 != bot_bot_set_nat_a )
=> ( ~ ( member_nat_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_761_finite__ne__induct,axiom,
! [F2: set_nat_a_a2,P: set_nat_a_a2 > $o] :
( ( finite7239108116303828181at_a_a @ F2 )
=> ( ( F2 != bot_bot_set_nat_a_a )
=> ( ! [X2: ( nat > a ) > a] : ( P @ ( insert_nat_a_a @ X2 @ bot_bot_set_nat_a_a ) )
=> ( ! [X2: ( nat > a ) > a,F3: set_nat_a_a2] :
( ( finite7239108116303828181at_a_a @ F3 )
=> ( ( F3 != bot_bot_set_nat_a_a )
=> ( ~ ( member_nat_a_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_762_finite__ne__induct,axiom,
! [F2: set_a_a,P: set_a_a > $o] :
( ( finite_finite_a_a @ F2 )
=> ( ( F2 != bot_bot_set_a_a )
=> ( ! [X2: a > a] : ( P @ ( insert_a_a @ X2 @ bot_bot_set_a_a ) )
=> ( ! [X2: a > a,F3: set_a_a] :
( ( finite_finite_a_a @ F3 )
=> ( ( F3 != bot_bot_set_a_a )
=> ( ~ ( member_a_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_763_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X2: nat] : ( P @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_764_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_765_finite__induct,axiom,
! [F2: set_nat_a,P: set_nat_a > $o] :
( ( finite_finite_nat_a @ F2 )
=> ( ( P @ bot_bot_set_nat_a )
=> ( ! [X2: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ~ ( member_nat_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_766_finite__induct,axiom,
! [F2: set_nat_a_a2,P: set_nat_a_a2 > $o] :
( ( finite7239108116303828181at_a_a @ F2 )
=> ( ( P @ bot_bot_set_nat_a_a )
=> ( ! [X2: ( nat > a ) > a,F3: set_nat_a_a2] :
( ( finite7239108116303828181at_a_a @ F3 )
=> ( ~ ( member_nat_a_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_767_finite__induct,axiom,
! [F2: set_a_a,P: set_a_a > $o] :
( ( finite_finite_a_a @ F2 )
=> ( ( P @ bot_bot_set_a_a )
=> ( ! [X2: a > a,F3: set_a_a] :
( ( finite_finite_a_a @ F3 )
=> ( ~ ( member_a_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_768_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_769_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A3: set_nat] :
( ( A3 = bot_bot_set_nat )
| ? [A6: set_nat,B3: nat] :
( ( A3
= ( insert_nat @ B3 @ A6 ) )
& ( finite_finite_nat @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_770_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A7: set_nat] :
( ? [A4: nat] :
( A
= ( insert_nat @ A4 @ A7 ) )
=> ~ ( finite_finite_nat @ A7 ) ) ) ) ).
% finite.cases
thf(fact_771_subset__singleton__iff,axiom,
! [X4: set_a,A: a] :
( ( ord_less_eq_set_a @ X4 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X4 = bot_bot_set_a )
| ( X4
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_772_subset__singleton__iff,axiom,
! [X4: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X4 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X4 = bot_bot_set_nat )
| ( X4
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_773_subset__singletonD,axiom,
! [A2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_774_subset__singletonD,axiom,
! [A2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_775_finite__subset__induct_H,axiom,
! [F2: set_nat_a,A2: set_nat_a,P: set_nat_a > $o] :
( ( finite_finite_nat_a @ F2 )
=> ( ( ord_le871467723717165285_nat_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat_a )
=> ( ! [A4: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ( member_nat_a @ A4 @ A2 )
=> ( ( ord_le871467723717165285_nat_a @ F3 @ A2 )
=> ( ~ ( member_nat_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_776_finite__subset__induct_H,axiom,
! [F2: set_nat_a_a2,A2: set_nat_a_a2,P: set_nat_a_a2 > $o] :
( ( finite7239108116303828181at_a_a @ F2 )
=> ( ( ord_le3509452538356653652at_a_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat_a_a )
=> ( ! [A4: ( nat > a ) > a,F3: set_nat_a_a2] :
( ( finite7239108116303828181at_a_a @ F3 )
=> ( ( member_nat_a_a @ A4 @ A2 )
=> ( ( ord_le3509452538356653652at_a_a @ F3 @ A2 )
=> ( ~ ( member_nat_a_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a_a @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_777_finite__subset__induct_H,axiom,
! [F2: set_a_a,A2: set_a_a,P: set_a_a > $o] :
( ( finite_finite_a_a @ F2 )
=> ( ( ord_less_eq_set_a_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a_a )
=> ( ! [A4: a > a,F3: set_a_a] :
( ( finite_finite_a_a @ F3 )
=> ( ( member_a_a @ A4 @ A2 )
=> ( ( ord_less_eq_set_a_a @ F3 @ A2 )
=> ( ~ ( member_a_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a_a @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_778_finite__subset__induct_H,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A4 @ A2 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ~ ( member_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_779_finite__subset__induct_H,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A4 @ A2 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ~ ( member_nat @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_780_finite__subset__induct,axiom,
! [F2: set_nat_a,A2: set_nat_a,P: set_nat_a > $o] :
( ( finite_finite_nat_a @ F2 )
=> ( ( ord_le871467723717165285_nat_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat_a )
=> ( ! [A4: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ( member_nat_a @ A4 @ A2 )
=> ( ~ ( member_nat_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_781_finite__subset__induct,axiom,
! [F2: set_nat_a_a2,A2: set_nat_a_a2,P: set_nat_a_a2 > $o] :
( ( finite7239108116303828181at_a_a @ F2 )
=> ( ( ord_le3509452538356653652at_a_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat_a_a )
=> ( ! [A4: ( nat > a ) > a,F3: set_nat_a_a2] :
( ( finite7239108116303828181at_a_a @ F3 )
=> ( ( member_nat_a_a @ A4 @ A2 )
=> ( ~ ( member_nat_a_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a_a @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_782_finite__subset__induct,axiom,
! [F2: set_a_a,A2: set_a_a,P: set_a_a > $o] :
( ( finite_finite_a_a @ F2 )
=> ( ( ord_less_eq_set_a_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a_a )
=> ( ! [A4: a > a,F3: set_a_a] :
( ( finite_finite_a_a @ F3 )
=> ( ( member_a_a @ A4 @ A2 )
=> ( ~ ( member_a_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a_a @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_783_finite__subset__induct,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A4 @ A2 )
=> ( ~ ( member_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_784_finite__subset__induct,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A4 @ A2 )
=> ( ~ ( member_nat @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_785_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y6: a] :
( ( member_a @ Y6 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_786_finite__ranking__induct,axiom,
! [S: set_nat_a,P: set_nat_a > $o,F: ( nat > a ) > nat] :
( ( finite_finite_nat_a @ S )
=> ( ( P @ bot_bot_set_nat_a )
=> ( ! [X2: nat > a,S2: set_nat_a] :
( ( finite_finite_nat_a @ S2 )
=> ( ! [Y6: nat > a] :
( ( member_nat_a @ Y6 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat_a @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_787_finite__ranking__induct,axiom,
! [S: set_nat_a_a2,P: set_nat_a_a2 > $o,F: ( ( nat > a ) > a ) > nat] :
( ( finite7239108116303828181at_a_a @ S )
=> ( ( P @ bot_bot_set_nat_a_a )
=> ( ! [X2: ( nat > a ) > a,S2: set_nat_a_a2] :
( ( finite7239108116303828181at_a_a @ S2 )
=> ( ! [Y6: ( nat > a ) > a] :
( ( member_nat_a_a @ Y6 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat_a_a @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_788_finite__ranking__induct,axiom,
! [S: set_a_a,P: set_a_a > $o,F: ( a > a ) > nat] :
( ( finite_finite_a_a @ S )
=> ( ( P @ bot_bot_set_a_a )
=> ( ! [X2: a > a,S2: set_a_a] :
( ( finite_finite_a_a @ S2 )
=> ( ! [Y6: a > a] :
( ( member_a_a @ Y6 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a_a @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_789_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y6: nat] :
( ( member_nat @ Y6 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_790_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( P @ M5 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less
thf(fact_791_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less_eq_nat @ M5 @ N )
& ( P @ M5 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less
thf(fact_792_the__elem__eq,axiom,
! [X: nat] :
( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% the_elem_eq
thf(fact_793_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_794_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_795_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_796_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_797_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_798_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_799_is__singletonI,axiom,
! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_800_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_801_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_802_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_803_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A6: set_nat] :
( A6
= ( insert_nat @ ( the_elem_nat @ A6 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_804_is__singletonI_H,axiom,
! [A2: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ A2 )
=> ( ( member_a @ Y3 @ A2 )
=> ( X2 = Y3 ) ) )
=> ( is_singleton_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_805_is__singletonI_H,axiom,
! [A2: set_nat_a] :
( ( A2 != bot_bot_set_nat_a )
=> ( ! [X2: nat > a,Y3: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( ( member_nat_a @ Y3 @ A2 )
=> ( X2 = Y3 ) ) )
=> ( is_singleton_nat_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_806_is__singletonI_H,axiom,
! [A2: set_nat_a_a2] :
( ( A2 != bot_bot_set_nat_a_a )
=> ( ! [X2: ( nat > a ) > a,Y3: ( nat > a ) > a] :
( ( member_nat_a_a @ X2 @ A2 )
=> ( ( member_nat_a_a @ Y3 @ A2 )
=> ( X2 = Y3 ) ) )
=> ( is_singleton_nat_a_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_807_is__singletonI_H,axiom,
! [A2: set_a_a] :
( ( A2 != bot_bot_set_a_a )
=> ( ! [X2: a > a,Y3: a > a] :
( ( member_a_a @ X2 @ A2 )
=> ( ( member_a_a @ Y3 @ A2 )
=> ( X2 = Y3 ) ) )
=> ( is_singleton_a_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_808_is__singletonI_H,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat,Y3: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( X2 = Y3 ) ) )
=> ( is_singleton_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_809_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_810_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_811_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_812_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_813_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z2: set_nat] : ( Y5 = Z2 ) )
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ( ord_less_eq_set_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_814_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_815_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_816_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_817_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_818_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_819_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_820_order__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_821_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_822_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_823_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_824_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_825_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_826_order__trans,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_827_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_828_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_829_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_830_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
& ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_831_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_832_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_nat,Z2: set_nat] : ( Y5 = Z2 ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
& ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_833_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_834_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_835_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_836_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_837_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_838_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_839_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_840_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_841_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_842_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_843_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_844_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z2: set_nat] : ( Y5 = Z2 ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_845_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_846_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_847_order__subst1,axiom,
! [A: set_a,F: set_nat > set_a,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_848_order__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_849_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_850_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_851_order__subst1,axiom,
! [A: set_nat,F: set_a > set_nat,B: set_a,C: set_a] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_852_order__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_853_order__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_854_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_855_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_856_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_nat,C: set_nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_857_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_858_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_859_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_860_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_a,C: set_a] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_861_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_862_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_863_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_864_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_865_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_866_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_867_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_868_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_869_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_a > set_nat,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_870_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_871_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_872_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_873_ord__eq__le__subst,axiom,
! [A: set_a,F: set_nat > set_a,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_874_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_875_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_876_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_877_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_878_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_nat,C: set_nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_879_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_880_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_881_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_882_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_a,C: set_a] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_883_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_884_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_885_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_886_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_887_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_888_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_889_is__singletonE,axiom,
! [A2: set_nat] :
( ( is_singleton_nat @ A2 )
=> ~ ! [X2: nat] :
( A2
!= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_890_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A6: set_nat] :
? [X3: nat] :
( A6
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_891_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_892_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_893_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_894_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_895_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_896_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_897_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_898_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_899_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_900_pred__subset__eq,axiom,
! [R: set_nat_a,S: set_nat_a] :
( ( ord_less_eq_nat_a_o
@ ^ [X3: nat > a] : ( member_nat_a @ X3 @ R )
@ ^ [X3: nat > a] : ( member_nat_a @ X3 @ S ) )
= ( ord_le871467723717165285_nat_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_901_pred__subset__eq,axiom,
! [R: set_nat_a_a2,S: set_nat_a_a2] :
( ( ord_le3623034401944517937_a_a_o
@ ^ [X3: ( nat > a ) > a] : ( member_nat_a_a @ X3 @ R )
@ ^ [X3: ( nat > a ) > a] : ( member_nat_a_a @ X3 @ S ) )
= ( ord_le3509452538356653652at_a_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_902_pred__subset__eq,axiom,
! [R: set_a_a,S: set_a_a] :
( ( ord_less_eq_a_a_o
@ ^ [X3: a > a] : ( member_a_a @ X3 @ R )
@ ^ [X3: a > a] : ( member_a_a @ X3 @ S ) )
= ( ord_less_eq_set_a_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_903_pred__subset__eq,axiom,
! [R: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X3: a] : ( member_a @ X3 @ R )
@ ^ [X3: a] : ( member_a @ X3 @ S ) )
= ( ord_less_eq_set_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_904_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ R )
@ ^ [X3: nat] : ( member_nat @ X3 @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_905_arg__min__least,axiom,
! [S: set_a,Y: a,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ( ( member_a @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic6340287419671400565_a_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_906_arg__min__least,axiom,
! [S: set_nat_a,Y: nat > a,F: ( nat > a ) > nat] :
( ( finite_finite_nat_a @ S )
=> ( ( S != bot_bot_set_nat_a )
=> ( ( member_nat_a @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic6419734799033661276_a_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_907_arg__min__least,axiom,
! [S: set_nat_a_a2,Y: ( nat > a ) > a,F: ( ( nat > a ) > a ) > nat] :
( ( finite7239108116303828181at_a_a @ S )
=> ( ( S != bot_bot_set_nat_a_a )
=> ( ( member_nat_a_a @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7650876701696636585_a_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_908_arg__min__least,axiom,
! [S: set_a_a,Y: a > a,F: ( a > a ) > nat] :
( ( finite_finite_a_a @ S )
=> ( ( S != bot_bot_set_a_a )
=> ( ( member_a_a @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic5687691984537318480_a_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_909_arg__min__least,axiom,
! [S: set_nat,Y: nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_910_insert__subsetI,axiom,
! [X: nat > a,A2: set_nat_a,X4: set_nat_a] :
( ( member_nat_a @ X @ A2 )
=> ( ( ord_le871467723717165285_nat_a @ X4 @ A2 )
=> ( ord_le871467723717165285_nat_a @ ( insert_nat_a @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_911_insert__subsetI,axiom,
! [X: ( nat > a ) > a,A2: set_nat_a_a2,X4: set_nat_a_a2] :
( ( member_nat_a_a @ X @ A2 )
=> ( ( ord_le3509452538356653652at_a_a @ X4 @ A2 )
=> ( ord_le3509452538356653652at_a_a @ ( insert_nat_a_a @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_912_insert__subsetI,axiom,
! [X: a > a,A2: set_a_a,X4: set_a_a] :
( ( member_a_a @ X @ A2 )
=> ( ( ord_less_eq_set_a_a @ X4 @ A2 )
=> ( ord_less_eq_set_a_a @ ( insert_a_a @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_913_insert__subsetI,axiom,
! [X: a,A2: set_a,X4: set_a] :
( ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ X4 @ A2 )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_914_insert__subsetI,axiom,
! [X: nat,A2: set_nat,X4: set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_915_fincomp__Un__disjoint,axiom,
! [A2: set_nat,B2: set_nat,G: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : m ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : m ) )
=> ( ( commut6741328216151336360_a_nat @ m @ composition @ unit @ G @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( composition @ ( commut6741328216151336360_a_nat @ m @ composition @ unit @ G @ A2 ) @ ( commut6741328216151336360_a_nat @ m @ composition @ unit @ G @ B2 ) ) ) ) ) ) ) ) ).
% fincomp_Un_disjoint
thf(fact_916_fincomp__Un__disjoint,axiom,
! [A2: set_nat_a,B2: set_nat_a,G: ( nat > a ) > a] :
( ( finite_finite_nat_a @ A2 )
=> ( ( finite_finite_nat_a @ B2 )
=> ( ( ( inf_inf_set_nat_a @ A2 @ B2 )
= bot_bot_set_nat_a )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : m ) )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ B2
@ ^ [Uu: nat > a] : m ) )
=> ( ( commut5242989786243415821_nat_a @ m @ composition @ unit @ G @ ( sup_sup_set_nat_a @ A2 @ B2 ) )
= ( composition @ ( commut5242989786243415821_nat_a @ m @ composition @ unit @ G @ A2 ) @ ( commut5242989786243415821_nat_a @ m @ composition @ unit @ G @ B2 ) ) ) ) ) ) ) ) ).
% fincomp_Un_disjoint
thf(fact_917_fincomp__Un__disjoint,axiom,
! [A2: set_a,B2: set_a,G: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A2
@ ^ [Uu: a] : m ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : m ) )
=> ( ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ G @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( composition @ ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ G @ A2 ) @ ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ G @ B2 ) ) ) ) ) ) ) ) ).
% fincomp_Un_disjoint
thf(fact_918_subset__emptyI,axiom,
! [A2: set_nat_a] :
( ! [X2: nat > a] :
~ ( member_nat_a @ X2 @ A2 )
=> ( ord_le871467723717165285_nat_a @ A2 @ bot_bot_set_nat_a ) ) ).
% subset_emptyI
thf(fact_919_subset__emptyI,axiom,
! [A2: set_nat_a_a2] :
( ! [X2: ( nat > a ) > a] :
~ ( member_nat_a_a @ X2 @ A2 )
=> ( ord_le3509452538356653652at_a_a @ A2 @ bot_bot_set_nat_a_a ) ) ).
% subset_emptyI
thf(fact_920_subset__emptyI,axiom,
! [A2: set_a_a] :
( ! [X2: a > a] :
~ ( member_a_a @ X2 @ A2 )
=> ( ord_less_eq_set_a_a @ A2 @ bot_bot_set_a_a ) ) ).
% subset_emptyI
thf(fact_921_subset__emptyI,axiom,
! [A2: set_a] :
( ! [X2: a] :
~ ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ A2 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_922_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_923_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_924_IntI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ( member_a @ C @ B2 )
=> ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_925_IntI,axiom,
! [C: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ A2 )
=> ( ( member_nat_a @ C @ B2 )
=> ( member_nat_a @ C @ ( inf_inf_set_nat_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_926_IntI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_927_IntI,axiom,
! [C: ( nat > a ) > a,A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ( member_nat_a_a @ C @ A2 )
=> ( ( member_nat_a_a @ C @ B2 )
=> ( member_nat_a_a @ C @ ( inf_inf_set_nat_a_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_928_IntI,axiom,
! [C: a > a,A2: set_a_a,B2: set_a_a] :
( ( member_a_a @ C @ A2 )
=> ( ( member_a_a @ C @ B2 )
=> ( member_a_a @ C @ ( inf_inf_set_a_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_929_Int__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ( member_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_930_Int__iff,axiom,
! [C: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( inf_inf_set_nat_a @ A2 @ B2 ) )
= ( ( member_nat_a @ C @ A2 )
& ( member_nat_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_931_Int__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ( member_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_932_Int__iff,axiom,
! [C: ( nat > a ) > a,A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ( member_nat_a_a @ C @ ( inf_inf_set_nat_a_a @ A2 @ B2 ) )
= ( ( member_nat_a_a @ C @ A2 )
& ( member_nat_a_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_933_Int__iff,axiom,
! [C: a > a,A2: set_a_a,B2: set_a_a] :
( ( member_a_a @ C @ ( inf_inf_set_a_a @ A2 @ B2 ) )
= ( ( member_a_a @ C @ A2 )
& ( member_a_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_934_UnCI,axiom,
! [C: a,B2: set_a,A2: set_a] :
( ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ A2 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_935_UnCI,axiom,
! [C: nat > a,B2: set_nat_a,A2: set_nat_a] :
( ( ~ ( member_nat_a @ C @ B2 )
=> ( member_nat_a @ C @ A2 ) )
=> ( member_nat_a @ C @ ( sup_sup_set_nat_a @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_936_UnCI,axiom,
! [C: nat,B2: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ A2 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_937_UnCI,axiom,
! [C: ( nat > a ) > a,B2: set_nat_a_a2,A2: set_nat_a_a2] :
( ( ~ ( member_nat_a_a @ C @ B2 )
=> ( member_nat_a_a @ C @ A2 ) )
=> ( member_nat_a_a @ C @ ( sup_sup_set_nat_a_a @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_938_UnCI,axiom,
! [C: a > a,B2: set_a_a,A2: set_a_a] :
( ( ~ ( member_a_a @ C @ B2 )
=> ( member_a_a @ C @ A2 ) )
=> ( member_a_a @ C @ ( sup_sup_set_a_a @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_939_Un__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
| ( member_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_940_Un__iff,axiom,
! [C: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( sup_sup_set_nat_a @ A2 @ B2 ) )
= ( ( member_nat_a @ C @ A2 )
| ( member_nat_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_941_Un__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
| ( member_nat @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_942_Un__iff,axiom,
! [C: ( nat > a ) > a,A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ( member_nat_a_a @ C @ ( sup_sup_set_nat_a_a @ A2 @ B2 ) )
= ( ( member_nat_a_a @ C @ A2 )
| ( member_nat_a_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_943_Un__iff,axiom,
! [C: a > a,A2: set_a_a,B2: set_a_a] :
( ( member_a_a @ C @ ( sup_sup_set_a_a @ A2 @ B2 ) )
= ( ( member_a_a @ C @ A2 )
| ( member_a_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_944_DiffI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_945_DiffI,axiom,
! [C: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ A2 )
=> ( ~ ( member_nat_a @ C @ B2 )
=> ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_946_DiffI,axiom,
! [C: ( nat > a ) > a,A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ( member_nat_a_a @ C @ A2 )
=> ( ~ ( member_nat_a_a @ C @ B2 )
=> ( member_nat_a_a @ C @ ( minus_1482667089342205261at_a_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_947_DiffI,axiom,
! [C: a > a,A2: set_a_a,B2: set_a_a] :
( ( member_a_a @ C @ A2 )
=> ( ~ ( member_a_a @ C @ B2 )
=> ( member_a_a @ C @ ( minus_minus_set_a_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_948_DiffI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_949_Diff__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_950_Diff__iff,axiom,
! [C: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) )
= ( ( member_nat_a @ C @ A2 )
& ~ ( member_nat_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_951_Diff__iff,axiom,
! [C: ( nat > a ) > a,A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ( member_nat_a_a @ C @ ( minus_1482667089342205261at_a_a @ A2 @ B2 ) )
= ( ( member_nat_a_a @ C @ A2 )
& ~ ( member_nat_a_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_952_Diff__iff,axiom,
! [C: a > a,A2: set_a_a,B2: set_a_a] :
( ( member_a_a @ C @ ( minus_minus_set_a_a @ A2 @ B2 ) )
= ( ( member_a_a @ C @ A2 )
& ~ ( member_a_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_953_Diff__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_954_Diff__idemp,axiom,
! [A2: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_955_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_956_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_957_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_958_finite__Int,axiom,
! [F2: set_nat,G2: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G2 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_959_Un__empty,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_960_Int__subset__iff,axiom,
! [C4: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C4 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( ord_less_eq_set_a @ C4 @ A2 )
& ( ord_less_eq_set_a @ C4 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_961_Int__subset__iff,axiom,
! [C4: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_set_nat @ C4 @ A2 )
& ( ord_less_eq_set_nat @ C4 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_962_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_963_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_964_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_965_finite__Un,axiom,
! [F2: set_nat,G2: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_966_finite__Diff2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_967_finite__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_968_Int__insert__right__if1,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_969_Int__insert__right__if1,axiom,
! [A: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ A @ A2 )
=> ( ( inf_inf_set_nat_a @ A2 @ ( insert_nat_a @ A @ B2 ) )
= ( insert_nat_a @ A @ ( inf_inf_set_nat_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_970_Int__insert__right__if1,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_971_Int__insert__right__if1,axiom,
! [A: ( nat > a ) > a,A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ( member_nat_a_a @ A @ A2 )
=> ( ( inf_inf_set_nat_a_a @ A2 @ ( insert_nat_a_a @ A @ B2 ) )
= ( insert_nat_a_a @ A @ ( inf_inf_set_nat_a_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_972_Int__insert__right__if1,axiom,
! [A: a > a,A2: set_a_a,B2: set_a_a] :
( ( member_a_a @ A @ A2 )
=> ( ( inf_inf_set_a_a @ A2 @ ( insert_a_a @ A @ B2 ) )
= ( insert_a_a @ A @ ( inf_inf_set_a_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_973_Int__insert__right__if0,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_974_Int__insert__right__if0,axiom,
! [A: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ~ ( member_nat_a @ A @ A2 )
=> ( ( inf_inf_set_nat_a @ A2 @ ( insert_nat_a @ A @ B2 ) )
= ( inf_inf_set_nat_a @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_975_Int__insert__right__if0,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_976_Int__insert__right__if0,axiom,
! [A: ( nat > a ) > a,A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ~ ( member_nat_a_a @ A @ A2 )
=> ( ( inf_inf_set_nat_a_a @ A2 @ ( insert_nat_a_a @ A @ B2 ) )
= ( inf_inf_set_nat_a_a @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_977_Int__insert__right__if0,axiom,
! [A: a > a,A2: set_a_a,B2: set_a_a] :
( ~ ( member_a_a @ A @ A2 )
=> ( ( inf_inf_set_a_a @ A2 @ ( insert_a_a @ A @ B2 ) )
= ( inf_inf_set_a_a @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_978_insert__inter__insert,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_979_Int__insert__left__if1,axiom,
! [A: a,C4: set_a,B2: set_a] :
( ( member_a @ A @ C4 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C4 )
= ( insert_a @ A @ ( inf_inf_set_a @ B2 @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_980_Int__insert__left__if1,axiom,
! [A: nat > a,C4: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ A @ C4 )
=> ( ( inf_inf_set_nat_a @ ( insert_nat_a @ A @ B2 ) @ C4 )
= ( insert_nat_a @ A @ ( inf_inf_set_nat_a @ B2 @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_981_Int__insert__left__if1,axiom,
! [A: nat,C4: set_nat,B2: set_nat] :
( ( member_nat @ A @ C4 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C4 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B2 @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_982_Int__insert__left__if1,axiom,
! [A: ( nat > a ) > a,C4: set_nat_a_a2,B2: set_nat_a_a2] :
( ( member_nat_a_a @ A @ C4 )
=> ( ( inf_inf_set_nat_a_a @ ( insert_nat_a_a @ A @ B2 ) @ C4 )
= ( insert_nat_a_a @ A @ ( inf_inf_set_nat_a_a @ B2 @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_983_Int__insert__left__if1,axiom,
! [A: a > a,C4: set_a_a,B2: set_a_a] :
( ( member_a_a @ A @ C4 )
=> ( ( inf_inf_set_a_a @ ( insert_a_a @ A @ B2 ) @ C4 )
= ( insert_a_a @ A @ ( inf_inf_set_a_a @ B2 @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_984_Int__insert__left__if0,axiom,
! [A: a,C4: set_a,B2: set_a] :
( ~ ( member_a @ A @ C4 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C4 )
= ( inf_inf_set_a @ B2 @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_985_Int__insert__left__if0,axiom,
! [A: nat > a,C4: set_nat_a,B2: set_nat_a] :
( ~ ( member_nat_a @ A @ C4 )
=> ( ( inf_inf_set_nat_a @ ( insert_nat_a @ A @ B2 ) @ C4 )
= ( inf_inf_set_nat_a @ B2 @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_986_Int__insert__left__if0,axiom,
! [A: nat,C4: set_nat,B2: set_nat] :
( ~ ( member_nat @ A @ C4 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C4 )
= ( inf_inf_set_nat @ B2 @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_987_Int__insert__left__if0,axiom,
! [A: ( nat > a ) > a,C4: set_nat_a_a2,B2: set_nat_a_a2] :
( ~ ( member_nat_a_a @ A @ C4 )
=> ( ( inf_inf_set_nat_a_a @ ( insert_nat_a_a @ A @ B2 ) @ C4 )
= ( inf_inf_set_nat_a_a @ B2 @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_988_Int__insert__left__if0,axiom,
! [A: a > a,C4: set_a_a,B2: set_a_a] :
( ~ ( member_a_a @ A @ C4 )
=> ( ( inf_inf_set_a_a @ ( insert_a_a @ A @ B2 ) @ C4 )
= ( inf_inf_set_a_a @ B2 @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_989_Un__subset__iff,axiom,
! [A2: set_a,B2: set_a,C4: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C4 )
= ( ( ord_less_eq_set_a @ A2 @ C4 )
& ( ord_less_eq_set_a @ B2 @ C4 ) ) ) ).
% Un_subset_iff
thf(fact_990_Un__subset__iff,axiom,
! [A2: set_nat,B2: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C4 )
= ( ( ord_less_eq_set_nat @ A2 @ C4 )
& ( ord_less_eq_set_nat @ B2 @ C4 ) ) ) ).
% Un_subset_iff
thf(fact_991_Un__insert__right,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_992_Un__insert__left,axiom,
! [A: nat,B2: set_nat,C4: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A @ B2 ) @ C4 )
= ( insert_nat @ A @ ( sup_sup_set_nat @ B2 @ C4 ) ) ) ).
% Un_insert_left
thf(fact_993_insert__Diff1,axiom,
! [X: a,B2: set_a,A2: set_a] :
( ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_994_insert__Diff1,axiom,
! [X: nat > a,B2: set_nat_a,A2: set_nat_a] :
( ( member_nat_a @ X @ B2 )
=> ( ( minus_490503922182417452_nat_a @ ( insert_nat_a @ X @ A2 ) @ B2 )
= ( minus_490503922182417452_nat_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_995_insert__Diff1,axiom,
! [X: ( nat > a ) > a,B2: set_nat_a_a2,A2: set_nat_a_a2] :
( ( member_nat_a_a @ X @ B2 )
=> ( ( minus_1482667089342205261at_a_a @ ( insert_nat_a_a @ X @ A2 ) @ B2 )
= ( minus_1482667089342205261at_a_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_996_insert__Diff1,axiom,
! [X: a > a,B2: set_a_a,A2: set_a_a] :
( ( member_a_a @ X @ B2 )
=> ( ( minus_minus_set_a_a @ ( insert_a_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_a_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_997_insert__Diff1,axiom,
! [X: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_998_Diff__insert0,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_999_Diff__insert0,axiom,
! [X: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ~ ( member_nat_a @ X @ A2 )
=> ( ( minus_490503922182417452_nat_a @ A2 @ ( insert_nat_a @ X @ B2 ) )
= ( minus_490503922182417452_nat_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1000_Diff__insert0,axiom,
! [X: ( nat > a ) > a,A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ~ ( member_nat_a_a @ X @ A2 )
=> ( ( minus_1482667089342205261at_a_a @ A2 @ ( insert_nat_a_a @ X @ B2 ) )
= ( minus_1482667089342205261at_a_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1001_Diff__insert0,axiom,
! [X: a > a,A2: set_a_a,B2: set_a_a] :
( ~ ( member_a_a @ X @ A2 )
=> ( ( minus_minus_set_a_a @ A2 @ ( insert_a_a @ X @ B2 ) )
= ( minus_minus_set_a_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1002_Diff__insert0,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1003_Un__Diff__cancel,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_1004_Un__Diff__cancel2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B2 @ A2 ) @ A2 )
= ( sup_sup_set_nat @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_1005_Pi__split__domain,axiom,
! [X: nat > a,I4: set_nat,J2: set_nat,X4: nat > set_a] :
( ( member_nat_a @ X @ ( pi_nat_a @ ( sup_sup_set_nat @ I4 @ J2 ) @ X4 ) )
= ( ( member_nat_a @ X @ ( pi_nat_a @ I4 @ X4 ) )
& ( member_nat_a @ X @ ( pi_nat_a @ J2 @ X4 ) ) ) ) ).
% Pi_split_domain
thf(fact_1006_Pi__split__domain,axiom,
! [X: ( nat > a ) > a,I4: set_nat_a,J2: set_nat_a,X4: ( nat > a ) > set_a] :
( ( member_nat_a_a @ X @ ( pi_nat_a_a @ ( sup_sup_set_nat_a @ I4 @ J2 ) @ X4 ) )
= ( ( member_nat_a_a @ X @ ( pi_nat_a_a @ I4 @ X4 ) )
& ( member_nat_a_a @ X @ ( pi_nat_a_a @ J2 @ X4 ) ) ) ) ).
% Pi_split_domain
thf(fact_1007_Pi__split__domain,axiom,
! [X: a > a,I4: set_a,J2: set_a,X4: a > set_a] :
( ( member_a_a @ X @ ( pi_a_a @ ( sup_sup_set_a @ I4 @ J2 ) @ X4 ) )
= ( ( member_a_a @ X @ ( pi_a_a @ I4 @ X4 ) )
& ( member_a_a @ X @ ( pi_a_a @ J2 @ X4 ) ) ) ) ).
% Pi_split_domain
thf(fact_1008_funcset__Int__left,axiom,
! [F: nat > a,A2: set_nat,C4: set_a,B2: set_nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : C4 ) )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : C4 ) )
=> ( member_nat_a @ F
@ ( pi_nat_a @ ( inf_inf_set_nat @ A2 @ B2 )
@ ^ [Uu: nat] : C4 ) ) ) ) ).
% funcset_Int_left
thf(fact_1009_funcset__Int__left,axiom,
! [F: ( nat > a ) > a,A2: set_nat_a,C4: set_a,B2: set_nat_a] :
( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : C4 ) )
=> ( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ B2
@ ^ [Uu: nat > a] : C4 ) )
=> ( member_nat_a_a @ F
@ ( pi_nat_a_a @ ( inf_inf_set_nat_a @ A2 @ B2 )
@ ^ [Uu: nat > a] : C4 ) ) ) ) ).
% funcset_Int_left
thf(fact_1010_funcset__Int__left,axiom,
! [F: a > a,A2: set_a,C4: set_a,B2: set_a] :
( ( member_a_a @ F
@ ( pi_a_a @ A2
@ ^ [Uu: a] : C4 ) )
=> ( ( member_a_a @ F
@ ( pi_a_a @ B2
@ ^ [Uu: a] : C4 ) )
=> ( member_a_a @ F
@ ( pi_a_a @ ( inf_inf_set_a @ A2 @ B2 )
@ ^ [Uu: a] : C4 ) ) ) ) ).
% funcset_Int_left
thf(fact_1011_funcset__Un__left,axiom,
! [F: nat > a,A2: set_nat,B2: set_nat,C4: set_a] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( sup_sup_set_nat @ A2 @ B2 )
@ ^ [Uu: nat] : C4 ) )
= ( ( member_nat_a @ F
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : C4 ) )
& ( member_nat_a @ F
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : C4 ) ) ) ) ).
% funcset_Un_left
thf(fact_1012_funcset__Un__left,axiom,
! [F: ( nat > a ) > a,A2: set_nat_a,B2: set_nat_a,C4: set_a] :
( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ ( sup_sup_set_nat_a @ A2 @ B2 )
@ ^ [Uu: nat > a] : C4 ) )
= ( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : C4 ) )
& ( member_nat_a_a @ F
@ ( pi_nat_a_a @ B2
@ ^ [Uu: nat > a] : C4 ) ) ) ) ).
% funcset_Un_left
thf(fact_1013_funcset__Un__left,axiom,
! [F: a > a,A2: set_a,B2: set_a,C4: set_a] :
( ( member_a_a @ F
@ ( pi_a_a @ ( sup_sup_set_a @ A2 @ B2 )
@ ^ [Uu: a] : C4 ) )
= ( ( member_a_a @ F
@ ( pi_a_a @ A2
@ ^ [Uu: a] : C4 ) )
& ( member_a_a @ F
@ ( pi_a_a @ B2
@ ^ [Uu: a] : C4 ) ) ) ) ).
% funcset_Un_left
thf(fact_1014_fincomp__Un__Int,axiom,
! [A2: set_nat,B2: set_nat,G: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A2
@ ^ [Uu: nat] : m ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : m ) )
=> ( ( composition @ ( commut6741328216151336360_a_nat @ m @ composition @ unit @ G @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ ( commut6741328216151336360_a_nat @ m @ composition @ unit @ G @ ( inf_inf_set_nat @ A2 @ B2 ) ) )
= ( composition @ ( commut6741328216151336360_a_nat @ m @ composition @ unit @ G @ A2 ) @ ( commut6741328216151336360_a_nat @ m @ composition @ unit @ G @ B2 ) ) ) ) ) ) ) ).
% fincomp_Un_Int
thf(fact_1015_fincomp__Un__Int,axiom,
! [A2: set_nat_a,B2: set_nat_a,G: ( nat > a ) > a] :
( ( finite_finite_nat_a @ A2 )
=> ( ( finite_finite_nat_a @ B2 )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ A2
@ ^ [Uu: nat > a] : m ) )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ B2
@ ^ [Uu: nat > a] : m ) )
=> ( ( composition @ ( commut5242989786243415821_nat_a @ m @ composition @ unit @ G @ ( sup_sup_set_nat_a @ A2 @ B2 ) ) @ ( commut5242989786243415821_nat_a @ m @ composition @ unit @ G @ ( inf_inf_set_nat_a @ A2 @ B2 ) ) )
= ( composition @ ( commut5242989786243415821_nat_a @ m @ composition @ unit @ G @ A2 ) @ ( commut5242989786243415821_nat_a @ m @ composition @ unit @ G @ B2 ) ) ) ) ) ) ) ).
% fincomp_Un_Int
thf(fact_1016_fincomp__Un__Int,axiom,
! [A2: set_a,B2: set_a,G: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A2
@ ^ [Uu: a] : m ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : m ) )
=> ( ( composition @ ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ G @ ( sup_sup_set_a @ A2 @ B2 ) ) @ ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ G @ ( inf_inf_set_a @ A2 @ B2 ) ) )
= ( composition @ ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ G @ A2 ) @ ( commut5005951359559292710mp_a_a @ m @ composition @ unit @ G @ B2 ) ) ) ) ) ) ) ).
% fincomp_Un_Int
thf(fact_1017_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B2 )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B2 )
& ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1018_insert__disjoint_I1_J,axiom,
! [A: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( ( inf_inf_set_nat_a @ ( insert_nat_a @ A @ A2 ) @ B2 )
= bot_bot_set_nat_a )
= ( ~ ( member_nat_a @ A @ B2 )
& ( ( inf_inf_set_nat_a @ A2 @ B2 )
= bot_bot_set_nat_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1019_insert__disjoint_I1_J,axiom,
! [A: ( nat > a ) > a,A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ( ( inf_inf_set_nat_a_a @ ( insert_nat_a_a @ A @ A2 ) @ B2 )
= bot_bot_set_nat_a_a )
= ( ~ ( member_nat_a_a @ A @ B2 )
& ( ( inf_inf_set_nat_a_a @ A2 @ B2 )
= bot_bot_set_nat_a_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1020_insert__disjoint_I1_J,axiom,
! [A: a > a,A2: set_a_a,B2: set_a_a] :
( ( ( inf_inf_set_a_a @ ( insert_a_a @ A @ A2 ) @ B2 )
= bot_bot_set_a_a )
= ( ~ ( member_a_a @ A @ B2 )
& ( ( inf_inf_set_a_a @ A2 @ B2 )
= bot_bot_set_a_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1021_insert__disjoint_I1_J,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B2 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B2 )
& ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_1022_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B2 ) )
= ( ~ ( member_a @ A @ B2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1023_insert__disjoint_I2_J,axiom,
! [A: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( bot_bot_set_nat_a
= ( inf_inf_set_nat_a @ ( insert_nat_a @ A @ A2 ) @ B2 ) )
= ( ~ ( member_nat_a @ A @ B2 )
& ( bot_bot_set_nat_a
= ( inf_inf_set_nat_a @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1024_insert__disjoint_I2_J,axiom,
! [A: ( nat > a ) > a,A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ( bot_bot_set_nat_a_a
= ( inf_inf_set_nat_a_a @ ( insert_nat_a_a @ A @ A2 ) @ B2 ) )
= ( ~ ( member_nat_a_a @ A @ B2 )
& ( bot_bot_set_nat_a_a
= ( inf_inf_set_nat_a_a @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1025_insert__disjoint_I2_J,axiom,
! [A: a > a,A2: set_a_a,B2: set_a_a] :
( ( bot_bot_set_a_a
= ( inf_inf_set_a_a @ ( insert_a_a @ A @ A2 ) @ B2 ) )
= ( ~ ( member_a_a @ A @ B2 )
& ( bot_bot_set_a_a
= ( inf_inf_set_a_a @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1026_insert__disjoint_I2_J,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B2 ) )
= ( ~ ( member_nat @ A @ B2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1027_disjoint__insert_I1_J,axiom,
! [B2: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B2 @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B2 )
& ( ( inf_inf_set_a @ B2 @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1028_disjoint__insert_I1_J,axiom,
! [B2: set_nat_a,A: nat > a,A2: set_nat_a] :
( ( ( inf_inf_set_nat_a @ B2 @ ( insert_nat_a @ A @ A2 ) )
= bot_bot_set_nat_a )
= ( ~ ( member_nat_a @ A @ B2 )
& ( ( inf_inf_set_nat_a @ B2 @ A2 )
= bot_bot_set_nat_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1029_disjoint__insert_I1_J,axiom,
! [B2: set_nat_a_a2,A: ( nat > a ) > a,A2: set_nat_a_a2] :
( ( ( inf_inf_set_nat_a_a @ B2 @ ( insert_nat_a_a @ A @ A2 ) )
= bot_bot_set_nat_a_a )
= ( ~ ( member_nat_a_a @ A @ B2 )
& ( ( inf_inf_set_nat_a_a @ B2 @ A2 )
= bot_bot_set_nat_a_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1030_disjoint__insert_I1_J,axiom,
! [B2: set_a_a,A: a > a,A2: set_a_a] :
( ( ( inf_inf_set_a_a @ B2 @ ( insert_a_a @ A @ A2 ) )
= bot_bot_set_a_a )
= ( ~ ( member_a_a @ A @ B2 )
& ( ( inf_inf_set_a_a @ B2 @ A2 )
= bot_bot_set_a_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1031_disjoint__insert_I1_J,axiom,
! [B2: set_nat,A: nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ B2 @ ( insert_nat @ A @ A2 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B2 )
& ( ( inf_inf_set_nat @ B2 @ A2 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_1032_disjoint__insert_I2_J,axiom,
! [A2: set_a,B: a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B @ B2 ) ) )
= ( ~ ( member_a @ B @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1033_disjoint__insert_I2_J,axiom,
! [A2: set_nat_a,B: nat > a,B2: set_nat_a] :
( ( bot_bot_set_nat_a
= ( inf_inf_set_nat_a @ A2 @ ( insert_nat_a @ B @ B2 ) ) )
= ( ~ ( member_nat_a @ B @ A2 )
& ( bot_bot_set_nat_a
= ( inf_inf_set_nat_a @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1034_disjoint__insert_I2_J,axiom,
! [A2: set_nat_a_a2,B: ( nat > a ) > a,B2: set_nat_a_a2] :
( ( bot_bot_set_nat_a_a
= ( inf_inf_set_nat_a_a @ A2 @ ( insert_nat_a_a @ B @ B2 ) ) )
= ( ~ ( member_nat_a_a @ B @ A2 )
& ( bot_bot_set_nat_a_a
= ( inf_inf_set_nat_a_a @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1035_disjoint__insert_I2_J,axiom,
! [A2: set_a_a,B: a > a,B2: set_a_a] :
( ( bot_bot_set_a_a
= ( inf_inf_set_a_a @ A2 @ ( insert_a_a @ B @ B2 ) ) )
= ( ~ ( member_a_a @ B @ A2 )
& ( bot_bot_set_a_a
= ( inf_inf_set_a_a @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1036_disjoint__insert_I2_J,axiom,
! [A2: set_nat,B: nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) )
= ( ~ ( member_nat @ B @ A2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1037_Diff__eq__empty__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( minus_minus_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_1038_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_1039_insert__Diff__single,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_1040_finite__Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_1041_Diff__disjoint,axiom,
! [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_1042_Diff__triv,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_1043_Diff__partition,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_1044_Diff__partition,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_1045_Un__Int__assoc__eq,axiom,
! [A2: set_a,B2: set_a,C4: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C4 )
= ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C4 ) ) )
= ( ord_less_eq_set_a @ C4 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_1046_Un__Int__assoc__eq,axiom,
! [A2: set_nat,B2: set_nat,C4: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C4 )
= ( inf_inf_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C4 ) ) )
= ( ord_less_eq_set_nat @ C4 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_1047_Diff__subset__conv,axiom,
! [A2: set_a,B2: set_a,C4: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ C4 )
= ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C4 ) ) ) ).
% Diff_subset_conv
thf(fact_1048_Diff__subset__conv,axiom,
! [A2: set_nat,B2: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ C4 )
= ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C4 ) ) ) ).
% Diff_subset_conv
thf(fact_1049_Int__Diff__disjoint,axiom,
! [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ B2 ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_1050_Pi__Int,axiom,
! [I4: set_nat,E: nat > set_a,F2: nat > set_a] :
( ( inf_inf_set_nat_a @ ( pi_nat_a @ I4 @ E ) @ ( pi_nat_a @ I4 @ F2 ) )
= ( pi_nat_a @ I4
@ ^ [I2: nat] : ( inf_inf_set_a @ ( E @ I2 ) @ ( F2 @ I2 ) ) ) ) ).
% Pi_Int
thf(fact_1051_Pi__Int,axiom,
! [I4: set_nat_a,E: ( nat > a ) > set_a,F2: ( nat > a ) > set_a] :
( ( inf_inf_set_nat_a_a @ ( pi_nat_a_a @ I4 @ E ) @ ( pi_nat_a_a @ I4 @ F2 ) )
= ( pi_nat_a_a @ I4
@ ^ [I2: nat > a] : ( inf_inf_set_a @ ( E @ I2 ) @ ( F2 @ I2 ) ) ) ) ).
% Pi_Int
thf(fact_1052_Pi__Int,axiom,
! [I4: set_a,E: a > set_a,F2: a > set_a] :
( ( inf_inf_set_a_a @ ( pi_a_a @ I4 @ E ) @ ( pi_a_a @ I4 @ F2 ) )
= ( pi_a_a @ I4
@ ^ [I2: a] : ( inf_inf_set_a @ ( E @ I2 ) @ ( F2 @ I2 ) ) ) ) ).
% Pi_Int
thf(fact_1053_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_1054_UnE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) )
=> ( ~ ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_1055_UnE,axiom,
! [C: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( sup_sup_set_nat_a @ A2 @ B2 ) )
=> ( ~ ( member_nat_a @ C @ A2 )
=> ( member_nat_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_1056_UnE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% UnE
thf(fact_1057_UnE,axiom,
! [C: ( nat > a ) > a,A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ( member_nat_a_a @ C @ ( sup_sup_set_nat_a_a @ A2 @ B2 ) )
=> ( ~ ( member_nat_a_a @ C @ A2 )
=> ( member_nat_a_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_1058_UnE,axiom,
! [C: a > a,A2: set_a_a,B2: set_a_a] :
( ( member_a_a @ C @ ( sup_sup_set_a_a @ A2 @ B2 ) )
=> ( ~ ( member_a_a @ C @ A2 )
=> ( member_a_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_1059_IntE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ~ ( member_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_1060_IntE,axiom,
! [C: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( inf_inf_set_nat_a @ A2 @ B2 ) )
=> ~ ( ( member_nat_a @ C @ A2 )
=> ~ ( member_nat_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_1061_IntE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ~ ( member_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_1062_IntE,axiom,
! [C: ( nat > a ) > a,A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ( member_nat_a_a @ C @ ( inf_inf_set_nat_a_a @ A2 @ B2 ) )
=> ~ ( ( member_nat_a_a @ C @ A2 )
=> ~ ( member_nat_a_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_1063_IntE,axiom,
! [C: a > a,A2: set_a_a,B2: set_a_a] :
( ( member_a_a @ C @ ( inf_inf_set_a_a @ A2 @ B2 ) )
=> ~ ( ( member_a_a @ C @ A2 )
=> ~ ( member_a_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_1064_UnI1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_1065_UnI1,axiom,
! [C: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ A2 )
=> ( member_nat_a @ C @ ( sup_sup_set_nat_a @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_1066_UnI1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_1067_UnI1,axiom,
! [C: ( nat > a ) > a,A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ( member_nat_a_a @ C @ A2 )
=> ( member_nat_a_a @ C @ ( sup_sup_set_nat_a_a @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_1068_UnI1,axiom,
! [C: a > a,A2: set_a_a,B2: set_a_a] :
( ( member_a_a @ C @ A2 )
=> ( member_a_a @ C @ ( sup_sup_set_a_a @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_1069_UnI2,axiom,
! [C: a,B2: set_a,A2: set_a] :
( ( member_a @ C @ B2 )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_1070_UnI2,axiom,
! [C: nat > a,B2: set_nat_a,A2: set_nat_a] :
( ( member_nat_a @ C @ B2 )
=> ( member_nat_a @ C @ ( sup_sup_set_nat_a @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_1071_UnI2,axiom,
! [C: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_1072_UnI2,axiom,
! [C: ( nat > a ) > a,B2: set_nat_a_a2,A2: set_nat_a_a2] :
( ( member_nat_a_a @ C @ B2 )
=> ( member_nat_a_a @ C @ ( sup_sup_set_nat_a_a @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_1073_UnI2,axiom,
! [C: a > a,B2: set_a_a,A2: set_a_a] :
( ( member_a_a @ C @ B2 )
=> ( member_a_a @ C @ ( sup_sup_set_a_a @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_1074_DiffE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1075_DiffE,axiom,
! [C: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) )
=> ~ ( ( member_nat_a @ C @ A2 )
=> ( member_nat_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1076_DiffE,axiom,
! [C: ( nat > a ) > a,A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ( member_nat_a_a @ C @ ( minus_1482667089342205261at_a_a @ A2 @ B2 ) )
=> ~ ( ( member_nat_a_a @ C @ A2 )
=> ( member_nat_a_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1077_DiffE,axiom,
! [C: a > a,A2: set_a_a,B2: set_a_a] :
( ( member_a_a @ C @ ( minus_minus_set_a_a @ A2 @ B2 ) )
=> ~ ( ( member_a_a @ C @ A2 )
=> ( member_a_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1078_DiffE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1079_IntD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ A2 ) ) ).
% IntD1
thf(fact_1080_IntD1,axiom,
! [C: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( inf_inf_set_nat_a @ A2 @ B2 ) )
=> ( member_nat_a @ C @ A2 ) ) ).
% IntD1
thf(fact_1081_IntD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_1082_IntD1,axiom,
! [C: ( nat > a ) > a,A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ( member_nat_a_a @ C @ ( inf_inf_set_nat_a_a @ A2 @ B2 ) )
=> ( member_nat_a_a @ C @ A2 ) ) ).
% IntD1
thf(fact_1083_IntD1,axiom,
! [C: a > a,A2: set_a_a,B2: set_a_a] :
( ( member_a_a @ C @ ( inf_inf_set_a_a @ A2 @ B2 ) )
=> ( member_a_a @ C @ A2 ) ) ).
% IntD1
thf(fact_1084_IntD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ B2 ) ) ).
% IntD2
thf(fact_1085_IntD2,axiom,
! [C: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( inf_inf_set_nat_a @ A2 @ B2 ) )
=> ( member_nat_a @ C @ B2 ) ) ).
% IntD2
thf(fact_1086_IntD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_1087_IntD2,axiom,
! [C: ( nat > a ) > a,A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ( member_nat_a_a @ C @ ( inf_inf_set_nat_a_a @ A2 @ B2 ) )
=> ( member_nat_a_a @ C @ B2 ) ) ).
% IntD2
thf(fact_1088_IntD2,axiom,
! [C: a > a,A2: set_a_a,B2: set_a_a] :
( ( member_a_a @ C @ ( inf_inf_set_a_a @ A2 @ B2 ) )
=> ( member_a_a @ C @ B2 ) ) ).
% IntD2
thf(fact_1089_DiffD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_1090_DiffD1,axiom,
! [C: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) )
=> ( member_nat_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_1091_DiffD1,axiom,
! [C: ( nat > a ) > a,A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ( member_nat_a_a @ C @ ( minus_1482667089342205261at_a_a @ A2 @ B2 ) )
=> ( member_nat_a_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_1092_DiffD1,axiom,
! [C: a > a,A2: set_a_a,B2: set_a_a] :
( ( member_a_a @ C @ ( minus_minus_set_a_a @ A2 @ B2 ) )
=> ( member_a_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_1093_DiffD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_1094_DiffD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( member_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_1095_DiffD2,axiom,
! [C: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) )
=> ~ ( member_nat_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_1096_DiffD2,axiom,
! [C: ( nat > a ) > a,A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ( member_nat_a_a @ C @ ( minus_1482667089342205261at_a_a @ A2 @ B2 ) )
=> ~ ( member_nat_a_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_1097_DiffD2,axiom,
! [C: a > a,A2: set_a_a,B2: set_a_a] :
( ( member_a_a @ C @ ( minus_minus_set_a_a @ A2 @ B2 ) )
=> ~ ( member_a_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_1098_DiffD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( member_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_1099_Un__def,axiom,
( sup_sup_set_nat_a
= ( ^ [A6: set_nat_a,B5: set_nat_a] :
( collect_nat_a
@ ^ [X3: nat > a] :
( ( member_nat_a @ X3 @ A6 )
| ( member_nat_a @ X3 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_1100_Un__def,axiom,
( sup_sup_set_nat_a_a
= ( ^ [A6: set_nat_a_a2,B5: set_nat_a_a2] :
( collect_nat_a_a
@ ^ [X3: ( nat > a ) > a] :
( ( member_nat_a_a @ X3 @ A6 )
| ( member_nat_a_a @ X3 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_1101_Un__def,axiom,
( sup_sup_set_a_a
= ( ^ [A6: set_a_a,B5: set_a_a] :
( collect_a_a
@ ^ [X3: a > a] :
( ( member_a_a @ X3 @ A6 )
| ( member_a_a @ X3 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_1102_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B5: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A6 )
| ( member_a @ X3 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_1103_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A6: set_nat,B5: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A6 )
| ( member_nat @ X3 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_1104_Diff__Un,axiom,
! [A2: set_nat,B2: set_nat,C4: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C4 ) )
= ( inf_inf_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ C4 ) ) ) ).
% Diff_Un
thf(fact_1105_Int__def,axiom,
( inf_inf_set_nat_a
= ( ^ [A6: set_nat_a,B5: set_nat_a] :
( collect_nat_a
@ ^ [X3: nat > a] :
( ( member_nat_a @ X3 @ A6 )
& ( member_nat_a @ X3 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_1106_Int__def,axiom,
( inf_inf_set_nat_a_a
= ( ^ [A6: set_nat_a_a2,B5: set_nat_a_a2] :
( collect_nat_a_a
@ ^ [X3: ( nat > a ) > a] :
( ( member_nat_a_a @ X3 @ A6 )
& ( member_nat_a_a @ X3 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_1107_Int__def,axiom,
( inf_inf_set_a_a
= ( ^ [A6: set_a_a,B5: set_a_a] :
( collect_a_a
@ ^ [X3: a > a] :
( ( member_a_a @ X3 @ A6 )
& ( member_a_a @ X3 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_1108_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B5: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A6 )
& ( member_a @ X3 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_1109_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A6: set_nat,B5: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A6 )
& ( member_nat @ X3 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_1110_Un__Diff,axiom,
! [A2: set_nat,B2: set_nat,C4: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C4 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ C4 ) @ ( minus_minus_set_nat @ B2 @ C4 ) ) ) ).
% Un_Diff
thf(fact_1111_Diff__Int,axiom,
! [A2: set_nat,B2: set_nat,C4: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C4 ) )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ C4 ) ) ) ).
% Diff_Int
thf(fact_1112_Int__Diff,axiom,
! [A2: set_nat,B2: set_nat,C4: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C4 )
= ( inf_inf_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C4 ) ) ) ).
% Int_Diff
thf(fact_1113_Diff__Int2,axiom,
! [A2: set_nat,C4: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C4 ) @ ( inf_inf_set_nat @ B2 @ C4 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C4 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_1114_Int__Collect,axiom,
! [X: nat > a,A2: set_nat_a,P: ( nat > a ) > $o] :
( ( member_nat_a @ X @ ( inf_inf_set_nat_a @ A2 @ ( collect_nat_a @ P ) ) )
= ( ( member_nat_a @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1115_Int__Collect,axiom,
! [X: ( nat > a ) > a,A2: set_nat_a_a2,P: ( ( nat > a ) > a ) > $o] :
( ( member_nat_a_a @ X @ ( inf_inf_set_nat_a_a @ A2 @ ( collect_nat_a_a @ P ) ) )
= ( ( member_nat_a_a @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1116_Int__Collect,axiom,
! [X: a > a,A2: set_a_a,P: ( a > a ) > $o] :
( ( member_a_a @ X @ ( inf_inf_set_a_a @ A2 @ ( collect_a_a @ P ) ) )
= ( ( member_a_a @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1117_Int__Collect,axiom,
! [X: a,A2: set_a,P: a > $o] :
( ( member_a @ X @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) )
= ( ( member_a @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1118_Int__Collect,axiom,
! [X: nat,A2: set_nat,P: nat > $o] :
( ( member_nat @ X @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_1119_Int__Diff__Un,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ B2 ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_1120_Un__Diff__Int,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ A2 @ B2 ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_1121_set__diff__eq,axiom,
( minus_490503922182417452_nat_a
= ( ^ [A6: set_nat_a,B5: set_nat_a] :
( collect_nat_a
@ ^ [X3: nat > a] :
( ( member_nat_a @ X3 @ A6 )
& ~ ( member_nat_a @ X3 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1122_set__diff__eq,axiom,
( minus_1482667089342205261at_a_a
= ( ^ [A6: set_nat_a_a2,B5: set_nat_a_a2] :
( collect_nat_a_a
@ ^ [X3: ( nat > a ) > a] :
( ( member_nat_a_a @ X3 @ A6 )
& ~ ( member_nat_a_a @ X3 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1123_set__diff__eq,axiom,
( minus_minus_set_a_a
= ( ^ [A6: set_a_a,B5: set_a_a] :
( collect_a_a
@ ^ [X3: a > a] :
( ( member_a_a @ X3 @ A6 )
& ~ ( member_a_a @ X3 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1124_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A6: set_a,B5: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A6 )
& ~ ( member_a @ X3 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1125_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A6: set_nat,B5: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A6 )
& ~ ( member_nat @ X3 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1126_Diff__Diff__Int,axiom,
! [A2: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_1127_Collect__conj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_1128_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_1129_Collect__disj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
| ( Q @ X3 ) ) )
= ( sup_sup_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_1130_Collect__disj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_1131_Diff__Int__distrib,axiom,
! [C4: set_nat,A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ C4 @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ C4 @ A2 ) @ ( inf_inf_set_nat @ C4 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_1132_Diff__Int__distrib2,axiom,
! [A2: set_nat,B2: set_nat,C4: set_nat] :
( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ C4 )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C4 ) @ ( inf_inf_set_nat @ B2 @ C4 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1133_Un__empty__left,axiom,
! [B2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_1134_Un__empty__right,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Un_empty_right
thf(fact_1135_finite__UnI,axiom,
! [F2: set_nat,G2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( finite_finite_nat @ G2 )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_1136_Un__infinite,axiom,
! [S: set_nat,T2: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_1137_infinite__Un,axiom,
! [S: set_nat,T2: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_1138_Int__emptyI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ~ ( member_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_1139_Int__emptyI,axiom,
! [A2: set_nat_a,B2: set_nat_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ~ ( member_nat_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_nat_a @ A2 @ B2 )
= bot_bot_set_nat_a ) ) ).
% Int_emptyI
thf(fact_1140_Int__emptyI,axiom,
! [A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ! [X2: ( nat > a ) > a] :
( ( member_nat_a_a @ X2 @ A2 )
=> ~ ( member_nat_a_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_nat_a_a @ A2 @ B2 )
= bot_bot_set_nat_a_a ) ) ).
% Int_emptyI
thf(fact_1141_Int__emptyI,axiom,
! [A2: set_a_a,B2: set_a_a] :
( ! [X2: a > a] :
( ( member_a_a @ X2 @ A2 )
=> ~ ( member_a_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_a_a @ A2 @ B2 )
= bot_bot_set_a_a ) ) ).
% Int_emptyI
thf(fact_1142_Int__emptyI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ( member_nat @ X2 @ B2 ) )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_1143_disjoint__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1144_disjoint__iff,axiom,
! [A2: set_nat_a,B2: set_nat_a] :
( ( ( inf_inf_set_nat_a @ A2 @ B2 )
= bot_bot_set_nat_a )
= ( ! [X3: nat > a] :
( ( member_nat_a @ X3 @ A2 )
=> ~ ( member_nat_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1145_disjoint__iff,axiom,
! [A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ( ( inf_inf_set_nat_a_a @ A2 @ B2 )
= bot_bot_set_nat_a_a )
= ( ! [X3: ( nat > a ) > a] :
( ( member_nat_a_a @ X3 @ A2 )
=> ~ ( member_nat_a_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1146_disjoint__iff,axiom,
! [A2: set_a_a,B2: set_a_a] :
( ( ( inf_inf_set_a_a @ A2 @ B2 )
= bot_bot_set_a_a )
= ( ! [X3: a > a] :
( ( member_a_a @ X3 @ A2 )
=> ~ ( member_a_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1147_disjoint__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ~ ( member_nat @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1148_Int__empty__left,axiom,
! [B2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B2 )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_1149_Int__empty__right,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_1150_disjoint__iff__not__equal,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ B2 )
=> ( X3 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1151_Un__mono,axiom,
! [A2: set_a,C4: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C4 )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C4 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_1152_Un__mono,axiom,
! [A2: set_nat,C4: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C4 )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ C4 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_1153_Un__least,axiom,
! [A2: set_a,C4: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C4 )
=> ( ( ord_less_eq_set_a @ B2 @ C4 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C4 ) ) ) ).
% Un_least
thf(fact_1154_Un__least,axiom,
! [A2: set_nat,C4: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C4 )
=> ( ( ord_less_eq_set_nat @ B2 @ C4 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C4 ) ) ) ).
% Un_least
thf(fact_1155_Un__upper1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_1156_Un__upper1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_1157_Un__upper2,axiom,
! [B2: set_a,A2: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_1158_Un__upper2,axiom,
! [B2: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_1159_Un__absorb1,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_1160_Un__absorb1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_1161_Un__absorb2,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_1162_Un__absorb2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_1163_subset__UnE,axiom,
! [C4: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C4 @ ( sup_sup_set_a @ A2 @ B2 ) )
=> ~ ! [A8: set_a] :
( ( ord_less_eq_set_a @ A8 @ A2 )
=> ! [B7: set_a] :
( ( ord_less_eq_set_a @ B7 @ B2 )
=> ( C4
!= ( sup_sup_set_a @ A8 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_1164_subset__UnE,axiom,
! [C4: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ~ ! [A8: set_nat] :
( ( ord_less_eq_set_nat @ A8 @ A2 )
=> ! [B7: set_nat] :
( ( ord_less_eq_set_nat @ B7 @ B2 )
=> ( C4
!= ( sup_sup_set_nat @ A8 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_1165_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
( ( sup_sup_set_a @ A6 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_1166_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B5: set_nat] :
( ( sup_sup_set_nat @ A6 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_1167_Int__Collect__mono,axiom,
! [A2: set_nat_a,B2: set_nat_a,P: ( nat > a ) > $o,Q: ( nat > a ) > $o] :
( ( ord_le871467723717165285_nat_a @ A2 @ B2 )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le871467723717165285_nat_a @ ( inf_inf_set_nat_a @ A2 @ ( collect_nat_a @ P ) ) @ ( inf_inf_set_nat_a @ B2 @ ( collect_nat_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1168_Int__Collect__mono,axiom,
! [A2: set_nat_a_a2,B2: set_nat_a_a2,P: ( ( nat > a ) > a ) > $o,Q: ( ( nat > a ) > a ) > $o] :
( ( ord_le3509452538356653652at_a_a @ A2 @ B2 )
=> ( ! [X2: ( nat > a ) > a] :
( ( member_nat_a_a @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le3509452538356653652at_a_a @ ( inf_inf_set_nat_a_a @ A2 @ ( collect_nat_a_a @ P ) ) @ ( inf_inf_set_nat_a_a @ B2 @ ( collect_nat_a_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1169_Int__Collect__mono,axiom,
! [A2: set_a_a,B2: set_a_a,P: ( a > a ) > $o,Q: ( a > a ) > $o] :
( ( ord_less_eq_set_a_a @ A2 @ B2 )
=> ( ! [X2: a > a] :
( ( member_a_a @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_a_a @ ( inf_inf_set_a_a @ A2 @ ( collect_a_a @ P ) ) @ ( inf_inf_set_a_a @ B2 @ ( collect_a_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1170_Int__Collect__mono,axiom,
! [A2: set_a,B2: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B2 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1171_Int__Collect__mono,axiom,
! [A2: set_nat,B2: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B2 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1172_Int__greatest,axiom,
! [C4: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C4 @ A2 )
=> ( ( ord_less_eq_set_a @ C4 @ B2 )
=> ( ord_less_eq_set_a @ C4 @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_1173_Int__greatest,axiom,
! [C4: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
=> ( ( ord_less_eq_set_nat @ C4 @ B2 )
=> ( ord_less_eq_set_nat @ C4 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_1174_Int__absorb2,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_1175_Int__absorb2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_1176_Int__absorb1,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_1177_Int__absorb1,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_1178_Int__lower2,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_1179_Int__lower2,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_1180_Int__lower1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_1181_Int__lower1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_1182_Int__mono,axiom,
! [A2: set_a,C4: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C4 )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C4 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1183_Int__mono,axiom,
! [A2: set_nat,C4: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C4 )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ C4 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1184_Diff__infinite__finite,axiom,
! [T2: set_nat,S: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1185_Int__insert__right,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B2 ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_1186_Int__insert__right,axiom,
! [A: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( ( member_nat_a @ A @ A2 )
=> ( ( inf_inf_set_nat_a @ A2 @ ( insert_nat_a @ A @ B2 ) )
= ( insert_nat_a @ A @ ( inf_inf_set_nat_a @ A2 @ B2 ) ) ) )
& ( ~ ( member_nat_a @ A @ A2 )
=> ( ( inf_inf_set_nat_a @ A2 @ ( insert_nat_a @ A @ B2 ) )
= ( inf_inf_set_nat_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_1187_Int__insert__right,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_1188_Int__insert__right,axiom,
! [A: ( nat > a ) > a,A2: set_nat_a_a2,B2: set_nat_a_a2] :
( ( ( member_nat_a_a @ A @ A2 )
=> ( ( inf_inf_set_nat_a_a @ A2 @ ( insert_nat_a_a @ A @ B2 ) )
= ( insert_nat_a_a @ A @ ( inf_inf_set_nat_a_a @ A2 @ B2 ) ) ) )
& ( ~ ( member_nat_a_a @ A @ A2 )
=> ( ( inf_inf_set_nat_a_a @ A2 @ ( insert_nat_a_a @ A @ B2 ) )
= ( inf_inf_set_nat_a_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_1189_Int__insert__right,axiom,
! [A: a > a,A2: set_a_a,B2: set_a_a] :
( ( ( member_a_a @ A @ A2 )
=> ( ( inf_inf_set_a_a @ A2 @ ( insert_a_a @ A @ B2 ) )
= ( insert_a_a @ A @ ( inf_inf_set_a_a @ A2 @ B2 ) ) ) )
& ( ~ ( member_a_a @ A @ A2 )
=> ( ( inf_inf_set_a_a @ A2 @ ( insert_a_a @ A @ B2 ) )
= ( inf_inf_set_a_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_1190_Int__insert__left,axiom,
! [A: a,C4: set_a,B2: set_a] :
( ( ( member_a @ A @ C4 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C4 )
= ( insert_a @ A @ ( inf_inf_set_a @ B2 @ C4 ) ) ) )
& ( ~ ( member_a @ A @ C4 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C4 )
= ( inf_inf_set_a @ B2 @ C4 ) ) ) ) ).
% Int_insert_left
thf(fact_1191_Int__insert__left,axiom,
! [A: nat > a,C4: set_nat_a,B2: set_nat_a] :
( ( ( member_nat_a @ A @ C4 )
=> ( ( inf_inf_set_nat_a @ ( insert_nat_a @ A @ B2 ) @ C4 )
= ( insert_nat_a @ A @ ( inf_inf_set_nat_a @ B2 @ C4 ) ) ) )
& ( ~ ( member_nat_a @ A @ C4 )
=> ( ( inf_inf_set_nat_a @ ( insert_nat_a @ A @ B2 ) @ C4 )
= ( inf_inf_set_nat_a @ B2 @ C4 ) ) ) ) ).
% Int_insert_left
thf(fact_1192_Int__insert__left,axiom,
! [A: nat,C4: set_nat,B2: set_nat] :
( ( ( member_nat @ A @ C4 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C4 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B2 @ C4 ) ) ) )
& ( ~ ( member_nat @ A @ C4 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C4 )
= ( inf_inf_set_nat @ B2 @ C4 ) ) ) ) ).
% Int_insert_left
thf(fact_1193_Int__insert__left,axiom,
! [A: ( nat > a ) > a,C4: set_nat_a_a2,B2: set_nat_a_a2] :
( ( ( member_nat_a_a @ A @ C4 )
=> ( ( inf_inf_set_nat_a_a @ ( insert_nat_a_a @ A @ B2 ) @ C4 )
= ( insert_nat_a_a @ A @ ( inf_inf_set_nat_a_a @ B2 @ C4 ) ) ) )
& ( ~ ( member_nat_a_a @ A @ C4 )
=> ( ( inf_inf_set_nat_a_a @ ( insert_nat_a_a @ A @ B2 ) @ C4 )
= ( inf_inf_set_nat_a_a @ B2 @ C4 ) ) ) ) ).
% Int_insert_left
thf(fact_1194_Int__insert__left,axiom,
! [A: a > a,C4: set_a_a,B2: set_a_a] :
( ( ( member_a_a @ A @ C4 )
=> ( ( inf_inf_set_a_a @ ( insert_a_a @ A @ B2 ) @ C4 )
= ( insert_a_a @ A @ ( inf_inf_set_a_a @ B2 @ C4 ) ) ) )
& ( ~ ( member_a_a @ A @ C4 )
=> ( ( inf_inf_set_a_a @ ( insert_a_a @ A @ B2 ) @ C4 )
= ( inf_inf_set_a_a @ B2 @ C4 ) ) ) ) ).
% Int_insert_left
thf(fact_1195_double__diff,axiom,
! [A2: set_a,B2: set_a,C4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C4 )
=> ( ( minus_minus_set_a @ B2 @ ( minus_minus_set_a @ C4 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_1196_double__diff,axiom,
! [A2: set_nat,B2: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C4 )
=> ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C4 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_1197_Diff__subset,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_1198_Diff__subset,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_1199_Diff__mono,axiom,
! [A2: set_a,C4: set_a,D2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C4 )
=> ( ( ord_less_eq_set_a @ D2 @ B2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ C4 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1200_Diff__mono,axiom,
! [A2: set_nat,C4: set_nat,D2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C4 )
=> ( ( ord_less_eq_set_nat @ D2 @ B2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ C4 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1201_insert__Diff__if,axiom,
! [X: a,B2: set_a,A2: set_a] :
( ( ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) )
& ( ~ ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1202_insert__Diff__if,axiom,
! [X: nat > a,B2: set_nat_a,A2: set_nat_a] :
( ( ( member_nat_a @ X @ B2 )
=> ( ( minus_490503922182417452_nat_a @ ( insert_nat_a @ X @ A2 ) @ B2 )
= ( minus_490503922182417452_nat_a @ A2 @ B2 ) ) )
& ( ~ ( member_nat_a @ X @ B2 )
=> ( ( minus_490503922182417452_nat_a @ ( insert_nat_a @ X @ A2 ) @ B2 )
= ( insert_nat_a @ X @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1203_insert__Diff__if,axiom,
! [X: ( nat > a ) > a,B2: set_nat_a_a2,A2: set_nat_a_a2] :
( ( ( member_nat_a_a @ X @ B2 )
=> ( ( minus_1482667089342205261at_a_a @ ( insert_nat_a_a @ X @ A2 ) @ B2 )
= ( minus_1482667089342205261at_a_a @ A2 @ B2 ) ) )
& ( ~ ( member_nat_a_a @ X @ B2 )
=> ( ( minus_1482667089342205261at_a_a @ ( insert_nat_a_a @ X @ A2 ) @ B2 )
= ( insert_nat_a_a @ X @ ( minus_1482667089342205261at_a_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1204_insert__Diff__if,axiom,
! [X: a > a,B2: set_a_a,A2: set_a_a] :
( ( ( member_a_a @ X @ B2 )
=> ( ( minus_minus_set_a_a @ ( insert_a_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_a_a @ A2 @ B2 ) ) )
& ( ~ ( member_a_a @ X @ B2 )
=> ( ( minus_minus_set_a_a @ ( insert_a_a @ X @ A2 ) @ B2 )
= ( insert_a_a @ X @ ( minus_minus_set_a_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1205_insert__Diff__if,axiom,
! [X: nat,B2: set_nat,A2: set_nat] :
( ( ( member_nat @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) )
& ( ~ ( member_nat @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1206_insert__def,axiom,
( insert_a
= ( ^ [A3: a] :
( sup_sup_set_a
@ ( collect_a
@ ^ [X3: a] : ( X3 = A3 ) ) ) ) ) ).
% insert_def
thf(fact_1207_insert__def,axiom,
( insert_nat
= ( ^ [A3: nat] :
( sup_sup_set_nat
@ ( collect_nat
@ ^ [X3: nat] : ( X3 = A3 ) ) ) ) ) ).
% insert_def
thf(fact_1208_ivl__disj__un__two__touch_I4_J,axiom,
! [L: nat,M3: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ U )
=> ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M3 ) @ ( set_or1269000886237332187st_nat @ M3 @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_1209_Suc__diff__diff,axiom,
! [M3: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M3 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M3 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1210_diff__Suc__Suc,axiom,
! [M3: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M3 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M3 @ N ) ) ).
% diff_Suc_Suc
thf(fact_1211_diff__self__eq__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ M3 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1212_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1213_diff__diff__cancel,axiom,
! [I3: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_1214_diff__is__0__eq_H,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1215_diff__is__0__eq,axiom,
! [M3: nat,N: nat] :
( ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% diff_is_0_eq
thf(fact_1216_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I3: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I3 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1217_diffs0__imp__equal,axiom,
! [M3: nat,N: nat] :
( ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M3 )
= zero_zero_nat )
=> ( M3 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1218_minus__nat_Odiff__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% minus_nat.diff_0
thf(fact_1219_eq__diff__iff,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M3 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M3 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1220_le__diff__iff,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1221_Nat_Odiff__diff__eq,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M3 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1222_diff__le__mono,axiom,
! [M3: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1223_diff__le__self,axiom,
! [M3: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N ) @ M3 ) ).
% diff_le_self
thf(fact_1224_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1225_diff__le__mono2,axiom,
! [M3: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_1226_Suc__diff__le,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_nat @ N @ M3 )
=> ( ( minus_minus_nat @ ( suc @ M3 ) @ N )
= ( suc @ ( minus_minus_nat @ M3 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1227_diff__commute,axiom,
! [I3: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K ) @ J ) ) ).
% diff_commute
thf(fact_1228_greaterThan__Suc,axiom,
! [K: nat] :
( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
= ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% greaterThan_Suc
thf(fact_1229_inverse__def,axiom,
( ( group_inverse_a @ m @ composition @ unit )
= ( restrict_a_a
@ ^ [U2: a] :
( the_a
@ ^ [V4: a] :
( ( member_a @ V4 @ m )
& ( ( composition @ U2 @ V4 )
= unit )
& ( ( composition @ V4 @ U2 )
= unit ) ) )
@ m ) ) ).
% inverse_def
thf(fact_1230_add__Suc__right,axiom,
! [M3: nat,N: nat] :
( ( plus_plus_nat @ M3 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M3 @ N ) ) ) ).
% add_Suc_right
thf(fact_1231_add__is__0,axiom,
! [M3: nat,N: nat] :
( ( ( plus_plus_nat @ M3 @ N )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1232_Nat_Oadd__0__right,axiom,
! [M3: nat] :
( ( plus_plus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% Nat.add_0_right
thf(fact_1233_nat__add__left__cancel__le,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1234_diff__diff__left,axiom,
! [I3: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J ) @ K )
= ( minus_minus_nat @ I3 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1235_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1236_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I3 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1237_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1238_image__Suc__atLeastAtMost,axiom,
! [I3: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I3 @ J ) )
= ( set_or1269000886237332187st_nat @ ( suc @ I3 ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_1239_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I3 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1240_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I3 )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I3 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1241_add__is__1,axiom,
! [M3: nat,N: nat] :
( ( ( plus_plus_nat @ M3 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M3 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1242_one__is__add,axiom,
! [M3: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M3 @ N ) )
= ( ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M3 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1243_diff__add__0,axiom,
! [N: nat,M3: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M3 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1244_Nat_Ole__imp__diff__is__add,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ( minus_minus_nat @ J @ I3 )
= K )
= ( J
= ( plus_plus_nat @ K @ I3 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1245_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I3 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I3 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1246_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J ) @ K )
= ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1247_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1248_le__diff__conv,axiom,
! [J: nat,K: nat,I3: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I3 @ K ) ) ) ).
% le_diff_conv
thf(fact_1249_Nat_Odiff__cancel,axiom,
! [K: nat,M3: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M3 @ N ) ) ).
% Nat.diff_cancel
thf(fact_1250_diff__cancel2,axiom,
! [M3: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M3 @ N ) ) ).
% diff_cancel2
thf(fact_1251_diff__add__inverse,axiom,
! [N: nat,M3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M3 ) @ N )
= M3 ) ).
% diff_add_inverse
thf(fact_1252_diff__add__inverse2,axiom,
! [M3: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ N ) @ N )
= M3 ) ).
% diff_add_inverse2
thf(fact_1253_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1254_trans__le__add2,axiom,
! [I3: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_le_add2
thf(fact_1255_trans__le__add1,axiom,
! [I3: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_le_add1
thf(fact_1256_add__le__mono1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1257_add__le__mono,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1258_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1259_add__leD2,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1260_add__leD1,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% add_leD1
thf(fact_1261_le__add2,axiom,
! [N: nat,M3: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M3 @ N ) ) ).
% le_add2
thf(fact_1262_le__add1,axiom,
! [N: nat,M3: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M3 ) ) ).
% le_add1
thf(fact_1263_add__leE,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M3 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1264_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1265_add__eq__self__zero,axiom,
! [M3: nat,N: nat] :
( ( ( plus_plus_nat @ M3 @ N )
= M3 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1266_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1267_add__Suc,axiom,
! [M3: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M3 ) @ N )
= ( suc @ ( plus_plus_nat @ M3 @ N ) ) ) ).
% add_Suc
thf(fact_1268_add__Suc__shift,axiom,
! [M3: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M3 ) @ N )
= ( plus_plus_nat @ M3 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1269_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_1270_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_1271_atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_atMost_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_1272_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
% Conjectures (1)
thf(conj_0,conjecture,
( ( commut6741328216151336360_a_nat @ m @ composition @ unit @ f @ ( set_ord_atMost_nat @ ( suc @ zero_zero_nat ) ) )
= ( composition
@ ( commut6741328216151336360_a_nat @ m @ composition @ unit
@ ^ [I2: nat] : ( f @ ( suc @ I2 ) )
@ ( set_ord_atMost_nat @ zero_zero_nat ) )
@ ( f @ zero_zero_nat ) ) ) ).
%------------------------------------------------------------------------------