TPTP Problem File: SLH0604^1.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_00066_001949__13295342_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1678 ( 357 unt; 392 typ; 0 def)
% Number of atoms : 4730 (1259 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 16518 ( 546 ~; 33 |; 401 &;12864 @)
% ( 0 <=>;2674 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Number of types : 36 ( 35 usr)
% Number of type conns : 3036 (3036 >; 0 *; 0 +; 0 <<)
% Number of symbols : 360 ( 357 usr; 30 con; 0-5 aty)
% Number of variables : 4750 ( 466 ^;4177 !; 107 ?;4750 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:09:42.370
%------------------------------------------------------------------------------
% Could-be-implicit typings (35)
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thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__b_Mtf__b_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__b_Mtf__a_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__b_M_Eo_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_I_Eo_Mtf__b_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_I_Eo_Mtf__a_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_Itf__b_J,type,
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thf(ty_n_t__Set__Oset_Itf__a_J,type,
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thf(ty_n_t__Set__Oset_I_Eo_J,type,
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thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
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% Explicit typings (357)
thf(sy_c_BNF__Wellorder__Constructions_OFunc_001_Eo_001tf__a,type,
bNF_We3495405677706807802nc_o_a: set_o > set_a > set_o_a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
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thf(sy_c_FiniteProduct_Ocommutative__monoid_OM__ify_001tf__a,type,
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thf(sy_c_FiniteProduct_Ocommutative__monoid_OM__ify_001tf__b,type,
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thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001t__Nat__Onat_001_Eo,type,
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thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001t__Nat__Onat_001tf__b,type,
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thf(sy_c_FiniteProduct_Ocommutative__monoid_Ofincomp_001tf__a_001_062_Itf__b_Mtf__a_J,type,
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thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Set__Oset_Itf__b_J,type,
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thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001tf__a,type,
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thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001tf__b,type,
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thf(sy_c_Fun_Ofun__upd_001tf__a_001_Eo,type,
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thf(sy_c_Fun_Ofun__upd_001tf__a_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Fun_Ofun__upd_001tf__a_001tf__a,type,
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thf(sy_c_Fun_Ofun__upd_001tf__a_001tf__b,type,
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thf(sy_c_Fun_Ofun__upd_001tf__b_001_Eo,type,
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thf(sy_c_Fun_Ofun__upd_001tf__b_001t__Nat__Onat,type,
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thf(sy_c_Fun_Ofun__upd_001tf__b_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Fun_Ofun__upd_001tf__b_001t__Set__Oset_Itf__b_J,type,
fun_upd_b_set_b: ( b > set_b ) > b > set_b > b > set_b ).
thf(sy_c_Fun_Ofun__upd_001tf__b_001tf__a,type,
fun_upd_b_a: ( b > a ) > b > a > b > a ).
thf(sy_c_Fun_Ofun__upd_001tf__b_001tf__b,type,
fun_upd_b_b: ( b > b ) > b > b > b > b ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001tf__b_001tf__a,type,
inj_on_b_a: ( b > a ) > set_b > $o ).
thf(sy_c_FuncSet_OPiE_001_062_Itf__b_Mtf__a_J_001_Eo,type,
piE_b_a_o: set_b_a > ( ( b > a ) > set_o ) > set_b_a_o ).
thf(sy_c_FuncSet_OPiE_001_062_Itf__b_Mtf__a_J_001t__Nat__Onat,type,
piE_b_a_nat: set_b_a > ( ( b > a ) > set_nat ) > set_b_a_nat ).
thf(sy_c_FuncSet_OPiE_001_062_Itf__b_Mtf__a_J_001tf__a,type,
piE_b_a_a: set_b_a > ( ( b > a ) > set_a ) > set_b_a_a ).
thf(sy_c_FuncSet_OPiE_001_062_Itf__b_Mtf__a_J_001tf__b,type,
piE_b_a_b: set_b_a > ( ( b > a ) > set_b ) > set_b_a_b ).
thf(sy_c_FuncSet_OPiE_001_Eo_001_062_Itf__b_Mtf__a_J,type,
piE_o_b_a: set_o > ( $o > set_b_a ) > set_o_b_a ).
thf(sy_c_FuncSet_OPiE_001_Eo_001_Eo,type,
piE_o_o: set_o > ( $o > set_o ) > set_o_o ).
thf(sy_c_FuncSet_OPiE_001_Eo_001t__Nat__Onat,type,
piE_o_nat: set_o > ( $o > set_nat ) > set_o_nat ).
thf(sy_c_FuncSet_OPiE_001_Eo_001tf__a,type,
piE_o_a: set_o > ( $o > set_a ) > set_o_a ).
thf(sy_c_FuncSet_OPiE_001_Eo_001tf__b,type,
piE_o_b: set_o > ( $o > set_b ) > set_o_b ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001_Eo,type,
piE_nat_o: set_nat > ( nat > set_o ) > set_nat_o ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001t__Nat__Onat,type,
piE_nat_nat: set_nat > ( nat > set_nat ) > set_nat_nat ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001tf__a,type,
piE_nat_a: set_nat > ( nat > set_a ) > set_nat_a ).
thf(sy_c_FuncSet_OPiE_001t__Nat__Onat_001tf__b,type,
piE_nat_b: set_nat > ( nat > set_b ) > set_nat_b ).
thf(sy_c_FuncSet_OPiE_001tf__a_001_Eo,type,
piE_a_o: set_a > ( a > set_o ) > set_a_o ).
thf(sy_c_FuncSet_OPiE_001tf__a_001t__Nat__Onat,type,
piE_a_nat: set_a > ( a > set_nat ) > set_a_nat ).
thf(sy_c_FuncSet_OPiE_001tf__a_001tf__a,type,
piE_a_a: set_a > ( a > set_a ) > set_a_a ).
thf(sy_c_FuncSet_OPiE_001tf__a_001tf__b,type,
piE_a_b: set_a > ( a > set_b ) > set_a_b ).
thf(sy_c_FuncSet_OPiE_001tf__b_001_062_Itf__b_Mtf__a_J,type,
piE_b_b_a: set_b > ( b > set_b_a ) > set_b_b_a ).
thf(sy_c_FuncSet_OPiE_001tf__b_001_Eo,type,
piE_b_o: set_b > ( b > set_o ) > set_b_o ).
thf(sy_c_FuncSet_OPiE_001tf__b_001t__Nat__Onat,type,
piE_b_nat: set_b > ( b > set_nat ) > set_b_nat ).
thf(sy_c_FuncSet_OPiE_001tf__b_001tf__a,type,
piE_b_a: set_b > ( b > set_a ) > set_b_a ).
thf(sy_c_FuncSet_OPiE_001tf__b_001tf__b,type,
piE_b_b: set_b > ( b > set_b ) > set_b_b ).
thf(sy_c_FuncSet_OPi_001_062_Itf__b_Mtf__a_J_001tf__a,type,
pi_b_a_a: set_b_a > ( ( b > a ) > set_a ) > set_b_a_a ).
thf(sy_c_FuncSet_OPi_001_062_Itf__b_Mtf__a_J_001tf__b,type,
pi_b_a_b: set_b_a > ( ( b > a ) > set_b ) > set_b_a_b ).
thf(sy_c_FuncSet_OPi_001_Eo_001_062_Itf__b_Mtf__a_J,type,
pi_o_b_a: set_o > ( $o > set_b_a ) > set_o_b_a ).
thf(sy_c_FuncSet_OPi_001_Eo_001t__Nat__Onat,type,
pi_o_nat: set_o > ( $o > set_nat ) > set_o_nat ).
thf(sy_c_FuncSet_OPi_001_Eo_001tf__a,type,
pi_o_a: set_o > ( $o > set_a ) > set_o_a ).
thf(sy_c_FuncSet_OPi_001_Eo_001tf__b,type,
pi_o_b: set_o > ( $o > set_b ) > set_o_b ).
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_001t__Nat__Onat_001tf__b,type,
pi_nat_b: set_nat > ( nat > set_b ) > set_nat_b ).
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_OPi_001tf__a_001tf__b,type,
pi_a_b: set_a > ( a > set_b ) > set_a_b ).
thf(sy_c_FuncSet_OPi_001tf__b_001_062_Itf__b_Mtf__a_J,type,
pi_b_b_a: set_b > ( b > set_b_a ) > set_b_b_a ).
thf(sy_c_FuncSet_OPi_001tf__b_001t__Nat__Onat,type,
pi_b_nat: set_b > ( b > set_nat ) > set_b_nat ).
thf(sy_c_FuncSet_OPi_001tf__b_001tf__a,type,
pi_b_a: set_b > ( b > set_a ) > set_b_a ).
thf(sy_c_FuncSet_OPi_001tf__b_001tf__b,type,
pi_b_b: set_b > ( b > set_b ) > set_b_b ).
thf(sy_c_FuncSet_Oextensional_001_062_Itf__b_Mtf__a_J_001tf__a,type,
extensional_b_a_a: set_b_a > set_b_a_a ).
thf(sy_c_FuncSet_Oextensional_001_Eo_001tf__a,type,
extensional_o_a: set_o > set_o_a ).
thf(sy_c_FuncSet_Oextensional_001t__Nat__Onat_001tf__a,type,
extensional_nat_a: set_nat > set_nat_a ).
thf(sy_c_FuncSet_Oextensional_001tf__a_001tf__a,type,
extensional_a_a: set_a > set_a_a ).
thf(sy_c_FuncSet_Oextensional_001tf__b_001tf__a,type,
extensional_b_a: set_b > set_b_a ).
thf(sy_c_FuncSet_Orestrict_001_062_Itf__b_Mtf__a_J_001tf__a,type,
restrict_b_a_a: ( ( b > a ) > a ) > set_b_a > ( b > a ) > a ).
thf(sy_c_FuncSet_Orestrict_001t__Nat__Onat_001t__Nat__Onat,type,
restrict_nat_nat: ( nat > nat ) > set_nat > nat > nat ).
thf(sy_c_FuncSet_Orestrict_001t__Nat__Onat_001tf__a,type,
restrict_nat_a: ( nat > a ) > set_nat > nat > a ).
thf(sy_c_FuncSet_Orestrict_001t__Nat__Onat_001tf__b,type,
restrict_nat_b: ( nat > b ) > set_nat > nat > b ).
thf(sy_c_FuncSet_Orestrict_001tf__a_001t__Nat__Onat,type,
restrict_a_nat: ( a > nat ) > set_a > a > nat ).
thf(sy_c_FuncSet_Orestrict_001tf__a_001tf__a,type,
restrict_a_a: ( a > a ) > set_a > a > a ).
thf(sy_c_FuncSet_Orestrict_001tf__a_001tf__b,type,
restrict_a_b: ( a > b ) > set_a > a > b ).
thf(sy_c_FuncSet_Orestrict_001tf__b_001_062_Itf__b_Mtf__a_J,type,
restrict_b_b_a: ( b > b > a ) > set_b > b > b > a ).
thf(sy_c_FuncSet_Orestrict_001tf__b_001t__Nat__Onat,type,
restrict_b_nat: ( b > nat ) > set_b > b > nat ).
thf(sy_c_FuncSet_Orestrict_001tf__b_001tf__a,type,
restrict_b_a: ( b > a ) > set_b > b > a ).
thf(sy_c_FuncSet_Orestrict_001tf__b_001tf__b,type,
restrict_b_b: ( b > b ) > set_b > b > b ).
thf(sy_c_Group__Theory_Oabelian__group_001tf__a,type,
group_201663378560352916roup_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid_001_062_Itf__b_Mtf__a_J,type,
group_4188790030012530981id_b_a: set_b_a > ( ( b > a ) > ( b > a ) > b > a ) > ( b > 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_001tf__b,type,
group_4866109990395492030noid_b: set_b > ( b > b > b ) > b > $o ).
thf(sy_c_Group__Theory_Ocommutative__monoid__axioms_001_062_Itf__b_Mtf__a_J,type,
group_4266494884492393160ms_b_a: set_b_a > ( ( b > a ) > ( b > a ) > b > 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_Ocommutative__monoid__axioms_001tf__b,type,
group_2081300317213596123ioms_b: set_b > ( b > b > b ) > $o ).
thf(sy_c_Group__Theory_Ogroup_001_062_Itf__b_Mtf__a_J,type,
group_group_b_a: set_b_a > ( ( b > a ) > ( b > a ) > b > a ) > ( b > 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_Ogroup_001tf__b,type,
group_group_b: set_b > ( b > b > b ) > b > $o ).
thf(sy_c_Group__Theory_Ogroup__axioms_001_062_Itf__b_Mtf__a_J,type,
group_3984435576162330991ms_b_a: set_b_a > ( ( b > a ) > ( b > a ) > b > a ) > ( b > a ) > $o ).
thf(sy_c_Group__Theory_Ogroup__axioms_001t__Nat__Onat,type,
group_661118103997438619ms_nat: set_nat > ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Group__Theory_Ogroup__axioms_001tf__a,type,
group_group_axioms_a: set_a > ( a > a > a ) > a > $o ).
thf(sy_c_Group__Theory_Ogroup__axioms_001tf__b,type,
group_group_axioms_b: set_b > ( b > b > b ) > b > $o ).
thf(sy_c_Group__Theory_Omonoid_001_062_Itf__b_Mtf__a_J,type,
group_monoid_b_a: set_b_a > ( ( b > a ) > ( b > a ) > b > a ) > ( b > 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_001tf__b,type,
group_monoid_b: set_b > ( b > b > b ) > b > $o ).
thf(sy_c_Group__Theory_Omonoid_OUnits_001_062_Itf__b_Mtf__a_J,type,
group_Units_b_a: set_b_a > ( ( b > a ) > ( b > a ) > b > a ) > ( b > a ) > set_b_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_OUnits_001tf__b,type,
group_Units_b: set_b > ( b > b > b ) > b > set_b ).
thf(sy_c_Group__Theory_Omonoid_Oinverse_001_062_Itf__b_Mtf__a_J,type,
group_inverse_b_a: set_b_a > ( ( b > a ) > ( b > a ) > b > a ) > ( b > a ) > ( b > a ) > b > 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_Oinverse_001tf__b,type,
group_inverse_b: set_b > ( b > b > b ) > b > b > b ).
thf(sy_c_Group__Theory_Omonoid_Oinvertible_001_062_Itf__b_Mtf__a_J,type,
group_invertible_b_a: set_b_a > ( ( b > a ) > ( b > a ) > b > a ) > ( b > a ) > ( b > 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_Omonoid_Oinvertible_001tf__b,type,
group_invertible_b: set_b > ( b > b > b ) > b > b > $o ).
thf(sy_c_Group__Theory_Osubgroup_001_062_Itf__b_Mtf__a_J,type,
group_subgroup_b_a: set_b_a > set_b_a > ( ( b > a ) > ( b > a ) > b > a ) > ( b > 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_Group__Theory_Osubgroup_001tf__b,type,
group_subgroup_b: set_b > set_b > ( b > b > b ) > b > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_Itf__b_Mtf__a_J_M_Eo_J,type,
minus_minus_b_a_o: ( ( b > a ) > $o ) > ( ( b > a ) > $o ) > ( b > a ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__a_M_Eo_J,type,
minus_minus_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__b_M_Eo_J,type,
minus_minus_b_o: ( b > $o ) > ( b > $o ) > b > $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_Itf__b_Mtf__a_J_J,type,
minus_minus_set_b_a: set_b_a > set_b_a > set_b_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_Eo_J,type,
minus_minus_set_o: set_o > set_o > set_o ).
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_Ominus__class_Ominus_001t__Set__Oset_Itf__b_J,type,
minus_minus_set_b: set_b > set_b > set_b ).
thf(sy_c_HOL_OThe_001tf__a,type,
the_a: ( a > $o ) > a ).
thf(sy_c_HOL_OUniq_001_062_Itf__b_Mtf__a_J,type,
uniq_b_a: ( ( b > a ) > $o ) > $o ).
thf(sy_c_HOL_OUniq_001_Eo,type,
uniq_o: ( $o > $o ) > $o ).
thf(sy_c_HOL_OUniq_001t__Nat__Onat,type,
uniq_nat: ( nat > $o ) > $o ).
thf(sy_c_HOL_OUniq_001tf__a,type,
uniq_a: ( a > $o ) > $o ).
thf(sy_c_HOL_OUniq_001tf__b,type,
uniq_b: ( b > $o ) > $o ).
thf(sy_c_HOL_Oundefined_001_062_Itf__b_Mtf__a_J,type,
undefined_b_a: b > a ).
thf(sy_c_HOL_Oundefined_001_Eo,type,
undefined_o: $o ).
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_HOL_Oundefined_001tf__b,type,
undefined_b: b ).
thf(sy_c_If_001t__Set__Oset_I_Eo_J,type,
if_set_o: $o > set_o > set_o > set_o ).
thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
if_set_nat: $o > set_nat > set_nat > set_nat ).
thf(sy_c_If_001t__Set__Oset_Itf__b_J,type,
if_set_b: $o > set_b > set_b > set_b ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_062_Itf__b_Mtf__a_J_001t__Nat__Onat,type,
lattic7032157021346806415_a_nat: ( ( b > a ) > nat ) > set_b_a > b > a ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_Eo_001t__Nat__Onat,type,
lattic2775856028456453135_o_nat: ( $o > nat ) > set_o > $o ).
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_Lattices__Big_Oord__class_Oarg__min__on_001tf__b_001t__Nat__Onat,type,
lattic7575731748627795062_b_nat: ( b > nat ) > set_b > b ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_Itf__b_Mtf__a_J_M_Eo_J,type,
bot_bot_b_a_o: ( b > a ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_Eo_M_Eo_J,type,
bot_bot_o_o: $o > $o ).
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_001_062_Itf__b_M_Eo_J,type,
bot_bot_b_o: b > $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_Itf__b_Mtf__a_J_M_Eo_J_J,type,
bot_bot_set_b_a_o: set_b_a_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_062_Itf__b_Mtf__a_J_Mtf__b_J_J,type,
bot_bot_set_b_a_b: set_b_a_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_Eo_Mtf__a_J_J,type,
bot_bot_set_o_a: set_o_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
bot_bot_set_nat_o: set_nat_o ).
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_It__Nat__Onat_Mtf__b_J_J,type,
bot_bot_set_nat_b: set_nat_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
bot_bot_set_a_o: set_a_o ).
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_I_062_Itf__a_Mtf__b_J_J,type,
bot_bot_set_a_b: set_a_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__b_M_Eo_J_J,type,
bot_bot_set_b_o: set_b_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__b_Mtf__a_J_J,type,
bot_bot_set_b_a: set_b_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__b_Mtf__b_J_J,type,
bot_bot_set_b_b: set_b_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
bot_bot_set_o: set_o ).
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_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_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
bot_bot_set_b: set_b ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_Itf__b_Mtf__a_J_M_Eo_J,type,
ord_less_eq_b_a_o: ( ( b > a ) > $o ) > ( ( b > 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_001_062_Itf__b_M_Eo_J,type,
ord_less_eq_b_o: ( b > $o ) > ( b > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $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_Itf__b_Mtf__a_J_Mtf__a_J_J,type,
ord_le4402886750609172241_b_a_a: set_b_a_a > set_b_a_a > $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_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_I_062_Itf__b_Mtf__a_J_J,type,
ord_less_eq_set_b_a: set_b_a > set_b_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $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_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
ord_less_eq_set_b: set_b > set_b > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
top_top_set_o: set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Set_OCollect_001_062_Itf__b_Mtf__a_J,type,
collect_b_a: ( ( b > a ) > $o ) > set_b_a ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
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_001t__Set__Oset_Itf__b_J,type,
collect_set_b: ( set_b > $o ) > set_set_b ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_Set_Ofilter_001_062_Itf__b_Mtf__a_J,type,
filter_b_a: ( ( b > a ) > $o ) > set_b_a > set_b_a ).
thf(sy_c_Set_Ofilter_001_Eo,type,
filter_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Ofilter_001t__Nat__Onat,type,
filter_nat: ( nat > $o ) > set_nat > set_nat ).
thf(sy_c_Set_Ofilter_001tf__a,type,
filter_a: ( a > $o ) > set_a > set_a ).
thf(sy_c_Set_Ofilter_001tf__b,type,
filter_b: ( b > $o ) > set_b > set_b ).
thf(sy_c_Set_Oimage_001_062_I_062_Itf__b_Mtf__a_J_Mtf__b_J_001tf__b,type,
image_b_a_b_b: ( ( ( b > a ) > b ) > b ) > set_b_a_b > set_b ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_Eo_J_001_Eo,type,
image_nat_o_o: ( ( nat > $o ) > $o ) > set_nat_o > set_o ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mtf__a_J_001tf__a,type,
image_nat_a_a: ( ( nat > a ) > a ) > set_nat_a > set_a ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mtf__b_J_001tf__b,type,
image_nat_b_b: ( ( nat > b ) > b ) > set_nat_b > set_b ).
thf(sy_c_Set_Oimage_001_062_Itf__a_M_Eo_J_001_Eo,type,
image_a_o_o: ( ( a > $o ) > $o ) > set_a_o > set_o ).
thf(sy_c_Set_Oimage_001_062_Itf__a_Mtf__a_J_001tf__a,type,
image_a_a_a: ( ( a > a ) > a ) > set_a_a > set_a ).
thf(sy_c_Set_Oimage_001_062_Itf__a_Mtf__b_J_001tf__b,type,
image_a_b_b: ( ( a > b ) > b ) > set_a_b > set_b ).
thf(sy_c_Set_Oimage_001_062_Itf__b_M_Eo_J_001_Eo,type,
image_b_o_o: ( ( b > $o ) > $o ) > set_b_o > set_o ).
thf(sy_c_Set_Oimage_001_062_Itf__b_Mtf__a_J_001_062_Itf__b_Mtf__a_J,type,
image_b_a_b_a: ( ( b > a ) > b > a ) > set_b_a > set_b_a ).
thf(sy_c_Set_Oimage_001_062_Itf__b_Mtf__a_J_001_Eo,type,
image_b_a_o: ( ( b > a ) > $o ) > set_b_a > set_o ).
thf(sy_c_Set_Oimage_001_062_Itf__b_Mtf__a_J_001t__Nat__Onat,type,
image_b_a_nat: ( ( b > a ) > nat ) > set_b_a > set_nat ).
thf(sy_c_Set_Oimage_001_062_Itf__b_Mtf__a_J_001tf__a,type,
image_b_a_a: ( ( b > a ) > a ) > set_b_a > set_a ).
thf(sy_c_Set_Oimage_001_062_Itf__b_Mtf__a_J_001tf__b,type,
image_b_a_b: ( ( b > a ) > b ) > set_b_a > set_b ).
thf(sy_c_Set_Oimage_001_062_Itf__b_Mtf__b_J_001tf__b,type,
image_b_b_b: ( ( b > b ) > b ) > set_b_b > set_b ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001tf__b,type,
image_o_b: ( $o > b ) > set_o > set_b ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_Itf__b_Mtf__a_J,type,
image_nat_b_a: ( nat > b > a ) > set_nat > set_b_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
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_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__b,type,
image_nat_b: ( nat > b ) > set_nat > set_b ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_Itf__b_Mtf__a_J_J_001t__Set__Oset_I_062_Itf__b_Mtf__a_J_J,type,
image_8428841068488964283et_b_a: ( set_b_a > set_b_a ) > set_set_b_a > set_set_b_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
image_set_o_set_o: ( set_o > set_o ) > set_set_o > set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__b_J,type,
image_set_b_set_b: ( set_b > set_b ) > set_set_b > set_set_b ).
thf(sy_c_Set_Oimage_001tf__a_001_062_Itf__b_Mtf__a_J,type,
image_a_b_a: ( a > b > a ) > set_a > set_b_a ).
thf(sy_c_Set_Oimage_001tf__a_001_Eo,type,
image_a_o: ( a > $o ) > set_a > set_o ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
image_a_set_a_a: ( a > set_a_a ) > set_a > set_set_a_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__b,type,
image_a_b: ( a > b ) > set_a > set_b ).
thf(sy_c_Set_Oimage_001tf__b_001_062_Itf__b_Mtf__a_J,type,
image_b_b_a: ( b > b > a ) > set_b > set_b_a ).
thf(sy_c_Set_Oimage_001tf__b_001_Eo,type,
image_b_o: ( b > $o ) > set_b > set_o ).
thf(sy_c_Set_Oimage_001tf__b_001t__Nat__Onat,type,
image_b_nat: ( b > nat ) > set_b > set_nat ).
thf(sy_c_Set_Oimage_001tf__b_001tf__a,type,
image_b_a: ( b > a ) > set_b > set_a ).
thf(sy_c_Set_Oimage_001tf__b_001tf__b,type,
image_b_b: ( b > b ) > set_b > set_b ).
thf(sy_c_Set_Oinsert_001_062_I_Eo_Mtf__a_J,type,
insert_o_a: ( $o > a ) > set_o_a > set_o_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_001_062_Itf__b_Mtf__a_J,type,
insert_b_a: ( b > a ) > set_b_a > set_b_a ).
thf(sy_c_Set_Oinsert_001_Eo,type,
insert_o: $o > set_o > set_o ).
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_Oinsert_001tf__b,type,
insert_b: b > set_b > set_b ).
thf(sy_c_Set_Ois__empty_001_Eo,type,
is_empty_o: set_o > $o ).
thf(sy_c_Set_Ois__singleton_001_062_Itf__b_Mtf__a_J,type,
is_singleton_b_a: set_b_a > $o ).
thf(sy_c_Set_Ois__singleton_001_Eo,type,
is_singleton_o: set_o > $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_Ois__singleton_001tf__b,type,
is_singleton_b: set_b > $o ).
thf(sy_c_Set_Oremove_001_062_Itf__b_Mtf__a_J,type,
remove_b_a: ( b > a ) > set_b_a > set_b_a ).
thf(sy_c_Set_Oremove_001_Eo,type,
remove_o: $o > set_o > set_o ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001tf__a,type,
remove_a: a > set_a > set_a ).
thf(sy_c_Set_Oremove_001tf__b,type,
remove_b: b > set_b > set_b ).
thf(sy_c_Set_Othe__elem_001_Eo,type,
the_elem_o: set_o > $o ).
thf(sy_c_Set_Othe__elem_001tf__b,type,
the_elem_b: set_b > b ).
thf(sy_c_member_001_062_I_062_Itf__b_Mtf__a_J_Mtf__a_J,type,
member_b_a_a: ( ( b > a ) > a ) > set_b_a_a > $o ).
thf(sy_c_member_001_062_I_062_Itf__b_Mtf__a_J_Mtf__b_J,type,
member_b_a_b: ( ( b > a ) > b ) > set_b_a_b > $o ).
thf(sy_c_member_001_062_I_Eo_M_062_Itf__b_Mtf__a_J_J,type,
member_o_b_a: ( $o > b > a ) > set_o_b_a > $o ).
thf(sy_c_member_001_062_I_Eo_M_Eo_J,type,
member_o_o: ( $o > $o ) > set_o_o > $o ).
thf(sy_c_member_001_062_I_Eo_Mt__Nat__Onat_J,type,
member_o_nat: ( $o > nat ) > set_o_nat > $o ).
thf(sy_c_member_001_062_I_Eo_Mtf__a_J,type,
member_o_a: ( $o > a ) > set_o_a > $o ).
thf(sy_c_member_001_062_I_Eo_Mtf__b_J,type,
member_o_b: ( $o > b ) > set_o_b > $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_It__Nat__Onat_Mtf__b_J,type,
member_nat_b: ( nat > b ) > set_nat_b > $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_001_062_Itf__a_Mtf__b_J,type,
member_a_b: ( a > b ) > set_a_b > $o ).
thf(sy_c_member_001_062_Itf__b_M_062_Itf__b_Mtf__a_J_J,type,
member_b_b_a: ( b > b > a ) > set_b_b_a > $o ).
thf(sy_c_member_001_062_Itf__b_M_Eo_J,type,
member_b_o: ( b > $o ) > set_b_o > $o ).
thf(sy_c_member_001_062_Itf__b_Mt__Nat__Onat_J,type,
member_b_nat: ( b > nat ) > set_b_nat > $o ).
thf(sy_c_member_001_062_Itf__b_Mtf__a_J,type,
member_b_a: ( b > a ) > set_b_a > $o ).
thf(sy_c_member_001_062_Itf__b_Mtf__b_J,type,
member_b_b: ( b > b ) > set_b_b > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_Itf__b_Mtf__a_J_J,type,
member_set_b_a: set_b_a > set_set_b_a > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $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_001t__Set__Oset_Itf__b_J,type,
member_set_b: set_b > set_set_b > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_Aa____,type,
aa: set_b ).
thf(sy_v_M,type,
m: set_a ).
thf(sy_v_a____,type,
a2: b ).
thf(sy_v_composition,type,
composition: a > a > a ).
thf(sy_v_f,type,
f: b > a ).
thf(sy_v_unit,type,
unit: a ).
% Relevant facts (1276)
thf(fact_0_insert_Ohyps_I1_J,axiom,
finite_finite_b @ aa ).
% insert.hyps(1)
thf(fact_1_insert_Ohyps_I2_J,axiom,
~ ( member_b @ a2 @ aa ) ).
% insert.hyps(2)
thf(fact_2__092_060open_062_I_092_060lambda_062i_O_A_092_060one_062_J_A_092_060in_062_AA_A_092_060rightarrow_062_AM_092_060close_062,axiom,
( member_b_a
@ ^ [I: b] : unit
@ ( pi_b_a @ aa
@ ^ [Uu: b] : m ) ) ).
% \<open>(\<lambda>i. \<one>) \<in> A \<rightarrow> M\<close>
thf(fact_3_commutative,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ m )
=> ( ( member_a @ Y @ m )
=> ( ( composition @ X @ Y )
= ( composition @ Y @ X ) ) ) ) ).
% commutative
thf(fact_4_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_5_insert_Oprems,axiom,
! [X: b] :
( ( member_b @ X @ ( insert_b @ a2 @ aa ) )
=> ( ( f @ X )
= unit ) ) ).
% insert.prems
thf(fact_6_insert_Ohyps_I3_J,axiom,
( ! [X2: b] :
( ( member_b @ X2 @ aa )
=> ( ( f @ X2 )
= unit ) )
=> ( ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ f @ aa )
= unit ) ) ).
% insert.hyps(3)
thf(fact_7_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_8_commutative__monoid__axioms,axiom,
group_4866109990395492029noid_a @ m @ composition @ unit ).
% commutative_monoid_axioms
thf(fact_9_fincomp__closed,axiom,
! [F: b > a,F2: set_b] :
( ( member_b_a @ F
@ ( pi_b_a @ F2
@ ^ [Uu: b] : m ) )
=> ( member_a @ ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ F @ F2 ) @ m ) ) ).
% fincomp_closed
thf(fact_10_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_11_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_12_unit__closed,axiom,
member_a @ unit @ m ).
% unit_closed
thf(fact_13_left__unit,axiom,
! [A: a] :
( ( member_a @ A @ m )
=> ( ( composition @ unit @ A )
= A ) ) ).
% left_unit
thf(fact_14_right__unit,axiom,
! [A: a] :
( ( member_a @ A @ m )
=> ( ( composition @ A @ unit )
= A ) ) ).
% right_unit
thf(fact_15_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_16_commutative__monoid_Ofincomp_Ocong,axiom,
commut5005951359559292711mp_a_b = commut5005951359559292711mp_a_b ).
% commutative_monoid.fincomp.cong
thf(fact_17_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_18_fincomp__infinite,axiom,
! [A2: set_b,F: b > a] :
( ~ ( finite_finite_b @ A2 )
=> ( ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ F @ A2 )
= unit ) ) ).
% fincomp_infinite
thf(fact_19_fincomp__insert,axiom,
! [F2: set_o,A: $o,F: $o > a] :
( ( finite_finite_o @ F2 )
=> ( ~ ( member_o @ A @ F2 )
=> ( ( member_o_a @ F
@ ( pi_o_a @ F2
@ ^ [Uu: $o] : m ) )
=> ( ( member_a @ ( F @ A ) @ m )
=> ( ( commut1011387283630023616mp_a_o @ m @ composition @ unit @ F @ ( insert_o @ A @ F2 ) )
= ( composition @ ( F @ A ) @ ( commut1011387283630023616mp_a_o @ m @ composition @ unit @ F @ F2 ) ) ) ) ) ) ) ).
% fincomp_insert
thf(fact_20_fincomp__insert,axiom,
! [F2: set_b_a,A: b > a,F: ( b > a ) > a] :
( ( finite_finite_b_a @ F2 )
=> ( ~ ( member_b_a @ A @ F2 )
=> ( ( member_b_a_a @ F
@ ( pi_b_a_a @ F2
@ ^ [Uu: b > a] : m ) )
=> ( ( member_a @ ( F @ A ) @ m )
=> ( ( commut3556868347779488380_a_b_a @ m @ composition @ unit @ F @ ( insert_b_a @ A @ F2 ) )
= ( composition @ ( F @ A ) @ ( commut3556868347779488380_a_b_a @ m @ composition @ unit @ F @ F2 ) ) ) ) ) ) ) ).
% fincomp_insert
thf(fact_21_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_22_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_23_fincomp__insert,axiom,
! [F2: set_b,A: b,F: b > a] :
( ( finite_finite_b @ F2 )
=> ( ~ ( member_b @ A @ F2 )
=> ( ( member_b_a @ F
@ ( pi_b_a @ F2
@ ^ [Uu: b] : m ) )
=> ( ( member_a @ ( F @ A ) @ m )
=> ( ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ F @ ( insert_b @ A @ F2 ) )
= ( composition @ ( F @ A ) @ ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ F @ F2 ) ) ) ) ) ) ) ).
% fincomp_insert
thf(fact_24_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_25_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_26_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_27_unit__invertible,axiom,
group_invertible_a @ m @ composition @ unit @ unit ).
% unit_invertible
thf(fact_28_monoid__axioms,axiom,
group_monoid_a @ m @ composition @ unit ).
% monoid_axioms
thf(fact_29_insertCI,axiom,
! [A: $o,B2: set_o,B: $o] :
( ( ~ ( member_o @ A @ B2 )
=> ( A = B ) )
=> ( member_o @ A @ ( insert_o @ B @ B2 ) ) ) ).
% insertCI
thf(fact_30_insertCI,axiom,
! [A: b,B2: set_b,B: b] :
( ( ~ ( member_b @ A @ B2 )
=> ( A = B ) )
=> ( member_b @ A @ ( insert_b @ B @ B2 ) ) ) ).
% insertCI
thf(fact_31_insertCI,axiom,
! [A: b > a,B2: set_b_a,B: b > a] :
( ( ~ ( member_b_a @ A @ B2 )
=> ( A = B ) )
=> ( member_b_a @ A @ ( insert_b_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_32_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_33_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_34_insert__iff,axiom,
! [A: $o,B: $o,A2: set_o] :
( ( member_o @ A @ ( insert_o @ B @ A2 ) )
= ( ( A = B )
| ( member_o @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_35_insert__iff,axiom,
! [A: b,B: b,A2: set_b] :
( ( member_b @ A @ ( insert_b @ B @ A2 ) )
= ( ( A = B )
| ( member_b @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_36_insert__iff,axiom,
! [A: b > a,B: b > a,A2: set_b_a] :
( ( member_b_a @ A @ ( insert_b_a @ B @ A2 ) )
= ( ( A = B )
| ( member_b_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_37_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_38_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_39_insert__absorb2,axiom,
! [X: b,A2: set_b] :
( ( insert_b @ X @ ( insert_b @ X @ A2 ) )
= ( insert_b @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_40_insert__absorb2,axiom,
! [X: $o,A2: set_o] :
( ( insert_o @ X @ ( insert_o @ X @ A2 ) )
= ( insert_o @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_41_fincomp__empty,axiom,
! [F: $o > a] :
( ( commut1011387283630023616mp_a_o @ m @ composition @ unit @ F @ bot_bot_set_o )
= unit ) ).
% fincomp_empty
thf(fact_42_fincomp__empty,axiom,
! [F: b > a] :
( ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ F @ bot_bot_set_b )
= unit ) ).
% fincomp_empty
thf(fact_43_inverse__unit,axiom,
( ( group_inverse_a @ m @ composition @ unit @ unit )
= unit ) ).
% inverse_unit
thf(fact_44_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_45_empty__iff,axiom,
! [C: b > a] :
~ ( member_b_a @ C @ bot_bot_set_b_a ) ).
% empty_iff
thf(fact_46_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_47_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_48_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_49_all__not__in__conv,axiom,
! [A2: set_b] :
( ( ! [X3: b] :
~ ( member_b @ X3 @ A2 ) )
= ( A2 = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_50_all__not__in__conv,axiom,
! [A2: set_b_a] :
( ( ! [X3: b > a] :
~ ( member_b_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_b_a ) ) ).
% all_not_in_conv
thf(fact_51_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_52_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_53_all__not__in__conv,axiom,
! [A2: set_o] :
( ( ! [X3: $o] :
~ ( member_o @ X3 @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_54_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_55_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_56_Collect__empty__eq,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( ! [X3: $o] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_57_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_58_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_59_empty__Collect__eq,axiom,
! [P: $o > $o] :
( ( bot_bot_set_o
= ( collect_o @ P ) )
= ( ! [X3: $o] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_60_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_61_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_62_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_63_singletonI,axiom,
! [A: b] : ( member_b @ A @ ( insert_b @ A @ bot_bot_set_b ) ) ).
% singletonI
thf(fact_64_singletonI,axiom,
! [A: b > a] : ( member_b_a @ A @ ( insert_b_a @ A @ bot_bot_set_b_a ) ) ).
% singletonI
thf(fact_65_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_66_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_67_singletonI,axiom,
! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).
% singletonI
thf(fact_68_singleton__conv2,axiom,
! [A: b] :
( ( collect_b
@ ( ^ [Y2: b,Z: b] : ( Y2 = Z )
@ A ) )
= ( insert_b @ A @ bot_bot_set_b ) ) ).
% singleton_conv2
thf(fact_69_singleton__conv2,axiom,
! [A: a] :
( ( collect_a
@ ( ^ [Y2: a,Z: a] : ( Y2 = Z )
@ A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_70_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y2: nat,Z: nat] : ( Y2 = Z )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_71_singleton__conv2,axiom,
! [A: $o] :
( ( collect_o
@ ( ^ [Y2: $o,Z: $o] : ( Y2 = Z )
@ A ) )
= ( insert_o @ A @ bot_bot_set_o ) ) ).
% singleton_conv2
thf(fact_72_singleton__conv,axiom,
! [A: b] :
( ( collect_b
@ ^ [X3: b] : ( X3 = A ) )
= ( insert_b @ A @ bot_bot_set_b ) ) ).
% singleton_conv
thf(fact_73_singleton__conv,axiom,
! [A: a] :
( ( collect_a
@ ^ [X3: a] : ( X3 = A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_74_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X3: nat] : ( X3 = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_75_singleton__conv,axiom,
! [A: $o] :
( ( collect_o
@ ^ [X3: $o] : ( X3 = A ) )
= ( insert_o @ A @ bot_bot_set_o ) ) ).
% singleton_conv
thf(fact_76_invertible__right__cancel,axiom,
! [X: a,Y: a,Z2: a] :
( ( group_invertible_a @ m @ composition @ unit @ X )
=> ( ( member_a @ X @ m )
=> ( ( member_a @ Y @ m )
=> ( ( member_a @ Z2 @ m )
=> ( ( ( composition @ Y @ X )
= ( composition @ Z2 @ X ) )
= ( Y = Z2 ) ) ) ) ) ) ).
% invertible_right_cancel
thf(fact_77_invertible__left__cancel,axiom,
! [X: a,Y: a,Z2: a] :
( ( group_invertible_a @ m @ composition @ unit @ X )
=> ( ( member_a @ X @ m )
=> ( ( member_a @ Y @ m )
=> ( ( member_a @ Z2 @ m )
=> ( ( ( composition @ X @ Y )
= ( composition @ X @ Z2 ) )
= ( Y = Z2 ) ) ) ) ) ) ).
% invertible_left_cancel
thf(fact_78_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_79_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_80_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_81_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_82_mem__Collect__eq,axiom,
! [A: b,P: b > $o] :
( ( member_b @ A @ ( collect_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_83_mem__Collect__eq,axiom,
! [A: b > a,P: ( b > a ) > $o] :
( ( member_b_a @ A @ ( collect_b_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_84_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_85_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_86_Collect__mem__eq,axiom,
! [A2: set_b] :
( ( collect_b
@ ^ [X3: b] : ( member_b @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_87_Collect__mem__eq,axiom,
! [A2: set_b_a] :
( ( collect_b_a
@ ^ [X3: b > a] : ( member_b_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_88_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_89_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_90_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_91_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_92_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_93_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_94_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_95_commutative__monoid_OM__ify__def,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,X: b] :
( ( group_4866109990395492030noid_b @ M @ Composition @ Unit )
=> ( ( ( member_b @ X @ M )
=> ( ( commutative_M_ify_b @ M @ Unit @ X )
= X ) )
& ( ~ ( member_b @ X @ M )
=> ( ( commutative_M_ify_b @ M @ Unit @ X )
= Unit ) ) ) ) ).
% commutative_monoid.M_ify_def
thf(fact_96_commutative__monoid_OM__ify__def,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,X: b > a] :
( ( group_4188790030012530981id_b_a @ M @ Composition @ Unit )
=> ( ( ( member_b_a @ X @ M )
=> ( ( commut3325098377247325640fy_b_a @ M @ Unit @ X )
= X ) )
& ( ~ ( member_b_a @ X @ M )
=> ( ( commut3325098377247325640fy_b_a @ M @ Unit @ X )
= Unit ) ) ) ) ).
% commutative_monoid.M_ify_def
thf(fact_97_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_98_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_99_commutative__monoid_Oleft__commute,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,A: b,B: b,C: b] :
( ( group_4866109990395492030noid_b @ M @ Composition @ Unit )
=> ( ( member_b @ A @ M )
=> ( ( member_b @ B @ M )
=> ( ( member_b @ C @ M )
=> ( ( Composition @ B @ ( Composition @ A @ C ) )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% commutative_monoid.left_commute
thf(fact_100_commutative__monoid_Oleft__commute,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,A: b > a,B: b > a,C: b > a] :
( ( group_4188790030012530981id_b_a @ M @ Composition @ Unit )
=> ( ( member_b_a @ A @ M )
=> ( ( member_b_a @ B @ M )
=> ( ( member_b_a @ C @ M )
=> ( ( Composition @ B @ ( Composition @ A @ C ) )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% commutative_monoid.left_commute
thf(fact_101_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_102_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_103_emptyE,axiom,
! [A: b] :
~ ( member_b @ A @ bot_bot_set_b ) ).
% emptyE
thf(fact_104_emptyE,axiom,
! [A: b > a] :
~ ( member_b_a @ A @ bot_bot_set_b_a ) ).
% emptyE
thf(fact_105_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_106_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_107_emptyE,axiom,
! [A: $o] :
~ ( member_o @ A @ bot_bot_set_o ) ).
% emptyE
thf(fact_108_equals0D,axiom,
! [A2: set_b,A: b] :
( ( A2 = bot_bot_set_b )
=> ~ ( member_b @ A @ A2 ) ) ).
% equals0D
thf(fact_109_equals0D,axiom,
! [A2: set_b_a,A: b > a] :
( ( A2 = bot_bot_set_b_a )
=> ~ ( member_b_a @ A @ A2 ) ) ).
% equals0D
thf(fact_110_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_111_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_112_equals0D,axiom,
! [A2: set_o,A: $o] :
( ( A2 = bot_bot_set_o )
=> ~ ( member_o @ A @ A2 ) ) ).
% equals0D
thf(fact_113_equals0I,axiom,
! [A2: set_b] :
( ! [Y3: b] :
~ ( member_b @ Y3 @ A2 )
=> ( A2 = bot_bot_set_b ) ) ).
% equals0I
thf(fact_114_equals0I,axiom,
! [A2: set_b_a] :
( ! [Y3: b > a] :
~ ( member_b_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_b_a ) ) ).
% equals0I
thf(fact_115_equals0I,axiom,
! [A2: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_116_equals0I,axiom,
! [A2: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_117_equals0I,axiom,
! [A2: set_o] :
( ! [Y3: $o] :
~ ( member_o @ Y3 @ A2 )
=> ( A2 = bot_bot_set_o ) ) ).
% equals0I
thf(fact_118_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X3: a] : $false ) ) ).
% empty_def
thf(fact_119_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X3: nat] : $false ) ) ).
% empty_def
thf(fact_120_empty__def,axiom,
( bot_bot_set_o
= ( collect_o
@ ^ [X3: $o] : $false ) ) ).
% empty_def
thf(fact_121_ex__in__conv,axiom,
! [A2: set_b] :
( ( ? [X3: b] : ( member_b @ X3 @ A2 ) )
= ( A2 != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_122_ex__in__conv,axiom,
! [A2: set_b_a] :
( ( ? [X3: b > a] : ( member_b_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_b_a ) ) ).
% ex_in_conv
thf(fact_123_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_124_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_125_ex__in__conv,axiom,
! [A2: set_o] :
( ( ? [X3: $o] : ( member_o @ X3 @ A2 ) )
= ( A2 != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_126_singleton__inject,axiom,
! [A: b,B: b] :
( ( ( insert_b @ A @ bot_bot_set_b )
= ( insert_b @ B @ bot_bot_set_b ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_127_singleton__inject,axiom,
! [A: $o,B: $o] :
( ( ( insert_o @ A @ bot_bot_set_o )
= ( insert_o @ B @ bot_bot_set_o ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_128_insert__not__empty,axiom,
! [A: b,A2: set_b] :
( ( insert_b @ A @ A2 )
!= bot_bot_set_b ) ).
% insert_not_empty
thf(fact_129_insert__not__empty,axiom,
! [A: $o,A2: set_o] :
( ( insert_o @ A @ A2 )
!= bot_bot_set_o ) ).
% insert_not_empty
thf(fact_130_doubleton__eq__iff,axiom,
! [A: b,B: b,C: b,D: b] :
( ( ( insert_b @ A @ ( insert_b @ B @ bot_bot_set_b ) )
= ( insert_b @ C @ ( insert_b @ D @ bot_bot_set_b ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_131_doubleton__eq__iff,axiom,
! [A: $o,B: $o,C: $o,D: $o] :
( ( ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) )
= ( insert_o @ C @ ( insert_o @ D @ bot_bot_set_o ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_132_singleton__iff,axiom,
! [B: b,A: b] :
( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_133_singleton__iff,axiom,
! [B: b > a,A: b > a] :
( ( member_b_a @ B @ ( insert_b_a @ A @ bot_bot_set_b_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_134_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_135_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_136_singleton__iff,axiom,
! [B: $o,A: $o] :
( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_137_singletonD,axiom,
! [B: b,A: b] :
( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_138_singletonD,axiom,
! [B: b > a,A: b > a] :
( ( member_b_a @ B @ ( insert_b_a @ A @ bot_bot_set_b_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_139_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_140_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_141_singletonD,axiom,
! [B: $o,A: $o] :
( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_142_commutative__monoid_OM__ify_Ocong,axiom,
commutative_M_ify_a = commutative_M_ify_a ).
% commutative_monoid.M_ify.cong
thf(fact_143_commutative__monoid_Ofincomp__empty,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: $o > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( commut1011387283630023616mp_a_o @ M @ Composition @ Unit @ F @ bot_bot_set_o )
= Unit ) ) ).
% commutative_monoid.fincomp_empty
thf(fact_144_commutative__monoid_Ofincomp__empty,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: b > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ F @ bot_bot_set_b )
= Unit ) ) ).
% commutative_monoid.fincomp_empty
thf(fact_145_Collect__conv__if2,axiom,
! [P: b > $o,A: b] :
( ( ( P @ A )
=> ( ( collect_b
@ ^ [X3: b] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_b @ A @ bot_bot_set_b ) ) )
& ( ~ ( P @ A )
=> ( ( collect_b
@ ^ [X3: b] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_b ) ) ) ).
% Collect_conv_if2
thf(fact_146_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_147_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_148_Collect__conv__if2,axiom,
! [P: $o > $o,A: $o] :
( ( ( P @ A )
=> ( ( collect_o
@ ^ [X3: $o] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ~ ( P @ A )
=> ( ( collect_o
@ ^ [X3: $o] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if2
thf(fact_149_Collect__conv__if,axiom,
! [P: b > $o,A: b] :
( ( ( P @ A )
=> ( ( collect_b
@ ^ [X3: b] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_b @ A @ bot_bot_set_b ) ) )
& ( ~ ( P @ A )
=> ( ( collect_b
@ ^ [X3: b] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bot_set_b ) ) ) ).
% Collect_conv_if
thf(fact_150_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_151_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_152_Collect__conv__if,axiom,
! [P: $o > $o,A: $o] :
( ( ( P @ A )
=> ( ( collect_o
@ ^ [X3: $o] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ~ ( P @ A )
=> ( ( collect_o
@ ^ [X3: $o] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if
thf(fact_153_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_154_commutative__monoid_Ofincomp__infinite,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_b,F: b > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ~ ( finite_finite_b @ A2 )
=> ( ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ).
% commutative_monoid.fincomp_infinite
thf(fact_155_commutative__monoid_Ofincomp__closed,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: b > a,F2: set_b] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_b_a @ F
@ ( pi_b_a @ F2
@ ^ [Uu: b] : M ) )
=> ( member_a @ ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ F @ F2 ) @ M ) ) ) ).
% commutative_monoid.fincomp_closed
thf(fact_156_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,F2: set_o,A: $o,F: $o > b] :
( ( group_4866109990395492030noid_b @ M @ Composition @ Unit )
=> ( ( finite_finite_o @ F2 )
=> ( ~ ( member_o @ A @ F2 )
=> ( ( member_o_b @ F
@ ( pi_o_b @ F2
@ ^ [Uu: $o] : M ) )
=> ( ( member_b @ ( F @ A ) @ M )
=> ( ( commut7335536921020405119mp_b_o @ M @ Composition @ Unit @ F @ ( insert_o @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut7335536921020405119mp_b_o @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_157_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,F2: set_o,A: $o,F: $o > nat] :
( ( group_6791354081887936081id_nat @ M @ Composition @ Unit )
=> ( ( finite_finite_o @ F2 )
=> ( ~ ( member_o @ A @ F2 )
=> ( ( member_o_nat @ F
@ ( pi_o_nat @ F2
@ ^ [Uu: $o] : M ) )
=> ( ( member_nat @ ( F @ A ) @ M )
=> ( ( commut6500812589713462176_nat_o @ M @ Composition @ Unit @ F @ ( insert_o @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut6500812589713462176_nat_o @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_158_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,F2: set_a,A: a,F: a > b] :
( ( group_4866109990395492030noid_b @ M @ Composition @ Unit )
=> ( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ A @ F2 )
=> ( ( member_a_b @ F
@ ( pi_a_b @ F2
@ ^ [Uu: a] : M ) )
=> ( ( member_b @ ( F @ A ) @ M )
=> ( ( commut2218495777586616677mp_b_a @ M @ Composition @ Unit @ F @ ( insert_a @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut2218495777586616677mp_b_a @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_159_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_160_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,F2: set_b,A: b,F: b > b] :
( ( group_4866109990395492030noid_b @ M @ Composition @ Unit )
=> ( ( finite_finite_b @ F2 )
=> ( ~ ( member_b @ A @ F2 )
=> ( ( member_b_b @ F
@ ( pi_b_b @ F2
@ ^ [Uu: b] : M ) )
=> ( ( member_b @ ( F @ A ) @ M )
=> ( ( commut2218495777586616678mp_b_b @ M @ Composition @ Unit @ F @ ( insert_b @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut2218495777586616678mp_b_b @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_161_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,F2: set_b,A: b,F: b > nat] :
( ( group_6791354081887936081id_nat @ M @ Composition @ Unit )
=> ( ( finite_finite_b @ F2 )
=> ( ~ ( member_b @ A @ F2 )
=> ( ( member_b_nat @ F
@ ( pi_b_nat @ F2
@ ^ [Uu: b] : M ) )
=> ( ( member_nat @ ( F @ A ) @ M )
=> ( ( commut1549887680474846983_nat_b @ M @ Composition @ Unit @ F @ ( insert_b @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut1549887680474846983_nat_b @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_162_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,F2: set_nat,A: nat,F: nat > b] :
( ( group_4866109990395492030noid_b @ M @ Composition @ Unit )
=> ( ( finite_finite_nat @ F2 )
=> ( ~ ( member_nat @ A @ F2 )
=> ( ( member_nat_b @ F
@ ( pi_nat_b @ F2
@ ^ [Uu: nat] : M ) )
=> ( ( member_b @ ( F @ A ) @ M )
=> ( ( commut7976772545107730857_b_nat @ M @ Composition @ Unit @ F @ ( insert_nat @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut7976772545107730857_b_nat @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_163_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_164_commutative__monoid_Ofincomp__insert,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F2: set_o,A: $o,F: $o > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( finite_finite_o @ F2 )
=> ( ~ ( member_o @ A @ F2 )
=> ( ( member_o_a @ F
@ ( pi_o_a @ F2
@ ^ [Uu: $o] : M ) )
=> ( ( member_a @ ( F @ A ) @ M )
=> ( ( commut1011387283630023616mp_a_o @ M @ Composition @ Unit @ F @ ( insert_o @ A @ F2 ) )
= ( Composition @ ( F @ A ) @ ( commut1011387283630023616mp_a_o @ M @ Composition @ Unit @ F @ F2 ) ) ) ) ) ) ) ) ).
% commutative_monoid.fincomp_insert
thf(fact_165_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_166_mk__disjoint__insert,axiom,
! [A: $o,A2: set_o] :
( ( member_o @ A @ A2 )
=> ? [B3: set_o] :
( ( A2
= ( insert_o @ A @ B3 ) )
& ~ ( member_o @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_167_mk__disjoint__insert,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ? [B3: set_b] :
( ( A2
= ( insert_b @ A @ B3 ) )
& ~ ( member_b @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_168_mk__disjoint__insert,axiom,
! [A: b > a,A2: set_b_a] :
( ( member_b_a @ A @ A2 )
=> ? [B3: set_b_a] :
( ( A2
= ( insert_b_a @ A @ B3 ) )
& ~ ( member_b_a @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_169_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B3: set_a] :
( ( A2
= ( insert_a @ A @ B3 ) )
& ~ ( member_a @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_170_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B3: set_nat] :
( ( A2
= ( insert_nat @ A @ B3 ) )
& ~ ( member_nat @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_171_insert__commute,axiom,
! [X: b,Y: b,A2: set_b] :
( ( insert_b @ X @ ( insert_b @ Y @ A2 ) )
= ( insert_b @ Y @ ( insert_b @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_172_insert__commute,axiom,
! [X: $o,Y: $o,A2: set_o] :
( ( insert_o @ X @ ( insert_o @ Y @ A2 ) )
= ( insert_o @ Y @ ( insert_o @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_173_insert__eq__iff,axiom,
! [A: $o,A2: set_o,B: $o,B2: set_o] :
( ~ ( member_o @ A @ A2 )
=> ( ~ ( member_o @ B @ B2 )
=> ( ( ( insert_o @ A @ A2 )
= ( insert_o @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A = ~ B )
=> ? [C2: set_o] :
( ( A2
= ( insert_o @ B @ C2 ) )
& ~ ( member_o @ B @ C2 )
& ( B2
= ( insert_o @ A @ C2 ) )
& ~ ( member_o @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_174_insert__eq__iff,axiom,
! [A: b,A2: set_b,B: b,B2: set_b] :
( ~ ( member_b @ A @ A2 )
=> ( ~ ( member_b @ B @ B2 )
=> ( ( ( insert_b @ A @ A2 )
= ( insert_b @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C2: set_b] :
( ( A2
= ( insert_b @ B @ C2 ) )
& ~ ( member_b @ B @ C2 )
& ( B2
= ( insert_b @ A @ C2 ) )
& ~ ( member_b @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_175_insert__eq__iff,axiom,
! [A: b > a,A2: set_b_a,B: b > a,B2: set_b_a] :
( ~ ( member_b_a @ A @ A2 )
=> ( ~ ( member_b_a @ B @ B2 )
=> ( ( ( insert_b_a @ A @ A2 )
= ( insert_b_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C2: set_b_a] :
( ( A2
= ( insert_b_a @ B @ C2 ) )
& ~ ( member_b_a @ B @ C2 )
& ( B2
= ( insert_b_a @ A @ C2 ) )
& ~ ( member_b_a @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_176_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 )
=> ? [C2: set_a] :
( ( A2
= ( insert_a @ B @ C2 ) )
& ~ ( member_a @ B @ C2 )
& ( B2
= ( insert_a @ A @ C2 ) )
& ~ ( member_a @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_177_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 )
=> ? [C2: set_nat] :
( ( A2
= ( insert_nat @ B @ C2 ) )
& ~ ( member_nat @ B @ C2 )
& ( B2
= ( insert_nat @ A @ C2 ) )
& ~ ( member_nat @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_178_insert__absorb,axiom,
! [A: $o,A2: set_o] :
( ( member_o @ A @ A2 )
=> ( ( insert_o @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_179_insert__absorb,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ( ( insert_b @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_180_insert__absorb,axiom,
! [A: b > a,A2: set_b_a] :
( ( member_b_a @ A @ A2 )
=> ( ( insert_b_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_181_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_182_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_183_insert__ident,axiom,
! [X: $o,A2: set_o,B2: set_o] :
( ~ ( member_o @ X @ A2 )
=> ( ~ ( member_o @ X @ B2 )
=> ( ( ( insert_o @ X @ A2 )
= ( insert_o @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_184_insert__ident,axiom,
! [X: b,A2: set_b,B2: set_b] :
( ~ ( member_b @ X @ A2 )
=> ( ~ ( member_b @ X @ B2 )
=> ( ( ( insert_b @ X @ A2 )
= ( insert_b @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_185_insert__ident,axiom,
! [X: b > a,A2: set_b_a,B2: set_b_a] :
( ~ ( member_b_a @ X @ A2 )
=> ( ~ ( member_b_a @ X @ B2 )
=> ( ( ( insert_b_a @ X @ A2 )
= ( insert_b_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_186_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_187_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_188_Set_Oset__insert,axiom,
! [X: $o,A2: set_o] :
( ( member_o @ X @ A2 )
=> ~ ! [B3: set_o] :
( ( A2
= ( insert_o @ X @ B3 ) )
=> ( member_o @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_189_Set_Oset__insert,axiom,
! [X: b,A2: set_b] :
( ( member_b @ X @ A2 )
=> ~ ! [B3: set_b] :
( ( A2
= ( insert_b @ X @ B3 ) )
=> ( member_b @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_190_Set_Oset__insert,axiom,
! [X: b > a,A2: set_b_a] :
( ( member_b_a @ X @ A2 )
=> ~ ! [B3: set_b_a] :
( ( A2
= ( insert_b_a @ X @ B3 ) )
=> ( member_b_a @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_191_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B3: set_a] :
( ( A2
= ( insert_a @ X @ B3 ) )
=> ( member_a @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_192_Set_Oset__insert,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ~ ! [B3: set_nat] :
( ( A2
= ( insert_nat @ X @ B3 ) )
=> ( member_nat @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_193_insertI2,axiom,
! [A: $o,B2: set_o,B: $o] :
( ( member_o @ A @ B2 )
=> ( member_o @ A @ ( insert_o @ B @ B2 ) ) ) ).
% insertI2
thf(fact_194_insertI2,axiom,
! [A: b,B2: set_b,B: b] :
( ( member_b @ A @ B2 )
=> ( member_b @ A @ ( insert_b @ B @ B2 ) ) ) ).
% insertI2
thf(fact_195_insertI2,axiom,
! [A: b > a,B2: set_b_a,B: b > a] :
( ( member_b_a @ A @ B2 )
=> ( member_b_a @ A @ ( insert_b_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_196_insertI2,axiom,
! [A: a,B2: set_a,B: a] :
( ( member_a @ A @ B2 )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_197_insertI2,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( member_nat @ A @ B2 )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_198_insertI1,axiom,
! [A: $o,B2: set_o] : ( member_o @ A @ ( insert_o @ A @ B2 ) ) ).
% insertI1
thf(fact_199_insertI1,axiom,
! [A: b,B2: set_b] : ( member_b @ A @ ( insert_b @ A @ B2 ) ) ).
% insertI1
thf(fact_200_insertI1,axiom,
! [A: b > a,B2: set_b_a] : ( member_b_a @ A @ ( insert_b_a @ A @ B2 ) ) ).
% insertI1
thf(fact_201_insertI1,axiom,
! [A: a,B2: set_a] : ( member_a @ A @ ( insert_a @ A @ B2 ) ) ).
% insertI1
thf(fact_202_insertI1,axiom,
! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_203_insertE,axiom,
! [A: $o,B: $o,A2: set_o] :
( ( member_o @ A @ ( insert_o @ B @ A2 ) )
=> ( ( A = ~ B )
=> ( member_o @ A @ A2 ) ) ) ).
% insertE
thf(fact_204_insertE,axiom,
! [A: b,B: b,A2: set_b] :
( ( member_b @ A @ ( insert_b @ B @ A2 ) )
=> ( ( A != B )
=> ( member_b @ A @ A2 ) ) ) ).
% insertE
thf(fact_205_insertE,axiom,
! [A: b > a,B: b > a,A2: set_b_a] :
( ( member_b_a @ A @ ( insert_b_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_b_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_206_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_207_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_208_insert__Collect,axiom,
! [A: b,P: b > $o] :
( ( insert_b @ A @ ( collect_b @ P ) )
= ( collect_b
@ ^ [U2: b] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_209_insert__Collect,axiom,
! [A: $o,P: $o > $o] :
( ( insert_o @ A @ ( collect_o @ P ) )
= ( collect_o
@ ^ [U2: $o] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_210_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_211_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_212_insert__compr,axiom,
( insert_o
= ( ^ [A3: $o,B4: set_o] :
( collect_o
@ ^ [X3: $o] :
( ( X3 = A3 )
| ( member_o @ X3 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_213_insert__compr,axiom,
( insert_b
= ( ^ [A3: b,B4: set_b] :
( collect_b
@ ^ [X3: b] :
( ( X3 = A3 )
| ( member_b @ X3 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_214_insert__compr,axiom,
( insert_b_a
= ( ^ [A3: b > a,B4: set_b_a] :
( collect_b_a
@ ^ [X3: b > a] :
( ( X3 = A3 )
| ( member_b_a @ X3 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_215_insert__compr,axiom,
( insert_a
= ( ^ [A3: a,B4: set_a] :
( collect_a
@ ^ [X3: a] :
( ( X3 = A3 )
| ( member_a @ X3 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_216_insert__compr,axiom,
( insert_nat
= ( ^ [A3: nat,B4: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( X3 = A3 )
| ( member_nat @ X3 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_217_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_218_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_219_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_220_comp__fun__commute__onI,axiom,
! [F: b > a,F2: set_b] :
( ( member_b_a @ F
@ ( pi_b_a @ F2
@ ^ [Uu: b] : m ) )
=> ( finite9173194153363770127on_b_a @ F2
@ ^ [X3: b,Y4: a] : ( composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ m @ unit @ Y4 ) ) ) ) ).
% comp_fun_commute_onI
thf(fact_221_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_222_fincomp__def,axiom,
! [A2: set_b,F: b > a] :
( ( ( finite_finite_b @ A2 )
=> ( ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ F @ A2 )
= ( finite_fold_b_a
@ ^ [X3: b,Y4: a] : ( composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ m @ unit @ Y4 ) )
@ unit
@ A2 ) ) )
& ( ~ ( finite_finite_b @ A2 )
=> ( ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ F @ A2 )
= unit ) ) ) ).
% fincomp_def
thf(fact_223_Pi__eq__empty,axiom,
! [A2: set_b,B2: b > set_a] :
( ( ( pi_b_a @ A2 @ B2 )
= bot_bot_set_b_a )
= ( ? [X3: b] :
( ( member_b @ X3 @ A2 )
& ( ( B2 @ X3 )
= bot_bot_set_a ) ) ) ) ).
% Pi_eq_empty
thf(fact_224_Pi__split__insert__domain,axiom,
! [X: b > b,I2: b,I3: set_b,X4: b > set_b] :
( ( member_b_b @ X @ ( pi_b_b @ ( insert_b @ I2 @ I3 ) @ X4 ) )
= ( ( member_b_b @ X @ ( pi_b_b @ I3 @ X4 ) )
& ( member_b @ ( X @ I2 ) @ ( X4 @ I2 ) ) ) ) ).
% Pi_split_insert_domain
thf(fact_225_Pi__split__insert__domain,axiom,
! [X: $o > b,I2: $o,I3: set_o,X4: $o > set_b] :
( ( member_o_b @ X @ ( pi_o_b @ ( insert_o @ I2 @ I3 ) @ X4 ) )
= ( ( member_o_b @ X @ ( pi_o_b @ I3 @ X4 ) )
& ( member_b @ ( X @ I2 ) @ ( X4 @ I2 ) ) ) ) ).
% Pi_split_insert_domain
thf(fact_226_Pi__split__insert__domain,axiom,
! [X: b > b > a,I2: b,I3: set_b,X4: b > set_b_a] :
( ( member_b_b_a @ X @ ( pi_b_b_a @ ( insert_b @ I2 @ I3 ) @ X4 ) )
= ( ( member_b_b_a @ X @ ( pi_b_b_a @ I3 @ X4 ) )
& ( member_b_a @ ( X @ I2 ) @ ( X4 @ I2 ) ) ) ) ).
% Pi_split_insert_domain
thf(fact_227_Pi__split__insert__domain,axiom,
! [X: $o > b > a,I2: $o,I3: set_o,X4: $o > set_b_a] :
( ( member_o_b_a @ X @ ( pi_o_b_a @ ( insert_o @ I2 @ I3 ) @ X4 ) )
= ( ( member_o_b_a @ X @ ( pi_o_b_a @ I3 @ X4 ) )
& ( member_b_a @ ( X @ I2 ) @ ( X4 @ I2 ) ) ) ) ).
% Pi_split_insert_domain
thf(fact_228_Pi__split__insert__domain,axiom,
! [X: $o > a,I2: $o,I3: set_o,X4: $o > set_a] :
( ( member_o_a @ X @ ( pi_o_a @ ( insert_o @ I2 @ I3 ) @ X4 ) )
= ( ( member_o_a @ X @ ( pi_o_a @ I3 @ X4 ) )
& ( member_a @ ( X @ I2 ) @ ( X4 @ I2 ) ) ) ) ).
% Pi_split_insert_domain
thf(fact_229_Pi__split__insert__domain,axiom,
! [X: b > nat,I2: b,I3: set_b,X4: b > set_nat] :
( ( member_b_nat @ X @ ( pi_b_nat @ ( insert_b @ I2 @ I3 ) @ X4 ) )
= ( ( member_b_nat @ X @ ( pi_b_nat @ I3 @ X4 ) )
& ( member_nat @ ( X @ I2 ) @ ( X4 @ I2 ) ) ) ) ).
% Pi_split_insert_domain
thf(fact_230_Pi__split__insert__domain,axiom,
! [X: $o > nat,I2: $o,I3: set_o,X4: $o > set_nat] :
( ( member_o_nat @ X @ ( pi_o_nat @ ( insert_o @ I2 @ I3 ) @ X4 ) )
= ( ( member_o_nat @ X @ ( pi_o_nat @ I3 @ X4 ) )
& ( member_nat @ ( X @ I2 ) @ ( X4 @ I2 ) ) ) ) ).
% Pi_split_insert_domain
thf(fact_231_Pi__split__insert__domain,axiom,
! [X: b > a,I2: b,I3: set_b,X4: b > set_a] :
( ( member_b_a @ X @ ( pi_b_a @ ( insert_b @ I2 @ I3 ) @ X4 ) )
= ( ( member_b_a @ X @ ( pi_b_a @ I3 @ X4 ) )
& ( member_a @ ( X @ I2 ) @ ( X4 @ I2 ) ) ) ) ).
% Pi_split_insert_domain
thf(fact_232_finite__insert,axiom,
! [A: $o,A2: set_o] :
( ( finite_finite_o @ ( insert_o @ A @ A2 ) )
= ( finite_finite_o @ A2 ) ) ).
% finite_insert
thf(fact_233_finite__insert,axiom,
! [A: b,A2: set_b] :
( ( finite_finite_b @ ( insert_b @ A @ A2 ) )
= ( finite_finite_b @ A2 ) ) ).
% finite_insert
thf(fact_234_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_235_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_236_finite__Collect__disjI,axiom,
! [P: b > $o,Q: b > $o] :
( ( finite_finite_b
@ ( collect_b
@ ^ [X3: b] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_b @ ( collect_b @ P ) )
& ( finite_finite_b @ ( collect_b @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_237_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_238_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_239_finite__Collect__conjI,axiom,
! [P: b > $o,Q: b > $o] :
( ( ( finite_finite_b @ ( collect_b @ P ) )
| ( finite_finite_b @ ( collect_b @ Q ) ) )
=> ( finite_finite_b
@ ( collect_b
@ ^ [X3: b] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_240_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_241_monoid_Oinvertible__left__inverse,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ U )
=> ( ( member_b @ U @ M )
=> ( ( Composition @ ( group_inverse_b @ M @ Composition @ Unit @ U ) @ U )
= Unit ) ) ) ) ).
% monoid.invertible_left_inverse
thf(fact_242_monoid_Oinvertible__left__inverse,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ U )
=> ( ( member_b_a @ U @ M )
=> ( ( Composition @ ( group_inverse_b_a @ M @ Composition @ Unit @ U ) @ U )
= Unit ) ) ) ) ).
% monoid.invertible_left_inverse
thf(fact_243_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_244_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_245_Pi__I,axiom,
! [A2: set_b,F: b > b,B2: b > set_b] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_b_b @ F @ ( pi_b_b @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_246_Pi__I,axiom,
! [A2: set_b,F: b > a,B2: b > set_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_b_a @ F @ ( pi_b_a @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_247_Pi__I,axiom,
! [A2: set_b,F: b > nat,B2: b > set_nat] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_b_nat @ F @ ( pi_b_nat @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_248_Pi__I,axiom,
! [A2: set_a,F: a > b,B2: a > set_b] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_a_b @ F @ ( pi_a_b @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_249_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_250_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_251_Pi__I,axiom,
! [A2: set_nat,F: nat > b,B2: nat > set_b] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_nat_b @ F @ ( pi_nat_b @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_252_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_253_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_254_Pi__I,axiom,
! [A2: set_b,F: b > b > a,B2: b > set_b_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_b_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_b_b_a @ F @ ( pi_b_b_a @ A2 @ B2 ) ) ) ).
% Pi_I
thf(fact_255_fold__closed__eq,axiom,
! [A2: set_b,B2: set_b,F: b > b > b,G: b > b > b,Z2: b] :
( ! [A4: b,B5: b] :
( ( member_b @ A4 @ A2 )
=> ( ( member_b @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: b,B5: b] :
( ( member_b @ A4 @ A2 )
=> ( ( member_b @ B5 @ B2 )
=> ( member_b @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_b @ Z2 @ B2 )
=> ( ( finite_fold_b_b @ F @ Z2 @ A2 )
= ( finite_fold_b_b @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_256_fold__closed__eq,axiom,
! [A2: set_b,B2: set_a,F: b > a > a,G: b > a > a,Z2: a] :
( ! [A4: b,B5: a] :
( ( member_b @ A4 @ A2 )
=> ( ( member_a @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: b,B5: a] :
( ( member_b @ A4 @ A2 )
=> ( ( member_a @ B5 @ B2 )
=> ( member_a @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_a @ Z2 @ B2 )
=> ( ( finite_fold_b_a @ F @ Z2 @ A2 )
= ( finite_fold_b_a @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_257_fold__closed__eq,axiom,
! [A2: set_b,B2: set_nat,F: b > nat > nat,G: b > nat > nat,Z2: nat] :
( ! [A4: b,B5: nat] :
( ( member_b @ A4 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: b,B5: nat] :
( ( member_b @ A4 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( member_nat @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_nat @ Z2 @ B2 )
=> ( ( finite_fold_b_nat @ F @ Z2 @ A2 )
= ( finite_fold_b_nat @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_258_fold__closed__eq,axiom,
! [A2: set_a,B2: set_b,F: a > b > b,G: a > b > b,Z2: b] :
( ! [A4: a,B5: b] :
( ( member_a @ A4 @ A2 )
=> ( ( member_b @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: a,B5: b] :
( ( member_a @ A4 @ A2 )
=> ( ( member_b @ B5 @ B2 )
=> ( member_b @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_b @ Z2 @ B2 )
=> ( ( finite_fold_a_b @ F @ Z2 @ A2 )
= ( finite_fold_a_b @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_259_fold__closed__eq,axiom,
! [A2: set_a,B2: set_a,F: a > a > a,G: a > a > a,Z2: a] :
( ! [A4: a,B5: a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_a @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: a,B5: a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_a @ B5 @ B2 )
=> ( member_a @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_a @ Z2 @ B2 )
=> ( ( finite_fold_a_a @ F @ Z2 @ A2 )
= ( finite_fold_a_a @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_260_fold__closed__eq,axiom,
! [A2: set_a,B2: set_nat,F: a > nat > nat,G: a > nat > nat,Z2: nat] :
( ! [A4: a,B5: nat] :
( ( member_a @ A4 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: a,B5: nat] :
( ( member_a @ A4 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( member_nat @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_nat @ Z2 @ B2 )
=> ( ( finite_fold_a_nat @ F @ Z2 @ A2 )
= ( finite_fold_a_nat @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_261_fold__closed__eq,axiom,
! [A2: set_nat,B2: set_b,F: nat > b > b,G: nat > b > b,Z2: b] :
( ! [A4: nat,B5: b] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_b @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: nat,B5: b] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_b @ B5 @ B2 )
=> ( member_b @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_b @ Z2 @ B2 )
=> ( ( finite_fold_nat_b @ F @ Z2 @ A2 )
= ( finite_fold_nat_b @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_262_fold__closed__eq,axiom,
! [A2: set_nat,B2: set_a,F: nat > a > a,G: nat > a > a,Z2: a] :
( ! [A4: nat,B5: a] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_a @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: nat,B5: a] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_a @ B5 @ B2 )
=> ( member_a @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_a @ Z2 @ B2 )
=> ( ( finite_fold_nat_a @ F @ Z2 @ A2 )
= ( finite_fold_nat_a @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_263_fold__closed__eq,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat > nat,G: nat > nat > nat,Z2: nat] :
( ! [A4: nat,B5: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: nat,B5: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( member_nat @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_nat @ Z2 @ B2 )
=> ( ( finite_fold_nat_nat @ F @ Z2 @ A2 )
= ( finite_fold_nat_nat @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_264_fold__closed__eq,axiom,
! [A2: set_b,B2: set_b_a,F: b > ( b > a ) > b > a,G: b > ( b > a ) > b > a,Z2: b > a] :
( ! [A4: b,B5: b > a] :
( ( member_b @ A4 @ A2 )
=> ( ( member_b_a @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: b,B5: b > a] :
( ( member_b @ A4 @ A2 )
=> ( ( member_b_a @ B5 @ B2 )
=> ( member_b_a @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_b_a @ Z2 @ B2 )
=> ( ( finite_fold_b_b_a @ F @ Z2 @ A2 )
= ( finite_fold_b_b_a @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_265_monoid_OUnits_Ocong,axiom,
group_Units_a = group_Units_a ).
% monoid.Units.cong
thf(fact_266_monoid_Omem__UnitsI,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ U )
=> ( ( member_b @ U @ M )
=> ( member_b @ U @ ( group_Units_b @ M @ Composition @ Unit ) ) ) ) ) ).
% monoid.mem_UnitsI
thf(fact_267_monoid_Omem__UnitsI,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ U )
=> ( ( member_b_a @ U @ M )
=> ( member_b_a @ U @ ( group_Units_b_a @ M @ Composition @ Unit ) ) ) ) ) ).
% monoid.mem_UnitsI
thf(fact_268_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_269_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_270_monoid_Omem__UnitsD,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( member_b @ U @ ( group_Units_b @ M @ Composition @ Unit ) )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ U )
& ( member_b @ U @ M ) ) ) ) ).
% monoid.mem_UnitsD
thf(fact_271_monoid_Omem__UnitsD,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( member_b_a @ U @ ( group_Units_b_a @ M @ Composition @ Unit ) )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ U )
& ( member_b_a @ U @ M ) ) ) ) ).
% monoid.mem_UnitsD
thf(fact_272_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_273_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_274_monoid_OUnits__def,axiom,
! [M: set_b,Composition: b > b > b,Unit: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_Units_b @ M @ Composition @ Unit )
= ( collect_b
@ ^ [U2: b] :
( ( member_b @ U2 @ M )
& ( group_invertible_b @ M @ Composition @ Unit @ U2 ) ) ) ) ) ).
% monoid.Units_def
thf(fact_275_monoid_OUnits__def,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_Units_b_a @ M @ Composition @ Unit )
= ( collect_b_a
@ ^ [U2: b > a] :
( ( member_b_a @ U2 @ M )
& ( group_invertible_b_a @ M @ Composition @ Unit @ U2 ) ) ) ) ) ).
% monoid.Units_def
thf(fact_276_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_277_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_278_Pi__cong,axiom,
! [A2: set_b,F: b > a,G: b > a,B2: b > set_a] :
( ! [W: b] :
( ( member_b @ W @ A2 )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( member_b_a @ F @ ( pi_b_a @ A2 @ B2 ) )
= ( member_b_a @ G @ ( pi_b_a @ A2 @ B2 ) ) ) ) ).
% Pi_cong
thf(fact_279_Pi__mem,axiom,
! [F: b > b,A2: set_b,B2: b > set_b,X: b] :
( ( member_b_b @ F @ ( pi_b_b @ A2 @ B2 ) )
=> ( ( member_b @ X @ A2 )
=> ( member_b @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_280_Pi__mem,axiom,
! [F: b > nat,A2: set_b,B2: b > set_nat,X: b] :
( ( member_b_nat @ F @ ( pi_b_nat @ A2 @ B2 ) )
=> ( ( member_b @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_281_Pi__mem,axiom,
! [F: a > b,A2: set_a,B2: a > set_b,X: a] :
( ( member_a_b @ F @ ( pi_a_b @ A2 @ B2 ) )
=> ( ( member_a @ X @ A2 )
=> ( member_b @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_282_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_283_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_284_Pi__mem,axiom,
! [F: nat > b,A2: set_nat,B2: nat > set_b,X: nat] :
( ( member_nat_b @ F @ ( pi_nat_b @ A2 @ B2 ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_b @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_285_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_286_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_287_Pi__mem,axiom,
! [F: b > a,A2: set_b,B2: b > set_a,X: b] :
( ( member_b_a @ F @ ( pi_b_a @ A2 @ B2 ) )
=> ( ( member_b @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_288_Pi__mem,axiom,
! [F: b > b > a,A2: set_b,B2: b > set_b_a,X: b] :
( ( member_b_b_a @ F @ ( pi_b_b_a @ A2 @ B2 ) )
=> ( ( member_b @ X @ A2 )
=> ( member_b_a @ ( F @ X ) @ ( B2 @ X ) ) ) ) ).
% Pi_mem
thf(fact_289_Pi__iff,axiom,
! [F: b > a,I3: set_b,X4: b > set_a] :
( ( member_b_a @ F @ ( pi_b_a @ I3 @ X4 ) )
= ( ! [X3: b] :
( ( member_b @ X3 @ I3 )
=> ( member_a @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% Pi_iff
thf(fact_290_Pi__I_H,axiom,
! [A2: set_b,F: b > b,B2: b > set_b] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_b_b @ F @ ( pi_b_b @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_291_Pi__I_H,axiom,
! [A2: set_b,F: b > a,B2: b > set_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_b_a @ F @ ( pi_b_a @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_292_Pi__I_H,axiom,
! [A2: set_b,F: b > nat,B2: b > set_nat] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_b_nat @ F @ ( pi_b_nat @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_293_Pi__I_H,axiom,
! [A2: set_a,F: a > b,B2: a > set_b] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_a_b @ F @ ( pi_a_b @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_294_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_295_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_296_Pi__I_H,axiom,
! [A2: set_nat,F: nat > b,B2: nat > set_b] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_nat_b @ F @ ( pi_nat_b @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_297_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_298_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_299_Pi__I_H,axiom,
! [A2: set_b,F: b > b > a,B2: b > set_b_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_b_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_b_b_a @ F @ ( pi_b_b_a @ A2 @ B2 ) ) ) ).
% Pi_I'
thf(fact_300_PiE,axiom,
! [F: b > b,A2: set_b,B2: b > set_b,X: b] :
( ( member_b_b @ F @ ( pi_b_b @ A2 @ B2 ) )
=> ( ~ ( member_b @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_b @ X @ A2 ) ) ) ).
% PiE
thf(fact_301_PiE,axiom,
! [F: a > b,A2: set_a,B2: a > set_b,X: a] :
( ( member_a_b @ F @ ( pi_a_b @ A2 @ B2 ) )
=> ( ~ ( member_b @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_a @ X @ A2 ) ) ) ).
% PiE
thf(fact_302_PiE,axiom,
! [F: nat > b,A2: set_nat,B2: nat > set_b,X: nat] :
( ( member_nat_b @ F @ ( pi_nat_b @ A2 @ B2 ) )
=> ( ~ ( member_b @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ).
% PiE
thf(fact_303_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_304_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_305_PiE,axiom,
! [F: b > nat,A2: set_b,B2: b > set_nat,X: b] :
( ( member_b_nat @ F @ ( pi_b_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_b @ X @ A2 ) ) ) ).
% PiE
thf(fact_306_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_307_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_308_PiE,axiom,
! [F: b > a,A2: set_b,B2: b > set_a,X: b] :
( ( member_b_a @ F @ ( pi_b_a @ A2 @ B2 ) )
=> ( ~ ( member_a @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_b @ X @ A2 ) ) ) ).
% PiE
thf(fact_309_PiE,axiom,
! [F: ( b > a ) > b,A2: set_b_a,B2: ( b > a ) > set_b,X: b > a] :
( ( member_b_a_b @ F @ ( pi_b_a_b @ A2 @ B2 ) )
=> ( ~ ( member_b @ ( F @ X ) @ ( B2 @ X ) )
=> ~ ( member_b_a @ X @ A2 ) ) ) ).
% PiE
thf(fact_310_Group__Theory_Omonoid__def,axiom,
( group_monoid_b
= ( ^ [M2: set_b,Composition2: b > b > b,Unit2: b] :
( ! [A3: b,B6: b] :
( ( member_b @ A3 @ M2 )
=> ( ( member_b @ B6 @ M2 )
=> ( member_b @ ( Composition2 @ A3 @ B6 ) @ M2 ) ) )
& ( member_b @ Unit2 @ M2 )
& ! [A3: b,B6: b,C3: b] :
( ( member_b @ A3 @ M2 )
=> ( ( member_b @ B6 @ M2 )
=> ( ( member_b @ C3 @ M2 )
=> ( ( Composition2 @ ( Composition2 @ A3 @ B6 ) @ C3 )
= ( Composition2 @ A3 @ ( Composition2 @ B6 @ C3 ) ) ) ) ) )
& ! [A3: b] :
( ( member_b @ A3 @ M2 )
=> ( ( Composition2 @ Unit2 @ A3 )
= A3 ) )
& ! [A3: b] :
( ( member_b @ A3 @ M2 )
=> ( ( Composition2 @ A3 @ Unit2 )
= A3 ) ) ) ) ) ).
% Group_Theory.monoid_def
thf(fact_311_Group__Theory_Omonoid__def,axiom,
( group_monoid_b_a
= ( ^ [M2: set_b_a,Composition2: ( b > a ) > ( b > a ) > b > a,Unit2: b > a] :
( ! [A3: b > a,B6: b > a] :
( ( member_b_a @ A3 @ M2 )
=> ( ( member_b_a @ B6 @ M2 )
=> ( member_b_a @ ( Composition2 @ A3 @ B6 ) @ M2 ) ) )
& ( member_b_a @ Unit2 @ M2 )
& ! [A3: b > a,B6: b > a,C3: b > a] :
( ( member_b_a @ A3 @ M2 )
=> ( ( member_b_a @ B6 @ M2 )
=> ( ( member_b_a @ C3 @ M2 )
=> ( ( Composition2 @ ( Composition2 @ A3 @ B6 ) @ C3 )
= ( Composition2 @ A3 @ ( Composition2 @ B6 @ C3 ) ) ) ) ) )
& ! [A3: b > a] :
( ( member_b_a @ A3 @ M2 )
=> ( ( Composition2 @ Unit2 @ A3 )
= A3 ) )
& ! [A3: b > a] :
( ( member_b_a @ A3 @ M2 )
=> ( ( Composition2 @ A3 @ Unit2 )
= A3 ) ) ) ) ) ).
% Group_Theory.monoid_def
thf(fact_312_Group__Theory_Omonoid__def,axiom,
( group_monoid_nat
= ( ^ [M2: set_nat,Composition2: nat > nat > nat,Unit2: nat] :
( ! [A3: nat,B6: nat] :
( ( member_nat @ A3 @ M2 )
=> ( ( member_nat @ B6 @ M2 )
=> ( member_nat @ ( Composition2 @ A3 @ B6 ) @ M2 ) ) )
& ( member_nat @ Unit2 @ M2 )
& ! [A3: nat,B6: nat,C3: nat] :
( ( member_nat @ A3 @ M2 )
=> ( ( member_nat @ B6 @ M2 )
=> ( ( member_nat @ C3 @ M2 )
=> ( ( Composition2 @ ( Composition2 @ A3 @ B6 ) @ C3 )
= ( Composition2 @ A3 @ ( Composition2 @ B6 @ C3 ) ) ) ) ) )
& ! [A3: nat] :
( ( member_nat @ A3 @ M2 )
=> ( ( Composition2 @ Unit2 @ A3 )
= A3 ) )
& ! [A3: nat] :
( ( member_nat @ A3 @ M2 )
=> ( ( Composition2 @ A3 @ Unit2 )
= A3 ) ) ) ) ) ).
% Group_Theory.monoid_def
thf(fact_313_Group__Theory_Omonoid__def,axiom,
( group_monoid_a
= ( ^ [M2: set_a,Composition2: a > a > a,Unit2: a] :
( ! [A3: a,B6: a] :
( ( member_a @ A3 @ M2 )
=> ( ( member_a @ B6 @ M2 )
=> ( member_a @ ( Composition2 @ A3 @ B6 ) @ M2 ) ) )
& ( member_a @ Unit2 @ M2 )
& ! [A3: a,B6: a,C3: a] :
( ( member_a @ A3 @ M2 )
=> ( ( member_a @ B6 @ M2 )
=> ( ( member_a @ C3 @ M2 )
=> ( ( Composition2 @ ( Composition2 @ A3 @ B6 ) @ C3 )
= ( Composition2 @ A3 @ ( Composition2 @ B6 @ C3 ) ) ) ) ) )
& ! [A3: a] :
( ( member_a @ A3 @ M2 )
=> ( ( Composition2 @ Unit2 @ A3 )
= A3 ) )
& ! [A3: a] :
( ( member_a @ A3 @ M2 )
=> ( ( Composition2 @ A3 @ Unit2 )
= A3 ) ) ) ) ) ).
% Group_Theory.monoid_def
thf(fact_314_monoid_Ocomposition__closed,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,A: b,B: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( member_b @ A @ M )
=> ( ( member_b @ B @ M )
=> ( member_b @ ( Composition @ A @ B ) @ M ) ) ) ) ).
% monoid.composition_closed
thf(fact_315_monoid_Ocomposition__closed,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,A: b > a,B: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( member_b_a @ A @ M )
=> ( ( member_b_a @ B @ M )
=> ( member_b_a @ ( Composition @ A @ B ) @ M ) ) ) ) ).
% monoid.composition_closed
thf(fact_316_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_317_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_318_monoid_Oinverse__unique,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b,V: b,V2: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_b @ U @ M )
=> ( ( member_b @ V2 @ M )
=> ( ( member_b @ V @ M )
=> ( V2 = V ) ) ) ) ) ) ) ).
% monoid.inverse_unique
thf(fact_319_monoid_Oinverse__unique,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a,V: b > a,V2: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_b_a @ U @ M )
=> ( ( member_b_a @ V2 @ M )
=> ( ( member_b_a @ V @ M )
=> ( V2 = V ) ) ) ) ) ) ) ).
% monoid.inverse_unique
thf(fact_320_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_321_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_322_monoid_Ounit__closed,axiom,
! [M: set_b,Composition: b > b > b,Unit: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( member_b @ Unit @ M ) ) ).
% monoid.unit_closed
thf(fact_323_monoid_Ounit__closed,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( member_b_a @ Unit @ M ) ) ).
% monoid.unit_closed
thf(fact_324_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_325_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_326_monoid_Oassociative,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,A: b,B: b,C: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( member_b @ A @ M )
=> ( ( member_b @ B @ M )
=> ( ( member_b @ C @ M )
=> ( ( Composition @ ( Composition @ A @ B ) @ C )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% monoid.associative
thf(fact_327_monoid_Oassociative,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,A: b > a,B: b > a,C: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( member_b_a @ A @ M )
=> ( ( member_b_a @ B @ M )
=> ( ( member_b_a @ C @ M )
=> ( ( Composition @ ( Composition @ A @ B ) @ C )
= ( Composition @ A @ ( Composition @ B @ C ) ) ) ) ) ) ) ).
% monoid.associative
thf(fact_328_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_329_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_330_monoid_Oright__unit,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,A: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( member_b @ A @ M )
=> ( ( Composition @ A @ Unit )
= A ) ) ) ).
% monoid.right_unit
thf(fact_331_monoid_Oright__unit,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,A: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( member_b_a @ A @ M )
=> ( ( Composition @ A @ Unit )
= A ) ) ) ).
% monoid.right_unit
thf(fact_332_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_333_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_334_monoid_Oleft__unit,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,A: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( member_b @ A @ M )
=> ( ( Composition @ Unit @ A )
= A ) ) ) ).
% monoid.left_unit
thf(fact_335_monoid_Oleft__unit,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,A: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( member_b_a @ A @ M )
=> ( ( Composition @ Unit @ A )
= A ) ) ) ).
% monoid.left_unit
thf(fact_336_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_337_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_338_Group__Theory_Omonoid_Ointro,axiom,
! [M: set_b,Composition: b > b > b,Unit: b] :
( ! [A4: b,B5: b] :
( ( member_b @ A4 @ M )
=> ( ( member_b @ B5 @ M )
=> ( member_b @ ( Composition @ A4 @ B5 ) @ M ) ) )
=> ( ( member_b @ Unit @ M )
=> ( ! [A4: b,B5: b,C4: b] :
( ( member_b @ A4 @ M )
=> ( ( member_b @ B5 @ M )
=> ( ( member_b @ C4 @ M )
=> ( ( Composition @ ( Composition @ A4 @ B5 ) @ C4 )
= ( Composition @ A4 @ ( Composition @ B5 @ C4 ) ) ) ) ) )
=> ( ! [A4: b] :
( ( member_b @ A4 @ M )
=> ( ( Composition @ Unit @ A4 )
= A4 ) )
=> ( ! [A4: b] :
( ( member_b @ A4 @ M )
=> ( ( Composition @ A4 @ Unit )
= A4 ) )
=> ( group_monoid_b @ M @ Composition @ Unit ) ) ) ) ) ) ).
% Group_Theory.monoid.intro
thf(fact_339_Group__Theory_Omonoid_Ointro,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a] :
( ! [A4: b > a,B5: b > a] :
( ( member_b_a @ A4 @ M )
=> ( ( member_b_a @ B5 @ M )
=> ( member_b_a @ ( Composition @ A4 @ B5 ) @ M ) ) )
=> ( ( member_b_a @ Unit @ M )
=> ( ! [A4: b > a,B5: b > a,C4: b > a] :
( ( member_b_a @ A4 @ M )
=> ( ( member_b_a @ B5 @ M )
=> ( ( member_b_a @ C4 @ M )
=> ( ( Composition @ ( Composition @ A4 @ B5 ) @ C4 )
= ( Composition @ A4 @ ( Composition @ B5 @ C4 ) ) ) ) ) )
=> ( ! [A4: b > a] :
( ( member_b_a @ A4 @ M )
=> ( ( Composition @ Unit @ A4 )
= A4 ) )
=> ( ! [A4: b > a] :
( ( member_b_a @ A4 @ M )
=> ( ( Composition @ A4 @ Unit )
= A4 ) )
=> ( group_monoid_b_a @ M @ Composition @ Unit ) ) ) ) ) ) ).
% Group_Theory.monoid.intro
thf(fact_340_Group__Theory_Omonoid_Ointro,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat] :
( ! [A4: nat,B5: nat] :
( ( member_nat @ A4 @ M )
=> ( ( member_nat @ B5 @ M )
=> ( member_nat @ ( Composition @ A4 @ B5 ) @ M ) ) )
=> ( ( member_nat @ Unit @ M )
=> ( ! [A4: nat,B5: nat,C4: nat] :
( ( member_nat @ A4 @ M )
=> ( ( member_nat @ B5 @ M )
=> ( ( member_nat @ C4 @ M )
=> ( ( Composition @ ( Composition @ A4 @ B5 ) @ C4 )
= ( Composition @ A4 @ ( Composition @ B5 @ C4 ) ) ) ) ) )
=> ( ! [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_341_Group__Theory_Omonoid_Ointro,axiom,
! [M: set_a,Composition: a > a > a,Unit: a] :
( ! [A4: a,B5: a] :
( ( member_a @ A4 @ M )
=> ( ( member_a @ B5 @ M )
=> ( member_a @ ( Composition @ A4 @ B5 ) @ M ) ) )
=> ( ( member_a @ Unit @ M )
=> ( ! [A4: a,B5: a,C4: a] :
( ( member_a @ A4 @ M )
=> ( ( member_a @ B5 @ M )
=> ( ( member_a @ C4 @ M )
=> ( ( Composition @ ( Composition @ A4 @ B5 ) @ C4 )
= ( Composition @ A4 @ ( Composition @ B5 @ C4 ) ) ) ) ) )
=> ( ! [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_342_monoid_Oinvertible_Ocong,axiom,
group_invertible_a = group_invertible_a ).
% monoid.invertible.cong
thf(fact_343_monoid_Oinverse_Ocong,axiom,
group_inverse_a = group_inverse_a ).
% monoid.inverse.cong
thf(fact_344_commutative__monoid_Ocommutative,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,X: b,Y: b] :
( ( group_4866109990395492030noid_b @ M @ Composition @ Unit )
=> ( ( member_b @ X @ M )
=> ( ( member_b @ Y @ M )
=> ( ( Composition @ X @ Y )
= ( Composition @ Y @ X ) ) ) ) ) ).
% commutative_monoid.commutative
thf(fact_345_commutative__monoid_Ocommutative,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,X: b > a,Y: b > a] :
( ( group_4188790030012530981id_b_a @ M @ Composition @ Unit )
=> ( ( member_b_a @ X @ M )
=> ( ( member_b_a @ Y @ M )
=> ( ( Composition @ X @ Y )
= ( Composition @ Y @ X ) ) ) ) ) ).
% commutative_monoid.commutative
thf(fact_346_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_347_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_348_not__finite__existsD,axiom,
! [P: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P ) )
=> ? [X_1: a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_349_not__finite__existsD,axiom,
! [P: b > $o] :
( ~ ( finite_finite_b @ ( collect_b @ P ) )
=> ? [X_1: b] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_350_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_351_pigeonhole__infinite__rel,axiom,
! [A2: set_b_a,B2: set_b,R: ( b > a ) > b > $o] :
( ~ ( finite_finite_b_a @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ! [X2: b > a] :
( ( member_b_a @ X2 @ A2 )
=> ? [Xa: b] :
( ( member_b @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: b] :
( ( member_b @ X2 @ B2 )
& ~ ( finite_finite_b_a
@ ( collect_b_a
@ ^ [A3: b > a] :
( ( member_b_a @ A3 @ A2 )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_352_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B2: set_b,R: a > b > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ? [Xa: b] :
( ( member_b @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: b] :
( ( member_b @ X2 @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A3: a] :
( ( member_a @ A3 @ A2 )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_353_pigeonhole__infinite__rel,axiom,
! [A2: set_b_a,B2: set_nat,R: ( b > a ) > nat > $o] :
( ~ ( finite_finite_b_a @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: b > a] :
( ( member_b_a @ X2 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite_finite_b_a
@ ( collect_b_a
@ ^ [A3: b > a] :
( ( member_b_a @ A3 @ A2 )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_354_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_355_pigeonhole__infinite__rel,axiom,
! [A2: set_b,B2: set_b,R: b > b > $o] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ? [Xa: b] :
( ( member_b @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: b] :
( ( member_b @ X2 @ B2 )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A3: b] :
( ( member_b @ A3 @ A2 )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_356_pigeonhole__infinite__rel,axiom,
! [A2: set_b,B2: set_nat,R: b > nat > $o] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A3: b] :
( ( member_b @ A3 @ A2 )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_357_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_b,R: nat > b > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ? [Xa: b] :
( ( member_b @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: b] :
( ( member_b @ X2 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_358_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_359_commutative__monoid_Ocomp__fun__commute__onI,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,F: b > a,F2: set_b] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( member_b_a @ F
@ ( pi_b_a @ F2
@ ^ [Uu: b] : M ) )
=> ( finite9173194153363770127on_b_a @ F2
@ ^ [X3: b,Y4: a] : ( Composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ M @ Unit @ Y4 ) ) ) ) ) ).
% commutative_monoid.comp_fun_commute_onI
thf(fact_360_funcsetI,axiom,
! [A2: set_b,F: b > b,B2: set_b] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ B2 ) )
=> ( member_b_b @ F
@ ( pi_b_b @ A2
@ ^ [Uu: b] : B2 ) ) ) ).
% funcsetI
thf(fact_361_funcsetI,axiom,
! [A2: set_b,F: b > a,B2: set_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ B2 ) )
=> ( member_b_a @ F
@ ( pi_b_a @ A2
@ ^ [Uu: b] : B2 ) ) ) ).
% funcsetI
thf(fact_362_funcsetI,axiom,
! [A2: set_b,F: b > nat,B2: set_nat] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( member_b_nat @ F
@ ( pi_b_nat @ A2
@ ^ [Uu: b] : B2 ) ) ) ).
% funcsetI
thf(fact_363_funcsetI,axiom,
! [A2: set_a,F: a > b,B2: set_b] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ B2 ) )
=> ( member_a_b @ F
@ ( pi_a_b @ A2
@ ^ [Uu: a] : B2 ) ) ) ).
% funcsetI
thf(fact_364_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_365_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_366_funcsetI,axiom,
! [A2: set_nat,F: nat > b,B2: set_b] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ B2 ) )
=> ( member_nat_b @ F
@ ( pi_nat_b @ A2
@ ^ [Uu: nat] : B2 ) ) ) ).
% funcsetI
thf(fact_367_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_368_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_369_funcsetI,axiom,
! [A2: set_b,F: b > b > a,B2: set_b_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_b_a @ ( F @ X2 ) @ B2 ) )
=> ( member_b_b_a @ F
@ ( pi_b_b_a @ A2
@ ^ [Uu: b] : B2 ) ) ) ).
% funcsetI
thf(fact_370_funcset__mem,axiom,
! [F: b > b,A2: set_b,B2: set_b,X: b] :
( ( member_b_b @ F
@ ( pi_b_b @ A2
@ ^ [Uu: b] : B2 ) )
=> ( ( member_b @ X @ A2 )
=> ( member_b @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_371_funcset__mem,axiom,
! [F: b > nat,A2: set_b,B2: set_nat,X: b] :
( ( member_b_nat @ F
@ ( pi_b_nat @ A2
@ ^ [Uu: b] : B2 ) )
=> ( ( member_b @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_372_funcset__mem,axiom,
! [F: a > b,A2: set_a,B2: set_b,X: a] :
( ( member_a_b @ F
@ ( pi_a_b @ A2
@ ^ [Uu: a] : B2 ) )
=> ( ( member_a @ X @ A2 )
=> ( member_b @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_373_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_374_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_375_funcset__mem,axiom,
! [F: nat > b,A2: set_nat,B2: set_b,X: nat] :
( ( member_nat_b @ F
@ ( pi_nat_b @ A2
@ ^ [Uu: nat] : B2 ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_b @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_376_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_377_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_378_funcset__mem,axiom,
! [F: b > a,A2: set_b,B2: set_a,X: b] :
( ( member_b_a @ F
@ ( pi_b_a @ A2
@ ^ [Uu: b] : B2 ) )
=> ( ( member_b @ X @ A2 )
=> ( member_a @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_379_funcset__mem,axiom,
! [F: b > b > a,A2: set_b,B2: set_b_a,X: b] :
( ( member_b_b_a @ F
@ ( pi_b_b_a @ A2
@ ^ [Uu: b] : B2 ) )
=> ( ( member_b @ X @ A2 )
=> ( member_b_a @ ( F @ X ) @ B2 ) ) ) ).
% funcset_mem
thf(fact_380_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_381_commutative__monoid_Ofincomp__def,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,A2: set_b,F: b > a] :
( ( group_4866109990395492029noid_a @ M @ Composition @ Unit )
=> ( ( ( finite_finite_b @ A2 )
=> ( ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ F @ A2 )
= ( finite_fold_b_a
@ ^ [X3: b,Y4: a] : ( Composition @ ( F @ X3 ) @ ( commutative_M_ify_a @ M @ Unit @ Y4 ) )
@ Unit
@ A2 ) ) )
& ( ~ ( finite_finite_b @ A2 )
=> ( ( commut5005951359559292711mp_a_b @ M @ Composition @ Unit @ F @ A2 )
= Unit ) ) ) ) ).
% commutative_monoid.fincomp_def
thf(fact_382_infinite__imp__nonempty,axiom,
! [S: set_b] :
( ~ ( finite_finite_b @ S )
=> ( S != bot_bot_set_b ) ) ).
% infinite_imp_nonempty
thf(fact_383_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_384_infinite__imp__nonempty,axiom,
! [S: set_o] :
( ~ ( finite_finite_o @ S )
=> ( S != bot_bot_set_o ) ) ).
% infinite_imp_nonempty
thf(fact_385_finite_OemptyI,axiom,
finite_finite_b @ bot_bot_set_b ).
% finite.emptyI
thf(fact_386_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_387_finite_OemptyI,axiom,
finite_finite_o @ bot_bot_set_o ).
% finite.emptyI
thf(fact_388_finite_OinsertI,axiom,
! [A2: set_o,A: $o] :
( ( finite_finite_o @ A2 )
=> ( finite_finite_o @ ( insert_o @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_389_finite_OinsertI,axiom,
! [A2: set_b,A: b] :
( ( finite_finite_b @ A2 )
=> ( finite_finite_b @ ( insert_b @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_390_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_391_monoid_Oinvertible__right__cancel,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,X: b,Y: b,Z2: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ X )
=> ( ( member_b @ X @ M )
=> ( ( member_b @ Y @ M )
=> ( ( member_b @ Z2 @ M )
=> ( ( ( Composition @ Y @ X )
= ( Composition @ Z2 @ X ) )
= ( Y = Z2 ) ) ) ) ) ) ) ).
% monoid.invertible_right_cancel
thf(fact_392_monoid_Oinvertible__right__cancel,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,X: b > a,Y: b > a,Z2: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ X )
=> ( ( member_b_a @ X @ M )
=> ( ( member_b_a @ Y @ M )
=> ( ( member_b_a @ Z2 @ M )
=> ( ( ( Composition @ Y @ X )
= ( Composition @ Z2 @ X ) )
= ( Y = Z2 ) ) ) ) ) ) ) ).
% monoid.invertible_right_cancel
thf(fact_393_monoid_Oinvertible__right__cancel,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat,Y: nat,Z2: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ X )
=> ( ( member_nat @ X @ M )
=> ( ( member_nat @ Y @ M )
=> ( ( member_nat @ Z2 @ M )
=> ( ( ( Composition @ Y @ X )
= ( Composition @ Z2 @ X ) )
= ( Y = Z2 ) ) ) ) ) ) ) ).
% monoid.invertible_right_cancel
thf(fact_394_monoid_Oinvertible__right__cancel,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a,Y: a,Z2: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ X )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( ( member_a @ Z2 @ M )
=> ( ( ( Composition @ Y @ X )
= ( Composition @ Z2 @ X ) )
= ( Y = Z2 ) ) ) ) ) ) ) ).
% monoid.invertible_right_cancel
thf(fact_395_monoid_Oinvertible__left__cancel,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,X: b,Y: b,Z2: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ X )
=> ( ( member_b @ X @ M )
=> ( ( member_b @ Y @ M )
=> ( ( member_b @ Z2 @ M )
=> ( ( ( Composition @ X @ Y )
= ( Composition @ X @ Z2 ) )
= ( Y = Z2 ) ) ) ) ) ) ) ).
% monoid.invertible_left_cancel
thf(fact_396_monoid_Oinvertible__left__cancel,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,X: b > a,Y: b > a,Z2: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ X )
=> ( ( member_b_a @ X @ M )
=> ( ( member_b_a @ Y @ M )
=> ( ( member_b_a @ Z2 @ M )
=> ( ( ( Composition @ X @ Y )
= ( Composition @ X @ Z2 ) )
= ( Y = Z2 ) ) ) ) ) ) ) ).
% monoid.invertible_left_cancel
thf(fact_397_monoid_Oinvertible__left__cancel,axiom,
! [M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat,Y: nat,Z2: nat] :
( ( group_monoid_nat @ M @ Composition @ Unit )
=> ( ( group_invertible_nat @ M @ Composition @ Unit @ X )
=> ( ( member_nat @ X @ M )
=> ( ( member_nat @ Y @ M )
=> ( ( member_nat @ Z2 @ M )
=> ( ( ( Composition @ X @ Y )
= ( Composition @ X @ Z2 ) )
= ( Y = Z2 ) ) ) ) ) ) ) ).
% monoid.invertible_left_cancel
thf(fact_398_monoid_Oinvertible__left__cancel,axiom,
! [M: set_a,Composition: a > a > a,Unit: a,X: a,Y: a,Z2: a] :
( ( group_monoid_a @ M @ Composition @ Unit )
=> ( ( group_invertible_a @ M @ Composition @ Unit @ X )
=> ( ( member_a @ X @ M )
=> ( ( member_a @ Y @ M )
=> ( ( member_a @ Z2 @ M )
=> ( ( ( Composition @ X @ Y )
= ( Composition @ X @ Z2 ) )
= ( Y = Z2 ) ) ) ) ) ) ) ).
% monoid.invertible_left_cancel
thf(fact_399_monoid_Ocomposition__invertible,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,X: b,Y: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ Y )
=> ( ( member_b @ X @ M )
=> ( ( member_b @ Y @ M )
=> ( group_invertible_b @ M @ Composition @ Unit @ ( Composition @ X @ Y ) ) ) ) ) ) ) ).
% monoid.composition_invertible
thf(fact_400_monoid_Ocomposition__invertible,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,X: b > a,Y: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ Y )
=> ( ( member_b_a @ X @ M )
=> ( ( member_b_a @ Y @ M )
=> ( group_invertible_b_a @ M @ Composition @ Unit @ ( Composition @ X @ Y ) ) ) ) ) ) ) ).
% monoid.composition_invertible
thf(fact_401_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_402_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_403_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_404_monoid_Oinvertible__def,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( member_b @ U @ M )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ U )
= ( ? [X3: b] :
( ( member_b @ X3 @ M )
& ( ( Composition @ U @ X3 )
= Unit )
& ( ( Composition @ X3 @ U )
= Unit ) ) ) ) ) ) ).
% monoid.invertible_def
thf(fact_405_monoid_Oinvertible__def,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( member_b_a @ U @ M )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ U )
= ( ? [X3: b > a] :
( ( member_b_a @ X3 @ M )
& ( ( Composition @ U @ X3 )
= Unit )
& ( ( Composition @ X3 @ U )
= Unit ) ) ) ) ) ) ).
% monoid.invertible_def
thf(fact_406_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_407_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_408_monoid_OinvertibleI,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b,V2: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_b @ U @ M )
=> ( ( member_b @ V2 @ M )
=> ( group_invertible_b @ M @ Composition @ Unit @ U ) ) ) ) ) ) ).
% monoid.invertibleI
thf(fact_409_monoid_OinvertibleI,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a,V2: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_b_a @ U @ M )
=> ( ( member_b_a @ V2 @ M )
=> ( group_invertible_b_a @ M @ Composition @ Unit @ U ) ) ) ) ) ) ).
% monoid.invertibleI
thf(fact_410_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_411_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_412_monoid_OinvertibleE,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ U )
=> ( ! [V3: b] :
( ( ( ( Composition @ U @ V3 )
= Unit )
& ( ( Composition @ V3 @ U )
= Unit ) )
=> ~ ( member_b @ V3 @ M ) )
=> ~ ( member_b @ U @ M ) ) ) ) ).
% monoid.invertibleE
thf(fact_413_monoid_OinvertibleE,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ U )
=> ( ! [V3: b > a] :
( ( ( ( Composition @ U @ V3 )
= Unit )
& ( ( Composition @ V3 @ U )
= Unit ) )
=> ~ ( member_b_a @ V3 @ M ) )
=> ~ ( member_b_a @ U @ M ) ) ) ) ).
% monoid.invertibleE
thf(fact_414_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_415_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_416_monoid_Oinverse__equality,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b,V2: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_b @ U @ M )
=> ( ( member_b @ V2 @ M )
=> ( ( group_inverse_b @ M @ Composition @ Unit @ U )
= V2 ) ) ) ) ) ) ).
% monoid.inverse_equality
thf(fact_417_monoid_Oinverse__equality,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a,V2: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( ( Composition @ U @ V2 )
= Unit )
=> ( ( ( Composition @ V2 @ U )
= Unit )
=> ( ( member_b_a @ U @ M )
=> ( ( member_b_a @ V2 @ M )
=> ( ( group_inverse_b_a @ M @ Composition @ Unit @ U )
= V2 ) ) ) ) ) ) ).
% monoid.inverse_equality
thf(fact_418_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_419_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_420_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_421_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_422_finite_Ocases,axiom,
! [A: set_b] :
( ( finite_finite_b @ A )
=> ( ( A != bot_bot_set_b )
=> ~ ! [A5: set_b] :
( ? [A4: b] :
( A
= ( insert_b @ A4 @ A5 ) )
=> ~ ( finite_finite_b @ A5 ) ) ) ) ).
% finite.cases
thf(fact_423_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A5: set_nat] :
( ? [A4: nat] :
( A
= ( insert_nat @ A4 @ A5 ) )
=> ~ ( finite_finite_nat @ A5 ) ) ) ) ).
% finite.cases
thf(fact_424_finite_Ocases,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ~ ! [A5: set_o] :
( ? [A4: $o] :
( A
= ( insert_o @ A4 @ A5 ) )
=> ~ ( finite_finite_o @ A5 ) ) ) ) ).
% finite.cases
thf(fact_425_finite_Osimps,axiom,
( finite_finite_b
= ( ^ [A3: set_b] :
( ( A3 = bot_bot_set_b )
| ? [A6: set_b,B6: b] :
( ( A3
= ( insert_b @ B6 @ A6 ) )
& ( finite_finite_b @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_426_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A3: set_nat] :
( ( A3 = bot_bot_set_nat )
| ? [A6: set_nat,B6: nat] :
( ( A3
= ( insert_nat @ B6 @ A6 ) )
& ( finite_finite_nat @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_427_finite_Osimps,axiom,
( finite_finite_o
= ( ^ [A3: set_o] :
( ( A3 = bot_bot_set_o )
| ? [A6: set_o,B6: $o] :
( ( A3
= ( insert_o @ B6 @ A6 ) )
& ( finite_finite_o @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_428_finite__induct,axiom,
! [F2: set_b_a,P: set_b_a > $o] :
( ( finite_finite_b_a @ F2 )
=> ( ( P @ bot_bot_set_b_a )
=> ( ! [X2: b > a,F3: set_b_a] :
( ( finite_finite_b_a @ F3 )
=> ( ~ ( member_b_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_429_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_430_finite__induct,axiom,
! [F2: set_b,P: set_b > $o] :
( ( finite_finite_b @ F2 )
=> ( ( P @ bot_bot_set_b )
=> ( ! [X2: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ~ ( member_b @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_431_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_432_finite__induct,axiom,
! [F2: set_o,P: set_o > $o] :
( ( finite_finite_o @ F2 )
=> ( ( P @ bot_bot_set_o )
=> ( ! [X2: $o,F3: set_o] :
( ( finite_finite_o @ F3 )
=> ( ~ ( member_o @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_o @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_433_finite__ne__induct,axiom,
! [F2: set_b_a,P: set_b_a > $o] :
( ( finite_finite_b_a @ F2 )
=> ( ( F2 != bot_bot_set_b_a )
=> ( ! [X2: b > a] : ( P @ ( insert_b_a @ X2 @ bot_bot_set_b_a ) )
=> ( ! [X2: b > a,F3: set_b_a] :
( ( finite_finite_b_a @ F3 )
=> ( ( F3 != bot_bot_set_b_a )
=> ( ~ ( member_b_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_434_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_435_finite__ne__induct,axiom,
! [F2: set_b,P: set_b > $o] :
( ( finite_finite_b @ F2 )
=> ( ( F2 != bot_bot_set_b )
=> ( ! [X2: b] : ( P @ ( insert_b @ X2 @ bot_bot_set_b ) )
=> ( ! [X2: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ( F3 != bot_bot_set_b )
=> ( ~ ( member_b @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_436_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_437_finite__ne__induct,axiom,
! [F2: set_o,P: set_o > $o] :
( ( finite_finite_o @ F2 )
=> ( ( F2 != bot_bot_set_o )
=> ( ! [X2: $o] : ( P @ ( insert_o @ X2 @ bot_bot_set_o ) )
=> ( ! [X2: $o,F3: set_o] :
( ( finite_finite_o @ F3 )
=> ( ( F3 != bot_bot_set_o )
=> ( ~ ( member_o @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_o @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_438_infinite__finite__induct,axiom,
! [P: set_b_a > $o,A2: set_b_a] :
( ! [A5: set_b_a] :
( ~ ( finite_finite_b_a @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bot_set_b_a )
=> ( ! [X2: b > a,F3: set_b_a] :
( ( finite_finite_b_a @ F3 )
=> ( ~ ( member_b_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_439_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A5: set_a] :
( ~ ( finite_finite_a @ A5 )
=> ( P @ A5 ) )
=> ( ( 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_440_infinite__finite__induct,axiom,
! [P: set_b > $o,A2: set_b] :
( ! [A5: set_b] :
( ~ ( finite_finite_b @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bot_set_b )
=> ( ! [X2: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ~ ( member_b @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_441_infinite__finite__induct,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ! [A5: set_nat] :
( ~ ( finite_finite_nat @ A5 )
=> ( P @ A5 ) )
=> ( ( 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_442_infinite__finite__induct,axiom,
! [P: set_o > $o,A2: set_o] :
( ! [A5: set_o] :
( ~ ( finite_finite_o @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bot_set_o )
=> ( ! [X2: $o,F3: set_o] :
( ( finite_finite_o @ F3 )
=> ( ~ ( member_o @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_o @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_443_monoid_Oinvertible__inverse__invertible,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ U )
=> ( ( member_b @ U @ M )
=> ( group_invertible_b @ M @ Composition @ Unit @ ( group_inverse_b @ M @ Composition @ Unit @ U ) ) ) ) ) ).
% monoid.invertible_inverse_invertible
thf(fact_444_monoid_Oinvertible__inverse__invertible,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ U )
=> ( ( member_b_a @ U @ M )
=> ( group_invertible_b_a @ M @ Composition @ Unit @ ( group_inverse_b_a @ M @ Composition @ Unit @ U ) ) ) ) ) ).
% monoid.invertible_inverse_invertible
thf(fact_445_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_446_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_447_monoid_Oinverse__composition__commute,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,X: b,Y: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ Y )
=> ( ( member_b @ X @ M )
=> ( ( member_b @ Y @ M )
=> ( ( group_inverse_b @ M @ Composition @ Unit @ ( Composition @ X @ Y ) )
= ( Composition @ ( group_inverse_b @ M @ Composition @ Unit @ Y ) @ ( group_inverse_b @ M @ Composition @ Unit @ X ) ) ) ) ) ) ) ) ).
% monoid.inverse_composition_commute
thf(fact_448_monoid_Oinverse__composition__commute,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,X: b > a,Y: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ X )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ Y )
=> ( ( member_b_a @ X @ M )
=> ( ( member_b_a @ Y @ M )
=> ( ( group_inverse_b_a @ M @ Composition @ Unit @ ( Composition @ X @ Y ) )
= ( Composition @ ( group_inverse_b_a @ M @ Composition @ Unit @ Y ) @ ( group_inverse_b_a @ M @ Composition @ Unit @ X ) ) ) ) ) ) ) ) ).
% monoid.inverse_composition_commute
thf(fact_449_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_450_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_451_monoid_Oinvertible__inverse__inverse,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ U )
=> ( ( member_b @ U @ M )
=> ( ( group_inverse_b @ M @ Composition @ Unit @ ( group_inverse_b @ M @ Composition @ Unit @ U ) )
= U ) ) ) ) ).
% monoid.invertible_inverse_inverse
thf(fact_452_monoid_Oinvertible__inverse__inverse,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ U )
=> ( ( member_b_a @ U @ M )
=> ( ( group_inverse_b_a @ M @ Composition @ Unit @ ( group_inverse_b_a @ M @ Composition @ Unit @ U ) )
= U ) ) ) ) ).
% monoid.invertible_inverse_inverse
thf(fact_453_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_454_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_455_monoid_Oinvertible__right__inverse2,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b,V2: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ U )
=> ( ( member_b @ U @ M )
=> ( ( member_b @ V2 @ M )
=> ( ( Composition @ U @ ( Composition @ ( group_inverse_b @ M @ Composition @ Unit @ U ) @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_right_inverse2
thf(fact_456_monoid_Oinvertible__right__inverse2,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a,V2: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ U )
=> ( ( member_b_a @ U @ M )
=> ( ( member_b_a @ V2 @ M )
=> ( ( Composition @ U @ ( Composition @ ( group_inverse_b_a @ M @ Composition @ Unit @ U ) @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_right_inverse2
thf(fact_457_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_458_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_459_monoid_Oinvertible__inverse__closed,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ U )
=> ( ( member_b @ U @ M )
=> ( member_b @ ( group_inverse_b @ M @ Composition @ Unit @ U ) @ M ) ) ) ) ).
% monoid.invertible_inverse_closed
thf(fact_460_monoid_Oinvertible__inverse__closed,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ U )
=> ( ( member_b_a @ U @ M )
=> ( member_b_a @ ( group_inverse_b_a @ M @ Composition @ Unit @ U ) @ M ) ) ) ) ).
% monoid.invertible_inverse_closed
thf(fact_461_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_462_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_463_monoid_Oinvertible__right__inverse,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ U )
=> ( ( member_b @ U @ M )
=> ( ( Composition @ U @ ( group_inverse_b @ M @ Composition @ Unit @ U ) )
= Unit ) ) ) ) ).
% monoid.invertible_right_inverse
thf(fact_464_monoid_Oinvertible__right__inverse,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ U )
=> ( ( member_b_a @ U @ M )
=> ( ( Composition @ U @ ( group_inverse_b_a @ M @ Composition @ Unit @ U ) )
= Unit ) ) ) ) ).
% monoid.invertible_right_inverse
thf(fact_465_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_466_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_467_monoid_Oinvertible__left__inverse2,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b,V2: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ U )
=> ( ( member_b @ U @ M )
=> ( ( member_b @ V2 @ M )
=> ( ( Composition @ ( group_inverse_b @ M @ Composition @ Unit @ U ) @ ( Composition @ U @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_left_inverse2
thf(fact_468_monoid_Oinvertible__left__inverse2,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a,V2: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ U )
=> ( ( member_b_a @ U @ M )
=> ( ( member_b_a @ V2 @ M )
=> ( ( Composition @ ( group_inverse_b_a @ M @ Composition @ Unit @ U ) @ ( Composition @ U @ V2 ) )
= V2 ) ) ) ) ) ).
% monoid.invertible_left_inverse2
thf(fact_469_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_470_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_471_group__of__Units,axiom,
group_group_a @ ( group_Units_a @ m @ composition @ unit ) @ composition @ unit ).
% group_of_Units
thf(fact_472_the__elem__eq,axiom,
! [X: b] :
( ( the_elem_b @ ( insert_b @ X @ bot_bot_set_b ) )
= X ) ).
% the_elem_eq
thf(fact_473_the__elem__eq,axiom,
! [X: $o] :
( ( the_elem_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% the_elem_eq
thf(fact_474_inverse__undefined,axiom,
! [U: a] :
( ~ ( member_a @ U @ m )
=> ( ( group_inverse_a @ m @ composition @ unit @ U )
= undefined_a ) ) ).
% inverse_undefined
thf(fact_475_is__singletonI,axiom,
! [X: b] : ( is_singleton_b @ ( insert_b @ X @ bot_bot_set_b ) ) ).
% is_singletonI
thf(fact_476_is__singletonI,axiom,
! [X: $o] : ( is_singleton_o @ ( insert_o @ X @ bot_bot_set_o ) ) ).
% is_singletonI
thf(fact_477_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_478_commutative__monoid__def,axiom,
( group_4866109990395492029noid_a
= ( ^ [M2: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_monoid_a @ M2 @ Composition2 @ Unit2 )
& ( group_2081300317213596122ioms_a @ M2 @ Composition2 ) ) ) ) ).
% commutative_monoid_def
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_Set_Ois__empty__def,axiom,
( is_empty_o
= ( ^ [A6: set_o] : ( A6 = bot_bot_set_o ) ) ) ).
% Set.is_empty_def
thf(fact_481_is__singletonE,axiom,
! [A2: set_b] :
( ( is_singleton_b @ A2 )
=> ~ ! [X2: b] :
( A2
!= ( insert_b @ X2 @ bot_bot_set_b ) ) ) ).
% is_singletonE
thf(fact_482_is__singletonE,axiom,
! [A2: set_o] :
( ( is_singleton_o @ A2 )
=> ~ ! [X2: $o] :
( A2
!= ( insert_o @ X2 @ bot_bot_set_o ) ) ) ).
% is_singletonE
thf(fact_483_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_484_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_485_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_486_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_487_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_488_subsetI,axiom,
! [A2: set_b,B2: set_b] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_b @ X2 @ B2 ) )
=> ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% subsetI
thf(fact_489_subsetI,axiom,
! [A2: set_b_a,B2: set_b_a] :
( ! [X2: b > a] :
( ( member_b_a @ X2 @ A2 )
=> ( member_b_a @ X2 @ B2 ) )
=> ( ord_less_eq_set_b_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_490_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_491_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_492_subset__empty,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% subset_empty
thf(fact_493_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_494_empty__subsetI,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).
% empty_subsetI
thf(fact_495_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_496_insert__subset,axiom,
! [X: $o,A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ ( insert_o @ X @ A2 ) @ B2 )
= ( ( member_o @ X @ B2 )
& ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_497_insert__subset,axiom,
! [X: b,A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ ( insert_b @ X @ A2 ) @ B2 )
= ( ( member_b @ X @ B2 )
& ( ord_less_eq_set_b @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_498_insert__subset,axiom,
! [X: b > a,A2: set_b_a,B2: set_b_a] :
( ( ord_less_eq_set_b_a @ ( insert_b_a @ X @ A2 ) @ B2 )
= ( ( member_b_a @ X @ B2 )
& ( ord_less_eq_set_b_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_499_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_500_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_501_finite__Collect__subsets,axiom,
! [A2: set_b] :
( ( finite_finite_b @ A2 )
=> ( finite_finite_set_b
@ ( collect_set_b
@ ^ [B4: set_b] : ( ord_less_eq_set_b @ B4 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_502_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B4: set_nat] : ( ord_less_eq_set_nat @ B4 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_503_finite__Collect__subsets,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B4: set_a] : ( ord_less_eq_set_a @ B4 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_504_singleton__insert__inj__eq,axiom,
! [B: b,A: b,A2: set_b] :
( ( ( insert_b @ B @ bot_bot_set_b )
= ( insert_b @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_505_singleton__insert__inj__eq,axiom,
! [B: $o,A: $o,A2: set_o] :
( ( ( insert_o @ B @ bot_bot_set_o )
= ( insert_o @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_506_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_507_singleton__insert__inj__eq_H,axiom,
! [A: b,A2: set_b,B: b] :
( ( ( insert_b @ A @ A2 )
= ( insert_b @ B @ bot_bot_set_b ) )
= ( ( A = B )
& ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_508_singleton__insert__inj__eq_H,axiom,
! [A: $o,A2: set_o,B: $o] :
( ( ( insert_o @ A @ A2 )
= ( insert_o @ B @ bot_bot_set_o ) )
= ( ( A = B )
& ( ord_less_eq_set_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_509_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_510_subgroup__transitive,axiom,
! [K: set_a,H2: set_a,Composition: a > a > a,Unit: a,G2: set_a] :
( ( group_subgroup_a @ K @ H2 @ Composition @ Unit )
=> ( ( group_subgroup_a @ H2 @ G2 @ Composition @ Unit )
=> ( group_subgroup_a @ K @ G2 @ Composition @ Unit ) ) ) ).
% subgroup_transitive
thf(fact_511_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_512_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_513_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_514_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [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_515_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_516_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_517_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_518_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_519_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_520_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_521_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_522_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_523_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_524_order__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_525_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_526_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
=> ( P @ A4 @ B5 ) )
=> ( ! [A4: nat,B5: nat] :
( ( P @ B5 @ A4 )
=> ( P @ A4 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_527_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [A3: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ A3 )
& ( ord_less_eq_set_a @ A3 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_528_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A3: nat,B6: nat] :
( ( ord_less_eq_nat @ B6 @ A3 )
& ( ord_less_eq_nat @ A3 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_529_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_530_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_531_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_532_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_533_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_534_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_535_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [A3: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A3 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_536_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A3: nat,B6: nat] :
( ( ord_less_eq_nat @ A3 @ B6 )
& ( ord_less_eq_nat @ B6 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_537_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_538_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_539_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_540_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_541_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_542_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_543_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_544_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_545_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_546_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_547_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_548_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_549_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_550_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_551_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_552_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_553_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_554_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_555_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_556_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_557_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_558_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_559_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_560_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_561_set__eq__subset,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [A6: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A6 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_562_subset__trans,axiom,
! [A2: set_a,B2: set_a,C5: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C5 )
=> ( ord_less_eq_set_a @ A2 @ C5 ) ) ) ).
% subset_trans
thf(fact_563_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_564_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_565_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_566_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B4: set_b] :
! [T: b] :
( ( member_b @ T @ A6 )
=> ( member_b @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_567_subset__iff,axiom,
( ord_less_eq_set_b_a
= ( ^ [A6: set_b_a,B4: set_b_a] :
! [T: b > a] :
( ( member_b_a @ T @ A6 )
=> ( member_b_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_568_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B4: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A6 )
=> ( member_nat @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_569_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B4: set_a] :
! [T: a] :
( ( member_a @ T @ A6 )
=> ( member_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_570_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_571_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_572_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B4: set_b] :
! [X3: b] :
( ( member_b @ X3 @ A6 )
=> ( member_b @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_573_subset__eq,axiom,
( ord_less_eq_set_b_a
= ( ^ [A6: set_b_a,B4: set_b_a] :
! [X3: b > a] :
( ( member_b_a @ X3 @ A6 )
=> ( member_b_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_574_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B4: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( member_nat @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_575_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B4: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A6 )
=> ( member_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_576_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_577_subsetD,axiom,
! [A2: set_b,B2: set_b,C: b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B2 ) ) ) ).
% subsetD
thf(fact_578_subsetD,axiom,
! [A2: set_b_a,B2: set_b_a,C: b > a] :
( ( ord_less_eq_set_b_a @ A2 @ B2 )
=> ( ( member_b_a @ C @ A2 )
=> ( member_b_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_579_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_580_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_581_in__mono,axiom,
! [A2: set_b,B2: set_b,X: b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( member_b @ X @ A2 )
=> ( member_b @ X @ B2 ) ) ) ).
% in_mono
thf(fact_582_in__mono,axiom,
! [A2: set_b_a,B2: set_b_a,X: b > a] :
( ( ord_less_eq_set_b_a @ A2 @ B2 )
=> ( ( member_b_a @ X @ A2 )
=> ( member_b_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_583_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_584_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_585_Pi__anti__mono,axiom,
! [A7: set_b,A2: set_b,B2: b > set_a] :
( ( ord_less_eq_set_b @ A7 @ A2 )
=> ( ord_less_eq_set_b_a @ ( pi_b_a @ A2 @ B2 ) @ ( pi_b_a @ A7 @ B2 ) ) ) ).
% Pi_anti_mono
thf(fact_586_Pi__mono,axiom,
! [A2: set_b,B2: b > set_a,C5: b > set_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( B2 @ X2 ) @ ( C5 @ X2 ) ) )
=> ( ord_less_eq_set_b_a @ ( pi_b_a @ A2 @ B2 ) @ ( pi_b_a @ A2 @ C5 ) ) ) ).
% Pi_mono
thf(fact_587_Pi__mono,axiom,
! [A2: set_b_a,B2: ( b > a ) > set_a,C5: ( b > a ) > set_a] :
( ! [X2: b > a] :
( ( member_b_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( B2 @ X2 ) @ ( C5 @ X2 ) ) )
=> ( ord_le4402886750609172241_b_a_a @ ( pi_b_a_a @ A2 @ B2 ) @ ( pi_b_a_a @ A2 @ C5 ) ) ) ).
% Pi_mono
thf(fact_588_Pi__mono,axiom,
! [A2: set_a,B2: a > set_a,C5: a > set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( B2 @ X2 ) @ ( C5 @ X2 ) ) )
=> ( ord_less_eq_set_a_a @ ( pi_a_a @ A2 @ B2 ) @ ( pi_a_a @ A2 @ C5 ) ) ) ).
% Pi_mono
thf(fact_589_Pi__mono,axiom,
! [A2: set_nat,B2: nat > set_a,C5: nat > set_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( B2 @ X2 ) @ ( C5 @ X2 ) ) )
=> ( ord_le871467723717165285_nat_a @ ( pi_nat_a @ A2 @ B2 ) @ ( pi_nat_a @ A2 @ C5 ) ) ) ).
% Pi_mono
thf(fact_590_bot_Oextremum,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% bot.extremum
thf(fact_591_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_592_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_593_bot_Oextremum__unique,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% bot.extremum_unique
thf(fact_594_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_595_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_596_bot_Oextremum__uniqueI,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
=> ( A = bot_bot_set_o ) ) ).
% bot.extremum_uniqueI
thf(fact_597_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_598_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_599_Collect__subset,axiom,
! [A2: set_b,P: b > $o] :
( ord_less_eq_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_600_Collect__subset,axiom,
! [A2: set_b_a,P: ( b > a ) > $o] :
( ord_less_eq_set_b_a
@ ( collect_b_a
@ ^ [X3: b > a] :
( ( member_b_a @ X3 @ A2 )
& ( P @ X3 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_601_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_602_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_603_commutative__monoid__axioms_Ointro,axiom,
! [M: set_b,Composition: b > b > b] :
( ! [X2: b,Y3: b] :
( ( member_b @ X2 @ M )
=> ( ( member_b @ Y3 @ M )
=> ( ( Composition @ X2 @ Y3 )
= ( Composition @ Y3 @ X2 ) ) ) )
=> ( group_2081300317213596123ioms_b @ M @ Composition ) ) ).
% commutative_monoid_axioms.intro
thf(fact_604_commutative__monoid__axioms_Ointro,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a] :
( ! [X2: b > a,Y3: b > a] :
( ( member_b_a @ X2 @ M )
=> ( ( member_b_a @ Y3 @ M )
=> ( ( Composition @ X2 @ Y3 )
= ( Composition @ Y3 @ X2 ) ) ) )
=> ( group_4266494884492393160ms_b_a @ M @ Composition ) ) ).
% commutative_monoid_axioms.intro
thf(fact_605_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_606_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_607_commutative__monoid__axioms__def,axiom,
( group_2081300317213596123ioms_b
= ( ^ [M2: set_b,Composition2: b > b > b] :
! [X3: b,Y4: b] :
( ( member_b @ X3 @ M2 )
=> ( ( member_b @ Y4 @ M2 )
=> ( ( Composition2 @ X3 @ Y4 )
= ( Composition2 @ Y4 @ X3 ) ) ) ) ) ) ).
% commutative_monoid_axioms_def
thf(fact_608_commutative__monoid__axioms__def,axiom,
( group_4266494884492393160ms_b_a
= ( ^ [M2: set_b_a,Composition2: ( b > a ) > ( b > a ) > b > a] :
! [X3: b > a,Y4: b > a] :
( ( member_b_a @ X3 @ M2 )
=> ( ( member_b_a @ Y4 @ M2 )
=> ( ( Composition2 @ X3 @ Y4 )
= ( Composition2 @ Y4 @ X3 ) ) ) ) ) ) ).
% commutative_monoid_axioms_def
thf(fact_609_commutative__monoid__axioms__def,axiom,
( group_2081300317213596122ioms_a
= ( ^ [M2: set_a,Composition2: a > a > a] :
! [X3: a,Y4: a] :
( ( member_a @ X3 @ M2 )
=> ( ( member_a @ Y4 @ M2 )
=> ( ( Composition2 @ X3 @ Y4 )
= ( Composition2 @ Y4 @ X3 ) ) ) ) ) ) ).
% commutative_monoid_axioms_def
thf(fact_610_commutative__monoid__axioms__def,axiom,
( group_5685275631618022900ms_nat
= ( ^ [M2: set_nat,Composition2: nat > nat > nat] :
! [X3: nat,Y4: nat] :
( ( member_nat @ X3 @ M2 )
=> ( ( member_nat @ Y4 @ M2 )
=> ( ( Composition2 @ X3 @ Y4 )
= ( Composition2 @ Y4 @ X3 ) ) ) ) ) ) ).
% commutative_monoid_axioms_def
thf(fact_611_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_612_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_613_bot__set__def,axiom,
( bot_bot_set_o
= ( collect_o @ bot_bot_o_o ) ) ).
% bot_set_def
thf(fact_614_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_615_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_616_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_617_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_618_finite__subset,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( finite_finite_b @ B2 )
=> ( finite_finite_b @ A2 ) ) ) ).
% finite_subset
thf(fact_619_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_620_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_621_infinite__super,axiom,
! [S: set_b,T2: set_b] :
( ( ord_less_eq_set_b @ S @ T2 )
=> ( ~ ( finite_finite_b @ S )
=> ~ ( finite_finite_b @ T2 ) ) ) ).
% infinite_super
thf(fact_622_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_623_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_624_rev__finite__subset,axiom,
! [B2: set_b,A2: set_b] :
( ( finite_finite_b @ B2 )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( finite_finite_b @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_625_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_626_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_627_subset__insertI2,axiom,
! [A2: set_b,B2: set_b,B: b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_628_subset__insertI2,axiom,
! [A2: set_o,B2: set_o,B: $o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ord_less_eq_set_o @ A2 @ ( insert_o @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_629_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_630_subset__insertI,axiom,
! [B2: set_b,A: b] : ( ord_less_eq_set_b @ B2 @ ( insert_b @ A @ B2 ) ) ).
% subset_insertI
thf(fact_631_subset__insertI,axiom,
! [B2: set_o,A: $o] : ( ord_less_eq_set_o @ B2 @ ( insert_o @ A @ B2 ) ) ).
% subset_insertI
thf(fact_632_subset__insertI,axiom,
! [B2: set_a,A: a] : ( ord_less_eq_set_a @ B2 @ ( insert_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_633_subset__insert,axiom,
! [X: $o,A2: set_o,B2: set_o] :
( ~ ( member_o @ X @ A2 )
=> ( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ B2 ) )
= ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_634_subset__insert,axiom,
! [X: b,A2: set_b,B2: set_b] :
( ~ ( member_b @ X @ A2 )
=> ( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X @ B2 ) )
= ( ord_less_eq_set_b @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_635_subset__insert,axiom,
! [X: b > a,A2: set_b_a,B2: set_b_a] :
( ~ ( member_b_a @ X @ A2 )
=> ( ( ord_less_eq_set_b_a @ A2 @ ( insert_b_a @ X @ B2 ) )
= ( ord_less_eq_set_b_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_636_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_637_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_638_insert__mono,axiom,
! [C5: set_b,D2: set_b,A: b] :
( ( ord_less_eq_set_b @ C5 @ D2 )
=> ( ord_less_eq_set_b @ ( insert_b @ A @ C5 ) @ ( insert_b @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_639_insert__mono,axiom,
! [C5: set_o,D2: set_o,A: $o] :
( ( ord_less_eq_set_o @ C5 @ D2 )
=> ( ord_less_eq_set_o @ ( insert_o @ A @ C5 ) @ ( insert_o @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_640_insert__mono,axiom,
! [C5: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C5 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C5 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_641_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_642_group_Oinvertible,axiom,
! [G2: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_group_b @ G2 @ Composition @ Unit )
=> ( ( member_b @ U @ G2 )
=> ( group_invertible_b @ G2 @ Composition @ Unit @ U ) ) ) ).
% group.invertible
thf(fact_643_group_Oinvertible,axiom,
! [G2: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_group_b_a @ G2 @ Composition @ Unit )
=> ( ( member_b_a @ U @ G2 )
=> ( group_invertible_b_a @ G2 @ Composition @ Unit @ U ) ) ) ).
% group.invertible
thf(fact_644_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_645_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_646_subgroup_Osubgroup__inverse__equality,axiom,
! [G2: set_b,M: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_subgroup_b @ G2 @ M @ Composition @ Unit )
=> ( ( member_b @ U @ G2 )
=> ( ( group_inverse_b @ M @ Composition @ Unit @ U )
= ( group_inverse_b @ G2 @ Composition @ Unit @ U ) ) ) ) ).
% subgroup.subgroup_inverse_equality
thf(fact_647_subgroup_Osubgroup__inverse__equality,axiom,
! [G2: set_b_a,M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_subgroup_b_a @ G2 @ M @ Composition @ Unit )
=> ( ( member_b_a @ U @ G2 )
=> ( ( group_inverse_b_a @ M @ Composition @ Unit @ U )
= ( group_inverse_b_a @ G2 @ Composition @ Unit @ U ) ) ) ) ).
% subgroup.subgroup_inverse_equality
thf(fact_648_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_649_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_650_monoid_OsubgroupI,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,G2: set_b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ( ord_less_eq_set_b @ G2 @ M )
=> ( ( member_b @ Unit @ G2 )
=> ( ! [G3: b,H: b] :
( ( member_b @ G3 @ G2 )
=> ( ( member_b @ H @ G2 )
=> ( member_b @ ( Composition @ G3 @ H ) @ G2 ) ) )
=> ( ! [G3: b] :
( ( member_b @ G3 @ G2 )
=> ( group_invertible_b @ M @ Composition @ Unit @ G3 ) )
=> ( ! [G3: b] :
( ( member_b @ G3 @ G2 )
=> ( member_b @ ( group_inverse_b @ M @ Composition @ Unit @ G3 ) @ G2 ) )
=> ( group_subgroup_b @ G2 @ M @ Composition @ Unit ) ) ) ) ) ) ) ).
% monoid.subgroupI
thf(fact_651_monoid_OsubgroupI,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,G2: set_b_a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ( ord_less_eq_set_b_a @ G2 @ M )
=> ( ( member_b_a @ Unit @ G2 )
=> ( ! [G3: b > a,H: b > a] :
( ( member_b_a @ G3 @ G2 )
=> ( ( member_b_a @ H @ G2 )
=> ( member_b_a @ ( Composition @ G3 @ H ) @ G2 ) ) )
=> ( ! [G3: b > a] :
( ( member_b_a @ G3 @ G2 )
=> ( group_invertible_b_a @ M @ Composition @ Unit @ G3 ) )
=> ( ! [G3: b > a] :
( ( member_b_a @ G3 @ G2 )
=> ( member_b_a @ ( group_inverse_b_a @ M @ Composition @ Unit @ G3 ) @ G2 ) )
=> ( group_subgroup_b_a @ G2 @ M @ Composition @ Unit ) ) ) ) ) ) ) ).
% monoid.subgroupI
thf(fact_652_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_653_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_654_finite__has__maximal,axiom,
! [A2: set_o] :
( ( finite_finite_o @ A2 )
=> ( ( A2 != bot_bot_set_o )
=> ? [X2: $o] :
( ( member_o @ X2 @ A2 )
& ! [Xa: $o] :
( ( member_o @ Xa @ A2 )
=> ( ( ord_less_eq_o @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_655_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_656_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_657_finite__has__minimal,axiom,
! [A2: set_o] :
( ( finite_finite_o @ A2 )
=> ( ( A2 != bot_bot_set_o )
=> ? [X2: $o] :
( ( member_o @ X2 @ A2 )
& ! [Xa: $o] :
( ( member_o @ Xa @ A2 )
=> ( ( ord_less_eq_o @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_658_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_659_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_660_subset__singletonD,axiom,
! [A2: set_b,X: b] :
( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) )
=> ( ( A2 = bot_bot_set_b )
| ( A2
= ( insert_b @ X @ bot_bot_set_b ) ) ) ) ).
% subset_singletonD
thf(fact_661_subset__singletonD,axiom,
! [A2: set_o,X: $o] :
( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) )
=> ( ( A2 = bot_bot_set_o )
| ( A2
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% subset_singletonD
thf(fact_662_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_663_subset__singleton__iff,axiom,
! [X4: set_b,A: b] :
( ( ord_less_eq_set_b @ X4 @ ( insert_b @ A @ bot_bot_set_b ) )
= ( ( X4 = bot_bot_set_b )
| ( X4
= ( insert_b @ A @ bot_bot_set_b ) ) ) ) ).
% subset_singleton_iff
thf(fact_664_subset__singleton__iff,axiom,
! [X4: set_o,A: $o] :
( ( ord_less_eq_set_o @ X4 @ ( insert_o @ A @ bot_bot_set_o ) )
= ( ( X4 = bot_bot_set_o )
| ( X4
= ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).
% subset_singleton_iff
thf(fact_665_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_666_is__singleton__the__elem,axiom,
( is_singleton_b
= ( ^ [A6: set_b] :
( A6
= ( insert_b @ ( the_elem_b @ A6 ) @ bot_bot_set_b ) ) ) ) ).
% is_singleton_the_elem
thf(fact_667_is__singleton__the__elem,axiom,
( is_singleton_o
= ( ^ [A6: set_o] :
( A6
= ( insert_o @ ( the_elem_o @ A6 ) @ bot_bot_set_o ) ) ) ) ).
% is_singleton_the_elem
thf(fact_668_monoid_Oinverse__undefined,axiom,
! [M: set_b,Composition: b > b > b,Unit: b,U: b] :
( ( group_monoid_b @ M @ Composition @ Unit )
=> ( ~ ( member_b @ U @ M )
=> ( ( group_inverse_b @ M @ Composition @ Unit @ U )
= undefined_b ) ) ) ).
% monoid.inverse_undefined
thf(fact_669_monoid_Oinverse__undefined,axiom,
! [M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,U: b > a] :
( ( group_monoid_b_a @ M @ Composition @ Unit )
=> ( ~ ( member_b_a @ U @ M )
=> ( ( group_inverse_b_a @ M @ Composition @ Unit @ U )
= undefined_b_a ) ) ) ).
% monoid.inverse_undefined
thf(fact_670_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_671_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_672_is__singletonI_H,axiom,
! [A2: set_b] :
( ( A2 != bot_bot_set_b )
=> ( ! [X2: b,Y3: b] :
( ( member_b @ X2 @ A2 )
=> ( ( member_b @ Y3 @ A2 )
=> ( X2 = Y3 ) ) )
=> ( is_singleton_b @ A2 ) ) ) ).
% is_singletonI'
thf(fact_673_is__singletonI_H,axiom,
! [A2: set_b_a] :
( ( A2 != bot_bot_set_b_a )
=> ( ! [X2: b > a,Y3: b > a] :
( ( member_b_a @ X2 @ A2 )
=> ( ( member_b_a @ Y3 @ A2 )
=> ( X2 = Y3 ) ) )
=> ( is_singleton_b_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_674_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_675_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_676_is__singletonI_H,axiom,
! [A2: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ! [X2: $o,Y3: $o] :
( ( member_o @ X2 @ A2 )
=> ( ( member_o @ Y3 @ A2 )
=> ( X2 = Y3 ) ) )
=> ( is_singleton_o @ A2 ) ) ) ).
% is_singletonI'
thf(fact_677_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_678_finite__subset__induct,axiom,
! [F2: set_b_a,A2: set_b_a,P: set_b_a > $o] :
( ( finite_finite_b_a @ F2 )
=> ( ( ord_less_eq_set_b_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_b_a )
=> ( ! [A4: b > a,F3: set_b_a] :
( ( finite_finite_b_a @ F3 )
=> ( ( member_b_a @ A4 @ A2 )
=> ( ~ ( member_b_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b_a @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_679_finite__subset__induct,axiom,
! [F2: set_b,A2: set_b,P: set_b > $o] :
( ( finite_finite_b @ F2 )
=> ( ( ord_less_eq_set_b @ F2 @ A2 )
=> ( ( P @ bot_bot_set_b )
=> ( ! [A4: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ( member_b @ A4 @ A2 )
=> ( ~ ( member_b @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_680_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_681_finite__subset__induct,axiom,
! [F2: set_o,A2: set_o,P: set_o > $o] :
( ( finite_finite_o @ F2 )
=> ( ( ord_less_eq_set_o @ F2 @ A2 )
=> ( ( P @ bot_bot_set_o )
=> ( ! [A4: $o,F3: set_o] :
( ( finite_finite_o @ F3 )
=> ( ( member_o @ A4 @ A2 )
=> ( ~ ( member_o @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_o @ A4 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_682_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_683_finite__subset__induct_H,axiom,
! [F2: set_b_a,A2: set_b_a,P: set_b_a > $o] :
( ( finite_finite_b_a @ F2 )
=> ( ( ord_less_eq_set_b_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_b_a )
=> ( ! [A4: b > a,F3: set_b_a] :
( ( finite_finite_b_a @ F3 )
=> ( ( member_b_a @ A4 @ A2 )
=> ( ( ord_less_eq_set_b_a @ F3 @ A2 )
=> ( ~ ( member_b_a @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b_a @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_684_finite__subset__induct_H,axiom,
! [F2: set_b,A2: set_b,P: set_b > $o] :
( ( finite_finite_b @ F2 )
=> ( ( ord_less_eq_set_b @ F2 @ A2 )
=> ( ( P @ bot_bot_set_b )
=> ( ! [A4: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ( member_b @ A4 @ A2 )
=> ( ( ord_less_eq_set_b @ F3 @ A2 )
=> ( ~ ( member_b @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_685_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_686_finite__subset__induct_H,axiom,
! [F2: set_o,A2: set_o,P: set_o > $o] :
( ( finite_finite_o @ F2 )
=> ( ( ord_less_eq_set_o @ F2 @ A2 )
=> ( ( P @ bot_bot_set_o )
=> ( ! [A4: $o,F3: set_o] :
( ( finite_finite_o @ F3 )
=> ( ( member_o @ A4 @ A2 )
=> ( ( ord_less_eq_set_o @ F3 @ A2 )
=> ( ~ ( member_o @ A4 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_o @ A4 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_687_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_688_subgroup_Osubgroup__inverse__iff,axiom,
! [G2: set_b,M: set_b,Composition: b > b > b,Unit: b,X: b] :
( ( group_subgroup_b @ G2 @ M @ Composition @ Unit )
=> ( ( group_invertible_b @ M @ Composition @ Unit @ X )
=> ( ( member_b @ X @ M )
=> ( ( member_b @ ( group_inverse_b @ M @ Composition @ Unit @ X ) @ G2 )
= ( member_b @ X @ G2 ) ) ) ) ) ).
% subgroup.subgroup_inverse_iff
thf(fact_689_subgroup_Osubgroup__inverse__iff,axiom,
! [G2: set_b_a,M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,X: b > a] :
( ( group_subgroup_b_a @ G2 @ M @ Composition @ Unit )
=> ( ( group_invertible_b_a @ M @ Composition @ Unit @ X )
=> ( ( member_b_a @ X @ M )
=> ( ( member_b_a @ ( group_inverse_b_a @ M @ Composition @ Unit @ X ) @ G2 )
= ( member_b_a @ X @ G2 ) ) ) ) ) ).
% subgroup.subgroup_inverse_iff
thf(fact_690_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_691_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_692_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_693_is__singleton__def,axiom,
( is_singleton_b
= ( ^ [A6: set_b] :
? [X3: b] :
( A6
= ( insert_b @ X3 @ bot_bot_set_b ) ) ) ) ).
% is_singleton_def
thf(fact_694_is__singleton__def,axiom,
( is_singleton_o
= ( ^ [A6: set_o] :
? [X3: $o] :
( A6
= ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ).
% is_singleton_def
thf(fact_695_finite__ranking__induct,axiom,
! [S: set_b_a,P: set_b_a > $o,F: ( b > a ) > nat] :
( ( finite_finite_b_a @ S )
=> ( ( P @ bot_bot_set_b_a )
=> ( ! [X2: b > a,S2: set_b_a] :
( ( finite_finite_b_a @ S2 )
=> ( ! [Y5: b > a] :
( ( member_b_a @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_b_a @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_696_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 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_697_finite__ranking__induct,axiom,
! [S: set_b,P: set_b > $o,F: b > nat] :
( ( finite_finite_b @ S )
=> ( ( P @ bot_bot_set_b )
=> ( ! [X2: b,S2: set_b] :
( ( finite_finite_b @ S2 )
=> ( ! [Y5: b] :
( ( member_b @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_b @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_698_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 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_699_finite__ranking__induct,axiom,
! [S: set_o,P: set_o > $o,F: $o > nat] :
( ( finite_finite_o @ S )
=> ( ( P @ bot_bot_set_o )
=> ( ! [X2: $o,S2: set_o] :
( ( finite_finite_o @ S2 )
=> ( ! [Y5: $o] :
( ( member_o @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_o @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_700_insert__subsetI,axiom,
! [X: $o,A2: set_o,X4: set_o] :
( ( member_o @ X @ A2 )
=> ( ( ord_less_eq_set_o @ X4 @ A2 )
=> ( ord_less_eq_set_o @ ( insert_o @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_701_insert__subsetI,axiom,
! [X: b,A2: set_b,X4: set_b] :
( ( member_b @ X @ A2 )
=> ( ( ord_less_eq_set_b @ X4 @ A2 )
=> ( ord_less_eq_set_b @ ( insert_b @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_702_insert__subsetI,axiom,
! [X: b > a,A2: set_b_a,X4: set_b_a] :
( ( member_b_a @ X @ A2 )
=> ( ( ord_less_eq_set_b_a @ X4 @ A2 )
=> ( ord_less_eq_set_b_a @ ( insert_b_a @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_703_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_704_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_705_subset__emptyI,axiom,
! [A2: set_b] :
( ! [X2: b] :
~ ( member_b @ X2 @ A2 )
=> ( ord_less_eq_set_b @ A2 @ bot_bot_set_b ) ) ).
% subset_emptyI
thf(fact_706_subset__emptyI,axiom,
! [A2: set_b_a] :
( ! [X2: b > a] :
~ ( member_b_a @ X2 @ A2 )
=> ( ord_less_eq_set_b_a @ A2 @ bot_bot_set_b_a ) ) ).
% subset_emptyI
thf(fact_707_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_708_subset__emptyI,axiom,
! [A2: set_o] :
( ! [X2: $o] :
~ ( member_o @ X2 @ A2 )
=> ( ord_less_eq_set_o @ A2 @ bot_bot_set_o ) ) ).
% subset_emptyI
thf(fact_709_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_710_Set__filter__fold,axiom,
! [A2: set_b,P: b > $o] :
( ( finite_finite_b @ A2 )
=> ( ( filter_b @ P @ A2 )
= ( finite_fold_b_set_b
@ ^ [X3: b,A8: set_b] : ( if_set_b @ ( P @ X3 ) @ ( insert_b @ X3 @ A8 ) @ A8 )
@ bot_bot_set_b
@ A2 ) ) ) ).
% Set_filter_fold
thf(fact_711_Set__filter__fold,axiom,
! [A2: set_nat,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( filter_nat @ P @ A2 )
= ( finite5529483035118572448et_nat
@ ^ [X3: nat,A8: set_nat] : ( if_set_nat @ ( P @ X3 ) @ ( insert_nat @ X3 @ A8 ) @ A8 )
@ bot_bot_set_nat
@ A2 ) ) ) ).
% Set_filter_fold
thf(fact_712_Set__filter__fold,axiom,
! [A2: set_o,P: $o > $o] :
( ( finite_finite_o @ A2 )
=> ( ( filter_o @ P @ A2 )
= ( finite_fold_o_set_o
@ ^ [X3: $o,A8: set_o] : ( if_set_o @ ( P @ X3 ) @ ( insert_o @ X3 @ A8 ) @ A8 )
@ bot_bot_set_o
@ A2 ) ) ) ).
% Set_filter_fold
thf(fact_713_DiffI,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ A2 )
=> ( ~ ( member_b @ C @ B2 )
=> ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_714_DiffI,axiom,
! [C: b > a,A2: set_b_a,B2: set_b_a] :
( ( member_b_a @ C @ A2 )
=> ( ~ ( member_b_a @ C @ B2 )
=> ( member_b_a @ C @ ( minus_minus_set_b_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_715_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_716_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_717_Diff__iff,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
= ( ( member_b @ C @ A2 )
& ~ ( member_b @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_718_Diff__iff,axiom,
! [C: b > a,A2: set_b_a,B2: set_b_a] :
( ( member_b_a @ C @ ( minus_minus_set_b_a @ A2 @ B2 ) )
= ( ( member_b_a @ C @ A2 )
& ~ ( member_b_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_719_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_720_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_721_member__filter,axiom,
! [X: b,P: b > $o,A2: set_b] :
( ( member_b @ X @ ( filter_b @ P @ A2 ) )
= ( ( member_b @ X @ A2 )
& ( P @ X ) ) ) ).
% member_filter
thf(fact_722_member__filter,axiom,
! [X: b > a,P: ( b > a ) > $o,A2: set_b_a] :
( ( member_b_a @ X @ ( filter_b_a @ P @ A2 ) )
= ( ( member_b_a @ X @ A2 )
& ( P @ X ) ) ) ).
% member_filter
thf(fact_723_member__filter,axiom,
! [X: a,P: a > $o,A2: set_a] :
( ( member_a @ X @ ( filter_a @ P @ A2 ) )
= ( ( member_a @ X @ A2 )
& ( P @ X ) ) ) ).
% member_filter
thf(fact_724_member__filter,axiom,
! [X: nat,P: nat > $o,A2: set_nat] :
( ( member_nat @ X @ ( filter_nat @ P @ A2 ) )
= ( ( member_nat @ X @ A2 )
& ( P @ X ) ) ) ).
% member_filter
thf(fact_725_Diff__cancel,axiom,
! [A2: set_o] :
( ( minus_minus_set_o @ A2 @ A2 )
= bot_bot_set_o ) ).
% Diff_cancel
thf(fact_726_empty__Diff,axiom,
! [A2: set_o] :
( ( minus_minus_set_o @ bot_bot_set_o @ A2 )
= bot_bot_set_o ) ).
% empty_Diff
thf(fact_727_Diff__empty,axiom,
! [A2: set_o] :
( ( minus_minus_set_o @ A2 @ bot_bot_set_o )
= A2 ) ).
% Diff_empty
thf(fact_728_finite__Diff2,axiom,
! [B2: set_b,A2: set_b] :
( ( finite_finite_b @ B2 )
=> ( ( finite_finite_b @ ( minus_minus_set_b @ A2 @ B2 ) )
= ( finite_finite_b @ A2 ) ) ) ).
% finite_Diff2
thf(fact_729_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_730_finite__Diff,axiom,
! [A2: set_b,B2: set_b] :
( ( finite_finite_b @ A2 )
=> ( finite_finite_b @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_731_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_732_Diff__insert0,axiom,
! [X: $o,A2: set_o,B2: set_o] :
( ~ ( member_o @ X @ A2 )
=> ( ( minus_minus_set_o @ A2 @ ( insert_o @ X @ B2 ) )
= ( minus_minus_set_o @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_733_Diff__insert0,axiom,
! [X: b,A2: set_b,B2: set_b] :
( ~ ( member_b @ X @ A2 )
=> ( ( minus_minus_set_b @ A2 @ ( insert_b @ X @ B2 ) )
= ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_734_Diff__insert0,axiom,
! [X: b > a,A2: set_b_a,B2: set_b_a] :
( ~ ( member_b_a @ X @ A2 )
=> ( ( minus_minus_set_b_a @ A2 @ ( insert_b_a @ X @ B2 ) )
= ( minus_minus_set_b_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_735_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_736_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_737_insert__Diff1,axiom,
! [X: $o,B2: set_o,A2: set_o] :
( ( member_o @ X @ B2 )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ B2 )
= ( minus_minus_set_o @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_738_insert__Diff1,axiom,
! [X: b,B2: set_b,A2: set_b] :
( ( member_b @ X @ B2 )
=> ( ( minus_minus_set_b @ ( insert_b @ X @ A2 ) @ B2 )
= ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_739_insert__Diff1,axiom,
! [X: b > a,B2: set_b_a,A2: set_b_a] :
( ( member_b_a @ X @ B2 )
=> ( ( minus_minus_set_b_a @ ( insert_b_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_b_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_740_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_741_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_742_Diff__eq__empty__iff,axiom,
! [A2: set_o,B2: set_o] :
( ( ( minus_minus_set_o @ A2 @ B2 )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_743_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_744_insert__Diff__single,axiom,
! [A: b,A2: set_b] :
( ( insert_b @ A @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
= ( insert_b @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_745_insert__Diff__single,axiom,
! [A: $o,A2: set_o] :
( ( insert_o @ A @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( insert_o @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_746_finite__Diff__insert,axiom,
! [A2: set_o,A: $o,B2: set_o] :
( ( finite_finite_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B2 ) ) )
= ( finite_finite_o @ ( minus_minus_set_o @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_747_finite__Diff__insert,axiom,
! [A2: set_b,A: b,B2: set_b] :
( ( finite_finite_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ B2 ) ) )
= ( finite_finite_b @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_748_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_749_set__diff__eq,axiom,
( minus_minus_set_b
= ( ^ [A6: set_b,B4: set_b] :
( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ A6 )
& ~ ( member_b @ X3 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_750_set__diff__eq,axiom,
( minus_minus_set_b_a
= ( ^ [A6: set_b_a,B4: set_b_a] :
( collect_b_a
@ ^ [X3: b > a] :
( ( member_b_a @ X3 @ A6 )
& ~ ( member_b_a @ X3 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_751_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A6: set_a,B4: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A6 )
& ~ ( member_a @ X3 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_752_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A6: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A6 )
& ~ ( member_nat @ X3 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_753_DiffE,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
=> ~ ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B2 ) ) ) ).
% DiffE
thf(fact_754_DiffE,axiom,
! [C: b > a,A2: set_b_a,B2: set_b_a] :
( ( member_b_a @ C @ ( minus_minus_set_b_a @ A2 @ B2 ) )
=> ~ ( ( member_b_a @ C @ A2 )
=> ( member_b_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_755_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_756_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_757_DiffD1,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
=> ( member_b @ C @ A2 ) ) ).
% DiffD1
thf(fact_758_DiffD1,axiom,
! [C: b > a,A2: set_b_a,B2: set_b_a] :
( ( member_b_a @ C @ ( minus_minus_set_b_a @ A2 @ B2 ) )
=> ( member_b_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_759_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_760_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_761_DiffD2,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
=> ~ ( member_b @ C @ B2 ) ) ).
% DiffD2
thf(fact_762_DiffD2,axiom,
! [C: b > a,A2: set_b_a,B2: set_b_a] :
( ( member_b_a @ C @ ( minus_minus_set_b_a @ A2 @ B2 ) )
=> ~ ( member_b_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_763_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_764_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_765_Diff__infinite__finite,axiom,
! [T2: set_b,S: set_b] :
( ( finite_finite_b @ T2 )
=> ( ~ ( finite_finite_b @ S )
=> ~ ( finite_finite_b @ ( minus_minus_set_b @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_766_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_767_Diff__mono,axiom,
! [A2: set_a,C5: set_a,D2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C5 )
=> ( ( ord_less_eq_set_a @ D2 @ B2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ C5 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_768_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_769_double__diff,axiom,
! [A2: set_a,B2: set_a,C5: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C5 )
=> ( ( minus_minus_set_a @ B2 @ ( minus_minus_set_a @ C5 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_770_insert__Diff__if,axiom,
! [X: $o,B2: set_o,A2: set_o] :
( ( ( member_o @ X @ B2 )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ B2 )
= ( minus_minus_set_o @ A2 @ B2 ) ) )
& ( ~ ( member_o @ X @ B2 )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ B2 )
= ( insert_o @ X @ ( minus_minus_set_o @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_771_insert__Diff__if,axiom,
! [X: b,B2: set_b,A2: set_b] :
( ( ( member_b @ X @ B2 )
=> ( ( minus_minus_set_b @ ( insert_b @ X @ A2 ) @ B2 )
= ( minus_minus_set_b @ A2 @ B2 ) ) )
& ( ~ ( member_b @ X @ B2 )
=> ( ( minus_minus_set_b @ ( insert_b @ X @ A2 ) @ B2 )
= ( insert_b @ X @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_772_insert__Diff__if,axiom,
! [X: b > a,B2: set_b_a,A2: set_b_a] :
( ( ( member_b_a @ X @ B2 )
=> ( ( minus_minus_set_b_a @ ( insert_b_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_b_a @ A2 @ B2 ) ) )
& ( ~ ( member_b_a @ X @ B2 )
=> ( ( minus_minus_set_b_a @ ( insert_b_a @ X @ A2 ) @ B2 )
= ( insert_b_a @ X @ ( minus_minus_set_b_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_773_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_774_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_775_Set_Ofilter__def,axiom,
( filter_b
= ( ^ [P2: b > $o,A6: set_b] :
( collect_b
@ ^ [A3: b] :
( ( member_b @ A3 @ A6 )
& ( P2 @ A3 ) ) ) ) ) ).
% Set.filter_def
thf(fact_776_Set_Ofilter__def,axiom,
( filter_b_a
= ( ^ [P2: ( b > a ) > $o,A6: set_b_a] :
( collect_b_a
@ ^ [A3: b > a] :
( ( member_b_a @ A3 @ A6 )
& ( P2 @ A3 ) ) ) ) ) ).
% Set.filter_def
thf(fact_777_Set_Ofilter__def,axiom,
( filter_a
= ( ^ [P2: a > $o,A6: set_a] :
( collect_a
@ ^ [A3: a] :
( ( member_a @ A3 @ A6 )
& ( P2 @ A3 ) ) ) ) ) ).
% Set.filter_def
thf(fact_778_Set_Ofilter__def,axiom,
( filter_nat
= ( ^ [P2: nat > $o,A6: set_nat] :
( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A6 )
& ( P2 @ A3 ) ) ) ) ) ).
% Set.filter_def
thf(fact_779_less__eq__set__def,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B4: set_b] :
( ord_less_eq_b_o
@ ^ [X3: b] : ( member_b @ X3 @ A6 )
@ ^ [X3: b] : ( member_b @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_780_less__eq__set__def,axiom,
( ord_less_eq_set_b_a
= ( ^ [A6: set_b_a,B4: set_b_a] :
( ord_less_eq_b_a_o
@ ^ [X3: b > a] : ( member_b_a @ X3 @ A6 )
@ ^ [X3: b > a] : ( member_b_a @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_781_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B4: set_nat] :
( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_782_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B4: set_a] :
( ord_less_eq_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A6 )
@ ^ [X3: a] : ( member_a @ X3 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_783_Diff__insert__absorb,axiom,
! [X: b,A2: set_b] :
( ~ ( member_b @ X @ A2 )
=> ( ( minus_minus_set_b @ ( insert_b @ X @ A2 ) @ ( insert_b @ X @ bot_bot_set_b ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_784_Diff__insert__absorb,axiom,
! [X: b > a,A2: set_b_a] :
( ~ ( member_b_a @ X @ A2 )
=> ( ( minus_minus_set_b_a @ ( insert_b_a @ X @ A2 ) @ ( insert_b_a @ X @ bot_bot_set_b_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_785_Diff__insert__absorb,axiom,
! [X: a,A2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_786_Diff__insert__absorb,axiom,
! [X: nat,A2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_787_Diff__insert__absorb,axiom,
! [X: $o,A2: set_o] :
( ~ ( member_o @ X @ A2 )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ ( insert_o @ X @ bot_bot_set_o ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_788_Diff__insert2,axiom,
! [A2: set_b,A: b,B2: set_b] :
( ( minus_minus_set_b @ A2 @ ( insert_b @ A @ B2 ) )
= ( minus_minus_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_789_Diff__insert2,axiom,
! [A2: set_o,A: $o,B2: set_o] :
( ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B2 ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_790_insert__Diff,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ( ( insert_b @ A @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_791_insert__Diff,axiom,
! [A: b > a,A2: set_b_a] :
( ( member_b_a @ A @ A2 )
=> ( ( insert_b_a @ A @ ( minus_minus_set_b_a @ A2 @ ( insert_b_a @ A @ bot_bot_set_b_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_792_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_793_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_794_insert__Diff,axiom,
! [A: $o,A2: set_o] :
( ( member_o @ A @ A2 )
=> ( ( insert_o @ A @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_795_Diff__insert,axiom,
! [A2: set_b,A: b,B2: set_b] :
( ( minus_minus_set_b @ A2 @ ( insert_b @ A @ B2 ) )
= ( minus_minus_set_b @ ( minus_minus_set_b @ A2 @ B2 ) @ ( insert_b @ A @ bot_bot_set_b ) ) ) ).
% Diff_insert
thf(fact_796_Diff__insert,axiom,
! [A2: set_o,A: $o,B2: set_o] :
( ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B2 ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ B2 ) @ ( insert_o @ A @ bot_bot_set_o ) ) ) ).
% Diff_insert
thf(fact_797_subset__Diff__insert,axiom,
! [A2: set_o,B2: set_o,X: $o,C5: set_o] :
( ( ord_less_eq_set_o @ A2 @ ( minus_minus_set_o @ B2 @ ( insert_o @ X @ C5 ) ) )
= ( ( ord_less_eq_set_o @ A2 @ ( minus_minus_set_o @ B2 @ C5 ) )
& ~ ( member_o @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_798_subset__Diff__insert,axiom,
! [A2: set_b,B2: set_b,X: b,C5: set_b] :
( ( ord_less_eq_set_b @ A2 @ ( minus_minus_set_b @ B2 @ ( insert_b @ X @ C5 ) ) )
= ( ( ord_less_eq_set_b @ A2 @ ( minus_minus_set_b @ B2 @ C5 ) )
& ~ ( member_b @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_799_subset__Diff__insert,axiom,
! [A2: set_b_a,B2: set_b_a,X: b > a,C5: set_b_a] :
( ( ord_less_eq_set_b_a @ A2 @ ( minus_minus_set_b_a @ B2 @ ( insert_b_a @ X @ C5 ) ) )
= ( ( ord_less_eq_set_b_a @ A2 @ ( minus_minus_set_b_a @ B2 @ C5 ) )
& ~ ( member_b_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_800_subset__Diff__insert,axiom,
! [A2: set_nat,B2: set_nat,X: nat,C5: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X @ C5 ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C5 ) )
& ~ ( member_nat @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_801_subset__Diff__insert,axiom,
! [A2: set_a,B2: set_a,X: a,C5: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ ( insert_a @ X @ C5 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ C5 ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_802_finite__filter,axiom,
! [S: set_b,P: b > $o] :
( ( finite_finite_b @ S )
=> ( finite_finite_b @ ( filter_b @ P @ S ) ) ) ).
% finite_filter
thf(fact_803_finite__filter,axiom,
! [S: set_nat,P: nat > $o] :
( ( finite_finite_nat @ S )
=> ( finite_finite_nat @ ( filter_nat @ P @ S ) ) ) ).
% finite_filter
thf(fact_804_finite__empty__induct,axiom,
! [A2: set_b_a,P: set_b_a > $o] :
( ( finite_finite_b_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A4: b > a,A5: set_b_a] :
( ( finite_finite_b_a @ A5 )
=> ( ( member_b_a @ A4 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_minus_set_b_a @ A5 @ ( insert_b_a @ A4 @ bot_bot_set_b_a ) ) ) ) ) )
=> ( P @ bot_bot_set_b_a ) ) ) ) ).
% finite_empty_induct
thf(fact_805_finite__empty__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A4: a,A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( member_a @ A4 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_minus_set_a @ A5 @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_806_finite__empty__induct,axiom,
! [A2: set_b,P: set_b > $o] :
( ( finite_finite_b @ A2 )
=> ( ( P @ A2 )
=> ( ! [A4: b,A5: set_b] :
( ( finite_finite_b @ A5 )
=> ( ( member_b @ A4 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_minus_set_b @ A5 @ ( insert_b @ A4 @ bot_bot_set_b ) ) ) ) ) )
=> ( P @ bot_bot_set_b ) ) ) ) ).
% finite_empty_induct
thf(fact_807_finite__empty__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A4: nat,A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ( member_nat @ A4 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_808_finite__empty__induct,axiom,
! [A2: set_o,P: set_o > $o] :
( ( finite_finite_o @ A2 )
=> ( ( P @ A2 )
=> ( ! [A4: $o,A5: set_o] :
( ( finite_finite_o @ A5 )
=> ( ( member_o @ A4 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_minus_set_o @ A5 @ ( insert_o @ A4 @ bot_bot_set_o ) ) ) ) ) )
=> ( P @ bot_bot_set_o ) ) ) ) ).
% finite_empty_induct
thf(fact_809_infinite__coinduct,axiom,
! [X4: set_b > $o,A2: set_b] :
( ( X4 @ A2 )
=> ( ! [A5: set_b] :
( ( X4 @ A5 )
=> ? [X5: b] :
( ( member_b @ X5 @ A5 )
& ( ( X4 @ ( minus_minus_set_b @ A5 @ ( insert_b @ X5 @ bot_bot_set_b ) ) )
| ~ ( finite_finite_b @ ( minus_minus_set_b @ A5 @ ( insert_b @ X5 @ bot_bot_set_b ) ) ) ) ) )
=> ~ ( finite_finite_b @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_810_infinite__coinduct,axiom,
! [X4: set_nat > $o,A2: set_nat] :
( ( X4 @ A2 )
=> ( ! [A5: set_nat] :
( ( X4 @ A5 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A5 )
& ( ( X4 @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_811_infinite__coinduct,axiom,
! [X4: set_o > $o,A2: set_o] :
( ( X4 @ A2 )
=> ( ! [A5: set_o] :
( ( X4 @ A5 )
=> ? [X5: $o] :
( ( member_o @ X5 @ A5 )
& ( ( X4 @ ( minus_minus_set_o @ A5 @ ( insert_o @ X5 @ bot_bot_set_o ) ) )
| ~ ( finite_finite_o @ ( minus_minus_set_o @ A5 @ ( insert_o @ X5 @ bot_bot_set_o ) ) ) ) ) )
=> ~ ( finite_finite_o @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_812_infinite__remove,axiom,
! [S: set_b,A: b] :
( ~ ( finite_finite_b @ S )
=> ~ ( finite_finite_b @ ( minus_minus_set_b @ S @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ).
% infinite_remove
thf(fact_813_infinite__remove,axiom,
! [S: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_814_infinite__remove,axiom,
! [S: set_o,A: $o] :
( ~ ( finite_finite_o @ S )
=> ~ ( finite_finite_o @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).
% infinite_remove
thf(fact_815_Diff__single__insert,axiom,
! [A2: set_b,X: b,B2: set_b] :
( ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) ) @ B2 )
=> ( ord_less_eq_set_b @ A2 @ ( insert_b @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_816_Diff__single__insert,axiom,
! [A2: set_o,X: $o,B2: set_o] :
( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B2 )
=> ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_817_Diff__single__insert,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_818_subset__insert__iff,axiom,
! [A2: set_b,X: b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X @ B2 ) )
= ( ( ( member_b @ X @ A2 )
=> ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) ) @ B2 ) )
& ( ~ ( member_b @ X @ A2 )
=> ( ord_less_eq_set_b @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_819_subset__insert__iff,axiom,
! [A2: set_b_a,X: b > a,B2: set_b_a] :
( ( ord_less_eq_set_b_a @ A2 @ ( insert_b_a @ X @ B2 ) )
= ( ( ( member_b_a @ X @ A2 )
=> ( ord_less_eq_set_b_a @ ( minus_minus_set_b_a @ A2 @ ( insert_b_a @ X @ bot_bot_set_b_a ) ) @ B2 ) )
& ( ~ ( member_b_a @ X @ A2 )
=> ( ord_less_eq_set_b_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_820_subset__insert__iff,axiom,
! [A2: set_nat,X: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
= ( ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_821_subset__insert__iff,axiom,
! [A2: set_o,X: $o,B2: set_o] :
( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ B2 ) )
= ( ( ( member_o @ X @ A2 )
=> ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B2 ) )
& ( ~ ( member_o @ X @ A2 )
=> ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_822_subset__insert__iff,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_823_finite__remove__induct,axiom,
! [B2: set_b_a,P: set_b_a > $o] :
( ( finite_finite_b_a @ B2 )
=> ( ( P @ bot_bot_set_b_a )
=> ( ! [A5: set_b_a] :
( ( finite_finite_b_a @ A5 )
=> ( ( A5 != bot_bot_set_b_a )
=> ( ( ord_less_eq_set_b_a @ A5 @ B2 )
=> ( ! [X5: b > a] :
( ( member_b_a @ X5 @ A5 )
=> ( P @ ( minus_minus_set_b_a @ A5 @ ( insert_b_a @ X5 @ bot_bot_set_b_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_824_finite__remove__induct,axiom,
! [B2: set_b,P: set_b > $o] :
( ( finite_finite_b @ B2 )
=> ( ( P @ bot_bot_set_b )
=> ( ! [A5: set_b] :
( ( finite_finite_b @ A5 )
=> ( ( A5 != bot_bot_set_b )
=> ( ( ord_less_eq_set_b @ A5 @ B2 )
=> ( ! [X5: b] :
( ( member_b @ X5 @ A5 )
=> ( P @ ( minus_minus_set_b @ A5 @ ( insert_b @ X5 @ bot_bot_set_b ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_825_finite__remove__induct,axiom,
! [B2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ( A5 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A5 @ B2 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A5 )
=> ( P @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_826_finite__remove__induct,axiom,
! [B2: set_o,P: set_o > $o] :
( ( finite_finite_o @ B2 )
=> ( ( P @ bot_bot_set_o )
=> ( ! [A5: set_o] :
( ( finite_finite_o @ A5 )
=> ( ( A5 != bot_bot_set_o )
=> ( ( ord_less_eq_set_o @ A5 @ B2 )
=> ( ! [X5: $o] :
( ( member_o @ X5 @ A5 )
=> ( P @ ( minus_minus_set_o @ A5 @ ( insert_o @ X5 @ bot_bot_set_o ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_827_finite__remove__induct,axiom,
! [B2: set_a,P: set_a > $o] :
( ( finite_finite_a @ B2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( A5 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A5 @ B2 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A5 )
=> ( P @ ( minus_minus_set_a @ A5 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_828_remove__induct,axiom,
! [P: set_b_a > $o,B2: set_b_a] :
( ( P @ bot_bot_set_b_a )
=> ( ( ~ ( finite_finite_b_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A5: set_b_a] :
( ( finite_finite_b_a @ A5 )
=> ( ( A5 != bot_bot_set_b_a )
=> ( ( ord_less_eq_set_b_a @ A5 @ B2 )
=> ( ! [X5: b > a] :
( ( member_b_a @ X5 @ A5 )
=> ( P @ ( minus_minus_set_b_a @ A5 @ ( insert_b_a @ X5 @ bot_bot_set_b_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_829_remove__induct,axiom,
! [P: set_b > $o,B2: set_b] :
( ( P @ bot_bot_set_b )
=> ( ( ~ ( finite_finite_b @ B2 )
=> ( P @ B2 ) )
=> ( ! [A5: set_b] :
( ( finite_finite_b @ A5 )
=> ( ( A5 != bot_bot_set_b )
=> ( ( ord_less_eq_set_b @ A5 @ B2 )
=> ( ! [X5: b] :
( ( member_b @ X5 @ A5 )
=> ( P @ ( minus_minus_set_b @ A5 @ ( insert_b @ X5 @ bot_bot_set_b ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_830_remove__induct,axiom,
! [P: set_nat > $o,B2: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B2 )
=> ( P @ B2 ) )
=> ( ! [A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ( A5 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A5 @ B2 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A5 )
=> ( P @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_831_remove__induct,axiom,
! [P: set_o > $o,B2: set_o] :
( ( P @ bot_bot_set_o )
=> ( ( ~ ( finite_finite_o @ B2 )
=> ( P @ B2 ) )
=> ( ! [A5: set_o] :
( ( finite_finite_o @ A5 )
=> ( ( A5 != bot_bot_set_o )
=> ( ( ord_less_eq_set_o @ A5 @ B2 )
=> ( ! [X5: $o] :
( ( member_o @ X5 @ A5 )
=> ( P @ ( minus_minus_set_o @ A5 @ ( insert_o @ X5 @ bot_bot_set_o ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_832_remove__induct,axiom,
! [P: set_a > $o,B2: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( A5 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A5 @ B2 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A5 )
=> ( P @ ( minus_minus_set_a @ A5 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_833_Collect__restrict,axiom,
! [X4: set_b,P: b > $o] :
( ord_less_eq_set_b
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ X4 )
& ( P @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_834_Collect__restrict,axiom,
! [X4: set_b_a,P: ( b > a ) > $o] :
( ord_less_eq_set_b_a
@ ( collect_b_a
@ ^ [X3: b > a] :
( ( member_b_a @ X3 @ X4 )
& ( P @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_835_Collect__restrict,axiom,
! [X4: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X4 )
& ( P @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_836_Collect__restrict,axiom,
! [X4: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ X4 )
& ( P @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_837_prop__restrict,axiom,
! [X: b,Z3: set_b,X4: set_b,P: b > $o] :
( ( member_b @ X @ Z3 )
=> ( ( ord_less_eq_set_b @ Z3
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ X4 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_838_prop__restrict,axiom,
! [X: b > a,Z3: set_b_a,X4: set_b_a,P: ( b > a ) > $o] :
( ( member_b_a @ X @ Z3 )
=> ( ( ord_less_eq_set_b_a @ Z3
@ ( collect_b_a
@ ^ [X3: b > a] :
( ( member_b_a @ X3 @ X4 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_839_prop__restrict,axiom,
! [X: nat,Z3: set_nat,X4: set_nat,P: nat > $o] :
( ( member_nat @ X @ Z3 )
=> ( ( ord_less_eq_set_nat @ Z3
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X4 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_840_prop__restrict,axiom,
! [X: a,Z3: set_a,X4: set_a,P: a > $o] :
( ( member_a @ X @ Z3 )
=> ( ( ord_less_eq_set_a @ Z3
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ X4 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_841_diff__shunt__var,axiom,
! [X: set_o,Y: set_o] :
( ( ( minus_minus_set_o @ X @ Y )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_842_diff__shunt__var,axiom,
! [X: set_a,Y: set_a] :
( ( ( minus_minus_set_a @ X @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_843_bot__empty__eq,axiom,
( bot_bot_b_o
= ( ^ [X3: b] : ( member_b @ X3 @ bot_bot_set_b ) ) ) ).
% bot_empty_eq
thf(fact_844_bot__empty__eq,axiom,
( bot_bot_b_a_o
= ( ^ [X3: b > a] : ( member_b_a @ X3 @ bot_bot_set_b_a ) ) ) ).
% bot_empty_eq
thf(fact_845_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_846_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_847_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X3: $o] : ( member_o @ X3 @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_848_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_849_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_850_Collect__empty__eq__bot,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( P = bot_bot_o_o ) ) ).
% Collect_empty_eq_bot
thf(fact_851_arg__min__least,axiom,
! [S: set_b_a,Y: b > a,F: ( b > a ) > nat] :
( ( finite_finite_b_a @ S )
=> ( ( S != bot_bot_set_b_a )
=> ( ( member_b_a @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7032157021346806415_a_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_852_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_853_arg__min__least,axiom,
! [S: set_b,Y: b,F: b > nat] :
( ( finite_finite_b @ S )
=> ( ( S != bot_bot_set_b )
=> ( ( member_b @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7575731748627795062_b_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_854_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_855_arg__min__least,axiom,
! [S: set_o,Y: $o,F: $o > nat] :
( ( finite_finite_o @ S )
=> ( ( S != bot_bot_set_o )
=> ( ( member_o @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic2775856028456453135_o_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_856_minus__fold__remove,axiom,
! [A2: set_b,B2: set_b] :
( ( finite_finite_b @ A2 )
=> ( ( minus_minus_set_b @ B2 @ A2 )
= ( finite_fold_b_set_b @ remove_b @ B2 @ A2 ) ) ) ).
% minus_fold_remove
thf(fact_857_minus__fold__remove,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( minus_minus_set_nat @ B2 @ A2 )
= ( finite5529483035118572448et_nat @ remove_nat @ B2 @ A2 ) ) ) ).
% minus_fold_remove
thf(fact_858_member__remove,axiom,
! [X: b,Y: b,A2: set_b] :
( ( member_b @ X @ ( remove_b @ Y @ A2 ) )
= ( ( member_b @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_859_member__remove,axiom,
! [X: b > a,Y: b > a,A2: set_b_a] :
( ( member_b_a @ X @ ( remove_b_a @ Y @ A2 ) )
= ( ( member_b_a @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_860_member__remove,axiom,
! [X: a,Y: a,A2: set_a] :
( ( member_a @ X @ ( remove_a @ Y @ A2 ) )
= ( ( member_a @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_861_member__remove,axiom,
! [X: nat,Y: nat,A2: set_nat] :
( ( member_nat @ X @ ( remove_nat @ Y @ A2 ) )
= ( ( member_nat @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_862_minus__set__def,axiom,
( minus_minus_set_b
= ( ^ [A6: set_b,B4: set_b] :
( collect_b
@ ( minus_minus_b_o
@ ^ [X3: b] : ( member_b @ X3 @ A6 )
@ ^ [X3: b] : ( member_b @ X3 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_863_minus__set__def,axiom,
( minus_minus_set_b_a
= ( ^ [A6: set_b_a,B4: set_b_a] :
( collect_b_a
@ ( minus_minus_b_a_o
@ ^ [X3: b > a] : ( member_b_a @ X3 @ A6 )
@ ^ [X3: b > a] : ( member_b_a @ X3 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_864_minus__set__def,axiom,
( minus_minus_set_a
= ( ^ [A6: set_a,B4: set_a] :
( collect_a
@ ( minus_minus_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A6 )
@ ^ [X3: a] : ( member_a @ X3 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_865_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A6: set_nat,B4: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_866_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_867_pred__subset__eq,axiom,
! [R: set_b,S: set_b] :
( ( ord_less_eq_b_o
@ ^ [X3: b] : ( member_b @ X3 @ R )
@ ^ [X3: b] : ( member_b @ X3 @ S ) )
= ( ord_less_eq_set_b @ R @ S ) ) ).
% pred_subset_eq
thf(fact_868_pred__subset__eq,axiom,
! [R: set_b_a,S: set_b_a] :
( ( ord_less_eq_b_a_o
@ ^ [X3: b > a] : ( member_b_a @ X3 @ R )
@ ^ [X3: b > a] : ( member_b_a @ X3 @ S ) )
= ( ord_less_eq_set_b_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_869_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_870_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_871_remove__def,axiom,
( remove_b
= ( ^ [X3: b,A6: set_b] : ( minus_minus_set_b @ A6 @ ( insert_b @ X3 @ bot_bot_set_b ) ) ) ) ).
% remove_def
thf(fact_872_remove__def,axiom,
( remove_o
= ( ^ [X3: $o,A6: set_o] : ( minus_minus_set_o @ A6 @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ).
% remove_def
thf(fact_873_extensional__empty,axiom,
( ( extensional_o_a @ bot_bot_set_o )
= ( insert_o_a
@ ^ [X3: $o] : undefined_a
@ bot_bot_set_o_a ) ) ).
% extensional_empty
thf(fact_874_Group__Theory_Ogroup_Ointro,axiom,
! [G2: set_a,Composition: a > a > a,Unit: a] :
( ( group_monoid_a @ G2 @ Composition @ Unit )
=> ( ( group_group_axioms_a @ G2 @ Composition @ Unit )
=> ( group_group_a @ G2 @ Composition @ Unit ) ) ) ).
% Group_Theory.group.intro
thf(fact_875_Group__Theory_Ogroup__def,axiom,
( group_group_a
= ( ^ [G4: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_monoid_a @ G4 @ Composition2 @ Unit2 )
& ( group_group_axioms_a @ G4 @ Composition2 @ Unit2 ) ) ) ) ).
% Group_Theory.group_def
thf(fact_876_extensional__insert__cancel,axiom,
! [A: b > a,I3: set_b,I2: b] :
( ( member_b_a @ A @ ( extensional_b_a @ I3 ) )
=> ( member_b_a @ A @ ( extensional_b_a @ ( insert_b @ I2 @ I3 ) ) ) ) ).
% extensional_insert_cancel
thf(fact_877_extensionalityI,axiom,
! [F: b > a,A2: set_b,G: b > a] :
( ( member_b_a @ F @ ( extensional_b_a @ A2 ) )
=> ( ( member_b_a @ G @ ( extensional_b_a @ A2 ) )
=> ( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( F = G ) ) ) ) ).
% extensionalityI
thf(fact_878_fold__graph__closed__eq,axiom,
! [A2: set_b,B2: set_b,F: b > b > b,G: b > b > b,Z2: b] :
( ! [A4: b,B5: b] :
( ( member_b @ A4 @ A2 )
=> ( ( member_b @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: b,B5: b] :
( ( member_b @ A4 @ A2 )
=> ( ( member_b @ B5 @ B2 )
=> ( member_b @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_b @ Z2 @ B2 )
=> ( ( finite5086552502106613254ph_b_b @ F @ Z2 @ A2 )
= ( finite5086552502106613254ph_b_b @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_879_fold__graph__closed__eq,axiom,
! [A2: set_b,B2: set_a,F: b > a > a,G: b > a > a,Z2: a] :
( ! [A4: b,B5: a] :
( ( member_b @ A4 @ A2 )
=> ( ( member_a @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: b,B5: a] :
( ( member_b @ A4 @ A2 )
=> ( ( member_a @ B5 @ B2 )
=> ( member_a @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_a @ Z2 @ B2 )
=> ( ( finite5086552502106613253ph_b_a @ F @ Z2 @ A2 )
= ( finite5086552502106613253ph_b_a @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_880_fold__graph__closed__eq,axiom,
! [A2: set_b,B2: set_nat,F: b > nat > nat,G: b > nat > nat,Z2: nat] :
( ! [A4: b,B5: nat] :
( ( member_b @ A4 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: b,B5: nat] :
( ( member_b @ A4 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( member_nat @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_nat @ Z2 @ B2 )
=> ( ( finite6345878069334568201_b_nat @ F @ Z2 @ A2 )
= ( finite6345878069334568201_b_nat @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_881_fold__graph__closed__eq,axiom,
! [A2: set_a,B2: set_b,F: a > b > b,G: a > b > b,Z2: b] :
( ! [A4: a,B5: b] :
( ( member_a @ A4 @ A2 )
=> ( ( member_b @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: a,B5: b] :
( ( member_a @ A4 @ A2 )
=> ( ( member_b @ B5 @ B2 )
=> ( member_b @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_b @ Z2 @ B2 )
=> ( ( finite7874008084079289287ph_a_b @ F @ Z2 @ A2 )
= ( finite7874008084079289287ph_a_b @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_882_fold__graph__closed__eq,axiom,
! [A2: set_a,B2: set_a,F: a > a > a,G: a > a > a,Z2: a] :
( ! [A4: a,B5: a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_a @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: a,B5: a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_a @ B5 @ B2 )
=> ( member_a @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_a @ Z2 @ B2 )
=> ( ( finite7874008084079289286ph_a_a @ F @ Z2 @ A2 )
= ( finite7874008084079289286ph_a_a @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_883_fold__graph__closed__eq,axiom,
! [A2: set_a,B2: set_nat,F: a > nat > nat,G: a > nat > nat,Z2: nat] :
( ! [A4: a,B5: nat] :
( ( member_a @ A4 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: a,B5: nat] :
( ( member_a @ A4 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( member_nat @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_nat @ Z2 @ B2 )
=> ( ( finite5110433740378173704_a_nat @ F @ Z2 @ A2 )
= ( finite5110433740378173704_a_nat @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_884_fold__graph__closed__eq,axiom,
! [A2: set_nat,B2: set_b,F: nat > b > b,G: nat > b > b,Z2: b] :
( ! [A4: nat,B5: b] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_b @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: nat,B5: b] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_b @ B5 @ B2 )
=> ( member_b @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_b @ Z2 @ B2 )
=> ( ( finite9142365241556460135_nat_b @ F @ Z2 @ A2 )
= ( finite9142365241556460135_nat_b @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_885_fold__graph__closed__eq,axiom,
! [A2: set_nat,B2: set_a,F: nat > a > a,G: nat > a > a,Z2: a] :
( ! [A4: nat,B5: a] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_a @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: nat,B5: a] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_a @ B5 @ B2 )
=> ( member_a @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_a @ Z2 @ B2 )
=> ( ( finite9142365241556460134_nat_a @ F @ Z2 @ A2 )
= ( finite9142365241556460134_nat_a @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_886_fold__graph__closed__eq,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat > nat,G: nat > nat > nat,Z2: nat] :
( ! [A4: nat,B5: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: nat,B5: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( member_nat @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_nat @ Z2 @ B2 )
=> ( ( finite1441398328259824232at_nat @ F @ Z2 @ A2 )
= ( finite1441398328259824232at_nat @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_887_fold__graph__closed__eq,axiom,
! [A2: set_b,B2: set_b_a,F: b > ( b > a ) > b > a,G: b > ( b > a ) > b > a,Z2: b > a] :
( ! [A4: b,B5: b > a] :
( ( member_b @ A4 @ A2 )
=> ( ( member_b_a @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: b,B5: b > a] :
( ( member_b @ A4 @ A2 )
=> ( ( member_b_a @ B5 @ B2 )
=> ( member_b_a @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_b_a @ Z2 @ B2 )
=> ( ( finite7185950986694385629_b_b_a @ F @ Z2 @ A2 )
= ( finite7185950986694385629_b_b_a @ G @ Z2 @ A2 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_888_fold__graph__closed__lemma,axiom,
! [G: b > b > b,Z2: b,A2: set_b,X: b,B2: set_b,F: b > b > b] :
( ( finite5086552502106613254ph_b_b @ G @ Z2 @ A2 @ X )
=> ( ! [A4: b,B5: b] :
( ( member_b @ A4 @ A2 )
=> ( ( member_b @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: b,B5: b] :
( ( member_b @ A4 @ A2 )
=> ( ( member_b @ B5 @ B2 )
=> ( member_b @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_b @ Z2 @ B2 )
=> ( ( finite5086552502106613254ph_b_b @ F @ Z2 @ A2 @ X )
& ( member_b @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_889_fold__graph__closed__lemma,axiom,
! [G: b > a > a,Z2: a,A2: set_b,X: a,B2: set_a,F: b > a > a] :
( ( finite5086552502106613253ph_b_a @ G @ Z2 @ A2 @ X )
=> ( ! [A4: b,B5: a] :
( ( member_b @ A4 @ A2 )
=> ( ( member_a @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: b,B5: a] :
( ( member_b @ A4 @ A2 )
=> ( ( member_a @ B5 @ B2 )
=> ( member_a @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_a @ Z2 @ B2 )
=> ( ( finite5086552502106613253ph_b_a @ F @ Z2 @ A2 @ X )
& ( member_a @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_890_fold__graph__closed__lemma,axiom,
! [G: b > nat > nat,Z2: nat,A2: set_b,X: nat,B2: set_nat,F: b > nat > nat] :
( ( finite6345878069334568201_b_nat @ G @ Z2 @ A2 @ X )
=> ( ! [A4: b,B5: nat] :
( ( member_b @ A4 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: b,B5: nat] :
( ( member_b @ A4 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( member_nat @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_nat @ Z2 @ B2 )
=> ( ( finite6345878069334568201_b_nat @ F @ Z2 @ A2 @ X )
& ( member_nat @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_891_fold__graph__closed__lemma,axiom,
! [G: a > b > b,Z2: b,A2: set_a,X: b,B2: set_b,F: a > b > b] :
( ( finite7874008084079289287ph_a_b @ G @ Z2 @ A2 @ X )
=> ( ! [A4: a,B5: b] :
( ( member_a @ A4 @ A2 )
=> ( ( member_b @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: a,B5: b] :
( ( member_a @ A4 @ A2 )
=> ( ( member_b @ B5 @ B2 )
=> ( member_b @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_b @ Z2 @ B2 )
=> ( ( finite7874008084079289287ph_a_b @ F @ Z2 @ A2 @ X )
& ( member_b @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_892_fold__graph__closed__lemma,axiom,
! [G: a > a > a,Z2: a,A2: set_a,X: a,B2: set_a,F: a > a > a] :
( ( finite7874008084079289286ph_a_a @ G @ Z2 @ A2 @ X )
=> ( ! [A4: a,B5: a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_a @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: a,B5: a] :
( ( member_a @ A4 @ A2 )
=> ( ( member_a @ B5 @ B2 )
=> ( member_a @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_a @ Z2 @ B2 )
=> ( ( finite7874008084079289286ph_a_a @ F @ Z2 @ A2 @ X )
& ( member_a @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_893_fold__graph__closed__lemma,axiom,
! [G: a > nat > nat,Z2: nat,A2: set_a,X: nat,B2: set_nat,F: a > nat > nat] :
( ( finite5110433740378173704_a_nat @ G @ Z2 @ A2 @ X )
=> ( ! [A4: a,B5: nat] :
( ( member_a @ A4 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: a,B5: nat] :
( ( member_a @ A4 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( member_nat @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_nat @ Z2 @ B2 )
=> ( ( finite5110433740378173704_a_nat @ F @ Z2 @ A2 @ X )
& ( member_nat @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_894_fold__graph__closed__lemma,axiom,
! [G: nat > b > b,Z2: b,A2: set_nat,X: b,B2: set_b,F: nat > b > b] :
( ( finite9142365241556460135_nat_b @ G @ Z2 @ A2 @ X )
=> ( ! [A4: nat,B5: b] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_b @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: nat,B5: b] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_b @ B5 @ B2 )
=> ( member_b @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_b @ Z2 @ B2 )
=> ( ( finite9142365241556460135_nat_b @ F @ Z2 @ A2 @ X )
& ( member_b @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_895_fold__graph__closed__lemma,axiom,
! [G: nat > a > a,Z2: a,A2: set_nat,X: a,B2: set_a,F: nat > a > a] :
( ( finite9142365241556460134_nat_a @ G @ Z2 @ A2 @ X )
=> ( ! [A4: nat,B5: a] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_a @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: nat,B5: a] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_a @ B5 @ B2 )
=> ( member_a @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_a @ Z2 @ B2 )
=> ( ( finite9142365241556460134_nat_a @ F @ Z2 @ A2 @ X )
& ( member_a @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_896_fold__graph__closed__lemma,axiom,
! [G: nat > nat > nat,Z2: nat,A2: set_nat,X: nat,B2: set_nat,F: nat > nat > nat] :
( ( finite1441398328259824232at_nat @ G @ Z2 @ A2 @ X )
=> ( ! [A4: nat,B5: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: nat,B5: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( member_nat @ B5 @ B2 )
=> ( member_nat @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_nat @ Z2 @ B2 )
=> ( ( finite1441398328259824232at_nat @ F @ Z2 @ A2 @ X )
& ( member_nat @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_897_fold__graph__closed__lemma,axiom,
! [G: b > ( b > a ) > b > a,Z2: b > a,A2: set_b,X: b > a,B2: set_b_a,F: b > ( b > a ) > b > a] :
( ( finite7185950986694385629_b_b_a @ G @ Z2 @ A2 @ X )
=> ( ! [A4: b,B5: b > a] :
( ( member_b @ A4 @ A2 )
=> ( ( member_b_a @ B5 @ B2 )
=> ( ( F @ A4 @ B5 )
= ( G @ A4 @ B5 ) ) ) )
=> ( ! [A4: b,B5: b > a] :
( ( member_b @ A4 @ A2 )
=> ( ( member_b_a @ B5 @ B2 )
=> ( member_b_a @ ( G @ A4 @ B5 ) @ B2 ) ) )
=> ( ( member_b_a @ Z2 @ B2 )
=> ( ( finite7185950986694385629_b_b_a @ F @ Z2 @ A2 @ X )
& ( member_b_a @ X @ B2 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_898_extensional__subset,axiom,
! [F: b > a,A2: set_b,B2: set_b] :
( ( member_b_a @ F @ ( extensional_b_a @ A2 ) )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( member_b_a @ F @ ( extensional_b_a @ B2 ) ) ) ) ).
% extensional_subset
thf(fact_899_extensional__arb,axiom,
! [F: ( b > a ) > a,A2: set_b_a,X: b > a] :
( ( member_b_a_a @ F @ ( extensional_b_a_a @ A2 ) )
=> ( ~ ( member_b_a @ X @ A2 )
=> ( ( F @ X )
= undefined_a ) ) ) ).
% extensional_arb
thf(fact_900_extensional__arb,axiom,
! [F: a > a,A2: set_a,X: a] :
( ( member_a_a @ F @ ( extensional_a_a @ A2 ) )
=> ( ~ ( member_a @ X @ A2 )
=> ( ( F @ X )
= undefined_a ) ) ) ).
% extensional_arb
thf(fact_901_extensional__arb,axiom,
! [F: nat > a,A2: set_nat,X: nat] :
( ( member_nat_a @ F @ ( extensional_nat_a @ A2 ) )
=> ( ~ ( member_nat @ X @ A2 )
=> ( ( F @ X )
= undefined_a ) ) ) ).
% extensional_arb
thf(fact_902_extensional__arb,axiom,
! [F: b > a,A2: set_b,X: b] :
( ( member_b_a @ F @ ( extensional_b_a @ A2 ) )
=> ( ~ ( member_b @ X @ A2 )
=> ( ( F @ X )
= undefined_a ) ) ) ).
% extensional_arb
thf(fact_903_Group__Theory_Ogroup__axioms__def,axiom,
( group_group_axioms_b
= ( ^ [G4: set_b,Composition2: b > b > b,Unit2: b] :
! [U2: b] :
( ( member_b @ U2 @ G4 )
=> ( group_invertible_b @ G4 @ Composition2 @ Unit2 @ U2 ) ) ) ) ).
% Group_Theory.group_axioms_def
thf(fact_904_Group__Theory_Ogroup__axioms__def,axiom,
( group_3984435576162330991ms_b_a
= ( ^ [G4: set_b_a,Composition2: ( b > a ) > ( b > a ) > b > a,Unit2: b > a] :
! [U2: b > a] :
( ( member_b_a @ U2 @ G4 )
=> ( group_invertible_b_a @ G4 @ Composition2 @ Unit2 @ U2 ) ) ) ) ).
% Group_Theory.group_axioms_def
thf(fact_905_Group__Theory_Ogroup__axioms__def,axiom,
( group_661118103997438619ms_nat
= ( ^ [G4: set_nat,Composition2: nat > nat > nat,Unit2: nat] :
! [U2: nat] :
( ( member_nat @ U2 @ G4 )
=> ( group_invertible_nat @ G4 @ Composition2 @ Unit2 @ U2 ) ) ) ) ).
% Group_Theory.group_axioms_def
thf(fact_906_Group__Theory_Ogroup__axioms__def,axiom,
( group_group_axioms_a
= ( ^ [G4: set_a,Composition2: a > a > a,Unit2: a] :
! [U2: a] :
( ( member_a @ U2 @ G4 )
=> ( group_invertible_a @ G4 @ Composition2 @ Unit2 @ U2 ) ) ) ) ).
% Group_Theory.group_axioms_def
thf(fact_907_Group__Theory_Ogroup__axioms_Ointro,axiom,
! [G2: set_b,Composition: b > b > b,Unit: b] :
( ! [U3: b] :
( ( member_b @ U3 @ G2 )
=> ( group_invertible_b @ G2 @ Composition @ Unit @ U3 ) )
=> ( group_group_axioms_b @ G2 @ Composition @ Unit ) ) ).
% Group_Theory.group_axioms.intro
thf(fact_908_Group__Theory_Ogroup__axioms_Ointro,axiom,
! [G2: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a] :
( ! [U3: b > a] :
( ( member_b_a @ U3 @ G2 )
=> ( group_invertible_b_a @ G2 @ Composition @ Unit @ U3 ) )
=> ( group_3984435576162330991ms_b_a @ G2 @ Composition @ Unit ) ) ).
% Group_Theory.group_axioms.intro
thf(fact_909_Group__Theory_Ogroup__axioms_Ointro,axiom,
! [G2: set_nat,Composition: nat > nat > nat,Unit: nat] :
( ! [U3: nat] :
( ( member_nat @ U3 @ G2 )
=> ( group_invertible_nat @ G2 @ Composition @ Unit @ U3 ) )
=> ( group_661118103997438619ms_nat @ G2 @ Composition @ Unit ) ) ).
% Group_Theory.group_axioms.intro
thf(fact_910_Group__Theory_Ogroup__axioms_Ointro,axiom,
! [G2: set_a,Composition: a > a > a,Unit: a] :
( ! [U3: a] :
( ( member_a @ U3 @ G2 )
=> ( group_invertible_a @ G2 @ Composition @ Unit @ U3 ) )
=> ( group_group_axioms_a @ G2 @ Composition @ Unit ) ) ).
% Group_Theory.group_axioms.intro
thf(fact_911_Group__Theory_Ogroup_Oaxioms_I2_J,axiom,
! [G2: set_a,Composition: a > a > a,Unit: a] :
( ( group_group_a @ G2 @ Composition @ Unit )
=> ( group_group_axioms_a @ G2 @ Composition @ Unit ) ) ).
% Group_Theory.group.axioms(2)
thf(fact_912_PiE__singleton,axiom,
! [F: b > a,A2: set_b] :
( ( member_b_a @ F @ ( extensional_b_a @ A2 ) )
=> ( ( piE_b_a @ A2
@ ^ [X3: b] : ( insert_a @ ( F @ X3 ) @ bot_bot_set_a ) )
= ( insert_b_a @ F @ bot_bot_set_b_a ) ) ) ).
% PiE_singleton
thf(fact_913_group_Oinverse__subgroupD,axiom,
! [G2: set_a,Composition: a > a > a,Unit: a,H2: set_a] :
( ( group_group_a @ G2 @ Composition @ Unit )
=> ( ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ G2 @ Composition @ Unit ) @ H2 ) @ G2 @ Composition @ Unit )
=> ( ( ord_less_eq_set_a @ H2 @ ( group_Units_a @ G2 @ Composition @ Unit ) )
=> ( group_subgroup_a @ H2 @ G2 @ Composition @ Unit ) ) ) ) ).
% group.inverse_subgroupD
thf(fact_914_PiE__empty__domain,axiom,
! [T2: $o > set_a] :
( ( piE_o_a @ bot_bot_set_o @ T2 )
= ( insert_o_a
@ ^ [X3: $o] : undefined_a
@ bot_bot_set_o_a ) ) ).
% PiE_empty_domain
thf(fact_915_subset__singleton__iff__Uniq,axiom,
! [A2: set_b] :
( ( ? [A3: b] : ( ord_less_eq_set_b @ A2 @ ( insert_b @ A3 @ bot_bot_set_b ) ) )
= ( uniq_b
@ ^ [X3: b] : ( member_b @ X3 @ A2 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_916_subset__singleton__iff__Uniq,axiom,
! [A2: set_b_a] :
( ( ? [A3: b > a] : ( ord_less_eq_set_b_a @ A2 @ ( insert_b_a @ A3 @ bot_bot_set_b_a ) ) )
= ( uniq_b_a
@ ^ [X3: b > a] : ( member_b_a @ X3 @ A2 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_917_subset__singleton__iff__Uniq,axiom,
! [A2: set_nat] :
( ( ? [A3: nat] : ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) )
= ( uniq_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_918_subset__singleton__iff__Uniq,axiom,
! [A2: set_o] :
( ( ? [A3: $o] : ( ord_less_eq_set_o @ A2 @ ( insert_o @ A3 @ bot_bot_set_o ) ) )
= ( uniq_o
@ ^ [X3: $o] : ( member_o @ X3 @ A2 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_919_subset__singleton__iff__Uniq,axiom,
! [A2: set_a] :
( ( ? [A3: a] : ( ord_less_eq_set_a @ A2 @ ( insert_a @ A3 @ bot_bot_set_a ) ) )
= ( uniq_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_920_abelian__group__def,axiom,
( group_201663378560352916roup_a
= ( ^ [G4: set_a,Composition2: a > a > a,Unit2: a] :
( ( group_group_a @ G4 @ Composition2 @ Unit2 )
& ( group_4866109990395492029noid_a @ G4 @ Composition2 @ Unit2 ) ) ) ) ).
% abelian_group_def
thf(fact_921_image__eqI,axiom,
! [B: b,F: b > b,X: b,A2: set_b] :
( ( B
= ( F @ X ) )
=> ( ( member_b @ X @ A2 )
=> ( member_b @ B @ ( image_b_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_922_image__eqI,axiom,
! [B: a,F: b > a,X: b,A2: set_b] :
( ( B
= ( F @ X ) )
=> ( ( member_b @ X @ A2 )
=> ( member_a @ B @ ( image_b_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_923_image__eqI,axiom,
! [B: nat,F: b > nat,X: b,A2: set_b] :
( ( B
= ( F @ X ) )
=> ( ( member_b @ X @ A2 )
=> ( member_nat @ B @ ( image_b_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_924_image__eqI,axiom,
! [B: b,F: a > b,X: a,A2: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_b @ B @ ( image_a_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_925_image__eqI,axiom,
! [B: a,F: a > a,X: a,A2: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ B @ ( image_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_926_image__eqI,axiom,
! [B: nat,F: a > nat,X: a,A2: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_nat @ B @ ( image_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_927_image__eqI,axiom,
! [B: b,F: nat > b,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_b @ B @ ( image_nat_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_928_image__eqI,axiom,
! [B: a,F: nat > a,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_a @ B @ ( image_nat_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_929_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A2: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_930_image__eqI,axiom,
! [B: b > a,F: b > b > a,X: b,A2: set_b] :
( ( B
= ( F @ X ) )
=> ( ( member_b @ X @ A2 )
=> ( member_b_a @ B @ ( image_b_b_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_931_image__empty,axiom,
! [F: $o > $o] :
( ( image_o_o @ F @ bot_bot_set_o )
= bot_bot_set_o ) ).
% image_empty
thf(fact_932_empty__is__image,axiom,
! [F: $o > $o,A2: set_o] :
( ( bot_bot_set_o
= ( image_o_o @ F @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_933_image__is__empty,axiom,
! [F: $o > $o,A2: set_o] :
( ( ( image_o_o @ F @ A2 )
= bot_bot_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_934_finite__imageI,axiom,
! [F2: set_b,H3: b > b] :
( ( finite_finite_b @ F2 )
=> ( finite_finite_b @ ( image_b_b @ H3 @ F2 ) ) ) ).
% finite_imageI
thf(fact_935_finite__imageI,axiom,
! [F2: set_b,H3: b > nat] :
( ( finite_finite_b @ F2 )
=> ( finite_finite_nat @ ( image_b_nat @ H3 @ F2 ) ) ) ).
% finite_imageI
thf(fact_936_finite__imageI,axiom,
! [F2: set_nat,H3: nat > b] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_b @ ( image_nat_b @ H3 @ F2 ) ) ) ).
% finite_imageI
thf(fact_937_finite__imageI,axiom,
! [F2: set_nat,H3: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_nat @ ( image_nat_nat @ H3 @ F2 ) ) ) ).
% finite_imageI
thf(fact_938_insert__image,axiom,
! [X: b,A2: set_b,F: b > b] :
( ( member_b @ X @ A2 )
=> ( ( insert_b @ ( F @ X ) @ ( image_b_b @ F @ A2 ) )
= ( image_b_b @ F @ A2 ) ) ) ).
% insert_image
thf(fact_939_insert__image,axiom,
! [X: b,A2: set_b,F: b > $o] :
( ( member_b @ X @ A2 )
=> ( ( insert_o @ ( F @ X ) @ ( image_b_o @ F @ A2 ) )
= ( image_b_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_940_insert__image,axiom,
! [X: b > a,A2: set_b_a,F: ( b > a ) > b] :
( ( member_b_a @ X @ A2 )
=> ( ( insert_b @ ( F @ X ) @ ( image_b_a_b @ F @ A2 ) )
= ( image_b_a_b @ F @ A2 ) ) ) ).
% insert_image
thf(fact_941_insert__image,axiom,
! [X: b > a,A2: set_b_a,F: ( b > a ) > $o] :
( ( member_b_a @ X @ A2 )
=> ( ( insert_o @ ( F @ X ) @ ( image_b_a_o @ F @ A2 ) )
= ( image_b_a_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_942_insert__image,axiom,
! [X: a,A2: set_a,F: a > b] :
( ( member_a @ X @ A2 )
=> ( ( insert_b @ ( F @ X ) @ ( image_a_b @ F @ A2 ) )
= ( image_a_b @ F @ A2 ) ) ) ).
% insert_image
thf(fact_943_insert__image,axiom,
! [X: a,A2: set_a,F: a > $o] :
( ( member_a @ X @ A2 )
=> ( ( insert_o @ ( F @ X ) @ ( image_a_o @ F @ A2 ) )
= ( image_a_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_944_insert__image,axiom,
! [X: nat,A2: set_nat,F: nat > b] :
( ( member_nat @ X @ A2 )
=> ( ( insert_b @ ( F @ X ) @ ( image_nat_b @ F @ A2 ) )
= ( image_nat_b @ F @ A2 ) ) ) ).
% insert_image
thf(fact_945_insert__image,axiom,
! [X: nat,A2: set_nat,F: nat > $o] :
( ( member_nat @ X @ A2 )
=> ( ( insert_o @ ( F @ X ) @ ( image_nat_o @ F @ A2 ) )
= ( image_nat_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_946_image__insert,axiom,
! [F: b > b,A: b,B2: set_b] :
( ( image_b_b @ F @ ( insert_b @ A @ B2 ) )
= ( insert_b @ ( F @ A ) @ ( image_b_b @ F @ B2 ) ) ) ).
% image_insert
thf(fact_947_image__insert,axiom,
! [F: b > $o,A: b,B2: set_b] :
( ( image_b_o @ F @ ( insert_b @ A @ B2 ) )
= ( insert_o @ ( F @ A ) @ ( image_b_o @ F @ B2 ) ) ) ).
% image_insert
thf(fact_948_image__insert,axiom,
! [F: $o > b,A: $o,B2: set_o] :
( ( image_o_b @ F @ ( insert_o @ A @ B2 ) )
= ( insert_b @ ( F @ A ) @ ( image_o_b @ F @ B2 ) ) ) ).
% image_insert
thf(fact_949_image__insert,axiom,
! [F: $o > $o,A: $o,B2: set_o] :
( ( image_o_o @ F @ ( insert_o @ A @ B2 ) )
= ( insert_o @ ( F @ A ) @ ( image_o_o @ F @ B2 ) ) ) ).
% image_insert
thf(fact_950_PiE__I,axiom,
! [A2: set_b,F: b > b,B2: b > set_b] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( ! [X2: b] :
( ~ ( member_b @ X2 @ A2 )
=> ( ( F @ X2 )
= undefined_b ) )
=> ( member_b_b @ F @ ( piE_b_b @ A2 @ B2 ) ) ) ) ).
% PiE_I
thf(fact_951_PiE__I,axiom,
! [A2: set_b,F: b > nat,B2: b > set_nat] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( ! [X2: b] :
( ~ ( member_b @ X2 @ A2 )
=> ( ( F @ X2 )
= undefined_nat ) )
=> ( member_b_nat @ F @ ( piE_b_nat @ A2 @ B2 ) ) ) ) ).
% PiE_I
thf(fact_952_PiE__I,axiom,
! [A2: set_a,F: a > b,B2: a > set_b] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( ! [X2: a] :
( ~ ( member_a @ X2 @ A2 )
=> ( ( F @ X2 )
= undefined_b ) )
=> ( member_a_b @ F @ ( piE_a_b @ A2 @ B2 ) ) ) ) ).
% PiE_I
thf(fact_953_PiE__I,axiom,
! [A2: set_a,F: a > nat,B2: a > set_nat] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( ! [X2: a] :
( ~ ( member_a @ X2 @ A2 )
=> ( ( F @ X2 )
= undefined_nat ) )
=> ( member_a_nat @ F @ ( piE_a_nat @ A2 @ B2 ) ) ) ) ).
% PiE_I
thf(fact_954_PiE__I,axiom,
! [A2: set_nat,F: nat > b,B2: nat > set_b] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( ! [X2: nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( F @ X2 )
= undefined_b ) )
=> ( member_nat_b @ F @ ( piE_nat_b @ A2 @ B2 ) ) ) ) ).
% PiE_I
thf(fact_955_PiE__I,axiom,
! [A2: set_nat,F: nat > nat,B2: nat > set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( ! [X2: nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( F @ X2 )
= undefined_nat ) )
=> ( member_nat_nat @ F @ ( piE_nat_nat @ A2 @ B2 ) ) ) ) ).
% PiE_I
thf(fact_956_PiE__I,axiom,
! [A2: set_b,F: b > a,B2: b > set_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( ! [X2: b] :
( ~ ( member_b @ X2 @ A2 )
=> ( ( F @ X2 )
= undefined_a ) )
=> ( member_b_a @ F @ ( piE_b_a @ A2 @ B2 ) ) ) ) ).
% PiE_I
thf(fact_957_PiE__I,axiom,
! [A2: set_a,F: a > a,B2: a > set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( ! [X2: a] :
( ~ ( member_a @ X2 @ A2 )
=> ( ( F @ X2 )
= undefined_a ) )
=> ( member_a_a @ F @ ( piE_a_a @ A2 @ B2 ) ) ) ) ).
% PiE_I
thf(fact_958_PiE__I,axiom,
! [A2: set_nat,F: nat > a,B2: nat > set_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( ! [X2: nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( F @ X2 )
= undefined_a ) )
=> ( member_nat_a @ F @ ( piE_nat_a @ A2 @ B2 ) ) ) ) ).
% PiE_I
thf(fact_959_PiE__I,axiom,
! [A2: set_b,F: b > b > a,B2: b > set_b_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_b_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( ! [X2: b] :
( ~ ( member_b @ X2 @ A2 )
=> ( ( F @ X2 )
= undefined_b_a ) )
=> ( member_b_b_a @ F @ ( piE_b_b_a @ A2 @ B2 ) ) ) ) ).
% PiE_I
thf(fact_960_PiE__empty__range,axiom,
! [I2: b,I3: set_b,F2: b > set_o] :
( ( member_b @ I2 @ I3 )
=> ( ( ( F2 @ I2 )
= bot_bot_set_o )
=> ( ( piE_b_o @ I3 @ F2 )
= bot_bot_set_b_o ) ) ) ).
% PiE_empty_range
thf(fact_961_PiE__empty__range,axiom,
! [I2: b > a,I3: set_b_a,F2: ( b > a ) > set_o] :
( ( member_b_a @ I2 @ I3 )
=> ( ( ( F2 @ I2 )
= bot_bot_set_o )
=> ( ( piE_b_a_o @ I3 @ F2 )
= bot_bot_set_b_a_o ) ) ) ).
% PiE_empty_range
thf(fact_962_PiE__empty__range,axiom,
! [I2: a,I3: set_a,F2: a > set_o] :
( ( member_a @ I2 @ I3 )
=> ( ( ( F2 @ I2 )
= bot_bot_set_o )
=> ( ( piE_a_o @ I3 @ F2 )
= bot_bot_set_a_o ) ) ) ).
% PiE_empty_range
thf(fact_963_PiE__empty__range,axiom,
! [I2: nat,I3: set_nat,F2: nat > set_o] :
( ( member_nat @ I2 @ I3 )
=> ( ( ( F2 @ I2 )
= bot_bot_set_o )
=> ( ( piE_nat_o @ I3 @ F2 )
= bot_bot_set_nat_o ) ) ) ).
% PiE_empty_range
thf(fact_964_PiE__mem,axiom,
! [F: b > b,S: set_b,T2: b > set_b,X: b] :
( ( member_b_b @ F @ ( piE_b_b @ S @ T2 ) )
=> ( ( member_b @ X @ S )
=> ( member_b @ ( F @ X ) @ ( T2 @ X ) ) ) ) ).
% PiE_mem
thf(fact_965_PiE__mem,axiom,
! [F: b > nat,S: set_b,T2: b > set_nat,X: b] :
( ( member_b_nat @ F @ ( piE_b_nat @ S @ T2 ) )
=> ( ( member_b @ X @ S )
=> ( member_nat @ ( F @ X ) @ ( T2 @ X ) ) ) ) ).
% PiE_mem
thf(fact_966_PiE__mem,axiom,
! [F: a > b,S: set_a,T2: a > set_b,X: a] :
( ( member_a_b @ F @ ( piE_a_b @ S @ T2 ) )
=> ( ( member_a @ X @ S )
=> ( member_b @ ( F @ X ) @ ( T2 @ X ) ) ) ) ).
% PiE_mem
thf(fact_967_PiE__mem,axiom,
! [F: a > a,S: set_a,T2: a > set_a,X: a] :
( ( member_a_a @ F @ ( piE_a_a @ S @ T2 ) )
=> ( ( member_a @ X @ S )
=> ( member_a @ ( F @ X ) @ ( T2 @ X ) ) ) ) ).
% PiE_mem
thf(fact_968_PiE__mem,axiom,
! [F: a > nat,S: set_a,T2: a > set_nat,X: a] :
( ( member_a_nat @ F @ ( piE_a_nat @ S @ T2 ) )
=> ( ( member_a @ X @ S )
=> ( member_nat @ ( F @ X ) @ ( T2 @ X ) ) ) ) ).
% PiE_mem
thf(fact_969_PiE__mem,axiom,
! [F: nat > b,S: set_nat,T2: nat > set_b,X: nat] :
( ( member_nat_b @ F @ ( piE_nat_b @ S @ T2 ) )
=> ( ( member_nat @ X @ S )
=> ( member_b @ ( F @ X ) @ ( T2 @ X ) ) ) ) ).
% PiE_mem
thf(fact_970_PiE__mem,axiom,
! [F: nat > a,S: set_nat,T2: nat > set_a,X: nat] :
( ( member_nat_a @ F @ ( piE_nat_a @ S @ T2 ) )
=> ( ( member_nat @ X @ S )
=> ( member_a @ ( F @ X ) @ ( T2 @ X ) ) ) ) ).
% PiE_mem
thf(fact_971_PiE__mem,axiom,
! [F: nat > nat,S: set_nat,T2: nat > set_nat,X: nat] :
( ( member_nat_nat @ F @ ( piE_nat_nat @ S @ T2 ) )
=> ( ( member_nat @ X @ S )
=> ( member_nat @ ( F @ X ) @ ( T2 @ X ) ) ) ) ).
% PiE_mem
thf(fact_972_PiE__mem,axiom,
! [F: b > a,S: set_b,T2: b > set_a,X: b] :
( ( member_b_a @ F @ ( piE_b_a @ S @ T2 ) )
=> ( ( member_b @ X @ S )
=> ( member_a @ ( F @ X ) @ ( T2 @ X ) ) ) ) ).
% PiE_mem
thf(fact_973_PiE__mem,axiom,
! [F: b > b > a,S: set_b,T2: b > set_b_a,X: b] :
( ( member_b_b_a @ F @ ( piE_b_b_a @ S @ T2 ) )
=> ( ( member_b @ X @ S )
=> ( member_b_a @ ( F @ X ) @ ( T2 @ X ) ) ) ) ).
% PiE_mem
thf(fact_974_PiE__ext,axiom,
! [X: b > a,K2: set_b,S3: b > set_a,Y: b > a] :
( ( member_b_a @ X @ ( piE_b_a @ K2 @ S3 ) )
=> ( ( member_b_a @ Y @ ( piE_b_a @ K2 @ S3 ) )
=> ( ! [I4: b] :
( ( member_b @ I4 @ K2 )
=> ( ( X @ I4 )
= ( Y @ I4 ) ) )
=> ( X = Y ) ) ) ) ).
% PiE_ext
thf(fact_975_rev__image__eqI,axiom,
! [X: b,A2: set_b,B: b,F: b > b] :
( ( member_b @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_b @ B @ ( image_b_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_976_rev__image__eqI,axiom,
! [X: b,A2: set_b,B: a,F: b > a] :
( ( member_b @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_b_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_977_rev__image__eqI,axiom,
! [X: b,A2: set_b,B: nat,F: b > nat] :
( ( member_b @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_b_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_978_rev__image__eqI,axiom,
! [X: a,A2: set_a,B: b,F: a > b] :
( ( member_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_b @ B @ ( image_a_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_979_rev__image__eqI,axiom,
! [X: a,A2: set_a,B: a,F: a > a] :
( ( member_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_980_rev__image__eqI,axiom,
! [X: a,A2: set_a,B: nat,F: a > nat] :
( ( member_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_981_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: b,F: nat > b] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_b @ B @ ( image_nat_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_982_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: a,F: nat > a] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_nat_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_983_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_984_rev__image__eqI,axiom,
! [X: b,A2: set_b,B: b > a,F: b > b > a] :
( ( member_b @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_b_a @ B @ ( image_b_b_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_985_imageI,axiom,
! [X: b,A2: set_b,F: b > b] :
( ( member_b @ X @ A2 )
=> ( member_b @ ( F @ X ) @ ( image_b_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_986_imageI,axiom,
! [X: b,A2: set_b,F: b > a] :
( ( member_b @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( image_b_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_987_imageI,axiom,
! [X: b,A2: set_b,F: b > nat] :
( ( member_b @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_b_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_988_imageI,axiom,
! [X: a,A2: set_a,F: a > b] :
( ( member_a @ X @ A2 )
=> ( member_b @ ( F @ X ) @ ( image_a_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_989_imageI,axiom,
! [X: a,A2: set_a,F: a > a] :
( ( member_a @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_990_imageI,axiom,
! [X: a,A2: set_a,F: a > nat] :
( ( member_a @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_991_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > b] :
( ( member_nat @ X @ A2 )
=> ( member_b @ ( F @ X ) @ ( image_nat_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_992_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > a] :
( ( member_nat @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( image_nat_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_993_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_994_imageI,axiom,
! [X: b,A2: set_b,F: b > b > a] :
( ( member_b @ X @ A2 )
=> ( member_b_a @ ( F @ X ) @ ( image_b_b_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_995_Compr__image__eq,axiom,
! [F: b > b,A2: set_b,P: b > $o] :
( ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( image_b_b @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_b_b @ F
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_996_Compr__image__eq,axiom,
! [F: a > b,A2: set_a,P: b > $o] :
( ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( image_a_b @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_a_b @ F
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_997_Compr__image__eq,axiom,
! [F: nat > b,A2: set_nat,P: b > $o] :
( ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( image_nat_b @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_nat_b @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_998_Compr__image__eq,axiom,
! [F: b > a,A2: set_b,P: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( image_b_a @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_b_a @ F
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_999_Compr__image__eq,axiom,
! [F: a > a,A2: set_a,P: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( image_a_a @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_a_a @ F
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1000_Compr__image__eq,axiom,
! [F: nat > a,A2: set_nat,P: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( image_nat_a @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_nat_a @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1001_Compr__image__eq,axiom,
! [F: b > nat,A2: set_b,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_b_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_b_nat @ F
@ ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1002_Compr__image__eq,axiom,
! [F: a > nat,A2: set_a,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_a_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_a_nat @ F
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1003_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1004_Compr__image__eq,axiom,
! [F: ( b > a ) > b,A2: set_b_a,P: b > $o] :
( ( collect_b
@ ^ [X3: b] :
( ( member_b @ X3 @ ( image_b_a_b @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_b_a_b @ F
@ ( collect_b_a
@ ^ [X3: b > a] :
( ( member_b_a @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1005_imageE,axiom,
! [B: b,F: b > b,A2: set_b] :
( ( member_b @ B @ ( image_b_b @ F @ A2 ) )
=> ~ ! [X2: b] :
( ( B
= ( F @ X2 ) )
=> ~ ( member_b @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_1006_imageE,axiom,
! [B: b,F: a > b,A2: set_a] :
( ( member_b @ B @ ( image_a_b @ F @ A2 ) )
=> ~ ! [X2: a] :
( ( B
= ( F @ X2 ) )
=> ~ ( member_a @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_1007_imageE,axiom,
! [B: b,F: nat > b,A2: set_nat] :
( ( member_b @ B @ ( image_nat_b @ F @ A2 ) )
=> ~ ! [X2: nat] :
( ( B
= ( F @ X2 ) )
=> ~ ( member_nat @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_1008_imageE,axiom,
! [B: a,F: b > a,A2: set_b] :
( ( member_a @ B @ ( image_b_a @ F @ A2 ) )
=> ~ ! [X2: b] :
( ( B
= ( F @ X2 ) )
=> ~ ( member_b @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_1009_imageE,axiom,
! [B: a,F: a > a,A2: set_a] :
( ( member_a @ B @ ( image_a_a @ F @ A2 ) )
=> ~ ! [X2: a] :
( ( B
= ( F @ X2 ) )
=> ~ ( member_a @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_1010_imageE,axiom,
! [B: a,F: nat > a,A2: set_nat] :
( ( member_a @ B @ ( image_nat_a @ F @ A2 ) )
=> ~ ! [X2: nat] :
( ( B
= ( F @ X2 ) )
=> ~ ( member_nat @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_1011_imageE,axiom,
! [B: nat,F: b > nat,A2: set_b] :
( ( member_nat @ B @ ( image_b_nat @ F @ A2 ) )
=> ~ ! [X2: b] :
( ( B
= ( F @ X2 ) )
=> ~ ( member_b @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_1012_imageE,axiom,
! [B: nat,F: a > nat,A2: set_a] :
( ( member_nat @ B @ ( image_a_nat @ F @ A2 ) )
=> ~ ! [X2: a] :
( ( B
= ( F @ X2 ) )
=> ~ ( member_a @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_1013_imageE,axiom,
! [B: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [X2: nat] :
( ( B
= ( F @ X2 ) )
=> ~ ( member_nat @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_1014_imageE,axiom,
! [B: b,F: ( b > a ) > b,A2: set_b_a] :
( ( member_b @ B @ ( image_b_a_b @ F @ A2 ) )
=> ~ ! [X2: b > a] :
( ( B
= ( F @ X2 ) )
=> ~ ( member_b_a @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_1015_PiE__eq__iff__not__empty,axiom,
! [I3: set_b,F2: b > set_o,F4: b > set_o] :
( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_o ) )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( ( F4 @ I4 )
!= bot_bot_set_o ) )
=> ( ( ( piE_b_o @ I3 @ F2 )
= ( piE_b_o @ I3 @ F4 ) )
= ( ! [X3: b] :
( ( member_b @ X3 @ I3 )
=> ( ( F2 @ X3 )
= ( F4 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_1016_PiE__eq__iff__not__empty,axiom,
! [I3: set_b_a,F2: ( b > a ) > set_o,F4: ( b > a ) > set_o] :
( ! [I4: b > a] :
( ( member_b_a @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_o ) )
=> ( ! [I4: b > a] :
( ( member_b_a @ I4 @ I3 )
=> ( ( F4 @ I4 )
!= bot_bot_set_o ) )
=> ( ( ( piE_b_a_o @ I3 @ F2 )
= ( piE_b_a_o @ I3 @ F4 ) )
= ( ! [X3: b > a] :
( ( member_b_a @ X3 @ I3 )
=> ( ( F2 @ X3 )
= ( F4 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_1017_PiE__eq__iff__not__empty,axiom,
! [I3: set_a,F2: a > set_o,F4: a > set_o] :
( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_o ) )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( ( F4 @ I4 )
!= bot_bot_set_o ) )
=> ( ( ( piE_a_o @ I3 @ F2 )
= ( piE_a_o @ I3 @ F4 ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ I3 )
=> ( ( F2 @ X3 )
= ( F4 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_1018_PiE__eq__iff__not__empty,axiom,
! [I3: set_nat,F2: nat > set_o,F4: nat > set_o] :
( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_o ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( F4 @ I4 )
!= bot_bot_set_o ) )
=> ( ( ( piE_nat_o @ I3 @ F2 )
= ( piE_nat_o @ I3 @ F4 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I3 )
=> ( ( F2 @ X3 )
= ( F4 @ X3 ) ) ) ) ) ) ) ).
% PiE_eq_iff_not_empty
thf(fact_1019_all__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A2 )
=> ( P @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1020_subset__image__iff,axiom,
! [B2: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B2
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1021_subset__imageE,axiom,
! [B2: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
=> ~ ! [C6: set_a] :
( ( ord_less_eq_set_a @ C6 @ A2 )
=> ( B2
!= ( image_a_a @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_1022_image__subsetI,axiom,
! [A2: set_b,F: b > b,B2: set_b] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_b @ ( image_b_b @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1023_image__subsetI,axiom,
! [A2: set_b,F: b > nat,B2: set_nat] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_b_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1024_image__subsetI,axiom,
! [A2: set_a,F: a > b,B2: set_b] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_b @ ( image_a_b @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1025_image__subsetI,axiom,
! [A2: set_a,F: a > nat,B2: set_nat] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1026_image__subsetI,axiom,
! [A2: set_nat,F: nat > b,B2: set_b] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_b @ ( image_nat_b @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1027_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1028_image__subsetI,axiom,
! [A2: set_b,F: b > a,B2: set_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_b_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1029_image__subsetI,axiom,
! [A2: set_a,F: a > a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1030_image__subsetI,axiom,
! [A2: set_nat,F: nat > a,B2: set_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1031_image__subsetI,axiom,
! [A2: set_b,F: b > b > a,B2: set_b_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_b_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_b_a @ ( image_b_b_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1032_image__mono,axiom,
! [A2: set_a,B2: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_1033_PiE__arb,axiom,
! [F: ( b > a ) > a,S: set_b_a,T2: ( b > a ) > set_a,X: b > a] :
( ( member_b_a_a @ F @ ( piE_b_a_a @ S @ T2 ) )
=> ( ~ ( member_b_a @ X @ S )
=> ( ( F @ X )
= undefined_a ) ) ) ).
% PiE_arb
thf(fact_1034_PiE__arb,axiom,
! [F: a > a,S: set_a,T2: a > set_a,X: a] :
( ( member_a_a @ F @ ( piE_a_a @ S @ T2 ) )
=> ( ~ ( member_a @ X @ S )
=> ( ( F @ X )
= undefined_a ) ) ) ).
% PiE_arb
thf(fact_1035_PiE__arb,axiom,
! [F: nat > a,S: set_nat,T2: nat > set_a,X: nat] :
( ( member_nat_a @ F @ ( piE_nat_a @ S @ T2 ) )
=> ( ~ ( member_nat @ X @ S )
=> ( ( F @ X )
= undefined_a ) ) ) ).
% PiE_arb
thf(fact_1036_PiE__arb,axiom,
! [F: b > a,S: set_b,T2: b > set_a,X: b] :
( ( member_b_a @ F @ ( piE_b_a @ S @ T2 ) )
=> ( ~ ( member_b @ X @ S )
=> ( ( F @ X )
= undefined_a ) ) ) ).
% PiE_arb
thf(fact_1037_PiE__E,axiom,
! [F: b > b,A2: set_b,B2: b > set_b,X: b] :
( ( member_b_b @ F @ ( piE_b_b @ A2 @ B2 ) )
=> ( ( ( member_b @ X @ A2 )
=> ~ ( member_b @ ( F @ X ) @ ( B2 @ X ) ) )
=> ~ ( ~ ( member_b @ X @ A2 )
=> ( ( F @ X )
!= undefined_b ) ) ) ) ).
% PiE_E
thf(fact_1038_PiE__E,axiom,
! [F: b > nat,A2: set_b,B2: b > set_nat,X: b] :
( ( member_b_nat @ F @ ( piE_b_nat @ A2 @ B2 ) )
=> ( ( ( member_b @ X @ A2 )
=> ~ ( member_nat @ ( F @ X ) @ ( B2 @ X ) ) )
=> ~ ( ~ ( member_b @ X @ A2 )
=> ( ( F @ X )
!= undefined_nat ) ) ) ) ).
% PiE_E
thf(fact_1039_PiE__E,axiom,
! [F: a > b,A2: set_a,B2: a > set_b,X: a] :
( ( member_a_b @ F @ ( piE_a_b @ A2 @ B2 ) )
=> ( ( ( member_a @ X @ A2 )
=> ~ ( member_b @ ( F @ X ) @ ( B2 @ X ) ) )
=> ~ ( ~ ( member_a @ X @ A2 )
=> ( ( F @ X )
!= undefined_b ) ) ) ) ).
% PiE_E
thf(fact_1040_PiE__E,axiom,
! [F: a > nat,A2: set_a,B2: a > set_nat,X: a] :
( ( member_a_nat @ F @ ( piE_a_nat @ A2 @ B2 ) )
=> ( ( ( member_a @ X @ A2 )
=> ~ ( member_nat @ ( F @ X ) @ ( B2 @ X ) ) )
=> ~ ( ~ ( member_a @ X @ A2 )
=> ( ( F @ X )
!= undefined_nat ) ) ) ) ).
% PiE_E
thf(fact_1041_PiE__E,axiom,
! [F: nat > b,A2: set_nat,B2: nat > set_b,X: nat] :
( ( member_nat_b @ F @ ( piE_nat_b @ A2 @ B2 ) )
=> ( ( ( member_nat @ X @ A2 )
=> ~ ( member_b @ ( F @ X ) @ ( B2 @ X ) ) )
=> ~ ( ~ ( member_nat @ X @ A2 )
=> ( ( F @ X )
!= undefined_b ) ) ) ) ).
% PiE_E
thf(fact_1042_PiE__E,axiom,
! [F: nat > nat,A2: set_nat,B2: nat > set_nat,X: nat] :
( ( member_nat_nat @ F @ ( piE_nat_nat @ A2 @ B2 ) )
=> ( ( ( member_nat @ X @ A2 )
=> ~ ( member_nat @ ( F @ X ) @ ( B2 @ X ) ) )
=> ~ ( ~ ( member_nat @ X @ A2 )
=> ( ( F @ X )
!= undefined_nat ) ) ) ) ).
% PiE_E
thf(fact_1043_PiE__E,axiom,
! [F: a > a,A2: set_a,B2: a > set_a,X: a] :
( ( member_a_a @ F @ ( piE_a_a @ A2 @ B2 ) )
=> ( ( ( member_a @ X @ A2 )
=> ~ ( member_a @ ( F @ X ) @ ( B2 @ X ) ) )
=> ~ ( ~ ( member_a @ X @ A2 )
=> ( ( F @ X )
!= undefined_a ) ) ) ) ).
% PiE_E
thf(fact_1044_PiE__E,axiom,
! [F: nat > a,A2: set_nat,B2: nat > set_a,X: nat] :
( ( member_nat_a @ F @ ( piE_nat_a @ A2 @ B2 ) )
=> ( ( ( member_nat @ X @ A2 )
=> ~ ( member_a @ ( F @ X ) @ ( B2 @ X ) ) )
=> ~ ( ~ ( member_nat @ X @ A2 )
=> ( ( F @ X )
!= undefined_a ) ) ) ) ).
% PiE_E
thf(fact_1045_PiE__E,axiom,
! [F: b > a,A2: set_b,B2: b > set_a,X: b] :
( ( member_b_a @ F @ ( piE_b_a @ A2 @ B2 ) )
=> ( ( ( member_b @ X @ A2 )
=> ~ ( member_a @ ( F @ X ) @ ( B2 @ X ) ) )
=> ~ ( ~ ( member_b @ X @ A2 )
=> ( ( F @ X )
!= undefined_a ) ) ) ) ).
% PiE_E
thf(fact_1046_PiE__E,axiom,
! [F: b > b > a,A2: set_b,B2: b > set_b_a,X: b] :
( ( member_b_b_a @ F @ ( piE_b_b_a @ A2 @ B2 ) )
=> ( ( ( member_b @ X @ A2 )
=> ~ ( member_b_a @ ( F @ X ) @ ( B2 @ X ) ) )
=> ~ ( ~ ( member_b @ X @ A2 )
=> ( ( F @ X )
!= undefined_b_a ) ) ) ) ).
% PiE_E
thf(fact_1047_PiE__iff,axiom,
! [F: b > a,I3: set_b,X4: b > set_a] :
( ( member_b_a @ F @ ( piE_b_a @ I3 @ X4 ) )
= ( ! [X3: b] :
( ( member_b @ X3 @ I3 )
=> ( member_a @ ( F @ X3 ) @ ( X4 @ X3 ) ) )
& ( member_b_a @ F @ ( extensional_b_a @ I3 ) ) ) ) ).
% PiE_iff
thf(fact_1048_finite__PiE,axiom,
! [S: set_b_a,T2: ( b > a ) > set_b] :
( ( finite_finite_b_a @ S )
=> ( ! [I4: b > a] :
( ( member_b_a @ I4 @ S )
=> ( finite_finite_b @ ( T2 @ I4 ) ) )
=> ( finite_finite_b_a_b @ ( piE_b_a_b @ S @ T2 ) ) ) ) ).
% finite_PiE
thf(fact_1049_finite__PiE,axiom,
! [S: set_a,T2: a > set_b] :
( ( finite_finite_a @ S )
=> ( ! [I4: a] :
( ( member_a @ I4 @ S )
=> ( finite_finite_b @ ( T2 @ I4 ) ) )
=> ( finite_finite_a_b @ ( piE_a_b @ S @ T2 ) ) ) ) ).
% finite_PiE
thf(fact_1050_finite__PiE,axiom,
! [S: set_b_a,T2: ( b > a ) > set_nat] :
( ( finite_finite_b_a @ S )
=> ( ! [I4: b > a] :
( ( member_b_a @ I4 @ S )
=> ( finite_finite_nat @ ( T2 @ I4 ) ) )
=> ( finite5500941983950307626_a_nat @ ( piE_b_a_nat @ S @ T2 ) ) ) ) ).
% finite_PiE
thf(fact_1051_finite__PiE,axiom,
! [S: set_a,T2: a > set_nat] :
( ( finite_finite_a @ S )
=> ( ! [I4: a] :
( ( member_a @ I4 @ S )
=> ( finite_finite_nat @ ( T2 @ I4 ) ) )
=> ( finite_finite_a_nat @ ( piE_a_nat @ S @ T2 ) ) ) ) ).
% finite_PiE
thf(fact_1052_finite__PiE,axiom,
! [S: set_b,T2: b > set_b] :
( ( finite_finite_b @ S )
=> ( ! [I4: b] :
( ( member_b @ I4 @ S )
=> ( finite_finite_b @ ( T2 @ I4 ) ) )
=> ( finite_finite_b_b @ ( piE_b_b @ S @ T2 ) ) ) ) ).
% finite_PiE
thf(fact_1053_finite__PiE,axiom,
! [S: set_b,T2: b > set_nat] :
( ( finite_finite_b @ S )
=> ( ! [I4: b] :
( ( member_b @ I4 @ S )
=> ( finite_finite_nat @ ( T2 @ I4 ) ) )
=> ( finite_finite_b_nat @ ( piE_b_nat @ S @ T2 ) ) ) ) ).
% finite_PiE
thf(fact_1054_finite__PiE,axiom,
! [S: set_nat,T2: nat > set_b] :
( ( finite_finite_nat @ S )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ S )
=> ( finite_finite_b @ ( T2 @ I4 ) ) )
=> ( finite_finite_nat_b @ ( piE_nat_b @ S @ T2 ) ) ) ) ).
% finite_PiE
thf(fact_1055_finite__PiE,axiom,
! [S: set_nat,T2: nat > set_nat] :
( ( finite_finite_nat @ S )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ S )
=> ( finite_finite_nat @ ( T2 @ I4 ) ) )
=> ( finite2115694454571419734at_nat @ ( piE_nat_nat @ S @ T2 ) ) ) ) ).
% finite_PiE
thf(fact_1056_pigeonhole__infinite,axiom,
! [A2: set_b_a,F: ( b > a ) > b] :
( ~ ( finite_finite_b_a @ A2 )
=> ( ( finite_finite_b @ ( image_b_a_b @ F @ A2 ) )
=> ? [X2: b > a] :
( ( member_b_a @ X2 @ A2 )
& ~ ( finite_finite_b_a
@ ( collect_b_a
@ ^ [A3: b > a] :
( ( member_b_a @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1057_pigeonhole__infinite,axiom,
! [A2: set_a,F: a > b] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_b @ ( image_a_b @ F @ A2 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A3: a] :
( ( member_a @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1058_pigeonhole__infinite,axiom,
! [A2: set_b_a,F: ( b > a ) > nat] :
( ~ ( finite_finite_b_a @ A2 )
=> ( ( finite_finite_nat @ ( image_b_a_nat @ F @ A2 ) )
=> ? [X2: b > a] :
( ( member_b_a @ X2 @ A2 )
& ~ ( finite_finite_b_a
@ ( collect_b_a
@ ^ [A3: b > a] :
( ( member_b_a @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1059_pigeonhole__infinite,axiom,
! [A2: set_a,F: a > nat] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_nat @ ( image_a_nat @ F @ A2 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A3: a] :
( ( member_a @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1060_pigeonhole__infinite,axiom,
! [A2: set_b,F: b > b] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_b @ ( image_b_b @ F @ A2 ) )
=> ? [X2: b] :
( ( member_b @ X2 @ A2 )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A3: b] :
( ( member_b @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1061_pigeonhole__infinite,axiom,
! [A2: set_b,F: b > nat] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_nat @ ( image_b_nat @ F @ A2 ) )
=> ? [X2: b] :
( ( member_b @ X2 @ A2 )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A3: b] :
( ( member_b @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1062_pigeonhole__infinite,axiom,
! [A2: set_nat,F: nat > b] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_b @ ( image_nat_b @ F @ A2 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1063_pigeonhole__infinite,axiom,
! [A2: set_nat,F: nat > nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A2 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_1064_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > b,B2: set_b] :
( ! [X2: a] :
( ( P @ X2 )
=> ( member_b @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_b @ ( image_a_b @ F @ ( collect_a @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_1065_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > b > a,B2: set_b_a] :
( ! [X2: a] :
( ( P @ X2 )
=> ( member_b_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_b_a @ ( image_a_b_a @ F @ ( collect_a @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_1066_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > nat,B2: set_nat] :
( ! [X2: a] :
( ( P @ X2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ ( collect_a @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_1067_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > b,B2: set_b] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( member_b @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_b @ ( image_nat_b @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_1068_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > b > a,B2: set_b_a] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( member_b_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_b_a @ ( image_nat_b_a @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_1069_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat,B2: set_nat] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_1070_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > a,B2: set_a] :
( ! [X2: a] :
( ( P @ X2 )
=> ( member_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ ( collect_a @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_1071_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > a,B2: set_a] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( member_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ ( collect_nat @ P ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_1072_PiE__eq__subset,axiom,
! [I3: set_b,F2: b > set_o,F4: b > set_o,I2: b] :
( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_o ) )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( ( F4 @ I4 )
!= bot_bot_set_o ) )
=> ( ( ( piE_b_o @ I3 @ F2 )
= ( piE_b_o @ I3 @ F4 ) )
=> ( ( member_b @ I2 @ I3 )
=> ( ord_less_eq_set_o @ ( F2 @ I2 ) @ ( F4 @ I2 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_1073_PiE__eq__subset,axiom,
! [I3: set_b_a,F2: ( b > a ) > set_o,F4: ( b > a ) > set_o,I2: b > a] :
( ! [I4: b > a] :
( ( member_b_a @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_o ) )
=> ( ! [I4: b > a] :
( ( member_b_a @ I4 @ I3 )
=> ( ( F4 @ I4 )
!= bot_bot_set_o ) )
=> ( ( ( piE_b_a_o @ I3 @ F2 )
= ( piE_b_a_o @ I3 @ F4 ) )
=> ( ( member_b_a @ I2 @ I3 )
=> ( ord_less_eq_set_o @ ( F2 @ I2 ) @ ( F4 @ I2 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_1074_PiE__eq__subset,axiom,
! [I3: set_a,F2: a > set_o,F4: a > set_o,I2: a] :
( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_o ) )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( ( F4 @ I4 )
!= bot_bot_set_o ) )
=> ( ( ( piE_a_o @ I3 @ F2 )
= ( piE_a_o @ I3 @ F4 ) )
=> ( ( member_a @ I2 @ I3 )
=> ( ord_less_eq_set_o @ ( F2 @ I2 ) @ ( F4 @ I2 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_1075_PiE__eq__subset,axiom,
! [I3: set_nat,F2: nat > set_o,F4: nat > set_o,I2: nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_o ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( F4 @ I4 )
!= bot_bot_set_o ) )
=> ( ( ( piE_nat_o @ I3 @ F2 )
= ( piE_nat_o @ I3 @ F4 ) )
=> ( ( member_nat @ I2 @ I3 )
=> ( ord_less_eq_set_o @ ( F2 @ I2 ) @ ( F4 @ I2 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_1076_PiE__eq__subset,axiom,
! [I3: set_b,F2: b > set_a,F4: b > set_a,I2: b] :
( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_a ) )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( ( F4 @ I4 )
!= bot_bot_set_a ) )
=> ( ( ( piE_b_a @ I3 @ F2 )
= ( piE_b_a @ I3 @ F4 ) )
=> ( ( member_b @ I2 @ I3 )
=> ( ord_less_eq_set_a @ ( F2 @ I2 ) @ ( F4 @ I2 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_1077_PiE__eq__subset,axiom,
! [I3: set_b_a,F2: ( b > a ) > set_a,F4: ( b > a ) > set_a,I2: b > a] :
( ! [I4: b > a] :
( ( member_b_a @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_a ) )
=> ( ! [I4: b > a] :
( ( member_b_a @ I4 @ I3 )
=> ( ( F4 @ I4 )
!= bot_bot_set_a ) )
=> ( ( ( piE_b_a_a @ I3 @ F2 )
= ( piE_b_a_a @ I3 @ F4 ) )
=> ( ( member_b_a @ I2 @ I3 )
=> ( ord_less_eq_set_a @ ( F2 @ I2 ) @ ( F4 @ I2 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_1078_PiE__eq__subset,axiom,
! [I3: set_a,F2: a > set_a,F4: a > set_a,I2: a] :
( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_a ) )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( ( F4 @ I4 )
!= bot_bot_set_a ) )
=> ( ( ( piE_a_a @ I3 @ F2 )
= ( piE_a_a @ I3 @ F4 ) )
=> ( ( member_a @ I2 @ I3 )
=> ( ord_less_eq_set_a @ ( F2 @ I2 ) @ ( F4 @ I2 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_1079_PiE__eq__subset,axiom,
! [I3: set_nat,F2: nat > set_a,F4: nat > set_a,I2: nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( F2 @ I4 )
!= bot_bot_set_a ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( F4 @ I4 )
!= bot_bot_set_a ) )
=> ( ( ( piE_nat_a @ I3 @ F2 )
= ( piE_nat_a @ I3 @ F4 ) )
=> ( ( member_nat @ I2 @ I3 )
=> ( ord_less_eq_set_a @ ( F2 @ I2 ) @ ( F4 @ I2 ) ) ) ) ) ) ).
% PiE_eq_subset
thf(fact_1080_finite__surj,axiom,
! [A2: set_b,B2: set_b,F: b > b] :
( ( finite_finite_b @ A2 )
=> ( ( ord_less_eq_set_b @ B2 @ ( image_b_b @ F @ A2 ) )
=> ( finite_finite_b @ B2 ) ) ) ).
% finite_surj
thf(fact_1081_finite__surj,axiom,
! [A2: set_b,B2: set_nat,F: b > nat] :
( ( finite_finite_b @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_b_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_1082_finite__surj,axiom,
! [A2: set_nat,B2: set_b,F: nat > b] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_b @ B2 @ ( image_nat_b @ F @ A2 ) )
=> ( finite_finite_b @ B2 ) ) ) ).
% finite_surj
thf(fact_1083_finite__surj,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_1084_finite__surj,axiom,
! [A2: set_b,B2: set_a,F: b > a] :
( ( finite_finite_b @ A2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_b_a @ F @ A2 ) )
=> ( finite_finite_a @ B2 ) ) ) ).
% finite_surj
thf(fact_1085_finite__surj,axiom,
! [A2: set_nat,B2: set_a,F: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F @ A2 ) )
=> ( finite_finite_a @ B2 ) ) ) ).
% finite_surj
thf(fact_1086_finite__subset__image,axiom,
! [B2: set_b,F: b > b,A2: set_b] :
( ( finite_finite_b @ B2 )
=> ( ( ord_less_eq_set_b @ B2 @ ( image_b_b @ F @ A2 ) )
=> ? [C6: set_b] :
( ( ord_less_eq_set_b @ C6 @ A2 )
& ( finite_finite_b @ C6 )
& ( B2
= ( image_b_b @ F @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1087_finite__subset__image,axiom,
! [B2: set_b,F: nat > b,A2: set_nat] :
( ( finite_finite_b @ B2 )
=> ( ( ord_less_eq_set_b @ B2 @ ( image_nat_b @ F @ A2 ) )
=> ? [C6: set_nat] :
( ( ord_less_eq_set_nat @ C6 @ A2 )
& ( finite_finite_nat @ C6 )
& ( B2
= ( image_nat_b @ F @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1088_finite__subset__image,axiom,
! [B2: set_nat,F: b > nat,A2: set_b] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_b_nat @ F @ A2 ) )
=> ? [C6: set_b] :
( ( ord_less_eq_set_b @ C6 @ A2 )
& ( finite_finite_b @ C6 )
& ( B2
= ( image_b_nat @ F @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1089_finite__subset__image,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ? [C6: set_nat] :
( ( ord_less_eq_set_nat @ C6 @ A2 )
& ( finite_finite_nat @ C6 )
& ( B2
= ( image_nat_nat @ F @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1090_finite__subset__image,axiom,
! [B2: set_b,F: a > b,A2: set_a] :
( ( finite_finite_b @ B2 )
=> ( ( ord_less_eq_set_b @ B2 @ ( image_a_b @ F @ A2 ) )
=> ? [C6: set_a] :
( ( ord_less_eq_set_a @ C6 @ A2 )
& ( finite_finite_a @ C6 )
& ( B2
= ( image_a_b @ F @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1091_finite__subset__image,axiom,
! [B2: set_nat,F: a > nat,A2: set_a] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A2 ) )
=> ? [C6: set_a] :
( ( ord_less_eq_set_a @ C6 @ A2 )
& ( finite_finite_a @ C6 )
& ( B2
= ( image_a_nat @ F @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1092_finite__subset__image,axiom,
! [B2: set_a,F: b > a,A2: set_b] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_b_a @ F @ A2 ) )
=> ? [C6: set_b] :
( ( ord_less_eq_set_b @ C6 @ A2 )
& ( finite_finite_b @ C6 )
& ( B2
= ( image_b_a @ F @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1093_finite__subset__image,axiom,
! [B2: set_a,F: nat > a,A2: set_nat] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F @ A2 ) )
=> ? [C6: set_nat] :
( ( ord_less_eq_set_nat @ C6 @ A2 )
& ( finite_finite_nat @ C6 )
& ( B2
= ( image_nat_a @ F @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1094_finite__subset__image,axiom,
! [B2: set_a,F: a > a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
=> ? [C6: set_a] :
( ( ord_less_eq_set_a @ C6 @ A2 )
& ( finite_finite_a @ C6 )
& ( B2
= ( image_a_a @ F @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1095_ex__finite__subset__image,axiom,
! [F: b > b,A2: set_b,P: set_b > $o] :
( ( ? [B4: set_b] :
( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ ( image_b_b @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_b] :
( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ A2 )
& ( P @ ( image_b_b @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1096_ex__finite__subset__image,axiom,
! [F: nat > b,A2: set_nat,P: set_b > $o] :
( ( ? [B4: set_b] :
( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ ( image_nat_b @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A2 )
& ( P @ ( image_nat_b @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1097_ex__finite__subset__image,axiom,
! [F: b > nat,A2: set_b,P: set_nat > $o] :
( ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_b_nat @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_b] :
( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ A2 )
& ( P @ ( image_b_nat @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1098_ex__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A2 )
& ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1099_ex__finite__subset__image,axiom,
! [F: a > b,A2: set_a,P: set_b > $o] :
( ( ? [B4: set_b] :
( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ ( image_a_b @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 )
& ( P @ ( image_a_b @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1100_ex__finite__subset__image,axiom,
! [F: a > nat,A2: set_a,P: set_nat > $o] :
( ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_a_nat @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 )
& ( P @ ( image_a_nat @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1101_ex__finite__subset__image,axiom,
! [F: b > a,A2: set_b,P: set_a > $o] :
( ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_b_a @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_b] :
( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ A2 )
& ( P @ ( image_b_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1102_ex__finite__subset__image,axiom,
! [F: nat > a,A2: set_nat,P: set_a > $o] :
( ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_nat_a @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A2 )
& ( P @ ( image_nat_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1103_ex__finite__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 )
& ( P @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1104_all__finite__subset__image,axiom,
! [F: b > b,A2: set_b,P: set_b > $o] :
( ( ! [B4: set_b] :
( ( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ ( image_b_b @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_b] :
( ( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ A2 ) )
=> ( P @ ( image_b_b @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1105_all__finite__subset__image,axiom,
! [F: nat > b,A2: set_nat,P: set_b > $o] :
( ( ! [B4: set_b] :
( ( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ ( image_nat_b @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A2 ) )
=> ( P @ ( image_nat_b @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1106_all__finite__subset__image,axiom,
! [F: b > nat,A2: set_b,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_b_nat @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_b] :
( ( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ A2 ) )
=> ( P @ ( image_b_nat @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1107_all__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A2 ) )
=> ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1108_all__finite__subset__image,axiom,
! [F: a > b,A2: set_a,P: set_b > $o] :
( ( ! [B4: set_b] :
( ( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ ( image_a_b @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 ) )
=> ( P @ ( image_a_b @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1109_all__finite__subset__image,axiom,
! [F: a > nat,A2: set_a,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_a_nat @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 ) )
=> ( P @ ( image_a_nat @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1110_all__finite__subset__image,axiom,
! [F: b > a,A2: set_b,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_b_a @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_b] :
( ( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ A2 ) )
=> ( P @ ( image_b_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1111_all__finite__subset__image,axiom,
! [F: nat > a,A2: set_nat,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_nat_a @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A2 ) )
=> ( P @ ( image_nat_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1112_all__finite__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 ) )
=> ( P @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1113_image__projection__PiE,axiom,
! [I3: set_b,S: b > set_b,I2: b] :
( ( ( ( piE_b_b @ I3 @ S )
= bot_bot_set_b_b )
=> ( ( image_b_b_b
@ ^ [F5: b > b] : ( F5 @ I2 )
@ ( piE_b_b @ I3 @ S ) )
= bot_bot_set_b ) )
& ( ( ( piE_b_b @ I3 @ S )
!= bot_bot_set_b_b )
=> ( ( ( member_b @ I2 @ I3 )
=> ( ( image_b_b_b
@ ^ [F5: b > b] : ( F5 @ I2 )
@ ( piE_b_b @ I3 @ S ) )
= ( S @ I2 ) ) )
& ( ~ ( member_b @ I2 @ I3 )
=> ( ( image_b_b_b
@ ^ [F5: b > b] : ( F5 @ I2 )
@ ( piE_b_b @ I3 @ S ) )
= ( insert_b @ undefined_b @ bot_bot_set_b ) ) ) ) ) ) ).
% image_projection_PiE
thf(fact_1114_image__projection__PiE,axiom,
! [I3: set_a,S: a > set_b,I2: a] :
( ( ( ( piE_a_b @ I3 @ S )
= bot_bot_set_a_b )
=> ( ( image_a_b_b
@ ^ [F5: a > b] : ( F5 @ I2 )
@ ( piE_a_b @ I3 @ S ) )
= bot_bot_set_b ) )
& ( ( ( piE_a_b @ I3 @ S )
!= bot_bot_set_a_b )
=> ( ( ( member_a @ I2 @ I3 )
=> ( ( image_a_b_b
@ ^ [F5: a > b] : ( F5 @ I2 )
@ ( piE_a_b @ I3 @ S ) )
= ( S @ I2 ) ) )
& ( ~ ( member_a @ I2 @ I3 )
=> ( ( image_a_b_b
@ ^ [F5: a > b] : ( F5 @ I2 )
@ ( piE_a_b @ I3 @ S ) )
= ( insert_b @ undefined_b @ bot_bot_set_b ) ) ) ) ) ) ).
% image_projection_PiE
thf(fact_1115_image__projection__PiE,axiom,
! [I3: set_nat,S: nat > set_b,I2: nat] :
( ( ( ( piE_nat_b @ I3 @ S )
= bot_bot_set_nat_b )
=> ( ( image_nat_b_b
@ ^ [F5: nat > b] : ( F5 @ I2 )
@ ( piE_nat_b @ I3 @ S ) )
= bot_bot_set_b ) )
& ( ( ( piE_nat_b @ I3 @ S )
!= bot_bot_set_nat_b )
=> ( ( ( member_nat @ I2 @ I3 )
=> ( ( image_nat_b_b
@ ^ [F5: nat > b] : ( F5 @ I2 )
@ ( piE_nat_b @ I3 @ S ) )
= ( S @ I2 ) ) )
& ( ~ ( member_nat @ I2 @ I3 )
=> ( ( image_nat_b_b
@ ^ [F5: nat > b] : ( F5 @ I2 )
@ ( piE_nat_b @ I3 @ S ) )
= ( insert_b @ undefined_b @ bot_bot_set_b ) ) ) ) ) ) ).
% image_projection_PiE
thf(fact_1116_image__projection__PiE,axiom,
! [I3: set_b,S: b > set_a,I2: b] :
( ( ( ( piE_b_a @ I3 @ S )
= bot_bot_set_b_a )
=> ( ( image_b_a_a
@ ^ [F5: b > a] : ( F5 @ I2 )
@ ( piE_b_a @ I3 @ S ) )
= bot_bot_set_a ) )
& ( ( ( piE_b_a @ I3 @ S )
!= bot_bot_set_b_a )
=> ( ( ( member_b @ I2 @ I3 )
=> ( ( image_b_a_a
@ ^ [F5: b > a] : ( F5 @ I2 )
@ ( piE_b_a @ I3 @ S ) )
= ( S @ I2 ) ) )
& ( ~ ( member_b @ I2 @ I3 )
=> ( ( image_b_a_a
@ ^ [F5: b > a] : ( F5 @ I2 )
@ ( piE_b_a @ I3 @ S ) )
= ( insert_a @ undefined_a @ bot_bot_set_a ) ) ) ) ) ) ).
% image_projection_PiE
thf(fact_1117_image__projection__PiE,axiom,
! [I3: set_a,S: a > set_a,I2: a] :
( ( ( ( piE_a_a @ I3 @ S )
= bot_bot_set_a_a )
=> ( ( image_a_a_a
@ ^ [F5: a > a] : ( F5 @ I2 )
@ ( piE_a_a @ I3 @ S ) )
= bot_bot_set_a ) )
& ( ( ( piE_a_a @ I3 @ S )
!= bot_bot_set_a_a )
=> ( ( ( member_a @ I2 @ I3 )
=> ( ( image_a_a_a
@ ^ [F5: a > a] : ( F5 @ I2 )
@ ( piE_a_a @ I3 @ S ) )
= ( S @ I2 ) ) )
& ( ~ ( member_a @ I2 @ I3 )
=> ( ( image_a_a_a
@ ^ [F5: a > a] : ( F5 @ I2 )
@ ( piE_a_a @ I3 @ S ) )
= ( insert_a @ undefined_a @ bot_bot_set_a ) ) ) ) ) ) ).
% image_projection_PiE
thf(fact_1118_image__projection__PiE,axiom,
! [I3: set_nat,S: nat > set_a,I2: nat] :
( ( ( ( piE_nat_a @ I3 @ S )
= bot_bot_set_nat_a )
=> ( ( image_nat_a_a
@ ^ [F5: nat > a] : ( F5 @ I2 )
@ ( piE_nat_a @ I3 @ S ) )
= bot_bot_set_a ) )
& ( ( ( piE_nat_a @ I3 @ S )
!= bot_bot_set_nat_a )
=> ( ( ( member_nat @ I2 @ I3 )
=> ( ( image_nat_a_a
@ ^ [F5: nat > a] : ( F5 @ I2 )
@ ( piE_nat_a @ I3 @ S ) )
= ( S @ I2 ) ) )
& ( ~ ( member_nat @ I2 @ I3 )
=> ( ( image_nat_a_a
@ ^ [F5: nat > a] : ( F5 @ I2 )
@ ( piE_nat_a @ I3 @ S ) )
= ( insert_a @ undefined_a @ bot_bot_set_a ) ) ) ) ) ) ).
% image_projection_PiE
thf(fact_1119_image__projection__PiE,axiom,
! [I3: set_b,S: b > set_o,I2: b] :
( ( ( ( piE_b_o @ I3 @ S )
= bot_bot_set_b_o )
=> ( ( image_b_o_o
@ ^ [F5: b > $o] : ( F5 @ I2 )
@ ( piE_b_o @ I3 @ S ) )
= bot_bot_set_o ) )
& ( ( ( piE_b_o @ I3 @ S )
!= bot_bot_set_b_o )
=> ( ( ( member_b @ I2 @ I3 )
=> ( ( image_b_o_o
@ ^ [F5: b > $o] : ( F5 @ I2 )
@ ( piE_b_o @ I3 @ S ) )
= ( S @ I2 ) ) )
& ( ~ ( member_b @ I2 @ I3 )
=> ( ( image_b_o_o
@ ^ [F5: b > $o] : ( F5 @ I2 )
@ ( piE_b_o @ I3 @ S ) )
= ( insert_o @ undefined_o @ bot_bot_set_o ) ) ) ) ) ) ).
% image_projection_PiE
thf(fact_1120_image__projection__PiE,axiom,
! [I3: set_a,S: a > set_o,I2: a] :
( ( ( ( piE_a_o @ I3 @ S )
= bot_bot_set_a_o )
=> ( ( image_a_o_o
@ ^ [F5: a > $o] : ( F5 @ I2 )
@ ( piE_a_o @ I3 @ S ) )
= bot_bot_set_o ) )
& ( ( ( piE_a_o @ I3 @ S )
!= bot_bot_set_a_o )
=> ( ( ( member_a @ I2 @ I3 )
=> ( ( image_a_o_o
@ ^ [F5: a > $o] : ( F5 @ I2 )
@ ( piE_a_o @ I3 @ S ) )
= ( S @ I2 ) ) )
& ( ~ ( member_a @ I2 @ I3 )
=> ( ( image_a_o_o
@ ^ [F5: a > $o] : ( F5 @ I2 )
@ ( piE_a_o @ I3 @ S ) )
= ( insert_o @ undefined_o @ bot_bot_set_o ) ) ) ) ) ) ).
% image_projection_PiE
thf(fact_1121_image__projection__PiE,axiom,
! [I3: set_nat,S: nat > set_o,I2: nat] :
( ( ( ( piE_nat_o @ I3 @ S )
= bot_bot_set_nat_o )
=> ( ( image_nat_o_o
@ ^ [F5: nat > $o] : ( F5 @ I2 )
@ ( piE_nat_o @ I3 @ S ) )
= bot_bot_set_o ) )
& ( ( ( piE_nat_o @ I3 @ S )
!= bot_bot_set_nat_o )
=> ( ( ( member_nat @ I2 @ I3 )
=> ( ( image_nat_o_o
@ ^ [F5: nat > $o] : ( F5 @ I2 )
@ ( piE_nat_o @ I3 @ S ) )
= ( S @ I2 ) ) )
& ( ~ ( member_nat @ I2 @ I3 )
=> ( ( image_nat_o_o
@ ^ [F5: nat > $o] : ( F5 @ I2 )
@ ( piE_nat_o @ I3 @ S ) )
= ( insert_o @ undefined_o @ bot_bot_set_o ) ) ) ) ) ) ).
% image_projection_PiE
thf(fact_1122_image__projection__PiE,axiom,
! [I3: set_b_a,S: ( b > a ) > set_b,I2: b > a] :
( ( ( ( piE_b_a_b @ I3 @ S )
= bot_bot_set_b_a_b )
=> ( ( image_b_a_b_b
@ ^ [F5: ( b > a ) > b] : ( F5 @ I2 )
@ ( piE_b_a_b @ I3 @ S ) )
= bot_bot_set_b ) )
& ( ( ( piE_b_a_b @ I3 @ S )
!= bot_bot_set_b_a_b )
=> ( ( ( member_b_a @ I2 @ I3 )
=> ( ( image_b_a_b_b
@ ^ [F5: ( b > a ) > b] : ( F5 @ I2 )
@ ( piE_b_a_b @ I3 @ S ) )
= ( S @ I2 ) ) )
& ( ~ ( member_b_a @ I2 @ I3 )
=> ( ( image_b_a_b_b
@ ^ [F5: ( b > a ) > b] : ( F5 @ I2 )
@ ( piE_b_a_b @ I3 @ S ) )
= ( insert_b @ undefined_b @ bot_bot_set_b ) ) ) ) ) ) ).
% image_projection_PiE
thf(fact_1123_PiE__mono,axiom,
! [A2: set_b,B2: b > set_a,C5: b > set_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( B2 @ X2 ) @ ( C5 @ X2 ) ) )
=> ( ord_less_eq_set_b_a @ ( piE_b_a @ A2 @ B2 ) @ ( piE_b_a @ A2 @ C5 ) ) ) ).
% PiE_mono
thf(fact_1124_PiE__mono,axiom,
! [A2: set_b_a,B2: ( b > a ) > set_a,C5: ( b > a ) > set_a] :
( ! [X2: b > a] :
( ( member_b_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( B2 @ X2 ) @ ( C5 @ X2 ) ) )
=> ( ord_le4402886750609172241_b_a_a @ ( piE_b_a_a @ A2 @ B2 ) @ ( piE_b_a_a @ A2 @ C5 ) ) ) ).
% PiE_mono
thf(fact_1125_PiE__mono,axiom,
! [A2: set_a,B2: a > set_a,C5: a > set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( B2 @ X2 ) @ ( C5 @ X2 ) ) )
=> ( ord_less_eq_set_a_a @ ( piE_a_a @ A2 @ B2 ) @ ( piE_a_a @ A2 @ C5 ) ) ) ).
% PiE_mono
thf(fact_1126_PiE__mono,axiom,
! [A2: set_nat,B2: nat > set_a,C5: nat > set_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( B2 @ X2 ) @ ( C5 @ X2 ) ) )
=> ( ord_le871467723717165285_nat_a @ ( piE_nat_a @ A2 @ B2 ) @ ( piE_nat_a @ A2 @ C5 ) ) ) ).
% PiE_mono
thf(fact_1127_image__constant__conv,axiom,
! [A2: set_o,C: b] :
( ( ( A2 = bot_bot_set_o )
=> ( ( image_o_b
@ ^ [X3: $o] : C
@ A2 )
= bot_bot_set_b ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( image_o_b
@ ^ [X3: $o] : C
@ A2 )
= ( insert_b @ C @ bot_bot_set_b ) ) ) ) ).
% image_constant_conv
thf(fact_1128_image__constant__conv,axiom,
! [A2: set_o,C: $o] :
( ( ( A2 = bot_bot_set_o )
=> ( ( image_o_o
@ ^ [X3: $o] : C
@ A2 )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( image_o_o
@ ^ [X3: $o] : C
@ A2 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_1129_image__constant,axiom,
! [X: b,A2: set_b,C: b] :
( ( member_b @ X @ A2 )
=> ( ( image_b_b
@ ^ [X3: b] : C
@ A2 )
= ( insert_b @ C @ bot_bot_set_b ) ) ) ).
% image_constant
thf(fact_1130_image__constant,axiom,
! [X: b > a,A2: set_b_a,C: b] :
( ( member_b_a @ X @ A2 )
=> ( ( image_b_a_b
@ ^ [X3: b > a] : C
@ A2 )
= ( insert_b @ C @ bot_bot_set_b ) ) ) ).
% image_constant
thf(fact_1131_image__constant,axiom,
! [X: a,A2: set_a,C: b] :
( ( member_a @ X @ A2 )
=> ( ( image_a_b
@ ^ [X3: a] : C
@ A2 )
= ( insert_b @ C @ bot_bot_set_b ) ) ) ).
% image_constant
thf(fact_1132_image__constant,axiom,
! [X: nat,A2: set_nat,C: b] :
( ( member_nat @ X @ A2 )
=> ( ( image_nat_b
@ ^ [X3: nat] : C
@ A2 )
= ( insert_b @ C @ bot_bot_set_b ) ) ) ).
% image_constant
thf(fact_1133_image__constant,axiom,
! [X: b,A2: set_b,C: $o] :
( ( member_b @ X @ A2 )
=> ( ( image_b_o
@ ^ [X3: b] : C
@ A2 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_1134_image__constant,axiom,
! [X: b > a,A2: set_b_a,C: $o] :
( ( member_b_a @ X @ A2 )
=> ( ( image_b_a_o
@ ^ [X3: b > a] : C
@ A2 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_1135_image__constant,axiom,
! [X: a,A2: set_a,C: $o] :
( ( member_a @ X @ A2 )
=> ( ( image_a_o
@ ^ [X3: a] : C
@ A2 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_1136_image__constant,axiom,
! [X: nat,A2: set_nat,C: $o] :
( ( member_nat @ X @ A2 )
=> ( ( image_nat_o
@ ^ [X3: nat] : C
@ A2 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_1137_subgroup_Oimage__of__inverse,axiom,
! [G2: set_b,M: set_b,Composition: b > b > b,Unit: b,X: b] :
( ( group_subgroup_b @ G2 @ M @ Composition @ Unit )
=> ( ( member_b @ X @ G2 )
=> ( member_b @ X @ ( image_b_b @ ( group_inverse_b @ M @ Composition @ Unit ) @ G2 ) ) ) ) ).
% subgroup.image_of_inverse
thf(fact_1138_subgroup_Oimage__of__inverse,axiom,
! [G2: set_b_a,M: set_b_a,Composition: ( b > a ) > ( b > a ) > b > a,Unit: b > a,X: b > a] :
( ( group_subgroup_b_a @ G2 @ M @ Composition @ Unit )
=> ( ( member_b_a @ X @ G2 )
=> ( member_b_a @ X @ ( image_b_a_b_a @ ( group_inverse_b_a @ M @ Composition @ Unit ) @ G2 ) ) ) ) ).
% subgroup.image_of_inverse
thf(fact_1139_subgroup_Oimage__of__inverse,axiom,
! [G2: set_nat,M: set_nat,Composition: nat > nat > nat,Unit: nat,X: nat] :
( ( group_subgroup_nat @ G2 @ M @ Composition @ Unit )
=> ( ( member_nat @ X @ G2 )
=> ( member_nat @ X @ ( image_nat_nat @ ( group_inverse_nat @ M @ Composition @ Unit ) @ G2 ) ) ) ) ).
% subgroup.image_of_inverse
thf(fact_1140_subgroup_Oimage__of__inverse,axiom,
! [G2: set_a,M: set_a,Composition: a > a > a,Unit: a,X: a] :
( ( group_subgroup_a @ G2 @ M @ Composition @ Unit )
=> ( ( member_a @ X @ G2 )
=> ( member_a @ X @ ( image_a_a @ ( group_inverse_a @ M @ Composition @ Unit ) @ G2 ) ) ) ) ).
% subgroup.image_of_inverse
thf(fact_1141_funcset__image,axiom,
! [F: b > a,A2: set_b,B2: set_a] :
( ( member_b_a @ F
@ ( pi_b_a @ A2
@ ^ [Uu: b] : B2 ) )
=> ( ord_less_eq_set_a @ ( image_b_a @ F @ A2 ) @ B2 ) ) ).
% funcset_image
thf(fact_1142_image__subset__iff__funcset,axiom,
! [F2: b > a,A2: set_b,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_b_a @ F2 @ A2 ) @ B2 )
= ( member_b_a @ F2
@ ( pi_b_a @ A2
@ ^ [Uu: b] : B2 ) ) ) ).
% image_subset_iff_funcset
thf(fact_1143_abelian__group_Oaxioms_I1_J,axiom,
! [G2: set_a,Composition: a > a > a,Unit: a] :
( ( group_201663378560352916roup_a @ G2 @ Composition @ Unit )
=> ( group_group_a @ G2 @ Composition @ Unit ) ) ).
% abelian_group.axioms(1)
thf(fact_1144_abelian__group_Oaxioms_I2_J,axiom,
! [G2: set_a,Composition: a > a > a,Unit: a] :
( ( group_201663378560352916roup_a @ G2 @ Composition @ Unit )
=> ( group_4866109990395492029noid_a @ G2 @ Composition @ Unit ) ) ).
% abelian_group.axioms(2)
thf(fact_1145_in__image__insert__iff,axiom,
! [B2: set_set_b,X: b,A2: set_b] :
( ! [C6: set_b] :
( ( member_set_b @ C6 @ B2 )
=> ~ ( member_b @ X @ C6 ) )
=> ( ( member_set_b @ A2 @ ( image_set_b_set_b @ ( insert_b @ X ) @ B2 ) )
= ( ( member_b @ X @ A2 )
& ( member_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1146_in__image__insert__iff,axiom,
! [B2: set_set_b_a,X: b > a,A2: set_b_a] :
( ! [C6: set_b_a] :
( ( member_set_b_a @ C6 @ B2 )
=> ~ ( member_b_a @ X @ C6 ) )
=> ( ( member_set_b_a @ A2 @ ( image_8428841068488964283et_b_a @ ( insert_b_a @ X ) @ B2 ) )
= ( ( member_b_a @ X @ A2 )
& ( member_set_b_a @ ( minus_minus_set_b_a @ A2 @ ( insert_b_a @ X @ bot_bot_set_b_a ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1147_in__image__insert__iff,axiom,
! [B2: set_set_a,X: a,A2: set_a] :
( ! [C6: set_a] :
( ( member_set_a @ C6 @ B2 )
=> ~ ( member_a @ X @ C6 ) )
=> ( ( member_set_a @ A2 @ ( image_set_a_set_a @ ( insert_a @ X ) @ B2 ) )
= ( ( member_a @ X @ A2 )
& ( member_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1148_in__image__insert__iff,axiom,
! [B2: set_set_nat,X: nat,A2: set_nat] :
( ! [C6: set_nat] :
( ( member_set_nat @ C6 @ B2 )
=> ~ ( member_nat @ X @ C6 ) )
=> ( ( member_set_nat @ A2 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X ) @ B2 ) )
= ( ( member_nat @ X @ A2 )
& ( member_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1149_in__image__insert__iff,axiom,
! [B2: set_set_o,X: $o,A2: set_o] :
( ! [C6: set_o] :
( ( member_set_o @ C6 @ B2 )
=> ~ ( member_o @ X @ C6 ) )
=> ( ( member_set_o @ A2 @ ( image_set_o_set_o @ ( insert_o @ X ) @ B2 ) )
= ( ( member_o @ X @ A2 )
& ( member_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1150_group_Oinverse__subgroupI,axiom,
! [G2: set_a,Composition: a > a > a,Unit: a,H2: set_a] :
( ( group_group_a @ G2 @ Composition @ Unit )
=> ( ( group_subgroup_a @ H2 @ G2 @ Composition @ Unit )
=> ( group_subgroup_a @ ( image_a_a @ ( group_inverse_a @ G2 @ Composition @ Unit ) @ H2 ) @ G2 @ Composition @ Unit ) ) ) ).
% group.inverse_subgroupI
thf(fact_1151_image__fold__insert,axiom,
! [A2: set_b,F: b > b] :
( ( finite_finite_b @ A2 )
=> ( ( image_b_b @ F @ A2 )
= ( finite_fold_b_set_b
@ ^ [K3: b] : ( insert_b @ ( F @ K3 ) )
@ bot_bot_set_b
@ A2 ) ) ) ).
% image_fold_insert
thf(fact_1152_image__fold__insert,axiom,
! [A2: set_nat,F: nat > b] :
( ( finite_finite_nat @ A2 )
=> ( ( image_nat_b @ F @ A2 )
= ( finite4864421579114109509_set_b
@ ^ [K3: nat] : ( insert_b @ ( F @ K3 ) )
@ bot_bot_set_b
@ A2 ) ) ) ).
% image_fold_insert
thf(fact_1153_image__fold__insert,axiom,
! [A2: set_b,F: b > $o] :
( ( finite_finite_b @ A2 )
=> ( ( image_b_o @ F @ A2 )
= ( finite_fold_b_set_o
@ ^ [K3: b] : ( insert_o @ ( F @ K3 ) )
@ bot_bot_set_o
@ A2 ) ) ) ).
% image_fold_insert
thf(fact_1154_image__fold__insert,axiom,
! [A2: set_nat,F: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( image_nat_o @ F @ A2 )
= ( finite3217087857726763998_set_o
@ ^ [K3: nat] : ( insert_o @ ( F @ K3 ) )
@ bot_bot_set_o
@ A2 ) ) ) ).
% image_fold_insert
thf(fact_1155_abelian__group_Ointro,axiom,
! [G2: set_a,Composition: a > a > a,Unit: a] :
( ( group_group_a @ G2 @ Composition @ Unit )
=> ( ( group_4866109990395492029noid_a @ G2 @ Composition @ Unit )
=> ( group_201663378560352916roup_a @ G2 @ Composition @ Unit ) ) ) ).
% abelian_group.intro
thf(fact_1156_ext__funcset__to__sing__iff,axiom,
! [A2: set_a,A: a] :
( ( piE_a_a @ A2
@ ^ [I: a] : ( insert_a @ A @ bot_bot_set_a ) )
= ( insert_a_a
@ ( restrict_a_a
@ ^ [X3: a] : A
@ A2 )
@ bot_bot_set_a_a ) ) ).
% ext_funcset_to_sing_iff
thf(fact_1157_extensional__funcset__fun__upd__restricts__rangeI,axiom,
! [S: set_b,F: b > b,X: b,T2: set_b] :
( ! [X2: b] :
( ( member_b @ X2 @ S )
=> ( ( F @ X )
!= ( F @ X2 ) ) )
=> ( ( member_b_b @ F
@ ( piE_b_b @ ( insert_b @ X @ S )
@ ^ [I: b] : T2 ) )
=> ( member_b_b @ ( fun_upd_b_b @ F @ X @ undefined_b )
@ ( piE_b_b @ S
@ ^ [I: b] : ( minus_minus_set_b @ T2 @ ( insert_b @ ( F @ X ) @ bot_bot_set_b ) ) ) ) ) ) ).
% extensional_funcset_fun_upd_restricts_rangeI
thf(fact_1158_extensional__funcset__fun__upd__restricts__rangeI,axiom,
! [S: set_o,F: $o > b,X: $o,T2: set_b] :
( ! [X2: $o] :
( ( member_o @ X2 @ S )
=> ( ( F @ X )
!= ( F @ X2 ) ) )
=> ( ( member_o_b @ F
@ ( piE_o_b @ ( insert_o @ X @ S )
@ ^ [I: $o] : T2 ) )
=> ( member_o_b @ ( fun_upd_o_b @ F @ X @ undefined_b )
@ ( piE_o_b @ S
@ ^ [I: $o] : ( minus_minus_set_b @ T2 @ ( insert_b @ ( F @ X ) @ bot_bot_set_b ) ) ) ) ) ) ).
% extensional_funcset_fun_upd_restricts_rangeI
thf(fact_1159_extensional__funcset__fun__upd__restricts__rangeI,axiom,
! [S: set_o,F: $o > a,X: $o,T2: set_a] :
( ! [X2: $o] :
( ( member_o @ X2 @ S )
=> ( ( F @ X )
!= ( F @ X2 ) ) )
=> ( ( member_o_a @ F
@ ( piE_o_a @ ( insert_o @ X @ S )
@ ^ [I: $o] : T2 ) )
=> ( member_o_a @ ( fun_upd_o_a @ F @ X @ undefined_a )
@ ( piE_o_a @ S
@ ^ [I: $o] : ( minus_minus_set_a @ T2 @ ( insert_a @ ( F @ X ) @ bot_bot_set_a ) ) ) ) ) ) ).
% extensional_funcset_fun_upd_restricts_rangeI
thf(fact_1160_extensional__funcset__fun__upd__restricts__rangeI,axiom,
! [S: set_b,F: b > a,X: b,T2: set_a] :
( ! [X2: b] :
( ( member_b @ X2 @ S )
=> ( ( F @ X )
!= ( F @ X2 ) ) )
=> ( ( member_b_a @ F
@ ( piE_b_a @ ( insert_b @ X @ S )
@ ^ [I: b] : T2 ) )
=> ( member_b_a @ ( fun_upd_b_a @ F @ X @ undefined_a )
@ ( piE_b_a @ S
@ ^ [I: b] : ( minus_minus_set_a @ T2 @ ( insert_a @ ( F @ X ) @ bot_bot_set_a ) ) ) ) ) ) ).
% extensional_funcset_fun_upd_restricts_rangeI
thf(fact_1161_extensional__funcset__fun__upd__restricts__rangeI,axiom,
! [S: set_b,F: b > $o,X: b,T2: set_o] :
( ! [X2: b] :
( ( member_b @ X2 @ S )
=> ( ( F @ X )
= ( ~ ( F @ X2 ) ) ) )
=> ( ( member_b_o @ F
@ ( piE_b_o @ ( insert_b @ X @ S )
@ ^ [I: b] : T2 ) )
=> ( member_b_o @ ( fun_upd_b_o @ F @ X @ undefined_o )
@ ( piE_b_o @ S
@ ^ [I: b] : ( minus_minus_set_o @ T2 @ ( insert_o @ ( F @ X ) @ bot_bot_set_o ) ) ) ) ) ) ).
% extensional_funcset_fun_upd_restricts_rangeI
thf(fact_1162_extensional__funcset__fun__upd__restricts__rangeI,axiom,
! [S: set_o,F: $o > $o,X: $o,T2: set_o] :
( ! [X2: $o] :
( ( member_o @ X2 @ S )
=> ( ( F @ X )
= ( ~ ( F @ X2 ) ) ) )
=> ( ( member_o_o @ F
@ ( piE_o_o @ ( insert_o @ X @ S )
@ ^ [I: $o] : T2 ) )
=> ( member_o_o @ ( fun_upd_o_o @ F @ X @ undefined_o )
@ ( piE_o_o @ S
@ ^ [I: $o] : ( minus_minus_set_o @ T2 @ ( insert_o @ ( F @ X ) @ bot_bot_set_o ) ) ) ) ) ) ).
% extensional_funcset_fun_upd_restricts_rangeI
thf(fact_1163_PiE__eq__singleton,axiom,
! [I3: set_a,S: a > set_a,F: a > a] :
( ( ( piE_a_a @ I3 @ S )
= ( insert_a_a @ ( restrict_a_a @ F @ I3 ) @ bot_bot_set_a_a ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ I3 )
=> ( ( S @ X3 )
= ( insert_a @ ( F @ X3 ) @ bot_bot_set_a ) ) ) ) ) ).
% PiE_eq_singleton
thf(fact_1164_Pi__cancel__fupd__range,axiom,
! [I2: b,I3: set_b,X: b > a,B2: b > set_a,B: set_a] :
( ~ ( member_b @ I2 @ I3 )
=> ( ( member_b_a @ X @ ( pi_b_a @ I3 @ ( fun_upd_b_set_a @ B2 @ I2 @ B ) ) )
= ( member_b_a @ X @ ( pi_b_a @ I3 @ B2 ) ) ) ) ).
% Pi_cancel_fupd_range
thf(fact_1165_image__restrict__eq,axiom,
! [F: a > a,A2: set_a] :
( ( image_a_a @ ( restrict_a_a @ F @ A2 ) @ A2 )
= ( image_a_a @ F @ A2 ) ) ).
% image_restrict_eq
thf(fact_1166_restrict__fupd,axiom,
! [I2: a,I3: set_a,F: a > a,X: a] :
( ~ ( member_a @ I2 @ I3 )
=> ( ( restrict_a_a @ ( fun_upd_a_a @ F @ I2 @ X ) @ I3 )
= ( restrict_a_a @ F @ I3 ) ) ) ).
% restrict_fupd
thf(fact_1167_PiE__restrict,axiom,
! [F: b > a,A2: set_b,B2: b > set_a] :
( ( member_b_a @ F @ ( piE_b_a @ A2 @ B2 ) )
=> ( ( restrict_b_a @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_1168_PiE__restrict,axiom,
! [F: a > a,A2: set_a,B2: a > set_a] :
( ( member_a_a @ F @ ( piE_a_a @ A2 @ B2 ) )
=> ( ( restrict_a_a @ F @ A2 )
= F ) ) ).
% PiE_restrict
thf(fact_1169_Pi__cancel__fupd,axiom,
! [I2: b,I3: set_b,X: b > a,A: a,B2: b > set_a] :
( ~ ( member_b @ I2 @ I3 )
=> ( ( member_b_a @ ( fun_upd_b_a @ X @ I2 @ A ) @ ( pi_b_a @ I3 @ B2 ) )
= ( member_b_a @ X @ ( pi_b_a @ I3 @ B2 ) ) ) ) ).
% Pi_cancel_fupd
thf(fact_1170_restrictI,axiom,
! [A2: set_b,F: b > b,B2: b > set_b] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_b_b @ ( restrict_b_b @ F @ A2 ) @ ( pi_b_b @ A2 @ B2 ) ) ) ).
% restrictI
thf(fact_1171_restrictI,axiom,
! [A2: set_b,F: b > a,B2: b > set_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_b_a @ ( restrict_b_a @ F @ A2 ) @ ( pi_b_a @ A2 @ B2 ) ) ) ).
% restrictI
thf(fact_1172_restrictI,axiom,
! [A2: set_b,F: b > nat,B2: b > set_nat] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_b_nat @ ( restrict_b_nat @ F @ A2 ) @ ( pi_b_nat @ A2 @ B2 ) ) ) ).
% restrictI
thf(fact_1173_restrictI,axiom,
! [A2: set_a,F: a > b,B2: a > set_b] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_a_b @ ( restrict_a_b @ F @ A2 ) @ ( pi_a_b @ A2 @ B2 ) ) ) ).
% restrictI
thf(fact_1174_restrictI,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 @ ( restrict_a_nat @ F @ A2 ) @ ( pi_a_nat @ A2 @ B2 ) ) ) ).
% restrictI
thf(fact_1175_restrictI,axiom,
! [A2: set_nat,F: nat > b,B2: nat > set_b] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_nat_b @ ( restrict_nat_b @ F @ A2 ) @ ( pi_nat_b @ A2 @ B2 ) ) ) ).
% restrictI
thf(fact_1176_restrictI,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 @ ( restrict_nat_a @ F @ A2 ) @ ( pi_nat_a @ A2 @ B2 ) ) ) ).
% restrictI
thf(fact_1177_restrictI,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 @ ( restrict_nat_nat @ F @ A2 ) @ ( pi_nat_nat @ A2 @ B2 ) ) ) ).
% restrictI
thf(fact_1178_restrictI,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 @ ( restrict_a_a @ F @ A2 ) @ ( pi_a_a @ A2 @ B2 ) ) ) ).
% restrictI
thf(fact_1179_restrictI,axiom,
! [A2: set_b,F: b > b > a,B2: b > set_b_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_b_a @ ( F @ X2 ) @ ( B2 @ X2 ) ) )
=> ( member_b_b_a @ ( restrict_b_b_a @ F @ A2 ) @ ( pi_b_b_a @ A2 @ B2 ) ) ) ).
% restrictI
thf(fact_1180_restrict__apply,axiom,
( restrict_b_a
= ( ^ [F5: b > a,A6: set_b,X3: b] : ( if_a @ ( member_b @ X3 @ A6 ) @ ( F5 @ X3 ) @ undefined_a ) ) ) ).
% restrict_apply
thf(fact_1181_restrict__apply,axiom,
( restrict_b_a_a
= ( ^ [F5: ( b > a ) > a,A6: set_b_a,X3: b > a] : ( if_a @ ( member_b_a @ X3 @ A6 ) @ ( F5 @ X3 ) @ undefined_a ) ) ) ).
% restrict_apply
thf(fact_1182_restrict__apply,axiom,
( restrict_nat_a
= ( ^ [F5: nat > a,A6: set_nat,X3: nat] : ( if_a @ ( member_nat @ X3 @ A6 ) @ ( F5 @ X3 ) @ undefined_a ) ) ) ).
% restrict_apply
thf(fact_1183_restrict__apply,axiom,
( restrict_a_a
= ( ^ [F5: a > a,A6: set_a,X3: a] : ( if_a @ ( member_a @ X3 @ A6 ) @ ( F5 @ X3 ) @ undefined_a ) ) ) ).
% restrict_apply
thf(fact_1184_restrict__upd,axiom,
! [I2: a,I3: set_a,F: a > a,Y: a] :
( ~ ( member_a @ I2 @ I3 )
=> ( ( fun_upd_a_a @ ( restrict_a_a @ F @ I3 ) @ I2 @ Y )
= ( restrict_a_a @ ( fun_upd_a_a @ F @ I2 @ Y ) @ ( insert_a @ I2 @ I3 ) ) ) ) ).
% restrict_upd
thf(fact_1185_extensional__insert,axiom,
! [A: b > a,I2: b,I3: set_b,B: a] :
( ( member_b_a @ A @ ( extensional_b_a @ ( insert_b @ I2 @ I3 ) ) )
=> ( member_b_a @ ( fun_upd_b_a @ A @ I2 @ B ) @ ( extensional_b_a @ ( insert_b @ I2 @ I3 ) ) ) ) ).
% extensional_insert
thf(fact_1186_restrict__extensional__sub,axiom,
! [A2: set_b,B2: set_b,F: b > a] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( member_b_a @ ( restrict_b_a @ F @ A2 ) @ ( extensional_b_a @ B2 ) ) ) ).
% restrict_extensional_sub
thf(fact_1187_restrict__extensional__sub,axiom,
! [A2: set_a,B2: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( member_a_a @ ( restrict_a_a @ F @ A2 ) @ ( extensional_a_a @ B2 ) ) ) ).
% restrict_extensional_sub
thf(fact_1188_restrict__PiE,axiom,
! [F: b > a,I3: set_b,S: b > set_a] :
( ( member_b_a @ ( restrict_b_a @ F @ I3 ) @ ( piE_b_a @ I3 @ S ) )
= ( member_b_a @ F @ ( pi_b_a @ I3 @ S ) ) ) ).
% restrict_PiE
thf(fact_1189_restrict__PiE,axiom,
! [F: a > a,I3: set_a,S: a > set_a] :
( ( member_a_a @ ( restrict_a_a @ F @ I3 ) @ ( piE_a_a @ I3 @ S ) )
= ( member_a_a @ F @ ( pi_a_a @ I3 @ S ) ) ) ).
% restrict_PiE
thf(fact_1190_extensional__insert__undefined,axiom,
! [A: $o > a,I2: $o,I3: set_o] :
( ( member_o_a @ A @ ( extensional_o_a @ ( insert_o @ I2 @ I3 ) ) )
=> ( member_o_a @ ( fun_upd_o_a @ A @ I2 @ undefined_a ) @ ( extensional_o_a @ I3 ) ) ) ).
% extensional_insert_undefined
thf(fact_1191_extensional__insert__undefined,axiom,
! [A: b > a,I2: b,I3: set_b] :
( ( member_b_a @ A @ ( extensional_b_a @ ( insert_b @ I2 @ I3 ) ) )
=> ( member_b_a @ ( fun_upd_b_a @ A @ I2 @ undefined_a ) @ ( extensional_b_a @ I3 ) ) ) ).
% extensional_insert_undefined
thf(fact_1192_restrict__ext,axiom,
! [A2: set_a,F: a > a,G: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( restrict_a_a @ F @ A2 )
= ( restrict_a_a @ G @ A2 ) ) ) ).
% restrict_ext
thf(fact_1193_restrict__apply_H,axiom,
! [X: a,A2: set_a,F: a > a] :
( ( member_a @ X @ A2 )
=> ( ( restrict_a_a @ F @ A2 @ X )
= ( F @ X ) ) ) ).
% restrict_apply'
thf(fact_1194_restrict__PiE__iff,axiom,
! [F: b > a,I3: set_b,X4: b > set_a] :
( ( member_b_a @ ( restrict_b_a @ F @ I3 ) @ ( piE_b_a @ I3 @ X4 ) )
= ( ! [X3: b] :
( ( member_b @ X3 @ I3 )
=> ( member_a @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_1195_restrict__PiE__iff,axiom,
! [F: a > a,I3: set_a,X4: a > set_a] :
( ( member_a_a @ ( restrict_a_a @ F @ I3 ) @ ( piE_a_a @ I3 @ X4 ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ I3 )
=> ( member_a @ ( F @ X3 ) @ ( X4 @ X3 ) ) ) ) ) ).
% restrict_PiE_iff
thf(fact_1196_restrict__Pi__cancel,axiom,
! [X: b > a,I3: set_b,A2: b > set_a] :
( ( member_b_a @ ( restrict_b_a @ X @ I3 ) @ ( pi_b_a @ I3 @ A2 ) )
= ( member_b_a @ X @ ( pi_b_a @ I3 @ A2 ) ) ) ).
% restrict_Pi_cancel
thf(fact_1197_restrict__Pi__cancel,axiom,
! [X: a > a,I3: set_a,A2: a > set_a] :
( ( member_a_a @ ( restrict_a_a @ X @ I3 ) @ ( pi_a_a @ I3 @ A2 ) )
= ( member_a_a @ X @ ( pi_a_a @ I3 @ A2 ) ) ) ).
% restrict_Pi_cancel
thf(fact_1198_restrict__def,axiom,
( restrict_b_a
= ( ^ [F5: b > a,A6: set_b,X3: b] : ( if_a @ ( member_b @ X3 @ A6 ) @ ( F5 @ X3 ) @ undefined_a ) ) ) ).
% restrict_def
thf(fact_1199_restrict__def,axiom,
( restrict_b_a_a
= ( ^ [F5: ( b > a ) > a,A6: set_b_a,X3: b > a] : ( if_a @ ( member_b_a @ X3 @ A6 ) @ ( F5 @ X3 ) @ undefined_a ) ) ) ).
% restrict_def
thf(fact_1200_restrict__def,axiom,
( restrict_nat_a
= ( ^ [F5: nat > a,A6: set_nat,X3: nat] : ( if_a @ ( member_nat @ X3 @ A6 ) @ ( F5 @ X3 ) @ undefined_a ) ) ) ).
% restrict_def
thf(fact_1201_restrict__def,axiom,
( restrict_a_a
= ( ^ [F5: a > a,A6: set_a,X3: a] : ( if_a @ ( member_a @ X3 @ A6 ) @ ( F5 @ X3 ) @ undefined_a ) ) ) ).
% restrict_def
thf(fact_1202_extensional__restrict,axiom,
! [F: b > a,A2: set_b] :
( ( member_b_a @ F @ ( extensional_b_a @ A2 ) )
=> ( ( restrict_b_a @ F @ A2 )
= F ) ) ).
% extensional_restrict
thf(fact_1203_extensional__restrict,axiom,
! [F: a > a,A2: set_a] :
( ( member_a_a @ F @ ( extensional_a_a @ A2 ) )
=> ( ( restrict_a_a @ F @ A2 )
= F ) ) ).
% extensional_restrict
thf(fact_1204_restrict__extensional,axiom,
! [F: b > a,A2: set_b] : ( member_b_a @ ( restrict_b_a @ F @ A2 ) @ ( extensional_b_a @ A2 ) ) ).
% restrict_extensional
thf(fact_1205_restrict__extensional,axiom,
! [F: a > a,A2: set_a] : ( member_a_a @ ( restrict_a_a @ F @ A2 ) @ ( extensional_a_a @ A2 ) ) ).
% restrict_extensional
thf(fact_1206_PiE__fun__upd,axiom,
! [Y: b,T2: b > set_b,X: b,F: b > b,S: set_b] :
( ( member_b @ Y @ ( T2 @ X ) )
=> ( ( member_b_b @ F @ ( piE_b_b @ S @ T2 ) )
=> ( member_b_b @ ( fun_upd_b_b @ F @ X @ Y ) @ ( piE_b_b @ ( insert_b @ X @ S ) @ T2 ) ) ) ) ).
% PiE_fun_upd
thf(fact_1207_PiE__fun__upd,axiom,
! [Y: b,T2: $o > set_b,X: $o,F: $o > b,S: set_o] :
( ( member_b @ Y @ ( T2 @ X ) )
=> ( ( member_o_b @ F @ ( piE_o_b @ S @ T2 ) )
=> ( member_o_b @ ( fun_upd_o_b @ F @ X @ Y ) @ ( piE_o_b @ ( insert_o @ X @ S ) @ T2 ) ) ) ) ).
% PiE_fun_upd
thf(fact_1208_PiE__fun__upd,axiom,
! [Y: b > a,T2: b > set_b_a,X: b,F: b > b > a,S: set_b] :
( ( member_b_a @ Y @ ( T2 @ X ) )
=> ( ( member_b_b_a @ F @ ( piE_b_b_a @ S @ T2 ) )
=> ( member_b_b_a @ ( fun_upd_b_b_a @ F @ X @ Y ) @ ( piE_b_b_a @ ( insert_b @ X @ S ) @ T2 ) ) ) ) ).
% PiE_fun_upd
thf(fact_1209_PiE__fun__upd,axiom,
! [Y: b > a,T2: $o > set_b_a,X: $o,F: $o > b > a,S: set_o] :
( ( member_b_a @ Y @ ( T2 @ X ) )
=> ( ( member_o_b_a @ F @ ( piE_o_b_a @ S @ T2 ) )
=> ( member_o_b_a @ ( fun_upd_o_b_a @ F @ X @ Y ) @ ( piE_o_b_a @ ( insert_o @ X @ S ) @ T2 ) ) ) ) ).
% PiE_fun_upd
thf(fact_1210_PiE__fun__upd,axiom,
! [Y: a,T2: $o > set_a,X: $o,F: $o > a,S: set_o] :
( ( member_a @ Y @ ( T2 @ X ) )
=> ( ( member_o_a @ F @ ( piE_o_a @ S @ T2 ) )
=> ( member_o_a @ ( fun_upd_o_a @ F @ X @ Y ) @ ( piE_o_a @ ( insert_o @ X @ S ) @ T2 ) ) ) ) ).
% PiE_fun_upd
thf(fact_1211_PiE__fun__upd,axiom,
! [Y: a,T2: b > set_a,X: b,F: b > a,S: set_b] :
( ( member_a @ Y @ ( T2 @ X ) )
=> ( ( member_b_a @ F @ ( piE_b_a @ S @ T2 ) )
=> ( member_b_a @ ( fun_upd_b_a @ F @ X @ Y ) @ ( piE_b_a @ ( insert_b @ X @ S ) @ T2 ) ) ) ) ).
% PiE_fun_upd
thf(fact_1212_PiE__fun__upd,axiom,
! [Y: nat,T2: b > set_nat,X: b,F: b > nat,S: set_b] :
( ( member_nat @ Y @ ( T2 @ X ) )
=> ( ( member_b_nat @ F @ ( piE_b_nat @ S @ T2 ) )
=> ( member_b_nat @ ( fun_upd_b_nat @ F @ X @ Y ) @ ( piE_b_nat @ ( insert_b @ X @ S ) @ T2 ) ) ) ) ).
% PiE_fun_upd
thf(fact_1213_PiE__fun__upd,axiom,
! [Y: nat,T2: $o > set_nat,X: $o,F: $o > nat,S: set_o] :
( ( member_nat @ Y @ ( T2 @ X ) )
=> ( ( member_o_nat @ F @ ( piE_o_nat @ S @ T2 ) )
=> ( member_o_nat @ ( fun_upd_o_nat @ F @ X @ Y ) @ ( piE_o_nat @ ( insert_o @ X @ S ) @ T2 ) ) ) ) ).
% PiE_fun_upd
thf(fact_1214_fun__upd__image,axiom,
! [X: b,A2: set_b,F: b > b,Y: b] :
( ( ( member_b @ X @ A2 )
=> ( ( image_b_b @ ( fun_upd_b_b @ F @ X @ Y ) @ A2 )
= ( insert_b @ Y @ ( image_b_b @ F @ ( minus_minus_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) ) ) ) ) )
& ( ~ ( member_b @ X @ A2 )
=> ( ( image_b_b @ ( fun_upd_b_b @ F @ X @ Y ) @ A2 )
= ( image_b_b @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_1215_fun__upd__image,axiom,
! [X: b,A2: set_b,F: b > $o,Y: $o] :
( ( ( member_b @ X @ A2 )
=> ( ( image_b_o @ ( fun_upd_b_o @ F @ X @ Y ) @ A2 )
= ( insert_o @ Y @ ( image_b_o @ F @ ( minus_minus_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) ) ) ) ) )
& ( ~ ( member_b @ X @ A2 )
=> ( ( image_b_o @ ( fun_upd_b_o @ F @ X @ Y ) @ A2 )
= ( image_b_o @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_1216_fun__upd__image,axiom,
! [X: b > a,A2: set_b_a,F: ( b > a ) > b,Y: b] :
( ( ( member_b_a @ X @ A2 )
=> ( ( image_b_a_b @ ( fun_upd_b_a_b @ F @ X @ Y ) @ A2 )
= ( insert_b @ Y @ ( image_b_a_b @ F @ ( minus_minus_set_b_a @ A2 @ ( insert_b_a @ X @ bot_bot_set_b_a ) ) ) ) ) )
& ( ~ ( member_b_a @ X @ A2 )
=> ( ( image_b_a_b @ ( fun_upd_b_a_b @ F @ X @ Y ) @ A2 )
= ( image_b_a_b @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_1217_fun__upd__image,axiom,
! [X: b > a,A2: set_b_a,F: ( b > a ) > $o,Y: $o] :
( ( ( member_b_a @ X @ A2 )
=> ( ( image_b_a_o @ ( fun_upd_b_a_o @ F @ X @ Y ) @ A2 )
= ( insert_o @ Y @ ( image_b_a_o @ F @ ( minus_minus_set_b_a @ A2 @ ( insert_b_a @ X @ bot_bot_set_b_a ) ) ) ) ) )
& ( ~ ( member_b_a @ X @ A2 )
=> ( ( image_b_a_o @ ( fun_upd_b_a_o @ F @ X @ Y ) @ A2 )
= ( image_b_a_o @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_1218_fun__upd__image,axiom,
! [X: a,A2: set_a,F: a > b,Y: b] :
( ( ( member_a @ X @ A2 )
=> ( ( image_a_b @ ( fun_upd_a_b @ F @ X @ Y ) @ A2 )
= ( insert_b @ Y @ ( image_a_b @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( image_a_b @ ( fun_upd_a_b @ F @ X @ Y ) @ A2 )
= ( image_a_b @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_1219_fun__upd__image,axiom,
! [X: a,A2: set_a,F: a > $o,Y: $o] :
( ( ( member_a @ X @ A2 )
=> ( ( image_a_o @ ( fun_upd_a_o @ F @ X @ Y ) @ A2 )
= ( insert_o @ Y @ ( image_a_o @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( image_a_o @ ( fun_upd_a_o @ F @ X @ Y ) @ A2 )
= ( image_a_o @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_1220_fun__upd__image,axiom,
! [X: nat,A2: set_nat,F: nat > b,Y: b] :
( ( ( member_nat @ X @ A2 )
=> ( ( image_nat_b @ ( fun_upd_nat_b @ F @ X @ Y ) @ A2 )
= ( insert_b @ Y @ ( image_nat_b @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ( image_nat_b @ ( fun_upd_nat_b @ F @ X @ Y ) @ A2 )
= ( image_nat_b @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_1221_fun__upd__image,axiom,
! [X: nat,A2: set_nat,F: nat > $o,Y: $o] :
( ( ( member_nat @ X @ A2 )
=> ( ( image_nat_o @ ( fun_upd_nat_o @ F @ X @ Y ) @ A2 )
= ( insert_o @ Y @ ( image_nat_o @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ( image_nat_o @ ( fun_upd_nat_o @ F @ X @ Y ) @ A2 )
= ( image_nat_o @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_1222_fun__upd__image,axiom,
! [X: $o,A2: set_o,F: $o > b,Y: b] :
( ( ( member_o @ X @ A2 )
=> ( ( image_o_b @ ( fun_upd_o_b @ F @ X @ Y ) @ A2 )
= ( insert_b @ Y @ ( image_o_b @ F @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ X @ A2 )
=> ( ( image_o_b @ ( fun_upd_o_b @ F @ X @ Y ) @ A2 )
= ( image_o_b @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_1223_fun__upd__image,axiom,
! [X: $o,A2: set_o,F: $o > $o,Y: $o] :
( ( ( member_o @ X @ A2 )
=> ( ( image_o_o @ ( fun_upd_o_o @ F @ X @ Y ) @ A2 )
= ( insert_o @ Y @ ( image_o_o @ F @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ X @ A2 )
=> ( ( image_o_o @ ( fun_upd_o_o @ F @ X @ Y ) @ A2 )
= ( image_o_o @ F @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_1224_fun__upd__in__PiE,axiom,
! [X: $o,S: set_o,F: $o > a,T2: $o > set_a] :
( ~ ( member_o @ X @ S )
=> ( ( member_o_a @ F @ ( piE_o_a @ ( insert_o @ X @ S ) @ T2 ) )
=> ( member_o_a @ ( fun_upd_o_a @ F @ X @ undefined_a ) @ ( piE_o_a @ S @ T2 ) ) ) ) ).
% fun_upd_in_PiE
thf(fact_1225_fun__upd__in__PiE,axiom,
! [X: b,S: set_b,F: b > a,T2: b > set_a] :
( ~ ( member_b @ X @ S )
=> ( ( member_b_a @ F @ ( piE_b_a @ ( insert_b @ X @ S ) @ T2 ) )
=> ( member_b_a @ ( fun_upd_b_a @ F @ X @ undefined_a ) @ ( piE_b_a @ S @ T2 ) ) ) ) ).
% fun_upd_in_PiE
thf(fact_1226_fun__upd__in__PiE,axiom,
! [X: b > a,S: set_b_a,F: ( b > a ) > a,T2: ( b > a ) > set_a] :
( ~ ( member_b_a @ X @ S )
=> ( ( member_b_a_a @ F @ ( piE_b_a_a @ ( insert_b_a @ X @ S ) @ T2 ) )
=> ( member_b_a_a @ ( fun_upd_b_a_a @ F @ X @ undefined_a ) @ ( piE_b_a_a @ S @ T2 ) ) ) ) ).
% fun_upd_in_PiE
thf(fact_1227_fun__upd__in__PiE,axiom,
! [X: a,S: set_a,F: a > a,T2: a > set_a] :
( ~ ( member_a @ X @ S )
=> ( ( member_a_a @ F @ ( piE_a_a @ ( insert_a @ X @ S ) @ T2 ) )
=> ( member_a_a @ ( fun_upd_a_a @ F @ X @ undefined_a ) @ ( piE_a_a @ S @ T2 ) ) ) ) ).
% fun_upd_in_PiE
thf(fact_1228_fun__upd__in__PiE,axiom,
! [X: nat,S: set_nat,F: nat > a,T2: nat > set_a] :
( ~ ( member_nat @ X @ S )
=> ( ( member_nat_a @ F @ ( piE_nat_a @ ( insert_nat @ X @ S ) @ T2 ) )
=> ( member_nat_a @ ( fun_upd_nat_a @ F @ X @ undefined_a ) @ ( piE_nat_a @ S @ T2 ) ) ) ) ).
% fun_upd_in_PiE
thf(fact_1229_Pi__fupd__iff,axiom,
! [I2: b,I3: set_b,F: b > b,B2: b > set_b,A2: set_b] :
( ( member_b @ I2 @ I3 )
=> ( ( member_b_b @ F @ ( pi_b_b @ I3 @ ( fun_upd_b_set_b @ B2 @ I2 @ A2 ) ) )
= ( ( member_b_b @ F @ ( pi_b_b @ ( minus_minus_set_b @ I3 @ ( insert_b @ I2 @ bot_bot_set_b ) ) @ B2 ) )
& ( member_b @ ( F @ I2 ) @ A2 ) ) ) ) ).
% Pi_fupd_iff
thf(fact_1230_Pi__fupd__iff,axiom,
! [I2: b,I3: set_b,F: b > nat,B2: b > set_nat,A2: set_nat] :
( ( member_b @ I2 @ I3 )
=> ( ( member_b_nat @ F @ ( pi_b_nat @ I3 @ ( fun_upd_b_set_nat @ B2 @ I2 @ A2 ) ) )
= ( ( member_b_nat @ F @ ( pi_b_nat @ ( minus_minus_set_b @ I3 @ ( insert_b @ I2 @ bot_bot_set_b ) ) @ B2 ) )
& ( member_nat @ ( F @ I2 ) @ A2 ) ) ) ) ).
% Pi_fupd_iff
thf(fact_1231_Pi__fupd__iff,axiom,
! [I2: b,I3: set_b,F: b > a,B2: b > set_a,A2: set_a] :
( ( member_b @ I2 @ I3 )
=> ( ( member_b_a @ F @ ( pi_b_a @ I3 @ ( fun_upd_b_set_a @ B2 @ I2 @ A2 ) ) )
= ( ( member_b_a @ F @ ( pi_b_a @ ( minus_minus_set_b @ I3 @ ( insert_b @ I2 @ bot_bot_set_b ) ) @ B2 ) )
& ( member_a @ ( F @ I2 ) @ A2 ) ) ) ) ).
% Pi_fupd_iff
thf(fact_1232_Pi__fupd__iff,axiom,
! [I2: a,I3: set_a,F: a > b,B2: a > set_b,A2: set_b] :
( ( member_a @ I2 @ I3 )
=> ( ( member_a_b @ F @ ( pi_a_b @ I3 @ ( fun_upd_a_set_b @ B2 @ I2 @ A2 ) ) )
= ( ( member_a_b @ F @ ( pi_a_b @ ( minus_minus_set_a @ I3 @ ( insert_a @ I2 @ bot_bot_set_a ) ) @ B2 ) )
& ( member_b @ ( F @ I2 ) @ A2 ) ) ) ) ).
% Pi_fupd_iff
thf(fact_1233_Pi__fupd__iff,axiom,
! [I2: a,I3: set_a,F: a > a,B2: a > set_a,A2: set_a] :
( ( member_a @ I2 @ I3 )
=> ( ( member_a_a @ F @ ( pi_a_a @ I3 @ ( fun_upd_a_set_a @ B2 @ I2 @ A2 ) ) )
= ( ( member_a_a @ F @ ( pi_a_a @ ( minus_minus_set_a @ I3 @ ( insert_a @ I2 @ bot_bot_set_a ) ) @ B2 ) )
& ( member_a @ ( F @ I2 ) @ A2 ) ) ) ) ).
% Pi_fupd_iff
thf(fact_1234_Pi__fupd__iff,axiom,
! [I2: a,I3: set_a,F: a > nat,B2: a > set_nat,A2: set_nat] :
( ( member_a @ I2 @ I3 )
=> ( ( member_a_nat @ F @ ( pi_a_nat @ I3 @ ( fun_upd_a_set_nat @ B2 @ I2 @ A2 ) ) )
= ( ( member_a_nat @ F @ ( pi_a_nat @ ( minus_minus_set_a @ I3 @ ( insert_a @ I2 @ bot_bot_set_a ) ) @ B2 ) )
& ( member_nat @ ( F @ I2 ) @ A2 ) ) ) ) ).
% Pi_fupd_iff
thf(fact_1235_Pi__fupd__iff,axiom,
! [I2: nat,I3: set_nat,F: nat > b,B2: nat > set_b,A2: set_b] :
( ( member_nat @ I2 @ I3 )
=> ( ( member_nat_b @ F @ ( pi_nat_b @ I3 @ ( fun_upd_nat_set_b @ B2 @ I2 @ A2 ) ) )
= ( ( member_nat_b @ F @ ( pi_nat_b @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) @ B2 ) )
& ( member_b @ ( F @ I2 ) @ A2 ) ) ) ) ).
% Pi_fupd_iff
thf(fact_1236_Pi__fupd__iff,axiom,
! [I2: nat,I3: set_nat,F: nat > a,B2: nat > set_a,A2: set_a] :
( ( member_nat @ I2 @ I3 )
=> ( ( member_nat_a @ F @ ( pi_nat_a @ I3 @ ( fun_upd_nat_set_a @ B2 @ I2 @ A2 ) ) )
= ( ( member_nat_a @ F @ ( pi_nat_a @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) @ B2 ) )
& ( member_a @ ( F @ I2 ) @ A2 ) ) ) ) ).
% Pi_fupd_iff
thf(fact_1237_Pi__fupd__iff,axiom,
! [I2: nat,I3: set_nat,F: nat > nat,B2: nat > set_nat,A2: set_nat] :
( ( member_nat @ I2 @ I3 )
=> ( ( member_nat_nat @ F @ ( pi_nat_nat @ I3 @ ( fun_upd_nat_set_nat @ B2 @ I2 @ A2 ) ) )
= ( ( member_nat_nat @ F @ ( pi_nat_nat @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) @ B2 ) )
& ( member_nat @ ( F @ I2 ) @ A2 ) ) ) ) ).
% Pi_fupd_iff
thf(fact_1238_Pi__fupd__iff,axiom,
! [I2: $o,I3: set_o,F: $o > b,B2: $o > set_b,A2: set_b] :
( ( member_o @ I2 @ I3 )
=> ( ( member_o_b @ F @ ( pi_o_b @ I3 @ ( fun_upd_o_set_b @ B2 @ I2 @ A2 ) ) )
= ( ( member_o_b @ F @ ( pi_o_b @ ( minus_minus_set_o @ I3 @ ( insert_o @ I2 @ bot_bot_set_o ) ) @ B2 ) )
& ( member_b @ ( F @ I2 ) @ A2 ) ) ) ) ).
% Pi_fupd_iff
thf(fact_1239_extensional__funcset__fun__upd__extends__rangeI,axiom,
! [A: b,T2: set_b,F: b > b,S: set_b,X: b] :
( ( member_b @ A @ T2 )
=> ( ( member_b_b @ F
@ ( piE_b_b @ S
@ ^ [I: b] : ( minus_minus_set_b @ T2 @ ( insert_b @ A @ bot_bot_set_b ) ) ) )
=> ( member_b_b @ ( fun_upd_b_b @ F @ X @ A )
@ ( piE_b_b @ ( insert_b @ X @ S )
@ ^ [I: b] : T2 ) ) ) ) ).
% extensional_funcset_fun_upd_extends_rangeI
thf(fact_1240_extensional__funcset__fun__upd__extends__rangeI,axiom,
! [A: b,T2: set_b,F: $o > b,S: set_o,X: $o] :
( ( member_b @ A @ T2 )
=> ( ( member_o_b @ F
@ ( piE_o_b @ S
@ ^ [I: $o] : ( minus_minus_set_b @ T2 @ ( insert_b @ A @ bot_bot_set_b ) ) ) )
=> ( member_o_b @ ( fun_upd_o_b @ F @ X @ A )
@ ( piE_o_b @ ( insert_o @ X @ S )
@ ^ [I: $o] : T2 ) ) ) ) ).
% extensional_funcset_fun_upd_extends_rangeI
thf(fact_1241_extensional__funcset__fun__upd__extends__rangeI,axiom,
! [A: b > a,T2: set_b_a,F: b > b > a,S: set_b,X: b] :
( ( member_b_a @ A @ T2 )
=> ( ( member_b_b_a @ F
@ ( piE_b_b_a @ S
@ ^ [I: b] : ( minus_minus_set_b_a @ T2 @ ( insert_b_a @ A @ bot_bot_set_b_a ) ) ) )
=> ( member_b_b_a @ ( fun_upd_b_b_a @ F @ X @ A )
@ ( piE_b_b_a @ ( insert_b @ X @ S )
@ ^ [I: b] : T2 ) ) ) ) ).
% extensional_funcset_fun_upd_extends_rangeI
thf(fact_1242_extensional__funcset__fun__upd__extends__rangeI,axiom,
! [A: b > a,T2: set_b_a,F: $o > b > a,S: set_o,X: $o] :
( ( member_b_a @ A @ T2 )
=> ( ( member_o_b_a @ F
@ ( piE_o_b_a @ S
@ ^ [I: $o] : ( minus_minus_set_b_a @ T2 @ ( insert_b_a @ A @ bot_bot_set_b_a ) ) ) )
=> ( member_o_b_a @ ( fun_upd_o_b_a @ F @ X @ A )
@ ( piE_o_b_a @ ( insert_o @ X @ S )
@ ^ [I: $o] : T2 ) ) ) ) ).
% extensional_funcset_fun_upd_extends_rangeI
thf(fact_1243_extensional__funcset__fun__upd__extends__rangeI,axiom,
! [A: a,T2: set_a,F: $o > a,S: set_o,X: $o] :
( ( member_a @ A @ T2 )
=> ( ( member_o_a @ F
@ ( piE_o_a @ S
@ ^ [I: $o] : ( minus_minus_set_a @ T2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) )
=> ( member_o_a @ ( fun_upd_o_a @ F @ X @ A )
@ ( piE_o_a @ ( insert_o @ X @ S )
@ ^ [I: $o] : T2 ) ) ) ) ).
% extensional_funcset_fun_upd_extends_rangeI
thf(fact_1244_extensional__funcset__fun__upd__extends__rangeI,axiom,
! [A: a,T2: set_a,F: b > a,S: set_b,X: b] :
( ( member_a @ A @ T2 )
=> ( ( member_b_a @ F
@ ( piE_b_a @ S
@ ^ [I: b] : ( minus_minus_set_a @ T2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) )
=> ( member_b_a @ ( fun_upd_b_a @ F @ X @ A )
@ ( piE_b_a @ ( insert_b @ X @ S )
@ ^ [I: b] : T2 ) ) ) ) ).
% extensional_funcset_fun_upd_extends_rangeI
thf(fact_1245_extensional__funcset__fun__upd__extends__rangeI,axiom,
! [A: nat,T2: set_nat,F: b > nat,S: set_b,X: b] :
( ( member_nat @ A @ T2 )
=> ( ( member_b_nat @ F
@ ( piE_b_nat @ S
@ ^ [I: b] : ( minus_minus_set_nat @ T2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) )
=> ( member_b_nat @ ( fun_upd_b_nat @ F @ X @ A )
@ ( piE_b_nat @ ( insert_b @ X @ S )
@ ^ [I: b] : T2 ) ) ) ) ).
% extensional_funcset_fun_upd_extends_rangeI
thf(fact_1246_extensional__funcset__fun__upd__extends__rangeI,axiom,
! [A: nat,T2: set_nat,F: $o > nat,S: set_o,X: $o] :
( ( member_nat @ A @ T2 )
=> ( ( member_o_nat @ F
@ ( piE_o_nat @ S
@ ^ [I: $o] : ( minus_minus_set_nat @ T2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) )
=> ( member_o_nat @ ( fun_upd_o_nat @ F @ X @ A )
@ ( piE_o_nat @ ( insert_o @ X @ S )
@ ^ [I: $o] : T2 ) ) ) ) ).
% extensional_funcset_fun_upd_extends_rangeI
thf(fact_1247_extensional__funcset__fun__upd__extends__rangeI,axiom,
! [A: $o,T2: set_o,F: b > $o,S: set_b,X: b] :
( ( member_o @ A @ T2 )
=> ( ( member_b_o @ F
@ ( piE_b_o @ S
@ ^ [I: b] : ( minus_minus_set_o @ T2 @ ( insert_o @ A @ bot_bot_set_o ) ) ) )
=> ( member_b_o @ ( fun_upd_b_o @ F @ X @ A )
@ ( piE_b_o @ ( insert_b @ X @ S )
@ ^ [I: b] : T2 ) ) ) ) ).
% extensional_funcset_fun_upd_extends_rangeI
thf(fact_1248_extensional__funcset__fun__upd__extends__rangeI,axiom,
! [A: $o,T2: set_o,F: $o > $o,S: set_o,X: $o] :
( ( member_o @ A @ T2 )
=> ( ( member_o_o @ F
@ ( piE_o_o @ S
@ ^ [I: $o] : ( minus_minus_set_o @ T2 @ ( insert_o @ A @ bot_bot_set_o ) ) ) )
=> ( member_o_o @ ( fun_upd_o_o @ F @ X @ A )
@ ( piE_o_o @ ( insert_o @ X @ S )
@ ^ [I: $o] : T2 ) ) ) ) ).
% extensional_funcset_fun_upd_extends_rangeI
thf(fact_1249_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_1250_PiE__over__singleton__iff,axiom,
! [A: a,B2: a > set_a] :
( ( piE_a_a @ ( insert_a @ A @ bot_bot_set_a ) @ B2 )
= ( comple6518619711525350638et_a_a
@ ( image_a_set_a_a
@ ^ [B6: a] :
( insert_a_a
@ ( restrict_a_a
@ ^ [X3: a] : B6
@ ( insert_a @ A @ bot_bot_set_a ) )
@ bot_bot_set_a_a )
@ ( B2 @ A ) ) ) ) ).
% PiE_over_singleton_iff
thf(fact_1251_extensional__funcset__fun__upd__inj__onI,axiom,
! [F: nat > nat,S: set_nat,T2: set_nat,A: nat,X: nat] :
( ( member_nat_nat @ F
@ ( piE_nat_nat @ S
@ ^ [I: nat] : ( minus_minus_set_nat @ T2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) )
=> ( ( inj_on_nat_nat @ F @ S )
=> ( inj_on_nat_nat @ ( fun_upd_nat_nat @ F @ X @ A ) @ S ) ) ) ).
% extensional_funcset_fun_upd_inj_onI
thf(fact_1252_extensional__funcset__fun__upd__inj__onI,axiom,
! [F: b > a,S: set_b,T2: set_a,A: a,X: b] :
( ( member_b_a @ F
@ ( piE_b_a @ S
@ ^ [I: b] : ( minus_minus_set_a @ T2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) )
=> ( ( inj_on_b_a @ F @ S )
=> ( inj_on_b_a @ ( fun_upd_b_a @ F @ X @ A ) @ S ) ) ) ).
% extensional_funcset_fun_upd_inj_onI
thf(fact_1253_Func__empty,axiom,
! [B2: set_a] :
( ( bNF_We3495405677706807802nc_o_a @ bot_bot_set_o @ B2 )
= ( insert_o_a
@ ^ [X3: $o] : undefined_a
@ bot_bot_set_o_a ) ) ).
% Func_empty
thf(fact_1254_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ N @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_1255_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_1256_diff__diff__cancel,axiom,
! [I2: nat,N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_1257_diff__commute,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_1258_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_1259_le__trans,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I2 @ K2 ) ) ) ).
% le_trans
thf(fact_1260_eq__imp__le,axiom,
! [M3: nat,N2: nat] :
( ( M3 = N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% eq_imp_le
thf(fact_1261_le__antisym,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M3 )
=> ( M3 = N2 ) ) ) ).
% le_antisym
thf(fact_1262_nat__le__linear,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
| ( ord_less_eq_nat @ N2 @ M3 ) ) ).
% nat_le_linear
thf(fact_1263_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1264_diff__le__mono2,axiom,
! [M3: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_1265_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_1266_diff__le__self,axiom,
! [M3: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N2 ) @ M3 ) ).
% diff_le_self
thf(fact_1267_diff__le__mono,axiom,
! [M3: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1268_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( minus_minus_nat @ M3 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1269_le__diff__iff,axiom,
! [K2: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1270_eq__diff__iff,axiom,
! [K2: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ( minus_minus_nat @ M3 @ K2 )
= ( minus_minus_nat @ N2 @ K2 ) )
= ( M3 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1271_inj__on__diff__nat,axiom,
! [N3: set_nat,K2: nat] :
( ! [N4: nat] :
( ( member_nat @ N4 @ N3 )
=> ( ord_less_eq_nat @ K2 @ N4 ) )
=> ( inj_on_nat_nat
@ ^ [N: nat] : ( minus_minus_nat @ N @ K2 )
@ N3 ) ) ).
% inj_on_diff_nat
thf(fact_1272_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1273_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M4: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_eq_nat @ X3 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1274_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1275_UNIV__bool,axiom,
( top_top_set_o
= ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% UNIV_bool
% Helper facts (9)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_I_Eo_J_T,axiom,
! [X: set_o,Y: set_o] :
( ( if_set_o @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_I_Eo_J_T,axiom,
! [X: set_o,Y: set_o] :
( ( if_set_o @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__b_J_T,axiom,
! [X: set_b,Y: set_b] :
( ( if_set_b @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__b_J_T,axiom,
! [X: set_b,Y: set_b] :
( ( if_set_b @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y: set_nat] :
( ( if_set_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y: set_nat] :
( ( if_set_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( commut5005951359559292711mp_a_b @ m @ composition @ unit @ f @ ( insert_b @ a2 @ aa ) )
= unit ) ).
%------------------------------------------------------------------------------