TPTP Problem File: SLH0543^1.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 : Eval_FO/0005_Ailamazyan/prob_03477_145843__15969928_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1757 ( 707 unt; 475 typ; 0 def)
% Number of atoms : 3858 (1809 equ; 0 cnn)
% Maximal formula atoms : 46 ( 3 avg)
% Number of connectives : 13649 ( 819 ~; 43 |; 221 &;10705 @)
% ( 0 <=>;1861 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 7 avg)
% Number of types : 85 ( 84 usr)
% Number of type conns : 2428 (2428 >; 0 *; 0 +; 0 <<)
% Number of symbols : 394 ( 391 usr; 19 con; 0-6 aty)
% Number of variables : 4883 ( 133 ^;4649 !; 101 ?;4883 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:12:12.205
%------------------------------------------------------------------------------
% Could-be-implicit typings (84)
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(ty_n_t__FO__Ofo____fmla_Itf__a_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
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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__FO__Ofo____term_It__Nat__Onat_J,type,
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thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__FO__Ofo____term_Itf__b_J,type,
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thf(ty_n_t__FO__Ofo____term_Itf__a_J,type,
fo_term_a: $tType ).
thf(ty_n_t__List__Olist_Itf__b_J,type,
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thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_tf__b,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (391)
thf(sy_c_Ailamazyan_OSP_001tf__a_001tf__b,type,
sP_a_b: fo_fmla_a_b > set_nat ).
thf(sy_c_Ailamazyan_OSP__list_001tf__a_001tf__b,type,
sP_list_a_b: fo_fmla_a_b > list_nat ).
thf(sy_c_Ailamazyan_OSP__rel_001tf__a_001tf__b,type,
sP_rel_a_b: fo_fmla_a_b > fo_fmla_a_b > $o ).
thf(sy_c_Ailamazyan_Oact__edom_001tf__a_001tf__b,type,
act_edom_a_b: fo_fmla_a_b > ( product_prod_b_nat > set_list_a ) > set_a ).
thf(sy_c_Ailamazyan_Oad__terms_001tf__a,type,
ad_terms_a: list_fo_term_a > set_a ).
thf(sy_c_Ailamazyan_Oesat_001tf__a_001tf__b,type,
esat_a_b: fo_fmla_a_b > ( product_prod_b_nat > set_list_a ) > ( nat > sum_sum_a_nat ) > set_Sum_sum_a_nat > $o ).
thf(sy_c_Ailamazyan_Oeval__eterm_001tf__a_001t__Nat__Onat,type,
eval_eterm_a_nat: ( nat > sum_sum_a_nat ) > fo_term_a > sum_sum_a_nat ).
thf(sy_c_Ailamazyan_Oeval__eterms_001tf__a_001t__Nat__Onat,type,
eval_eterms_a_nat: ( nat > sum_sum_a_nat ) > list_fo_term_a > list_Sum_sum_a_nat ).
thf(sy_c_Ailamazyan_Osp__equiv__pair_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
sp_equ1372826846814931683_a_nat: produc4672180596006801056_a_nat > produc4672180596006801056_a_nat > $o ).
thf(sy_c_Ailamazyan_Osp__equiv__pair_001_062_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_001_062_It__Nat__Onat_Mtf__a_J,type,
sp_equ547821432705481458_nat_a: produc5835360497134304175_nat_a > produc5835360497134304175_nat_a > $o ).
thf(sy_c_Ailamazyan_Osp__equiv__pair_001_062_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
sp_equ4301142204835971757_a_nat: produc5433867242478491114_a_nat > produc5433867242478491114_a_nat > $o ).
thf(sy_c_Ailamazyan_Osp__equiv__pair_001t__Nat__Onat_001t__Nat__Onat,type,
sp_equ885134019442699404at_nat: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_Ailamazyan_Osp__equiv__pair_001tf__b_001t__Nat__Onat,type,
sp_equiv_pair_b_nat: product_prod_b_nat > product_prod_b_nat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
comple2307003609928055243_set_a: set_set_a > set_a ).
thf(sy_c_FO_Ofo__fmla_OBool_001t__Nat__Onat_001t__Nat__Onat,type,
fo_Bool_nat_nat: $o > fo_fmla_nat_nat ).
thf(sy_c_FO_Ofo__fmla_OBool_001tf__a_001tf__b,type,
fo_Bool_a_b: $o > fo_fmla_a_b ).
thf(sy_c_FO_Ofo__fmla_OConj_001t__Nat__Onat_001t__Nat__Onat,type,
fo_Conj_nat_nat: fo_fmla_nat_nat > fo_fmla_nat_nat > fo_fmla_nat_nat ).
thf(sy_c_FO_Ofo__fmla_OConj_001tf__a_001tf__b,type,
fo_Conj_a_b: fo_fmla_a_b > fo_fmla_a_b > fo_fmla_a_b ).
thf(sy_c_FO_Ofo__fmla_ODisj_001t__Nat__Onat_001t__Nat__Onat,type,
fo_Disj_nat_nat: fo_fmla_nat_nat > fo_fmla_nat_nat > fo_fmla_nat_nat ).
thf(sy_c_FO_Ofo__fmla_ODisj_001tf__a_001tf__b,type,
fo_Disj_a_b: fo_fmla_a_b > fo_fmla_a_b > fo_fmla_a_b ).
thf(sy_c_FO_Ofo__fmla_OEqa_001t__Nat__Onat_001t__Nat__Onat,type,
fo_Eqa_nat_nat: fo_term_nat > fo_term_nat > fo_fmla_nat_nat ).
thf(sy_c_FO_Ofo__fmla_OEqa_001tf__a_001tf__b,type,
fo_Eqa_a_b: fo_term_a > fo_term_a > fo_fmla_a_b ).
thf(sy_c_FO_Ofo__fmla_OExists_001t__Nat__Onat_001t__Nat__Onat,type,
fo_Exists_nat_nat: nat > fo_fmla_nat_nat > fo_fmla_nat_nat ).
thf(sy_c_FO_Ofo__fmla_OExists_001tf__a_001tf__b,type,
fo_Exists_a_b: nat > fo_fmla_a_b > fo_fmla_a_b ).
thf(sy_c_FO_Ofo__fmla_OForall_001t__Nat__Onat_001t__Nat__Onat,type,
fo_Forall_nat_nat: nat > fo_fmla_nat_nat > fo_fmla_nat_nat ).
thf(sy_c_FO_Ofo__fmla_OForall_001tf__a_001tf__b,type,
fo_Forall_a_b: nat > fo_fmla_a_b > fo_fmla_a_b ).
thf(sy_c_FO_Ofo__fmla_ONeg_001t__Nat__Onat_001t__Nat__Onat,type,
fo_Neg_nat_nat: fo_fmla_nat_nat > fo_fmla_nat_nat ).
thf(sy_c_FO_Ofo__fmla_ONeg_001tf__a_001tf__b,type,
fo_Neg_a_b: fo_fmla_a_b > fo_fmla_a_b ).
thf(sy_c_FO_Ofo__fmla_OPred_001t__Nat__Onat_001t__Nat__Onat,type,
fo_Pred_nat_nat: nat > list_fo_term_nat > fo_fmla_nat_nat ).
thf(sy_c_FO_Ofo__fmla_OPred_001tf__b_001tf__a,type,
fo_Pred_b_a: b > list_fo_term_a > fo_fmla_a_b ).
thf(sy_c_FO_Ofo__fmla_Orel__fo__fmla_001t__Nat__Onat_001t__Nat__Onat_001t__FO__Ofo____term_Itf__a_J_001t__FO__Ofo____term_Itf__a_J,type,
fo_rel3437815733934627446term_a: ( nat > nat > $o ) > ( fo_term_a > fo_term_a > $o ) > fo_fml4923465511750459541term_a > fo_fml4923465511750459541term_a > $o ).
thf(sy_c_FO_Ofo__fmla_Orel__fo__fmla_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
fo_rel4206018205878794776at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > fo_fmla_nat_nat > fo_fmla_nat_nat > $o ).
thf(sy_c_FO_Ofo__fmla_Orel__fo__fmla_001t__Nat__Onat_001t__Nat__Onat_001tf__a_001tf__a,type,
fo_rel2011848245745679222at_a_a: ( nat > nat > $o ) > ( a > a > $o ) > fo_fmla_nat_a > fo_fmla_nat_a > $o ).
thf(sy_c_FO_Ofo__fmla_Orel__fo__fmla_001t__Nat__Onat_001t__Nat__Onat_001tf__a_001tf__b,type,
fo_rel2011848245745679223at_a_b: ( nat > nat > $o ) > ( a > b > $o ) > fo_fmla_nat_a > fo_fmla_nat_b > $o ).
thf(sy_c_FO_Ofo__fmla_Orel__fo__fmla_001t__Nat__Onat_001t__Nat__Onat_001tf__b_001tf__a,type,
fo_rel8447764700627778997at_b_a: ( nat > nat > $o ) > ( b > a > $o ) > fo_fmla_nat_b > fo_fmla_nat_a > $o ).
thf(sy_c_FO_Ofo__fmla_Orel__fo__fmla_001t__Nat__Onat_001t__Nat__Onat_001tf__b_001tf__b,type,
fo_rel8447764700627778998at_b_b: ( nat > nat > $o ) > ( b > b > $o ) > fo_fmla_nat_b > fo_fmla_nat_b > $o ).
thf(sy_c_FO_Ofo__fmla_Orel__fo__fmla_001t__Nat__Onat_001tf__b_001tf__a_001t__Nat__Onat,type,
fo_rel8143554221338701365_a_nat: ( nat > b > $o ) > ( a > nat > $o ) > fo_fmla_nat_a > fo_fmla_b_nat > $o ).
thf(sy_c_FO_Ofo__fmla_Orel__fo__fmla_001t__Nat__Onat_001tf__b_001tf__a_001tf__b,type,
fo_rel861035770506876890_b_a_b: ( nat > b > $o ) > ( a > b > $o ) > fo_fmla_nat_a > fo_fmla_b_b > $o ).
thf(sy_c_FO_Ofo__fmla_Orel__fo__fmla_001t__Nat__Onat_001tf__b_001tf__b_001t__Nat__Onat,type,
fo_rel155626513440320054_b_nat: ( nat > b > $o ) > ( b > nat > $o ) > fo_fmla_nat_b > fo_fmla_b_nat > $o ).
thf(sy_c_FO_Ofo__fmla_Orel__fo__fmla_001t__Nat__Onat_001tf__b_001tf__b_001tf__a,type,
fo_rel7296952225388976664_b_b_a: ( nat > b > $o ) > ( b > a > $o ) > fo_fmla_nat_b > fo_fmla_b_a > $o ).
thf(sy_c_FO_Ofo__fmla_Orel__fo__fmla_001t__Nat__Onat_001tf__b_001tf__b_001tf__b,type,
fo_rel7296952225388976665_b_b_b: ( nat > b > $o ) > ( b > b > $o ) > fo_fmla_nat_b > fo_fmla_b_b > $o ).
thf(sy_c_FO_Ofo__fmla_Orel__fo__fmla_001tf__a_001tf__a_001t__Nat__Onat_001t__Nat__Onat,type,
fo_rel5500385131963405430at_nat: ( a > a > $o ) > ( nat > nat > $o ) > fo_fmla_a_nat > fo_fmla_a_nat > $o ).
thf(sy_c_FO_Ofo__fmla_Orel__fo__fmla_001tf__a_001tf__a_001tf__a_001tf__a,type,
fo_rel1667748303170485972_a_a_a: ( a > a > $o ) > ( a > a > $o ) > fo_fmla_a_a > fo_fmla_a_a > $o ).
thf(sy_c_FO_Ofo__fmla_Orel__fo__fmla_001tf__a_001tf__a_001tf__b_001tf__b,type,
fo_rel8103664758052585748_a_b_b: ( a > a > $o ) > ( b > b > $o ) > fo_fmla_a_b > fo_fmla_a_b > $o ).
thf(sy_c_FO_Ofo__fmla_Orel__fo__fmla_001tf__b_001tf__b_001t__Nat__Onat_001t__Nat__Onat,type,
fo_rel2016979423744071350at_nat: ( b > b > $o ) > ( nat > nat > $o ) > fo_fmla_b_nat > fo_fmla_b_nat > $o ).
thf(sy_c_FO_Ofo__fmla_Orel__fo__fmla_001tf__b_001tf__b_001tf__a_001tf__a,type,
fo_rel8751928955653415188_b_a_a: ( b > b > $o ) > ( a > a > $o ) > fo_fmla_b_a > fo_fmla_b_a > $o ).
thf(sy_c_FO_Ofo__fmla_Orel__fo__fmla_001tf__b_001tf__b_001tf__b_001tf__b,type,
fo_rel5964473373680739156_b_b_b: ( b > b > $o ) > ( b > b > $o ) > fo_fmla_b_b > fo_fmla_b_b > $o ).
thf(sy_c_FO_Ofo__fmla_Oset1__fo__fmla_001t__Nat__Onat_001t__FO__Ofo____term_Itf__a_J,type,
fo_set1740353235476592200term_a: fo_fml4923465511750459541term_a > set_nat ).
thf(sy_c_FO_Ofo__fmla_Oset1__fo__fmla_001t__Nat__Onat_001t__Nat__Onat,type,
fo_set7155282118041507904at_nat: fo_fmla_nat_nat > set_nat ).
thf(sy_c_FO_Ofo__fmla_Oset1__fo__fmla_001t__Nat__Onat_001tf__a,type,
fo_set3273380792390310542_nat_a: fo_fmla_nat_a > set_nat ).
thf(sy_c_FO_Ofo__fmla_Oset1__fo__fmla_001t__Nat__Onat_001tf__b,type,
fo_set3273380792390310543_nat_b: fo_fmla_nat_b > set_nat ).
thf(sy_c_FO_Ofo__fmla_Oset1__fo__fmla_001tf__a_001t__Nat__Onat,type,
fo_set8464821328066799920_a_nat: fo_fmla_a_nat > set_a ).
thf(sy_c_FO_Ofo__fmla_Oset1__fo__fmla_001tf__a_001tf__a,type,
fo_set1_fo_fmla_a_a: fo_fmla_a_a > set_a ).
thf(sy_c_FO_Ofo__fmla_Oset1__fo__fmla_001tf__a_001tf__b,type,
fo_set1_fo_fmla_a_b: fo_fmla_a_b > set_a ).
thf(sy_c_FO_Ofo__fmla_Oset1__fo__fmla_001tf__b_001t__Nat__Onat,type,
fo_set476893620168418609_b_nat: fo_fmla_b_nat > set_b ).
thf(sy_c_FO_Ofo__fmla_Oset1__fo__fmla_001tf__b_001tf__a,type,
fo_set1_fo_fmla_b_a: fo_fmla_b_a > set_b ).
thf(sy_c_FO_Ofo__fmla_Oset1__fo__fmla_001tf__b_001tf__b,type,
fo_set1_fo_fmla_b_b: fo_fmla_b_b > set_b ).
thf(sy_c_FO_Ofo__fmla_Oset2__fo__fmla_001t__Nat__Onat_001t__FO__Ofo____term_Itf__a_J,type,
fo_set7640127306678011015term_a: fo_fml4923465511750459541term_a > set_fo_term_a ).
thf(sy_c_FO_Ofo__fmla_Oset2__fo__fmla_001t__Nat__Onat_001t__Nat__Onat,type,
fo_set4065224739847495105at_nat: fo_fmla_nat_nat > set_nat ).
thf(sy_c_FO_Ofo__fmla_Oset2__fo__fmla_001t__Nat__Onat_001tf__a,type,
fo_set7690413055645365069_nat_a: fo_fmla_nat_a > set_a ).
thf(sy_c_FO_Ofo__fmla_Oset2__fo__fmla_001t__Nat__Onat_001tf__b,type,
fo_set7690413055645365070_nat_b: fo_fmla_nat_b > set_b ).
thf(sy_c_FO_Ofo__fmla_Oset2__fo__fmla_001tf__a_001t__Nat__Onat,type,
fo_set3658481554467078639_a_nat: fo_fmla_a_nat > set_nat ).
thf(sy_c_FO_Ofo__fmla_Oset2__fo__fmla_001tf__a_001tf__a,type,
fo_set2_fo_fmla_a_a: fo_fmla_a_a > set_a ).
thf(sy_c_FO_Ofo__fmla_Oset2__fo__fmla_001tf__a_001tf__b,type,
fo_set2_fo_fmla_a_b: fo_fmla_a_b > set_b ).
thf(sy_c_FO_Ofo__fmla_Oset2__fo__fmla_001tf__b_001t__Nat__Onat,type,
fo_set4893925883423473136_b_nat: fo_fmla_b_nat > set_nat ).
thf(sy_c_FO_Ofo__fmla_Oset2__fo__fmla_001tf__b_001tf__a,type,
fo_set2_fo_fmla_b_a: fo_fmla_b_a > set_a ).
thf(sy_c_FO_Ofo__fmla_Oset2__fo__fmla_001tf__b_001tf__b,type,
fo_set2_fo_fmla_b_b: fo_fmla_b_b > set_b ).
thf(sy_c_FO_Ofo__term_OConst_001t__FO__Ofo____term_Itf__a_J,type,
fo_Const_fo_term_a: fo_term_a > fo_term_fo_term_a ).
thf(sy_c_FO_Ofo__term_OConst_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
fo_Con7354235876602600827_a_nat: list_Sum_sum_a_nat > fo_ter518426593632715199_a_nat ).
thf(sy_c_FO_Ofo__term_OConst_001t__Nat__Onat,type,
fo_Const_nat: nat > fo_term_nat ).
thf(sy_c_FO_Ofo__term_OConst_001t__Set__Oset_It__Nat__Onat_J,type,
fo_Const_set_nat: set_nat > fo_term_set_nat ).
thf(sy_c_FO_Ofo__term_OConst_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
fo_Con399356768421508341_a_nat: sum_sum_a_nat > fo_ter3269878860578495673_a_nat ).
thf(sy_c_FO_Ofo__term_OConst_001tf__a,type,
fo_Const_a: a > fo_term_a ).
thf(sy_c_FO_Ofo__term_OConst_001tf__b,type,
fo_Const_b: b > fo_term_b ).
thf(sy_c_FO_Ofo__term_OVar_001t__Nat__Onat,type,
fo_Var_nat: nat > fo_term_nat ).
thf(sy_c_FO_Ofo__term_OVar_001tf__a,type,
fo_Var_a: nat > fo_term_a ).
thf(sy_c_FO_Ofo__term_OVar_001tf__b,type,
fo_Var_b: nat > fo_term_b ).
thf(sy_c_FO_Ofo__term_Orel__fo__term_001t__FO__Ofo____term_Itf__a_J_001t__FO__Ofo____term_Itf__a_J,type,
fo_rel5005517413957816468term_a: ( fo_term_a > fo_term_a > $o ) > fo_term_fo_term_a > fo_term_fo_term_a > $o ).
thf(sy_c_FO_Ofo__term_Orel__fo__term_001t__FO__Ofo____term_Itf__a_J_001tf__b,type,
fo_rel8200119315616323291rm_a_b: ( fo_term_a > b > $o ) > fo_term_fo_term_a > fo_term_b > $o ).
thf(sy_c_FO_Ofo__term_Orel__fo__term_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
fo_rel6364512094553371476_a_nat: ( list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o ) > fo_ter518426593632715199_a_nat > fo_ter518426593632715199_a_nat > $o ).
thf(sy_c_FO_Ofo__term_Orel__fo__term_001t__Nat__Onat_001t__Nat__Onat,type,
fo_rel1533169934621585718at_nat: ( nat > nat > $o ) > fo_term_nat > fo_term_nat > $o ).
thf(sy_c_FO_Ofo__term_Orel__fo__term_001t__Nat__Onat_001tf__a,type,
fo_rel_fo_term_nat_a: ( nat > a > $o ) > fo_term_nat > fo_term_a > $o ).
thf(sy_c_FO_Ofo__term_Orel__fo__term_001t__Nat__Onat_001tf__b,type,
fo_rel_fo_term_nat_b: ( nat > b > $o ) > fo_term_nat > fo_term_b > $o ).
thf(sy_c_FO_Ofo__term_Orel__fo__term_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
fo_rel2202147410123713698et_nat: ( set_nat > set_nat > $o ) > fo_term_set_nat > fo_term_set_nat > $o ).
thf(sy_c_FO_Ofo__term_Orel__fo__term_001tf__a_001t__Nat__Onat,type,
fo_rel_fo_term_a_nat: ( a > nat > $o ) > fo_term_a > fo_term_nat > $o ).
thf(sy_c_FO_Ofo__term_Orel__fo__term_001tf__a_001tf__a,type,
fo_rel_fo_term_a_a: ( a > a > $o ) > fo_term_a > fo_term_a > $o ).
thf(sy_c_FO_Ofo__term_Orel__fo__term_001tf__a_001tf__b,type,
fo_rel_fo_term_a_b: ( a > b > $o ) > fo_term_a > fo_term_b > $o ).
thf(sy_c_FO_Ofo__term_Orel__fo__term_001tf__b_001t__Nat__Onat,type,
fo_rel_fo_term_b_nat: ( b > nat > $o ) > fo_term_b > fo_term_nat > $o ).
thf(sy_c_FO_Ofo__term_Orel__fo__term_001tf__b_001tf__a,type,
fo_rel_fo_term_b_a: ( b > a > $o ) > fo_term_b > fo_term_a > $o ).
thf(sy_c_FO_Ofo__term_Orel__fo__term_001tf__b_001tf__b,type,
fo_rel_fo_term_b_b: ( b > b > $o ) > fo_term_b > fo_term_b > $o ).
thf(sy_c_FO_Ofo__term_Oset__fo__term_001t__FO__Ofo____term_Itf__a_J,type,
fo_set9081760999098759482term_a: fo_term_fo_term_a > set_fo_term_a ).
thf(sy_c_FO_Ofo__term_Oset__fo__term_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
fo_set5182096037969159431_a_nat: fo_ter518426593632715199_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_FO_Ofo__term_Oset__fo__term_001t__Nat__Onat,type,
fo_set_fo_term_nat: fo_term_nat > set_nat ).
thf(sy_c_FO_Ofo__term_Oset__fo__term_001t__Set__Oset_It__Nat__Onat_J,type,
fo_set3263689133970341252et_nat: fo_term_set_nat > set_set_nat ).
thf(sy_c_FO_Ofo__term_Oset__fo__term_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
fo_set5789020685998436993_a_nat: fo_ter3269878860578495673_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_FO_Ofo__term_Oset__fo__term_001tf__a,type,
fo_set_fo_term_a: fo_term_a > set_a ).
thf(sy_c_FO_Ofo__term_Oset__fo__term_001tf__b,type,
fo_set_fo_term_b: fo_term_b > set_b ).
thf(sy_c_FO_Ofv__fo__fmla_001tf__a_001tf__b,type,
fv_fo_fmla_a_b: fo_fmla_a_b > set_nat ).
thf(sy_c_FO_Ofv__fo__fmla__list_001tf__a_001tf__b,type,
fv_fo_fmla_list_a_b: fo_fmla_a_b > list_nat ).
thf(sy_c_FO_Ofv__fo__fmla__rel_001tf__a_001tf__b,type,
fv_fo_fmla_rel_a_b: fo_fmla_a_b > fo_fmla_a_b > $o ).
thf(sy_c_FO_Ofv__fo__term__set_001t__Nat__Onat,type,
fv_fo_term_set_nat: fo_term_nat > set_nat ).
thf(sy_c_FO_Ofv__fo__term__set_001tf__a,type,
fv_fo_term_set_a: fo_term_a > set_nat ).
thf(sy_c_FO_Ofv__fo__term__set__rel_001t__Nat__Onat,type,
fv_fo_5930752286079999714el_nat: fo_term_nat > fo_term_nat > $o ).
thf(sy_c_FO_Ofv__fo__term__set__rel_001tf__a,type,
fv_fo_term_set_rel_a: fo_term_a > fo_term_a > $o ).
thf(sy_c_FO_Ofv__fo__terms__list_001tf__a,type,
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thf(sy_c_FO_Ofv__fo__terms__set_001tf__a,type,
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thf(sy_c_FO_Olist__fo__term_001t__Nat__Onat,type,
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thf(sy_c_FO_Olist__fo__term_001tf__a,type,
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thf(sy_c_FO_Owf__fo__intp_001tf__a_001tf__b,type,
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thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_member_001t__Nat__Onat,type,
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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_I,type,
i: product_prod_b_nat > set_list_a ).
thf(sy_v_X,type,
x: set_Sum_sum_a_nat ).
thf(sy_v__092_060phi_062,type,
phi: fo_fmla_a_b ).
thf(sy_v__092_060sigma_062,type,
sigma: nat > sum_sum_a_nat ).
thf(sy_v_n,type,
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thf(sy_v_x____,type,
x2: sum_sum_a_nat ).
% Relevant facts (1278)
thf(fact_0_assms_I1_J,axiom,
~ ( member_nat @ n @ ( fv_fo_fmla_a_b @ phi ) ) ).
% assms(1)
thf(fact_1__092_060open_062esat_A_092_060phi_062_AI_A_I_092_060sigma_062_In_A_058_061_Ax_J_J_AX_092_060close_062,axiom,
esat_a_b @ phi @ i @ ( fun_up180537416982607344_a_nat @ sigma @ n @ x2 ) @ x ).
% \<open>esat \<phi> I (\<sigma>(n := x)) X\<close>
thf(fact_2__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_O_Aesat_A_092_060phi_062_AI_A_I_092_060sigma_062_In_A_058_061_Ax_J_J_AX_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X: sum_sum_a_nat] :
~ ( esat_a_b @ phi @ i @ ( fun_up180537416982607344_a_nat @ sigma @ n @ X ) @ x ) ).
% \<open>\<And>thesis. (\<And>x. esat \<phi> I (\<sigma>(n := x)) X \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_3_assms_I2_J,axiom,
esat_a_b @ ( fo_Exists_a_b @ n @ phi ) @ i @ sigma @ x ).
% assms(2)
thf(fact_4_esat__fv__cong,axiom,
! [Phi: fo_fmla_a_b,Sigma: nat > sum_sum_a_nat,Sigma2: nat > sum_sum_a_nat,I: product_prod_b_nat > set_list_a,X2: set_Sum_sum_a_nat] :
( ! [N: nat] :
( ( member_nat @ N @ ( fv_fo_fmla_a_b @ Phi ) )
=> ( ( Sigma @ N )
= ( Sigma2 @ N ) ) )
=> ( ( esat_a_b @ Phi @ I @ Sigma @ X2 )
= ( esat_a_b @ Phi @ I @ Sigma2 @ X2 ) ) ) ).
% esat_fv_cong
thf(fact_5_esat_Osimps_I5_J,axiom,
! [Phi: fo_fmla_a_b,Psi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > sum_sum_a_nat,X2: set_Sum_sum_a_nat] :
( ( esat_a_b @ ( fo_Conj_a_b @ Phi @ Psi ) @ I @ Sigma @ X2 )
= ( ( esat_a_b @ Phi @ I @ Sigma @ X2 )
& ( esat_a_b @ Psi @ I @ Sigma @ X2 ) ) ) ).
% esat.simps(5)
thf(fact_6_esat_Osimps_I6_J,axiom,
! [Phi: fo_fmla_a_b,Psi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > sum_sum_a_nat,X2: set_Sum_sum_a_nat] :
( ( esat_a_b @ ( fo_Disj_a_b @ Phi @ Psi ) @ I @ Sigma @ X2 )
= ( ( esat_a_b @ Phi @ I @ Sigma @ X2 )
| ( esat_a_b @ Psi @ I @ Sigma @ X2 ) ) ) ).
% esat.simps(6)
thf(fact_7_esat_Osimps_I4_J,axiom,
! [Phi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > sum_sum_a_nat,X2: set_Sum_sum_a_nat] :
( ( esat_a_b @ ( fo_Neg_a_b @ Phi ) @ I @ Sigma @ X2 )
= ( ~ ( esat_a_b @ Phi @ I @ Sigma @ X2 ) ) ) ).
% esat.simps(4)
thf(fact_8_esat_Osimps_I2_J,axiom,
! [B: $o,I: product_prod_b_nat > set_list_a,Sigma: nat > sum_sum_a_nat,X2: set_Sum_sum_a_nat] :
( ( esat_a_b @ ( fo_Bool_a_b @ B ) @ I @ Sigma @ X2 )
= B ) ).
% esat.simps(2)
thf(fact_9_esat_Osimps_I7_J,axiom,
! [N2: nat,Phi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > sum_sum_a_nat,X2: set_Sum_sum_a_nat] :
( ( esat_a_b @ ( fo_Exists_a_b @ N2 @ Phi ) @ I @ Sigma @ X2 )
= ( ? [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ X2 )
& ( esat_a_b @ Phi @ I @ ( fun_up180537416982607344_a_nat @ Sigma @ N2 @ X3 ) @ X2 ) ) ) ) ).
% esat.simps(7)
thf(fact_10_esat_Osimps_I8_J,axiom,
! [N2: nat,Phi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > sum_sum_a_nat,X2: set_Sum_sum_a_nat] :
( ( esat_a_b @ ( fo_Forall_a_b @ N2 @ Phi ) @ I @ Sigma @ X2 )
= ( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ X2 )
=> ( esat_a_b @ Phi @ I @ ( fun_up180537416982607344_a_nat @ Sigma @ N2 @ X3 ) @ X2 ) ) ) ) ).
% esat.simps(8)
thf(fact_11_esat_Osimps_I3_J,axiom,
! [T: fo_term_a,T2: fo_term_a,I: product_prod_b_nat > set_list_a,Sigma: nat > sum_sum_a_nat,X2: set_Sum_sum_a_nat] :
( ( esat_a_b @ ( fo_Eqa_a_b @ T @ T2 ) @ I @ Sigma @ X2 )
= ( ( eval_eterm_a_nat @ Sigma @ T )
= ( eval_eterm_a_nat @ Sigma @ T2 ) ) ) ).
% esat.simps(3)
thf(fact_12_fo__fmla_Oinject_I2_J,axiom,
! [X22: $o,Y2: $o] :
( ( ( fo_Bool_a_b @ X22 )
= ( fo_Bool_a_b @ Y2 ) )
= ( X22 = Y2 ) ) ).
% fo_fmla.inject(2)
thf(fact_13_fo__fmla_Oinject_I8_J,axiom,
! [X81: nat,X82: fo_fmla_a_b,Y81: nat,Y82: fo_fmla_a_b] :
( ( ( fo_Forall_a_b @ X81 @ X82 )
= ( fo_Forall_a_b @ Y81 @ Y82 ) )
= ( ( X81 = Y81 )
& ( X82 = Y82 ) ) ) ).
% fo_fmla.inject(8)
thf(fact_14_fo__fmla_Oinject_I7_J,axiom,
! [X71: nat,X72: fo_fmla_a_b,Y71: nat,Y72: fo_fmla_a_b] :
( ( ( fo_Exists_a_b @ X71 @ X72 )
= ( fo_Exists_a_b @ Y71 @ Y72 ) )
= ( ( X71 = Y71 )
& ( X72 = Y72 ) ) ) ).
% fo_fmla.inject(7)
thf(fact_15_fo__fmla_Oinject_I4_J,axiom,
! [X4: fo_fmla_a_b,Y4: fo_fmla_a_b] :
( ( ( fo_Neg_a_b @ X4 )
= ( fo_Neg_a_b @ Y4 ) )
= ( X4 = Y4 ) ) ).
% fo_fmla.inject(4)
thf(fact_16_fo__fmla_Oinject_I6_J,axiom,
! [X61: fo_fmla_a_b,X62: fo_fmla_a_b,Y61: fo_fmla_a_b,Y62: fo_fmla_a_b] :
( ( ( fo_Disj_a_b @ X61 @ X62 )
= ( fo_Disj_a_b @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fo_fmla.inject(6)
thf(fact_17_fo__fmla_Oinject_I5_J,axiom,
! [X51: fo_fmla_a_b,X52: fo_fmla_a_b,Y51: fo_fmla_a_b,Y52: fo_fmla_a_b] :
( ( ( fo_Conj_a_b @ X51 @ X52 )
= ( fo_Conj_a_b @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fo_fmla.inject(5)
thf(fact_18_fo__fmla_Oinject_I3_J,axiom,
! [X31: fo_term_a,X32: fo_term_a,Y31: fo_term_a,Y32: fo_term_a] :
( ( ( fo_Eqa_a_b @ X31 @ X32 )
= ( fo_Eqa_a_b @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% fo_fmla.inject(3)
thf(fact_19_fun__upd__upd,axiom,
! [F: nat > sum_sum_a_nat,X5: nat,Y: sum_sum_a_nat,Z: sum_sum_a_nat] :
( ( fun_up180537416982607344_a_nat @ ( fun_up180537416982607344_a_nat @ F @ X5 @ Y ) @ X5 @ Z )
= ( fun_up180537416982607344_a_nat @ F @ X5 @ Z ) ) ).
% fun_upd_upd
thf(fact_20_fun__upd__triv,axiom,
! [F: nat > sum_sum_a_nat,X5: nat] :
( ( fun_up180537416982607344_a_nat @ F @ X5 @ ( F @ X5 ) )
= F ) ).
% fun_upd_triv
thf(fact_21_fun__upd__apply,axiom,
( fun_up180537416982607344_a_nat
= ( ^ [F2: nat > sum_sum_a_nat,X3: nat,Y3: sum_sum_a_nat,Z2: nat] : ( if_Sum_sum_a_nat @ ( Z2 = X3 ) @ Y3 @ ( F2 @ Z2 ) ) ) ) ).
% fun_upd_apply
thf(fact_22_fv__fo__fmla_Osimps_I4_J,axiom,
! [Phi: fo_fmla_a_b] :
( ( fv_fo_fmla_a_b @ ( fo_Neg_a_b @ Phi ) )
= ( fv_fo_fmla_a_b @ Phi ) ) ).
% fv_fo_fmla.simps(4)
thf(fact_23_fo__fmla_Oinject_I1_J,axiom,
! [X11: b,X12: list_fo_term_a,Y11: b,Y12: list_fo_term_a] :
( ( ( fo_Pred_b_a @ X11 @ X12 )
= ( fo_Pred_b_a @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% fo_fmla.inject(1)
thf(fact_24_fo__fmla_Odistinct_I3_J,axiom,
! [X11: b,X12: list_fo_term_a,X31: fo_term_a,X32: fo_term_a] :
( ( fo_Pred_b_a @ X11 @ X12 )
!= ( fo_Eqa_a_b @ X31 @ X32 ) ) ).
% fo_fmla.distinct(3)
thf(fact_25_fo__fmla_Odistinct_I7_J,axiom,
! [X11: b,X12: list_fo_term_a,X51: fo_fmla_a_b,X52: fo_fmla_a_b] :
( ( fo_Pred_b_a @ X11 @ X12 )
!= ( fo_Conj_a_b @ X51 @ X52 ) ) ).
% fo_fmla.distinct(7)
thf(fact_26_fo__fmla_Odistinct_I9_J,axiom,
! [X11: b,X12: list_fo_term_a,X61: fo_fmla_a_b,X62: fo_fmla_a_b] :
( ( fo_Pred_b_a @ X11 @ X12 )
!= ( fo_Disj_a_b @ X61 @ X62 ) ) ).
% fo_fmla.distinct(9)
thf(fact_27_fo__fmla_Odistinct_I5_J,axiom,
! [X11: b,X12: list_fo_term_a,X4: fo_fmla_a_b] :
( ( fo_Pred_b_a @ X11 @ X12 )
!= ( fo_Neg_a_b @ X4 ) ) ).
% fo_fmla.distinct(5)
thf(fact_28_fo__fmla_Odistinct_I11_J,axiom,
! [X11: b,X12: list_fo_term_a,X71: nat,X72: fo_fmla_a_b] :
( ( fo_Pred_b_a @ X11 @ X12 )
!= ( fo_Exists_a_b @ X71 @ X72 ) ) ).
% fo_fmla.distinct(11)
thf(fact_29_fo__fmla_Odistinct_I13_J,axiom,
! [X11: b,X12: list_fo_term_a,X81: nat,X82: fo_fmla_a_b] :
( ( fo_Pred_b_a @ X11 @ X12 )
!= ( fo_Forall_a_b @ X81 @ X82 ) ) ).
% fo_fmla.distinct(13)
thf(fact_30_fo__fmla_Odistinct_I1_J,axiom,
! [X11: b,X12: list_fo_term_a,X22: $o] :
( ( fo_Pred_b_a @ X11 @ X12 )
!= ( fo_Bool_a_b @ X22 ) ) ).
% fo_fmla.distinct(1)
thf(fact_31_fun__upd__idem__iff,axiom,
! [F: nat > sum_sum_a_nat,X5: nat,Y: sum_sum_a_nat] :
( ( ( fun_up180537416982607344_a_nat @ F @ X5 @ Y )
= F )
= ( ( F @ X5 )
= Y ) ) ).
% fun_upd_idem_iff
thf(fact_32_fun__upd__twist,axiom,
! [A: nat,C: nat,M: nat > sum_sum_a_nat,B: sum_sum_a_nat,D: sum_sum_a_nat] :
( ( A != C )
=> ( ( fun_up180537416982607344_a_nat @ ( fun_up180537416982607344_a_nat @ M @ A @ B ) @ C @ D )
= ( fun_up180537416982607344_a_nat @ ( fun_up180537416982607344_a_nat @ M @ C @ D ) @ A @ B ) ) ) ).
% fun_upd_twist
thf(fact_33_fun__upd__other,axiom,
! [Z: nat,X5: nat,F: nat > sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( Z != X5 )
=> ( ( fun_up180537416982607344_a_nat @ F @ X5 @ Y @ Z )
= ( F @ Z ) ) ) ).
% fun_upd_other
thf(fact_34_fun__upd__same,axiom,
! [F: nat > sum_sum_a_nat,X5: nat,Y: sum_sum_a_nat] :
( ( fun_up180537416982607344_a_nat @ F @ X5 @ Y @ X5 )
= Y ) ).
% fun_upd_same
thf(fact_35_fun__upd__idem,axiom,
! [F: nat > sum_sum_a_nat,X5: nat,Y: sum_sum_a_nat] :
( ( ( F @ X5 )
= Y )
=> ( ( fun_up180537416982607344_a_nat @ F @ X5 @ Y )
= F ) ) ).
% fun_upd_idem
thf(fact_36_fun__upd__eqD,axiom,
! [F: nat > sum_sum_a_nat,X5: nat,Y: sum_sum_a_nat,G: nat > sum_sum_a_nat,Z: sum_sum_a_nat] :
( ( ( fun_up180537416982607344_a_nat @ F @ X5 @ Y )
= ( fun_up180537416982607344_a_nat @ G @ X5 @ Z ) )
=> ( Y = Z ) ) ).
% fun_upd_eqD
thf(fact_37_fun__upd__def,axiom,
( fun_up180537416982607344_a_nat
= ( ^ [F2: nat > sum_sum_a_nat,A2: nat,B2: sum_sum_a_nat,X3: nat] : ( if_Sum_sum_a_nat @ ( X3 = A2 ) @ B2 @ ( F2 @ X3 ) ) ) ) ).
% fun_upd_def
thf(fact_38_fv__fo__fmla__list__rec_Ocases,axiom,
! [X5: fo_fmla_a_b] :
( ! [Uu: b,Ts: list_fo_term_a] :
( X5
!= ( fo_Pred_b_a @ Uu @ Ts ) )
=> ( ! [B3: $o] :
( X5
!= ( fo_Bool_a_b @ B3 ) )
=> ( ! [T3: fo_term_a,T4: fo_term_a] :
( X5
!= ( fo_Eqa_a_b @ T3 @ T4 ) )
=> ( ! [Phi2: fo_fmla_a_b] :
( X5
!= ( fo_Neg_a_b @ Phi2 ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( X5
!= ( fo_Conj_a_b @ Phi2 @ Psi2 ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( X5
!= ( fo_Disj_a_b @ Phi2 @ Psi2 ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b] :
( X5
!= ( fo_Exists_a_b @ N @ Phi2 ) )
=> ~ ! [N: nat,Phi2: fo_fmla_a_b] :
( X5
!= ( fo_Forall_a_b @ N @ Phi2 ) ) ) ) ) ) ) ) ) ).
% fv_fo_fmla_list_rec.cases
thf(fact_39_fo__fmla_Oexhaust,axiom,
! [Y: fo_fmla_a_b] :
( ! [X112: b,X122: list_fo_term_a] :
( Y
!= ( fo_Pred_b_a @ X112 @ X122 ) )
=> ( ! [X23: $o] :
( Y
!= ( fo_Bool_a_b @ X23 ) )
=> ( ! [X312: fo_term_a,X322: fo_term_a] :
( Y
!= ( fo_Eqa_a_b @ X312 @ X322 ) )
=> ( ! [X42: fo_fmla_a_b] :
( Y
!= ( fo_Neg_a_b @ X42 ) )
=> ( ! [X512: fo_fmla_a_b,X522: fo_fmla_a_b] :
( Y
!= ( fo_Conj_a_b @ X512 @ X522 ) )
=> ( ! [X612: fo_fmla_a_b,X622: fo_fmla_a_b] :
( Y
!= ( fo_Disj_a_b @ X612 @ X622 ) )
=> ( ! [X712: nat,X722: fo_fmla_a_b] :
( Y
!= ( fo_Exists_a_b @ X712 @ X722 ) )
=> ~ ! [X812: nat,X822: fo_fmla_a_b] :
( Y
!= ( fo_Forall_a_b @ X812 @ X822 ) ) ) ) ) ) ) ) ) ).
% fo_fmla.exhaust
thf(fact_40_mem__Collect__eq,axiom,
! [A: fo_term_a,P: fo_term_a > $o] :
( ( member_fo_term_a @ A @ ( collect_fo_term_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_41_mem__Collect__eq,axiom,
! [A: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ A @ ( collec7555443234367654128_a_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_42_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_43_mem__Collect__eq,axiom,
! [A: b,P: b > $o] :
( ( member_b @ A @ ( collect_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_44_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_45_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A3: set_fo_term_a] :
( ( collect_fo_term_a
@ ^ [X3: fo_term_a] : ( member_fo_term_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A3: set_li6526943997496501093_a_nat] :
( ( collec7555443234367654128_a_nat
@ ^ [X3: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A3: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A3: set_b] :
( ( collect_b
@ ^ [X3: b] : ( member_b @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_52_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_53_sz__fmla__induct,axiom,
! [P: fo_fmla_a_b > $o,Phi: fo_fmla_a_b] :
( ! [R: b,Ts: list_fo_term_a] : ( P @ ( fo_Pred_b_a @ R @ Ts ) )
=> ( ! [B3: $o] : ( P @ ( fo_Bool_a_b @ B3 ) )
=> ( ! [T3: fo_term_a,T4: fo_term_a] : ( P @ ( fo_Eqa_a_b @ T3 @ T4 ) )
=> ( ! [Phi2: fo_fmla_a_b] :
( ( P @ Phi2 )
=> ( P @ ( fo_Neg_a_b @ Phi2 ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( P @ Phi2 )
=> ( ( P @ Psi2 )
=> ( P @ ( fo_Conj_a_b @ Phi2 @ Psi2 ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( P @ Phi2 )
=> ( ( P @ Psi2 )
=> ( P @ ( fo_Disj_a_b @ Phi2 @ Psi2 ) ) ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b] :
( ( P @ Phi2 )
=> ( P @ ( fo_Exists_a_b @ N @ Phi2 ) ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b] :
( ( P @ ( fo_Exists_a_b @ N @ ( fo_Neg_a_b @ Phi2 ) ) )
=> ( P @ ( fo_Forall_a_b @ N @ Phi2 ) ) )
=> ( P @ Phi ) ) ) ) ) ) ) ) ) ).
% sz_fmla_induct
thf(fact_54_sz__fmla_Ocases,axiom,
! [X5: fo_fmla_a_b] :
( ! [Phi2: fo_fmla_a_b] :
( X5
!= ( fo_Neg_a_b @ Phi2 ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( X5
!= ( fo_Conj_a_b @ Phi2 @ Psi2 ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( X5
!= ( fo_Disj_a_b @ Phi2 @ Psi2 ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b] :
( X5
!= ( fo_Exists_a_b @ N @ Phi2 ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b] :
( X5
!= ( fo_Forall_a_b @ N @ Phi2 ) )
=> ( ! [V: b,Va: list_fo_term_a] :
( X5
!= ( fo_Pred_b_a @ V @ Va ) )
=> ( ! [V: $o] :
( X5
!= ( fo_Bool_a_b @ V ) )
=> ~ ! [V: fo_term_a,Va: fo_term_a] :
( X5
!= ( fo_Eqa_a_b @ V @ Va ) ) ) ) ) ) ) ) ) ).
% sz_fmla.cases
thf(fact_55_fo__fmla_Odistinct_I29_J,axiom,
! [X31: fo_term_a,X32: fo_term_a,X51: fo_fmla_a_b,X52: fo_fmla_a_b] :
( ( fo_Eqa_a_b @ X31 @ X32 )
!= ( fo_Conj_a_b @ X51 @ X52 ) ) ).
% fo_fmla.distinct(29)
thf(fact_56_fo__fmla_Odistinct_I31_J,axiom,
! [X31: fo_term_a,X32: fo_term_a,X61: fo_fmla_a_b,X62: fo_fmla_a_b] :
( ( fo_Eqa_a_b @ X31 @ X32 )
!= ( fo_Disj_a_b @ X61 @ X62 ) ) ).
% fo_fmla.distinct(31)
thf(fact_57_fo__fmla_Odistinct_I27_J,axiom,
! [X31: fo_term_a,X32: fo_term_a,X4: fo_fmla_a_b] :
( ( fo_Eqa_a_b @ X31 @ X32 )
!= ( fo_Neg_a_b @ X4 ) ) ).
% fo_fmla.distinct(27)
thf(fact_58_fo__fmla_Odistinct_I33_J,axiom,
! [X31: fo_term_a,X32: fo_term_a,X71: nat,X72: fo_fmla_a_b] :
( ( fo_Eqa_a_b @ X31 @ X32 )
!= ( fo_Exists_a_b @ X71 @ X72 ) ) ).
% fo_fmla.distinct(33)
thf(fact_59_fo__fmla_Odistinct_I35_J,axiom,
! [X31: fo_term_a,X32: fo_term_a,X81: nat,X82: fo_fmla_a_b] :
( ( fo_Eqa_a_b @ X31 @ X32 )
!= ( fo_Forall_a_b @ X81 @ X82 ) ) ).
% fo_fmla.distinct(35)
thf(fact_60_fo__fmla_Odistinct_I45_J,axiom,
! [X51: fo_fmla_a_b,X52: fo_fmla_a_b,X61: fo_fmla_a_b,X62: fo_fmla_a_b] :
( ( fo_Conj_a_b @ X51 @ X52 )
!= ( fo_Disj_a_b @ X61 @ X62 ) ) ).
% fo_fmla.distinct(45)
thf(fact_61_fo__fmla_Odistinct_I37_J,axiom,
! [X4: fo_fmla_a_b,X51: fo_fmla_a_b,X52: fo_fmla_a_b] :
( ( fo_Neg_a_b @ X4 )
!= ( fo_Conj_a_b @ X51 @ X52 ) ) ).
% fo_fmla.distinct(37)
thf(fact_62_fo__fmla_Odistinct_I15_J,axiom,
! [X22: $o,X31: fo_term_a,X32: fo_term_a] :
( ( fo_Bool_a_b @ X22 )
!= ( fo_Eqa_a_b @ X31 @ X32 ) ) ).
% fo_fmla.distinct(15)
thf(fact_63_fo__fmla_Odistinct_I47_J,axiom,
! [X51: fo_fmla_a_b,X52: fo_fmla_a_b,X71: nat,X72: fo_fmla_a_b] :
( ( fo_Conj_a_b @ X51 @ X52 )
!= ( fo_Exists_a_b @ X71 @ X72 ) ) ).
% fo_fmla.distinct(47)
thf(fact_64_fo__fmla_Odistinct_I49_J,axiom,
! [X51: fo_fmla_a_b,X52: fo_fmla_a_b,X81: nat,X82: fo_fmla_a_b] :
( ( fo_Conj_a_b @ X51 @ X52 )
!= ( fo_Forall_a_b @ X81 @ X82 ) ) ).
% fo_fmla.distinct(49)
thf(fact_65_fo__fmla_Odistinct_I39_J,axiom,
! [X4: fo_fmla_a_b,X61: fo_fmla_a_b,X62: fo_fmla_a_b] :
( ( fo_Neg_a_b @ X4 )
!= ( fo_Disj_a_b @ X61 @ X62 ) ) ).
% fo_fmla.distinct(39)
thf(fact_66_fo__fmla_Odistinct_I51_J,axiom,
! [X61: fo_fmla_a_b,X62: fo_fmla_a_b,X71: nat,X72: fo_fmla_a_b] :
( ( fo_Disj_a_b @ X61 @ X62 )
!= ( fo_Exists_a_b @ X71 @ X72 ) ) ).
% fo_fmla.distinct(51)
thf(fact_67_fo__fmla_Odistinct_I53_J,axiom,
! [X61: fo_fmla_a_b,X62: fo_fmla_a_b,X81: nat,X82: fo_fmla_a_b] :
( ( fo_Disj_a_b @ X61 @ X62 )
!= ( fo_Forall_a_b @ X81 @ X82 ) ) ).
% fo_fmla.distinct(53)
thf(fact_68_fo__fmla_Odistinct_I41_J,axiom,
! [X4: fo_fmla_a_b,X71: nat,X72: fo_fmla_a_b] :
( ( fo_Neg_a_b @ X4 )
!= ( fo_Exists_a_b @ X71 @ X72 ) ) ).
% fo_fmla.distinct(41)
thf(fact_69_fo__fmla_Odistinct_I19_J,axiom,
! [X22: $o,X51: fo_fmla_a_b,X52: fo_fmla_a_b] :
( ( fo_Bool_a_b @ X22 )
!= ( fo_Conj_a_b @ X51 @ X52 ) ) ).
% fo_fmla.distinct(19)
thf(fact_70_fo__fmla_Odistinct_I43_J,axiom,
! [X4: fo_fmla_a_b,X81: nat,X82: fo_fmla_a_b] :
( ( fo_Neg_a_b @ X4 )
!= ( fo_Forall_a_b @ X81 @ X82 ) ) ).
% fo_fmla.distinct(43)
thf(fact_71_fo__fmla_Odistinct_I55_J,axiom,
! [X71: nat,X72: fo_fmla_a_b,X81: nat,X82: fo_fmla_a_b] :
( ( fo_Exists_a_b @ X71 @ X72 )
!= ( fo_Forall_a_b @ X81 @ X82 ) ) ).
% fo_fmla.distinct(55)
thf(fact_72_fo__fmla_Odistinct_I21_J,axiom,
! [X22: $o,X61: fo_fmla_a_b,X62: fo_fmla_a_b] :
( ( fo_Bool_a_b @ X22 )
!= ( fo_Disj_a_b @ X61 @ X62 ) ) ).
% fo_fmla.distinct(21)
thf(fact_73_fo__fmla_Odistinct_I17_J,axiom,
! [X22: $o,X4: fo_fmla_a_b] :
( ( fo_Bool_a_b @ X22 )
!= ( fo_Neg_a_b @ X4 ) ) ).
% fo_fmla.distinct(17)
thf(fact_74_fo__fmla_Odistinct_I23_J,axiom,
! [X22: $o,X71: nat,X72: fo_fmla_a_b] :
( ( fo_Bool_a_b @ X22 )
!= ( fo_Exists_a_b @ X71 @ X72 ) ) ).
% fo_fmla.distinct(23)
thf(fact_75_fo__fmla_Odistinct_I25_J,axiom,
! [X22: $o,X81: nat,X82: fo_fmla_a_b] :
( ( fo_Bool_a_b @ X22 )
!= ( fo_Forall_a_b @ X81 @ X82 ) ) ).
% fo_fmla.distinct(25)
thf(fact_76_act__edom_Ocases,axiom,
! [X5: produc7525339858011107362list_a] :
( ! [R: b,Ts: list_fo_term_a,I2: product_prod_b_nat > set_list_a] :
( X5
!= ( produc2741640913913151572list_a @ ( fo_Pred_b_a @ R @ Ts ) @ I2 ) )
=> ( ! [B3: $o,I2: product_prod_b_nat > set_list_a] :
( X5
!= ( produc2741640913913151572list_a @ ( fo_Bool_a_b @ B3 ) @ I2 ) )
=> ( ! [T3: fo_term_a,T4: fo_term_a,I2: product_prod_b_nat > set_list_a] :
( X5
!= ( produc2741640913913151572list_a @ ( fo_Eqa_a_b @ T3 @ T4 ) @ I2 ) )
=> ( ! [Phi2: fo_fmla_a_b,I2: product_prod_b_nat > set_list_a] :
( X5
!= ( produc2741640913913151572list_a @ ( fo_Neg_a_b @ Phi2 ) @ I2 ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b,I2: product_prod_b_nat > set_list_a] :
( X5
!= ( produc2741640913913151572list_a @ ( fo_Conj_a_b @ Phi2 @ Psi2 ) @ I2 ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b,I2: product_prod_b_nat > set_list_a] :
( X5
!= ( produc2741640913913151572list_a @ ( fo_Disj_a_b @ Phi2 @ Psi2 ) @ I2 ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b,I2: product_prod_b_nat > set_list_a] :
( X5
!= ( produc2741640913913151572list_a @ ( fo_Exists_a_b @ N @ Phi2 ) @ I2 ) )
=> ~ ! [N: nat,Phi2: fo_fmla_a_b,I2: product_prod_b_nat > set_list_a] :
( X5
!= ( produc2741640913913151572list_a @ ( fo_Forall_a_b @ N @ Phi2 ) @ I2 ) ) ) ) ) ) ) ) ) ).
% act_edom.cases
thf(fact_77_fo__fmla_Oset__cases_I2_J,axiom,
! [E: nat,A: fo_fmla_nat_nat] :
( ( member_nat @ E @ ( fo_set4065224739847495105at_nat @ A ) )
=> ( ! [Z22: list_fo_term_nat] :
( A
!= ( fo_Pred_nat_nat @ E @ Z22 ) )
=> ( ! [Z3: fo_fmla_nat_nat] :
( ( A
= ( fo_Neg_nat_nat @ Z3 ) )
=> ~ ( member_nat @ E @ ( fo_set4065224739847495105at_nat @ Z3 ) ) )
=> ( ! [Z1: fo_fmla_nat_nat] :
( ? [Z22: fo_fmla_nat_nat] :
( A
= ( fo_Conj_nat_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( fo_set4065224739847495105at_nat @ Z1 ) ) )
=> ( ! [Z1: fo_fmla_nat_nat,Z22: fo_fmla_nat_nat] :
( ( A
= ( fo_Conj_nat_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( fo_set4065224739847495105at_nat @ Z22 ) ) )
=> ( ! [Z1: fo_fmla_nat_nat] :
( ? [Z22: fo_fmla_nat_nat] :
( A
= ( fo_Disj_nat_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( fo_set4065224739847495105at_nat @ Z1 ) ) )
=> ( ! [Z1: fo_fmla_nat_nat,Z22: fo_fmla_nat_nat] :
( ( A
= ( fo_Disj_nat_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( fo_set4065224739847495105at_nat @ Z22 ) ) )
=> ( ! [Z1: nat,Z22: fo_fmla_nat_nat] :
( ( A
= ( fo_Exists_nat_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( fo_set4065224739847495105at_nat @ Z22 ) ) )
=> ~ ! [Z1: nat,Z22: fo_fmla_nat_nat] :
( ( A
= ( fo_Forall_nat_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( fo_set4065224739847495105at_nat @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fo_fmla.set_cases(2)
thf(fact_78_fo__fmla_Oset__cases_I2_J,axiom,
! [E: b,A: fo_fmla_a_b] :
( ( member_b @ E @ ( fo_set2_fo_fmla_a_b @ A ) )
=> ( ! [Z22: list_fo_term_a] :
( A
!= ( fo_Pred_b_a @ E @ Z22 ) )
=> ( ! [Z3: fo_fmla_a_b] :
( ( A
= ( fo_Neg_a_b @ Z3 ) )
=> ~ ( member_b @ E @ ( fo_set2_fo_fmla_a_b @ Z3 ) ) )
=> ( ! [Z1: fo_fmla_a_b] :
( ? [Z22: fo_fmla_a_b] :
( A
= ( fo_Conj_a_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( fo_set2_fo_fmla_a_b @ Z1 ) ) )
=> ( ! [Z1: fo_fmla_a_b,Z22: fo_fmla_a_b] :
( ( A
= ( fo_Conj_a_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( fo_set2_fo_fmla_a_b @ Z22 ) ) )
=> ( ! [Z1: fo_fmla_a_b] :
( ? [Z22: fo_fmla_a_b] :
( A
= ( fo_Disj_a_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( fo_set2_fo_fmla_a_b @ Z1 ) ) )
=> ( ! [Z1: fo_fmla_a_b,Z22: fo_fmla_a_b] :
( ( A
= ( fo_Disj_a_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( fo_set2_fo_fmla_a_b @ Z22 ) ) )
=> ( ! [Z1: nat,Z22: fo_fmla_a_b] :
( ( A
= ( fo_Exists_a_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( fo_set2_fo_fmla_a_b @ Z22 ) ) )
=> ~ ! [Z1: nat,Z22: fo_fmla_a_b] :
( ( A
= ( fo_Forall_a_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( fo_set2_fo_fmla_a_b @ Z22 ) ) ) ) ) ) ) ) ) ) ) ).
% fo_fmla.set_cases(2)
thf(fact_79_sat_Ocases,axiom,
! [X5: produc8907840021022247092_nat_a] :
( ! [R: b,Ts: list_fo_term_a,I2: product_prod_b_nat > set_list_a,Sigma3: nat > a] :
( X5
!= ( produc7243001479329598822_nat_a @ ( fo_Pred_b_a @ R @ Ts ) @ ( produc2895298938842563487_nat_a @ I2 @ Sigma3 ) ) )
=> ( ! [B3: $o,I2: product_prod_b_nat > set_list_a,Sigma3: nat > a] :
( X5
!= ( produc7243001479329598822_nat_a @ ( fo_Bool_a_b @ B3 ) @ ( produc2895298938842563487_nat_a @ I2 @ Sigma3 ) ) )
=> ( ! [T3: fo_term_a,T4: fo_term_a,I2: product_prod_b_nat > set_list_a,Sigma3: nat > a] :
( X5
!= ( produc7243001479329598822_nat_a @ ( fo_Eqa_a_b @ T3 @ T4 ) @ ( produc2895298938842563487_nat_a @ I2 @ Sigma3 ) ) )
=> ( ! [Phi2: fo_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma3: nat > a] :
( X5
!= ( produc7243001479329598822_nat_a @ ( fo_Neg_a_b @ Phi2 ) @ ( produc2895298938842563487_nat_a @ I2 @ Sigma3 ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma3: nat > a] :
( X5
!= ( produc7243001479329598822_nat_a @ ( fo_Conj_a_b @ Phi2 @ Psi2 ) @ ( produc2895298938842563487_nat_a @ I2 @ Sigma3 ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma3: nat > a] :
( X5
!= ( produc7243001479329598822_nat_a @ ( fo_Disj_a_b @ Phi2 @ Psi2 ) @ ( produc2895298938842563487_nat_a @ I2 @ Sigma3 ) ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma3: nat > a] :
( X5
!= ( produc7243001479329598822_nat_a @ ( fo_Exists_a_b @ N @ Phi2 ) @ ( produc2895298938842563487_nat_a @ I2 @ Sigma3 ) ) )
=> ~ ! [N: nat,Phi2: fo_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma3: nat > a] :
( X5
!= ( produc7243001479329598822_nat_a @ ( fo_Forall_a_b @ N @ Phi2 ) @ ( produc2895298938842563487_nat_a @ I2 @ Sigma3 ) ) ) ) ) ) ) ) ) ) ).
% sat.cases
thf(fact_80_SP__list__rec_Ocases,axiom,
! [X5: fo_fmla_a_b] :
( ! [N: nat,N3: nat] :
( X5
!= ( fo_Eqa_a_b @ ( fo_Var_a @ N ) @ ( fo_Var_a @ N3 ) ) )
=> ( ! [Phi2: fo_fmla_a_b] :
( X5
!= ( fo_Neg_a_b @ Phi2 ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( X5
!= ( fo_Conj_a_b @ Phi2 @ Psi2 ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( X5
!= ( fo_Disj_a_b @ Phi2 @ Psi2 ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b] :
( X5
!= ( fo_Exists_a_b @ N @ Phi2 ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b] :
( X5
!= ( fo_Forall_a_b @ N @ Phi2 ) )
=> ( ! [V: b,Va: list_fo_term_a] :
( X5
!= ( fo_Pred_b_a @ V @ Va ) )
=> ( ! [V: $o] :
( X5
!= ( fo_Bool_a_b @ V ) )
=> ( ! [Vb: a,Va: fo_term_a] :
( X5
!= ( fo_Eqa_a_b @ ( fo_Const_a @ Vb ) @ Va ) )
=> ~ ! [V: fo_term_a,Vb: a] :
( X5
!= ( fo_Eqa_a_b @ V @ ( fo_Const_a @ Vb ) ) ) ) ) ) ) ) ) ) ) ) ).
% SP_list_rec.cases
thf(fact_81_fv__fo__fmla_Osimps_I1_J,axiom,
! [Uu2: b,Ts2: list_fo_term_a] :
( ( fv_fo_fmla_a_b @ ( fo_Pred_b_a @ Uu2 @ Ts2 ) )
= ( fv_fo_terms_set_a @ Ts2 ) ) ).
% fv_fo_fmla.simps(1)
thf(fact_82_esat_Ocases,axiom,
! [X5: produc8277654585549370735_a_nat] :
( ! [R: b,Ts: list_fo_term_a,I2: product_prod_b_nat > set_list_a,Sigma3: nat > sum_sum_a_nat,X6: set_Sum_sum_a_nat] :
( X5
!= ( produc8250703347703957409_a_nat @ ( fo_Pred_b_a @ R @ Ts ) @ ( produc6651248262528101210_a_nat @ I2 @ ( produc3720304352952013712_a_nat @ Sigma3 @ X6 ) ) ) )
=> ( ! [B3: $o,I2: product_prod_b_nat > set_list_a,Sigma3: nat > sum_sum_a_nat,X6: set_Sum_sum_a_nat] :
( X5
!= ( produc8250703347703957409_a_nat @ ( fo_Bool_a_b @ B3 ) @ ( produc6651248262528101210_a_nat @ I2 @ ( produc3720304352952013712_a_nat @ Sigma3 @ X6 ) ) ) )
=> ( ! [T3: fo_term_a,T4: fo_term_a,I2: product_prod_b_nat > set_list_a,Sigma3: nat > sum_sum_a_nat,X6: set_Sum_sum_a_nat] :
( X5
!= ( produc8250703347703957409_a_nat @ ( fo_Eqa_a_b @ T3 @ T4 ) @ ( produc6651248262528101210_a_nat @ I2 @ ( produc3720304352952013712_a_nat @ Sigma3 @ X6 ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma3: nat > sum_sum_a_nat,X6: set_Sum_sum_a_nat] :
( X5
!= ( produc8250703347703957409_a_nat @ ( fo_Neg_a_b @ Phi2 ) @ ( produc6651248262528101210_a_nat @ I2 @ ( produc3720304352952013712_a_nat @ Sigma3 @ X6 ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma3: nat > sum_sum_a_nat,X6: set_Sum_sum_a_nat] :
( X5
!= ( produc8250703347703957409_a_nat @ ( fo_Conj_a_b @ Phi2 @ Psi2 ) @ ( produc6651248262528101210_a_nat @ I2 @ ( produc3720304352952013712_a_nat @ Sigma3 @ X6 ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma3: nat > sum_sum_a_nat,X6: set_Sum_sum_a_nat] :
( X5
!= ( produc8250703347703957409_a_nat @ ( fo_Disj_a_b @ Phi2 @ Psi2 ) @ ( produc6651248262528101210_a_nat @ I2 @ ( produc3720304352952013712_a_nat @ Sigma3 @ X6 ) ) ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma3: nat > sum_sum_a_nat,X6: set_Sum_sum_a_nat] :
( X5
!= ( produc8250703347703957409_a_nat @ ( fo_Exists_a_b @ N @ Phi2 ) @ ( produc6651248262528101210_a_nat @ I2 @ ( produc3720304352952013712_a_nat @ Sigma3 @ X6 ) ) ) )
=> ~ ! [N: nat,Phi2: fo_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma3: nat > sum_sum_a_nat,X6: set_Sum_sum_a_nat] :
( X5
!= ( produc8250703347703957409_a_nat @ ( fo_Forall_a_b @ N @ Phi2 ) @ ( produc6651248262528101210_a_nat @ I2 @ ( produc3720304352952013712_a_nat @ Sigma3 @ X6 ) ) ) ) ) ) ) ) ) ) ) ).
% esat.cases
thf(fact_83_fv__fo__fmla_Osimps_I2_J,axiom,
! [B: $o] :
( ( fv_fo_fmla_a_b @ ( fo_Bool_a_b @ B ) )
= bot_bot_set_nat ) ).
% fv_fo_fmla.simps(2)
thf(fact_84_fo__fmla_Oset__cases_I1_J,axiom,
! [E: nat,A: fo_fmla_nat_nat] :
( ( member_nat @ E @ ( fo_set7155282118041507904at_nat @ A ) )
=> ( ! [Z1: nat,Z22: list_fo_term_nat] :
( ( A
= ( fo_Pred_nat_nat @ Z1 @ Z22 ) )
=> ! [X: fo_term_nat] :
( ( member_fo_term_nat @ X @ ( set_fo_term_nat2 @ Z22 ) )
=> ~ ( member_nat @ E @ ( fo_set_fo_term_nat @ X ) ) ) )
=> ( ! [Z1: fo_term_nat] :
( ? [Z22: fo_term_nat] :
( A
= ( fo_Eqa_nat_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( fo_set_fo_term_nat @ Z1 ) ) )
=> ( ! [Z1: fo_term_nat,Z22: fo_term_nat] :
( ( A
= ( fo_Eqa_nat_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( fo_set_fo_term_nat @ Z22 ) ) )
=> ( ! [Z3: fo_fmla_nat_nat] :
( ( A
= ( fo_Neg_nat_nat @ Z3 ) )
=> ~ ( member_nat @ E @ ( fo_set7155282118041507904at_nat @ Z3 ) ) )
=> ( ! [Z1: fo_fmla_nat_nat] :
( ? [Z22: fo_fmla_nat_nat] :
( A
= ( fo_Conj_nat_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( fo_set7155282118041507904at_nat @ Z1 ) ) )
=> ( ! [Z1: fo_fmla_nat_nat,Z22: fo_fmla_nat_nat] :
( ( A
= ( fo_Conj_nat_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( fo_set7155282118041507904at_nat @ Z22 ) ) )
=> ( ! [Z1: fo_fmla_nat_nat] :
( ? [Z22: fo_fmla_nat_nat] :
( A
= ( fo_Disj_nat_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( fo_set7155282118041507904at_nat @ Z1 ) ) )
=> ( ! [Z1: fo_fmla_nat_nat,Z22: fo_fmla_nat_nat] :
( ( A
= ( fo_Disj_nat_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( fo_set7155282118041507904at_nat @ Z22 ) ) )
=> ( ! [Z1: nat,Z22: fo_fmla_nat_nat] :
( ( A
= ( fo_Exists_nat_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( fo_set7155282118041507904at_nat @ Z22 ) ) )
=> ~ ! [Z1: nat,Z22: fo_fmla_nat_nat] :
( ( A
= ( fo_Forall_nat_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( fo_set7155282118041507904at_nat @ Z22 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% fo_fmla.set_cases(1)
thf(fact_85_fo__fmla_Oset__cases_I1_J,axiom,
! [E: a,A: fo_fmla_a_b] :
( ( member_a @ E @ ( fo_set1_fo_fmla_a_b @ A ) )
=> ( ! [Z1: b,Z22: list_fo_term_a] :
( ( A
= ( fo_Pred_b_a @ Z1 @ Z22 ) )
=> ! [X: fo_term_a] :
( ( member_fo_term_a @ X @ ( set_fo_term_a2 @ Z22 ) )
=> ~ ( member_a @ E @ ( fo_set_fo_term_a @ X ) ) ) )
=> ( ! [Z1: fo_term_a] :
( ? [Z22: fo_term_a] :
( A
= ( fo_Eqa_a_b @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( fo_set_fo_term_a @ Z1 ) ) )
=> ( ! [Z1: fo_term_a,Z22: fo_term_a] :
( ( A
= ( fo_Eqa_a_b @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( fo_set_fo_term_a @ Z22 ) ) )
=> ( ! [Z3: fo_fmla_a_b] :
( ( A
= ( fo_Neg_a_b @ Z3 ) )
=> ~ ( member_a @ E @ ( fo_set1_fo_fmla_a_b @ Z3 ) ) )
=> ( ! [Z1: fo_fmla_a_b] :
( ? [Z22: fo_fmla_a_b] :
( A
= ( fo_Conj_a_b @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( fo_set1_fo_fmla_a_b @ Z1 ) ) )
=> ( ! [Z1: fo_fmla_a_b,Z22: fo_fmla_a_b] :
( ( A
= ( fo_Conj_a_b @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( fo_set1_fo_fmla_a_b @ Z22 ) ) )
=> ( ! [Z1: fo_fmla_a_b] :
( ? [Z22: fo_fmla_a_b] :
( A
= ( fo_Disj_a_b @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( fo_set1_fo_fmla_a_b @ Z1 ) ) )
=> ( ! [Z1: fo_fmla_a_b,Z22: fo_fmla_a_b] :
( ( A
= ( fo_Disj_a_b @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( fo_set1_fo_fmla_a_b @ Z22 ) ) )
=> ( ! [Z1: nat,Z22: fo_fmla_a_b] :
( ( A
= ( fo_Exists_a_b @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( fo_set1_fo_fmla_a_b @ Z22 ) ) )
=> ~ ! [Z1: nat,Z22: fo_fmla_a_b] :
( ( A
= ( fo_Forall_a_b @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( fo_set1_fo_fmla_a_b @ Z22 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% fo_fmla.set_cases(1)
thf(fact_86_fo__fmla_Orel__induct,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X5: fo_fmla_nat_nat,Y: fo_fmla_nat_nat,Q: fo_fmla_nat_nat > fo_fmla_nat_nat > $o] :
( ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ X5 @ Y )
=> ( ! [A11: nat,A12: list_fo_term_nat,B11: nat,B12: list_fo_term_nat] :
( ( R2 @ A11 @ B11 )
=> ( ( list_a3012696415560123489rm_nat @ ( fo_rel1533169934621585718at_nat @ R1 ) @ A12 @ B12 )
=> ( Q @ ( fo_Pred_nat_nat @ A11 @ A12 ) @ ( fo_Pred_nat_nat @ B11 @ B12 ) ) ) )
=> ( ! [A22: $o,B22: $o] :
( ( A22 = B22 )
=> ( Q @ ( fo_Bool_nat_nat @ A22 ) @ ( fo_Bool_nat_nat @ B22 ) ) )
=> ( ! [A31: fo_term_nat,A32: fo_term_nat,B31: fo_term_nat,B32: fo_term_nat] :
( ( fo_rel1533169934621585718at_nat @ R1 @ A31 @ B31 )
=> ( ( fo_rel1533169934621585718at_nat @ R1 @ A32 @ B32 )
=> ( Q @ ( fo_Eqa_nat_nat @ A31 @ A32 ) @ ( fo_Eqa_nat_nat @ B31 @ B32 ) ) ) )
=> ( ! [A4: fo_fmla_nat_nat,B4: fo_fmla_nat_nat] :
( ( Q @ A4 @ B4 )
=> ( Q @ ( fo_Neg_nat_nat @ A4 ) @ ( fo_Neg_nat_nat @ B4 ) ) )
=> ( ! [A51: fo_fmla_nat_nat,A52: fo_fmla_nat_nat,B51: fo_fmla_nat_nat,B52: fo_fmla_nat_nat] :
( ( Q @ A51 @ B51 )
=> ( ( Q @ A52 @ B52 )
=> ( Q @ ( fo_Conj_nat_nat @ A51 @ A52 ) @ ( fo_Conj_nat_nat @ B51 @ B52 ) ) ) )
=> ( ! [A61: fo_fmla_nat_nat,A62: fo_fmla_nat_nat,B61: fo_fmla_nat_nat,B62: fo_fmla_nat_nat] :
( ( Q @ A61 @ B61 )
=> ( ( Q @ A62 @ B62 )
=> ( Q @ ( fo_Disj_nat_nat @ A61 @ A62 ) @ ( fo_Disj_nat_nat @ B61 @ B62 ) ) ) )
=> ( ! [A71: nat,A72: fo_fmla_nat_nat,B71: nat,B72: fo_fmla_nat_nat] :
( ( A71 = B71 )
=> ( ( Q @ A72 @ B72 )
=> ( Q @ ( fo_Exists_nat_nat @ A71 @ A72 ) @ ( fo_Exists_nat_nat @ B71 @ B72 ) ) ) )
=> ( ! [A81: nat,A82: fo_fmla_nat_nat,B81: nat,B82: fo_fmla_nat_nat] :
( ( A81 = B81 )
=> ( ( Q @ A82 @ B82 )
=> ( Q @ ( fo_Forall_nat_nat @ A81 @ A82 ) @ ( fo_Forall_nat_nat @ B81 @ B82 ) ) ) )
=> ( Q @ X5 @ Y ) ) ) ) ) ) ) ) ) ) ).
% fo_fmla.rel_induct
thf(fact_87_fo__fmla_Orel__induct,axiom,
! [R1: a > a > $o,R2: b > b > $o,X5: fo_fmla_a_b,Y: fo_fmla_a_b,Q: fo_fmla_a_b > fo_fmla_a_b > $o] :
( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X5 @ Y )
=> ( ! [A11: b,A12: list_fo_term_a,B11: b,B12: list_fo_term_a] :
( ( R2 @ A11 @ B11 )
=> ( ( list_a2487961086254413575term_a @ ( fo_rel_fo_term_a_a @ R1 ) @ A12 @ B12 )
=> ( Q @ ( fo_Pred_b_a @ A11 @ A12 ) @ ( fo_Pred_b_a @ B11 @ B12 ) ) ) )
=> ( ! [A22: $o,B22: $o] :
( ( A22 = B22 )
=> ( Q @ ( fo_Bool_a_b @ A22 ) @ ( fo_Bool_a_b @ B22 ) ) )
=> ( ! [A31: fo_term_a,A32: fo_term_a,B31: fo_term_a,B32: fo_term_a] :
( ( fo_rel_fo_term_a_a @ R1 @ A31 @ B31 )
=> ( ( fo_rel_fo_term_a_a @ R1 @ A32 @ B32 )
=> ( Q @ ( fo_Eqa_a_b @ A31 @ A32 ) @ ( fo_Eqa_a_b @ B31 @ B32 ) ) ) )
=> ( ! [A4: fo_fmla_a_b,B4: fo_fmla_a_b] :
( ( Q @ A4 @ B4 )
=> ( Q @ ( fo_Neg_a_b @ A4 ) @ ( fo_Neg_a_b @ B4 ) ) )
=> ( ! [A51: fo_fmla_a_b,A52: fo_fmla_a_b,B51: fo_fmla_a_b,B52: fo_fmla_a_b] :
( ( Q @ A51 @ B51 )
=> ( ( Q @ A52 @ B52 )
=> ( Q @ ( fo_Conj_a_b @ A51 @ A52 ) @ ( fo_Conj_a_b @ B51 @ B52 ) ) ) )
=> ( ! [A61: fo_fmla_a_b,A62: fo_fmla_a_b,B61: fo_fmla_a_b,B62: fo_fmla_a_b] :
( ( Q @ A61 @ B61 )
=> ( ( Q @ A62 @ B62 )
=> ( Q @ ( fo_Disj_a_b @ A61 @ A62 ) @ ( fo_Disj_a_b @ B61 @ B62 ) ) ) )
=> ( ! [A71: nat,A72: fo_fmla_a_b,B71: nat,B72: fo_fmla_a_b] :
( ( A71 = B71 )
=> ( ( Q @ A72 @ B72 )
=> ( Q @ ( fo_Exists_a_b @ A71 @ A72 ) @ ( fo_Exists_a_b @ B71 @ B72 ) ) ) )
=> ( ! [A81: nat,A82: fo_fmla_a_b,B81: nat,B82: fo_fmla_a_b] :
( ( A81 = B81 )
=> ( ( Q @ A82 @ B82 )
=> ( Q @ ( fo_Forall_a_b @ A81 @ A82 ) @ ( fo_Forall_a_b @ B81 @ B82 ) ) ) )
=> ( Q @ X5 @ Y ) ) ) ) ) ) ) ) ) ) ).
% fo_fmla.rel_induct
thf(fact_88_fo__term_Oinject_I2_J,axiom,
! [X22: nat,Y2: nat] :
( ( ( fo_Var_a @ X22 )
= ( fo_Var_a @ Y2 ) )
= ( X22 = Y2 ) ) ).
% fo_term.inject(2)
thf(fact_89_fo__term_Oinject_I1_J,axiom,
! [X1: a,Y1: a] :
( ( ( fo_Const_a @ X1 )
= ( fo_Const_a @ Y1 ) )
= ( X1 = Y1 ) ) ).
% fo_term.inject(1)
thf(fact_90_fo__term_Oinject_I1_J,axiom,
! [X1: nat,Y1: nat] :
( ( ( fo_Const_nat @ X1 )
= ( fo_Const_nat @ Y1 ) )
= ( X1 = Y1 ) ) ).
% fo_term.inject(1)
thf(fact_91_fv__fo__terms__setI,axiom,
! [M: nat,Ts2: list_fo_term_a] :
( ( member_fo_term_a @ ( fo_Var_a @ M ) @ ( set_fo_term_a2 @ Ts2 ) )
=> ( member_nat @ M @ ( fv_fo_terms_set_a @ Ts2 ) ) ) ).
% fv_fo_terms_setI
thf(fact_92_fv__fo__terms__setD,axiom,
! [M: nat,Ts2: list_fo_term_a] :
( ( member_nat @ M @ ( fv_fo_terms_set_a @ Ts2 ) )
=> ( member_fo_term_a @ ( fo_Var_a @ M ) @ ( set_fo_term_a2 @ Ts2 ) ) ) ).
% fv_fo_terms_setD
thf(fact_93_fo__term_Orel__refl__strong,axiom,
! [X5: fo_term_fo_term_a,Ra: fo_term_a > fo_term_a > $o] :
( ! [Z3: fo_term_a] :
( ( member_fo_term_a @ Z3 @ ( fo_set9081760999098759482term_a @ X5 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( fo_rel5005517413957816468term_a @ Ra @ X5 @ X5 ) ) ).
% fo_term.rel_refl_strong
thf(fact_94_fo__term_Orel__refl__strong,axiom,
! [X5: fo_ter518426593632715199_a_nat,Ra: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
( ! [Z3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Z3 @ ( fo_set5182096037969159431_a_nat @ X5 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( fo_rel6364512094553371476_a_nat @ Ra @ X5 @ X5 ) ) ).
% fo_term.rel_refl_strong
thf(fact_95_fo__term_Orel__refl__strong,axiom,
! [X5: fo_term_set_nat,Ra: set_nat > set_nat > $o] :
( ! [Z3: set_nat] :
( ( member_set_nat @ Z3 @ ( fo_set3263689133970341252et_nat @ X5 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( fo_rel2202147410123713698et_nat @ Ra @ X5 @ X5 ) ) ).
% fo_term.rel_refl_strong
thf(fact_96_fo__term_Orel__refl__strong,axiom,
! [X5: fo_term_b,Ra: b > b > $o] :
( ! [Z3: b] :
( ( member_b @ Z3 @ ( fo_set_fo_term_b @ X5 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( fo_rel_fo_term_b_b @ Ra @ X5 @ X5 ) ) ).
% fo_term.rel_refl_strong
thf(fact_97_fo__term_Orel__refl__strong,axiom,
! [X5: fo_term_a,Ra: a > a > $o] :
( ! [Z3: a] :
( ( member_a @ Z3 @ ( fo_set_fo_term_a @ X5 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( fo_rel_fo_term_a_a @ Ra @ X5 @ X5 ) ) ).
% fo_term.rel_refl_strong
thf(fact_98_fo__term_Orel__refl__strong,axiom,
! [X5: fo_term_nat,Ra: nat > nat > $o] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( fo_set_fo_term_nat @ X5 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( fo_rel1533169934621585718at_nat @ Ra @ X5 @ X5 ) ) ).
% fo_term.rel_refl_strong
thf(fact_99_fo__term_Orel__mono__strong,axiom,
! [R3: nat > nat > $o,X5: fo_term_nat,Y: fo_term_nat,Ra: nat > nat > $o] :
( ( fo_rel1533169934621585718at_nat @ R3 @ X5 @ Y )
=> ( ! [Z3: nat,Yb: nat] :
( ( member_nat @ Z3 @ ( fo_set_fo_term_nat @ X5 ) )
=> ( ( member_nat @ Yb @ ( fo_set_fo_term_nat @ Y ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( fo_rel1533169934621585718at_nat @ Ra @ X5 @ Y ) ) ) ).
% fo_term.rel_mono_strong
thf(fact_100_fo__term_Orel__mono__strong,axiom,
! [R3: b > b > $o,X5: fo_term_b,Y: fo_term_b,Ra: b > b > $o] :
( ( fo_rel_fo_term_b_b @ R3 @ X5 @ Y )
=> ( ! [Z3: b,Yb: b] :
( ( member_b @ Z3 @ ( fo_set_fo_term_b @ X5 ) )
=> ( ( member_b @ Yb @ ( fo_set_fo_term_b @ Y ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( fo_rel_fo_term_b_b @ Ra @ X5 @ Y ) ) ) ).
% fo_term.rel_mono_strong
thf(fact_101_fo__term_Orel__mono__strong,axiom,
! [R3: b > a > $o,X5: fo_term_b,Y: fo_term_a,Ra: b > a > $o] :
( ( fo_rel_fo_term_b_a @ R3 @ X5 @ Y )
=> ( ! [Z3: b,Yb: a] :
( ( member_b @ Z3 @ ( fo_set_fo_term_b @ X5 ) )
=> ( ( member_a @ Yb @ ( fo_set_fo_term_a @ Y ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( fo_rel_fo_term_b_a @ Ra @ X5 @ Y ) ) ) ).
% fo_term.rel_mono_strong
thf(fact_102_fo__term_Orel__mono__strong,axiom,
! [R3: b > nat > $o,X5: fo_term_b,Y: fo_term_nat,Ra: b > nat > $o] :
( ( fo_rel_fo_term_b_nat @ R3 @ X5 @ Y )
=> ( ! [Z3: b,Yb: nat] :
( ( member_b @ Z3 @ ( fo_set_fo_term_b @ X5 ) )
=> ( ( member_nat @ Yb @ ( fo_set_fo_term_nat @ Y ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( fo_rel_fo_term_b_nat @ Ra @ X5 @ Y ) ) ) ).
% fo_term.rel_mono_strong
thf(fact_103_fo__term_Orel__mono__strong,axiom,
! [R3: a > b > $o,X5: fo_term_a,Y: fo_term_b,Ra: a > b > $o] :
( ( fo_rel_fo_term_a_b @ R3 @ X5 @ Y )
=> ( ! [Z3: a,Yb: b] :
( ( member_a @ Z3 @ ( fo_set_fo_term_a @ X5 ) )
=> ( ( member_b @ Yb @ ( fo_set_fo_term_b @ Y ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( fo_rel_fo_term_a_b @ Ra @ X5 @ Y ) ) ) ).
% fo_term.rel_mono_strong
thf(fact_104_fo__term_Orel__mono__strong,axiom,
! [R3: a > a > $o,X5: fo_term_a,Y: fo_term_a,Ra: a > a > $o] :
( ( fo_rel_fo_term_a_a @ R3 @ X5 @ Y )
=> ( ! [Z3: a,Yb: a] :
( ( member_a @ Z3 @ ( fo_set_fo_term_a @ X5 ) )
=> ( ( member_a @ Yb @ ( fo_set_fo_term_a @ Y ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( fo_rel_fo_term_a_a @ Ra @ X5 @ Y ) ) ) ).
% fo_term.rel_mono_strong
thf(fact_105_fo__term_Orel__mono__strong,axiom,
! [R3: a > nat > $o,X5: fo_term_a,Y: fo_term_nat,Ra: a > nat > $o] :
( ( fo_rel_fo_term_a_nat @ R3 @ X5 @ Y )
=> ( ! [Z3: a,Yb: nat] :
( ( member_a @ Z3 @ ( fo_set_fo_term_a @ X5 ) )
=> ( ( member_nat @ Yb @ ( fo_set_fo_term_nat @ Y ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( fo_rel_fo_term_a_nat @ Ra @ X5 @ Y ) ) ) ).
% fo_term.rel_mono_strong
thf(fact_106_fo__term_Orel__mono__strong,axiom,
! [R3: nat > b > $o,X5: fo_term_nat,Y: fo_term_b,Ra: nat > b > $o] :
( ( fo_rel_fo_term_nat_b @ R3 @ X5 @ Y )
=> ( ! [Z3: nat,Yb: b] :
( ( member_nat @ Z3 @ ( fo_set_fo_term_nat @ X5 ) )
=> ( ( member_b @ Yb @ ( fo_set_fo_term_b @ Y ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( fo_rel_fo_term_nat_b @ Ra @ X5 @ Y ) ) ) ).
% fo_term.rel_mono_strong
thf(fact_107_fo__term_Orel__mono__strong,axiom,
! [R3: nat > a > $o,X5: fo_term_nat,Y: fo_term_a,Ra: nat > a > $o] :
( ( fo_rel_fo_term_nat_a @ R3 @ X5 @ Y )
=> ( ! [Z3: nat,Yb: a] :
( ( member_nat @ Z3 @ ( fo_set_fo_term_nat @ X5 ) )
=> ( ( member_a @ Yb @ ( fo_set_fo_term_a @ Y ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( fo_rel_fo_term_nat_a @ Ra @ X5 @ Y ) ) ) ).
% fo_term.rel_mono_strong
thf(fact_108_fo__term_Orel__mono__strong,axiom,
! [R3: fo_term_a > b > $o,X5: fo_term_fo_term_a,Y: fo_term_b,Ra: fo_term_a > b > $o] :
( ( fo_rel8200119315616323291rm_a_b @ R3 @ X5 @ Y )
=> ( ! [Z3: fo_term_a,Yb: b] :
( ( member_fo_term_a @ Z3 @ ( fo_set9081760999098759482term_a @ X5 ) )
=> ( ( member_b @ Yb @ ( fo_set_fo_term_b @ Y ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( fo_rel8200119315616323291rm_a_b @ Ra @ X5 @ Y ) ) ) ).
% fo_term.rel_mono_strong
thf(fact_109_fo__fmla_Orel__refl__strong,axiom,
! [X5: fo_fmla_nat_nat,R1a: nat > nat > $o,R2a: nat > nat > $o] :
( ! [Z1: nat] :
( ( member_nat @ Z1 @ ( fo_set7155282118041507904at_nat @ X5 ) )
=> ( R1a @ Z1 @ Z1 ) )
=> ( ! [Z22: nat] :
( ( member_nat @ Z22 @ ( fo_set4065224739847495105at_nat @ X5 ) )
=> ( R2a @ Z22 @ Z22 ) )
=> ( fo_rel4206018205878794776at_nat @ R1a @ R2a @ X5 @ X5 ) ) ) ).
% fo_fmla.rel_refl_strong
thf(fact_110_fo__fmla_Orel__refl__strong,axiom,
! [X5: fo_fmla_nat_b,R1a: nat > nat > $o,R2a: b > b > $o] :
( ! [Z1: nat] :
( ( member_nat @ Z1 @ ( fo_set3273380792390310543_nat_b @ X5 ) )
=> ( R1a @ Z1 @ Z1 ) )
=> ( ! [Z22: b] :
( ( member_b @ Z22 @ ( fo_set7690413055645365070_nat_b @ X5 ) )
=> ( R2a @ Z22 @ Z22 ) )
=> ( fo_rel8447764700627778998at_b_b @ R1a @ R2a @ X5 @ X5 ) ) ) ).
% fo_fmla.rel_refl_strong
thf(fact_111_fo__fmla_Orel__refl__strong,axiom,
! [X5: fo_fmla_nat_a,R1a: nat > nat > $o,R2a: a > a > $o] :
( ! [Z1: nat] :
( ( member_nat @ Z1 @ ( fo_set3273380792390310542_nat_a @ X5 ) )
=> ( R1a @ Z1 @ Z1 ) )
=> ( ! [Z22: a] :
( ( member_a @ Z22 @ ( fo_set7690413055645365069_nat_a @ X5 ) )
=> ( R2a @ Z22 @ Z22 ) )
=> ( fo_rel2011848245745679222at_a_a @ R1a @ R2a @ X5 @ X5 ) ) ) ).
% fo_fmla.rel_refl_strong
thf(fact_112_fo__fmla_Orel__refl__strong,axiom,
! [X5: fo_fmla_b_nat,R1a: b > b > $o,R2a: nat > nat > $o] :
( ! [Z1: b] :
( ( member_b @ Z1 @ ( fo_set476893620168418609_b_nat @ X5 ) )
=> ( R1a @ Z1 @ Z1 ) )
=> ( ! [Z22: nat] :
( ( member_nat @ Z22 @ ( fo_set4893925883423473136_b_nat @ X5 ) )
=> ( R2a @ Z22 @ Z22 ) )
=> ( fo_rel2016979423744071350at_nat @ R1a @ R2a @ X5 @ X5 ) ) ) ).
% fo_fmla.rel_refl_strong
thf(fact_113_fo__fmla_Orel__refl__strong,axiom,
! [X5: fo_fmla_b_b,R1a: b > b > $o,R2a: b > b > $o] :
( ! [Z1: b] :
( ( member_b @ Z1 @ ( fo_set1_fo_fmla_b_b @ X5 ) )
=> ( R1a @ Z1 @ Z1 ) )
=> ( ! [Z22: b] :
( ( member_b @ Z22 @ ( fo_set2_fo_fmla_b_b @ X5 ) )
=> ( R2a @ Z22 @ Z22 ) )
=> ( fo_rel5964473373680739156_b_b_b @ R1a @ R2a @ X5 @ X5 ) ) ) ).
% fo_fmla.rel_refl_strong
thf(fact_114_fo__fmla_Orel__refl__strong,axiom,
! [X5: fo_fmla_b_a,R1a: b > b > $o,R2a: a > a > $o] :
( ! [Z1: b] :
( ( member_b @ Z1 @ ( fo_set1_fo_fmla_b_a @ X5 ) )
=> ( R1a @ Z1 @ Z1 ) )
=> ( ! [Z22: a] :
( ( member_a @ Z22 @ ( fo_set2_fo_fmla_b_a @ X5 ) )
=> ( R2a @ Z22 @ Z22 ) )
=> ( fo_rel8751928955653415188_b_a_a @ R1a @ R2a @ X5 @ X5 ) ) ) ).
% fo_fmla.rel_refl_strong
thf(fact_115_fo__fmla_Orel__refl__strong,axiom,
! [X5: fo_fmla_a_nat,R1a: a > a > $o,R2a: nat > nat > $o] :
( ! [Z1: a] :
( ( member_a @ Z1 @ ( fo_set8464821328066799920_a_nat @ X5 ) )
=> ( R1a @ Z1 @ Z1 ) )
=> ( ! [Z22: nat] :
( ( member_nat @ Z22 @ ( fo_set3658481554467078639_a_nat @ X5 ) )
=> ( R2a @ Z22 @ Z22 ) )
=> ( fo_rel5500385131963405430at_nat @ R1a @ R2a @ X5 @ X5 ) ) ) ).
% fo_fmla.rel_refl_strong
thf(fact_116_fo__fmla_Orel__refl__strong,axiom,
! [X5: fo_fmla_a_a,R1a: a > a > $o,R2a: a > a > $o] :
( ! [Z1: a] :
( ( member_a @ Z1 @ ( fo_set1_fo_fmla_a_a @ X5 ) )
=> ( R1a @ Z1 @ Z1 ) )
=> ( ! [Z22: a] :
( ( member_a @ Z22 @ ( fo_set2_fo_fmla_a_a @ X5 ) )
=> ( R2a @ Z22 @ Z22 ) )
=> ( fo_rel1667748303170485972_a_a_a @ R1a @ R2a @ X5 @ X5 ) ) ) ).
% fo_fmla.rel_refl_strong
thf(fact_117_fo__fmla_Orel__refl__strong,axiom,
! [X5: fo_fmla_a_b,R1a: a > a > $o,R2a: b > b > $o] :
( ! [Z1: a] :
( ( member_a @ Z1 @ ( fo_set1_fo_fmla_a_b @ X5 ) )
=> ( R1a @ Z1 @ Z1 ) )
=> ( ! [Z22: b] :
( ( member_b @ Z22 @ ( fo_set2_fo_fmla_a_b @ X5 ) )
=> ( R2a @ Z22 @ Z22 ) )
=> ( fo_rel8103664758052585748_a_b_b @ R1a @ R2a @ X5 @ X5 ) ) ) ).
% fo_fmla.rel_refl_strong
thf(fact_118_fo__fmla_Orel__refl__strong,axiom,
! [X5: fo_fml4923465511750459541term_a,R1a: nat > nat > $o,R2a: fo_term_a > fo_term_a > $o] :
( ! [Z1: nat] :
( ( member_nat @ Z1 @ ( fo_set1740353235476592200term_a @ X5 ) )
=> ( R1a @ Z1 @ Z1 ) )
=> ( ! [Z22: fo_term_a] :
( ( member_fo_term_a @ Z22 @ ( fo_set7640127306678011015term_a @ X5 ) )
=> ( R2a @ Z22 @ Z22 ) )
=> ( fo_rel3437815733934627446term_a @ R1a @ R2a @ X5 @ X5 ) ) ) ).
% fo_fmla.rel_refl_strong
thf(fact_119_fo__fmla_Orel__mono__strong,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X5: fo_fmla_nat_nat,Y: fo_fmla_nat_nat,R1a: nat > nat > $o,R2a: nat > nat > $o] :
( ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ X5 @ Y )
=> ( ! [Z1: nat,Y13: nat] :
( ( member_nat @ Z1 @ ( fo_set7155282118041507904at_nat @ X5 ) )
=> ( ( member_nat @ Y13 @ ( fo_set7155282118041507904at_nat @ Y ) )
=> ( ( R1 @ Z1 @ Y13 )
=> ( R1a @ Z1 @ Y13 ) ) ) )
=> ( ! [Z22: nat,Y22: nat] :
( ( member_nat @ Z22 @ ( fo_set4065224739847495105at_nat @ X5 ) )
=> ( ( member_nat @ Y22 @ ( fo_set4065224739847495105at_nat @ Y ) )
=> ( ( R2 @ Z22 @ Y22 )
=> ( R2a @ Z22 @ Y22 ) ) ) )
=> ( fo_rel4206018205878794776at_nat @ R1a @ R2a @ X5 @ Y ) ) ) ) ).
% fo_fmla.rel_mono_strong
thf(fact_120_fo__fmla_Orel__mono__strong,axiom,
! [R1: nat > nat > $o,R2: b > b > $o,X5: fo_fmla_nat_b,Y: fo_fmla_nat_b,R1a: nat > nat > $o,R2a: b > b > $o] :
( ( fo_rel8447764700627778998at_b_b @ R1 @ R2 @ X5 @ Y )
=> ( ! [Z1: nat,Y13: nat] :
( ( member_nat @ Z1 @ ( fo_set3273380792390310543_nat_b @ X5 ) )
=> ( ( member_nat @ Y13 @ ( fo_set3273380792390310543_nat_b @ Y ) )
=> ( ( R1 @ Z1 @ Y13 )
=> ( R1a @ Z1 @ Y13 ) ) ) )
=> ( ! [Z22: b,Y22: b] :
( ( member_b @ Z22 @ ( fo_set7690413055645365070_nat_b @ X5 ) )
=> ( ( member_b @ Y22 @ ( fo_set7690413055645365070_nat_b @ Y ) )
=> ( ( R2 @ Z22 @ Y22 )
=> ( R2a @ Z22 @ Y22 ) ) ) )
=> ( fo_rel8447764700627778998at_b_b @ R1a @ R2a @ X5 @ Y ) ) ) ) ).
% fo_fmla.rel_mono_strong
thf(fact_121_fo__fmla_Orel__mono__strong,axiom,
! [R1: nat > nat > $o,R2: b > a > $o,X5: fo_fmla_nat_b,Y: fo_fmla_nat_a,R1a: nat > nat > $o,R2a: b > a > $o] :
( ( fo_rel8447764700627778997at_b_a @ R1 @ R2 @ X5 @ Y )
=> ( ! [Z1: nat,Y13: nat] :
( ( member_nat @ Z1 @ ( fo_set3273380792390310543_nat_b @ X5 ) )
=> ( ( member_nat @ Y13 @ ( fo_set3273380792390310542_nat_a @ Y ) )
=> ( ( R1 @ Z1 @ Y13 )
=> ( R1a @ Z1 @ Y13 ) ) ) )
=> ( ! [Z22: b,Y22: a] :
( ( member_b @ Z22 @ ( fo_set7690413055645365070_nat_b @ X5 ) )
=> ( ( member_a @ Y22 @ ( fo_set7690413055645365069_nat_a @ Y ) )
=> ( ( R2 @ Z22 @ Y22 )
=> ( R2a @ Z22 @ Y22 ) ) ) )
=> ( fo_rel8447764700627778997at_b_a @ R1a @ R2a @ X5 @ Y ) ) ) ) ).
% fo_fmla.rel_mono_strong
thf(fact_122_fo__fmla_Orel__mono__strong,axiom,
! [R1: nat > nat > $o,R2: a > b > $o,X5: fo_fmla_nat_a,Y: fo_fmla_nat_b,R1a: nat > nat > $o,R2a: a > b > $o] :
( ( fo_rel2011848245745679223at_a_b @ R1 @ R2 @ X5 @ Y )
=> ( ! [Z1: nat,Y13: nat] :
( ( member_nat @ Z1 @ ( fo_set3273380792390310542_nat_a @ X5 ) )
=> ( ( member_nat @ Y13 @ ( fo_set3273380792390310543_nat_b @ Y ) )
=> ( ( R1 @ Z1 @ Y13 )
=> ( R1a @ Z1 @ Y13 ) ) ) )
=> ( ! [Z22: a,Y22: b] :
( ( member_a @ Z22 @ ( fo_set7690413055645365069_nat_a @ X5 ) )
=> ( ( member_b @ Y22 @ ( fo_set7690413055645365070_nat_b @ Y ) )
=> ( ( R2 @ Z22 @ Y22 )
=> ( R2a @ Z22 @ Y22 ) ) ) )
=> ( fo_rel2011848245745679223at_a_b @ R1a @ R2a @ X5 @ Y ) ) ) ) ).
% fo_fmla.rel_mono_strong
thf(fact_123_fo__fmla_Orel__mono__strong,axiom,
! [R1: nat > nat > $o,R2: a > a > $o,X5: fo_fmla_nat_a,Y: fo_fmla_nat_a,R1a: nat > nat > $o,R2a: a > a > $o] :
( ( fo_rel2011848245745679222at_a_a @ R1 @ R2 @ X5 @ Y )
=> ( ! [Z1: nat,Y13: nat] :
( ( member_nat @ Z1 @ ( fo_set3273380792390310542_nat_a @ X5 ) )
=> ( ( member_nat @ Y13 @ ( fo_set3273380792390310542_nat_a @ Y ) )
=> ( ( R1 @ Z1 @ Y13 )
=> ( R1a @ Z1 @ Y13 ) ) ) )
=> ( ! [Z22: a,Y22: a] :
( ( member_a @ Z22 @ ( fo_set7690413055645365069_nat_a @ X5 ) )
=> ( ( member_a @ Y22 @ ( fo_set7690413055645365069_nat_a @ Y ) )
=> ( ( R2 @ Z22 @ Y22 )
=> ( R2a @ Z22 @ Y22 ) ) ) )
=> ( fo_rel2011848245745679222at_a_a @ R1a @ R2a @ X5 @ Y ) ) ) ) ).
% fo_fmla.rel_mono_strong
thf(fact_124_fo__fmla_Orel__mono__strong,axiom,
! [R1: nat > b > $o,R2: b > nat > $o,X5: fo_fmla_nat_b,Y: fo_fmla_b_nat,R1a: nat > b > $o,R2a: b > nat > $o] :
( ( fo_rel155626513440320054_b_nat @ R1 @ R2 @ X5 @ Y )
=> ( ! [Z1: nat,Y13: b] :
( ( member_nat @ Z1 @ ( fo_set3273380792390310543_nat_b @ X5 ) )
=> ( ( member_b @ Y13 @ ( fo_set476893620168418609_b_nat @ Y ) )
=> ( ( R1 @ Z1 @ Y13 )
=> ( R1a @ Z1 @ Y13 ) ) ) )
=> ( ! [Z22: b,Y22: nat] :
( ( member_b @ Z22 @ ( fo_set7690413055645365070_nat_b @ X5 ) )
=> ( ( member_nat @ Y22 @ ( fo_set4893925883423473136_b_nat @ Y ) )
=> ( ( R2 @ Z22 @ Y22 )
=> ( R2a @ Z22 @ Y22 ) ) ) )
=> ( fo_rel155626513440320054_b_nat @ R1a @ R2a @ X5 @ Y ) ) ) ) ).
% fo_fmla.rel_mono_strong
thf(fact_125_fo__fmla_Orel__mono__strong,axiom,
! [R1: nat > b > $o,R2: b > b > $o,X5: fo_fmla_nat_b,Y: fo_fmla_b_b,R1a: nat > b > $o,R2a: b > b > $o] :
( ( fo_rel7296952225388976665_b_b_b @ R1 @ R2 @ X5 @ Y )
=> ( ! [Z1: nat,Y13: b] :
( ( member_nat @ Z1 @ ( fo_set3273380792390310543_nat_b @ X5 ) )
=> ( ( member_b @ Y13 @ ( fo_set1_fo_fmla_b_b @ Y ) )
=> ( ( R1 @ Z1 @ Y13 )
=> ( R1a @ Z1 @ Y13 ) ) ) )
=> ( ! [Z22: b,Y22: b] :
( ( member_b @ Z22 @ ( fo_set7690413055645365070_nat_b @ X5 ) )
=> ( ( member_b @ Y22 @ ( fo_set2_fo_fmla_b_b @ Y ) )
=> ( ( R2 @ Z22 @ Y22 )
=> ( R2a @ Z22 @ Y22 ) ) ) )
=> ( fo_rel7296952225388976665_b_b_b @ R1a @ R2a @ X5 @ Y ) ) ) ) ).
% fo_fmla.rel_mono_strong
thf(fact_126_fo__fmla_Orel__mono__strong,axiom,
! [R1: nat > b > $o,R2: b > a > $o,X5: fo_fmla_nat_b,Y: fo_fmla_b_a,R1a: nat > b > $o,R2a: b > a > $o] :
( ( fo_rel7296952225388976664_b_b_a @ R1 @ R2 @ X5 @ Y )
=> ( ! [Z1: nat,Y13: b] :
( ( member_nat @ Z1 @ ( fo_set3273380792390310543_nat_b @ X5 ) )
=> ( ( member_b @ Y13 @ ( fo_set1_fo_fmla_b_a @ Y ) )
=> ( ( R1 @ Z1 @ Y13 )
=> ( R1a @ Z1 @ Y13 ) ) ) )
=> ( ! [Z22: b,Y22: a] :
( ( member_b @ Z22 @ ( fo_set7690413055645365070_nat_b @ X5 ) )
=> ( ( member_a @ Y22 @ ( fo_set2_fo_fmla_b_a @ Y ) )
=> ( ( R2 @ Z22 @ Y22 )
=> ( R2a @ Z22 @ Y22 ) ) ) )
=> ( fo_rel7296952225388976664_b_b_a @ R1a @ R2a @ X5 @ Y ) ) ) ) ).
% fo_fmla.rel_mono_strong
thf(fact_127_fo__fmla_Orel__mono__strong,axiom,
! [R1: nat > b > $o,R2: a > nat > $o,X5: fo_fmla_nat_a,Y: fo_fmla_b_nat,R1a: nat > b > $o,R2a: a > nat > $o] :
( ( fo_rel8143554221338701365_a_nat @ R1 @ R2 @ X5 @ Y )
=> ( ! [Z1: nat,Y13: b] :
( ( member_nat @ Z1 @ ( fo_set3273380792390310542_nat_a @ X5 ) )
=> ( ( member_b @ Y13 @ ( fo_set476893620168418609_b_nat @ Y ) )
=> ( ( R1 @ Z1 @ Y13 )
=> ( R1a @ Z1 @ Y13 ) ) ) )
=> ( ! [Z22: a,Y22: nat] :
( ( member_a @ Z22 @ ( fo_set7690413055645365069_nat_a @ X5 ) )
=> ( ( member_nat @ Y22 @ ( fo_set4893925883423473136_b_nat @ Y ) )
=> ( ( R2 @ Z22 @ Y22 )
=> ( R2a @ Z22 @ Y22 ) ) ) )
=> ( fo_rel8143554221338701365_a_nat @ R1a @ R2a @ X5 @ Y ) ) ) ) ).
% fo_fmla.rel_mono_strong
thf(fact_128_fo__fmla_Orel__mono__strong,axiom,
! [R1: nat > b > $o,R2: a > b > $o,X5: fo_fmla_nat_a,Y: fo_fmla_b_b,R1a: nat > b > $o,R2a: a > b > $o] :
( ( fo_rel861035770506876890_b_a_b @ R1 @ R2 @ X5 @ Y )
=> ( ! [Z1: nat,Y13: b] :
( ( member_nat @ Z1 @ ( fo_set3273380792390310542_nat_a @ X5 ) )
=> ( ( member_b @ Y13 @ ( fo_set1_fo_fmla_b_b @ Y ) )
=> ( ( R1 @ Z1 @ Y13 )
=> ( R1a @ Z1 @ Y13 ) ) ) )
=> ( ! [Z22: a,Y22: b] :
( ( member_a @ Z22 @ ( fo_set7690413055645365069_nat_a @ X5 ) )
=> ( ( member_b @ Y22 @ ( fo_set2_fo_fmla_b_b @ Y ) )
=> ( ( R2 @ Z22 @ Y22 )
=> ( R2a @ Z22 @ Y22 ) ) ) )
=> ( fo_rel861035770506876890_b_a_b @ R1a @ R2a @ X5 @ Y ) ) ) ) ).
% fo_fmla.rel_mono_strong
thf(fact_129_fv__fo__term__list_Ocases,axiom,
! [X5: fo_term_a] :
( ! [N: nat] :
( X5
!= ( fo_Var_a @ N ) )
=> ~ ! [V: a] :
( X5
!= ( fo_Const_a @ V ) ) ) ).
% fv_fo_term_list.cases
thf(fact_130_fv__fo__term__list_Ocases,axiom,
! [X5: fo_term_nat] :
( ! [N: nat] :
( X5
!= ( fo_Var_nat @ N ) )
=> ~ ! [V: nat] :
( X5
!= ( fo_Const_nat @ V ) ) ) ).
% fv_fo_term_list.cases
thf(fact_131_fo__term_Orel__refl,axiom,
! [Ra: a > a > $o,X5: fo_term_a] :
( ! [X: a] : ( Ra @ X @ X )
=> ( fo_rel_fo_term_a_a @ Ra @ X5 @ X5 ) ) ).
% fo_term.rel_refl
thf(fact_132_fo__term_Orel__refl,axiom,
! [Ra: nat > nat > $o,X5: fo_term_nat] :
( ! [X: nat] : ( Ra @ X @ X )
=> ( fo_rel1533169934621585718at_nat @ Ra @ X5 @ X5 ) ) ).
% fo_term.rel_refl
thf(fact_133_fo__term_Orel__cong,axiom,
! [X5: fo_term_nat,Ya: fo_term_nat,Y: fo_term_nat,Xa: fo_term_nat,R3: nat > nat > $o,Ra: nat > nat > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: nat,Yb: nat] :
( ( member_nat @ Z3 @ ( fo_set_fo_term_nat @ Ya ) )
=> ( ( member_nat @ Yb @ ( fo_set_fo_term_nat @ Xa ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( fo_rel1533169934621585718at_nat @ R3 @ X5 @ Y )
= ( fo_rel1533169934621585718at_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fo_term.rel_cong
thf(fact_134_fo__term_Orel__cong,axiom,
! [X5: fo_term_b,Ya: fo_term_b,Y: fo_term_b,Xa: fo_term_b,R3: b > b > $o,Ra: b > b > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: b,Yb: b] :
( ( member_b @ Z3 @ ( fo_set_fo_term_b @ Ya ) )
=> ( ( member_b @ Yb @ ( fo_set_fo_term_b @ Xa ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( fo_rel_fo_term_b_b @ R3 @ X5 @ Y )
= ( fo_rel_fo_term_b_b @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fo_term.rel_cong
thf(fact_135_fo__term_Orel__cong,axiom,
! [X5: fo_term_b,Ya: fo_term_b,Y: fo_term_a,Xa: fo_term_a,R3: b > a > $o,Ra: b > a > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: b,Yb: a] :
( ( member_b @ Z3 @ ( fo_set_fo_term_b @ Ya ) )
=> ( ( member_a @ Yb @ ( fo_set_fo_term_a @ Xa ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( fo_rel_fo_term_b_a @ R3 @ X5 @ Y )
= ( fo_rel_fo_term_b_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fo_term.rel_cong
thf(fact_136_fo__term_Orel__cong,axiom,
! [X5: fo_term_b,Ya: fo_term_b,Y: fo_term_nat,Xa: fo_term_nat,R3: b > nat > $o,Ra: b > nat > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: b,Yb: nat] :
( ( member_b @ Z3 @ ( fo_set_fo_term_b @ Ya ) )
=> ( ( member_nat @ Yb @ ( fo_set_fo_term_nat @ Xa ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( fo_rel_fo_term_b_nat @ R3 @ X5 @ Y )
= ( fo_rel_fo_term_b_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fo_term.rel_cong
thf(fact_137_fo__term_Orel__cong,axiom,
! [X5: fo_term_a,Ya: fo_term_a,Y: fo_term_b,Xa: fo_term_b,R3: a > b > $o,Ra: a > b > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: a,Yb: b] :
( ( member_a @ Z3 @ ( fo_set_fo_term_a @ Ya ) )
=> ( ( member_b @ Yb @ ( fo_set_fo_term_b @ Xa ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( fo_rel_fo_term_a_b @ R3 @ X5 @ Y )
= ( fo_rel_fo_term_a_b @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fo_term.rel_cong
thf(fact_138_fo__term_Orel__cong,axiom,
! [X5: fo_term_a,Ya: fo_term_a,Y: fo_term_a,Xa: fo_term_a,R3: a > a > $o,Ra: a > a > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: a,Yb: a] :
( ( member_a @ Z3 @ ( fo_set_fo_term_a @ Ya ) )
=> ( ( member_a @ Yb @ ( fo_set_fo_term_a @ Xa ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( fo_rel_fo_term_a_a @ R3 @ X5 @ Y )
= ( fo_rel_fo_term_a_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fo_term.rel_cong
thf(fact_139_fo__term_Orel__cong,axiom,
! [X5: fo_term_a,Ya: fo_term_a,Y: fo_term_nat,Xa: fo_term_nat,R3: a > nat > $o,Ra: a > nat > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: a,Yb: nat] :
( ( member_a @ Z3 @ ( fo_set_fo_term_a @ Ya ) )
=> ( ( member_nat @ Yb @ ( fo_set_fo_term_nat @ Xa ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( fo_rel_fo_term_a_nat @ R3 @ X5 @ Y )
= ( fo_rel_fo_term_a_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fo_term.rel_cong
thf(fact_140_fo__term_Orel__cong,axiom,
! [X5: fo_term_nat,Ya: fo_term_nat,Y: fo_term_b,Xa: fo_term_b,R3: nat > b > $o,Ra: nat > b > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: nat,Yb: b] :
( ( member_nat @ Z3 @ ( fo_set_fo_term_nat @ Ya ) )
=> ( ( member_b @ Yb @ ( fo_set_fo_term_b @ Xa ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( fo_rel_fo_term_nat_b @ R3 @ X5 @ Y )
= ( fo_rel_fo_term_nat_b @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fo_term.rel_cong
thf(fact_141_fo__term_Orel__cong,axiom,
! [X5: fo_term_nat,Ya: fo_term_nat,Y: fo_term_a,Xa: fo_term_a,R3: nat > a > $o,Ra: nat > a > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: nat,Yb: a] :
( ( member_nat @ Z3 @ ( fo_set_fo_term_nat @ Ya ) )
=> ( ( member_a @ Yb @ ( fo_set_fo_term_a @ Xa ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( fo_rel_fo_term_nat_a @ R3 @ X5 @ Y )
= ( fo_rel_fo_term_nat_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fo_term.rel_cong
thf(fact_142_fo__term_Orel__cong,axiom,
! [X5: fo_term_fo_term_a,Ya: fo_term_fo_term_a,Y: fo_term_b,Xa: fo_term_b,R3: fo_term_a > b > $o,Ra: fo_term_a > b > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: fo_term_a,Yb: b] :
( ( member_fo_term_a @ Z3 @ ( fo_set9081760999098759482term_a @ Ya ) )
=> ( ( member_b @ Yb @ ( fo_set_fo_term_b @ Xa ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( fo_rel8200119315616323291rm_a_b @ R3 @ X5 @ Y )
= ( fo_rel8200119315616323291rm_a_b @ Ra @ Ya @ Xa ) ) ) ) ) ).
% fo_term.rel_cong
thf(fact_143_fo__fmla_Orel__refl,axiom,
! [R1a: a > a > $o,R2a: b > b > $o,X5: fo_fmla_a_b] :
( ! [X: a] : ( R1a @ X @ X )
=> ( ! [X: b] : ( R2a @ X @ X )
=> ( fo_rel8103664758052585748_a_b_b @ R1a @ R2a @ X5 @ X5 ) ) ) ).
% fo_fmla.rel_refl
thf(fact_144_fo__fmla_Orel__refl,axiom,
! [R1a: nat > nat > $o,R2a: nat > nat > $o,X5: fo_fmla_nat_nat] :
( ! [X: nat] : ( R1a @ X @ X )
=> ( ! [X: nat] : ( R2a @ X @ X )
=> ( fo_rel4206018205878794776at_nat @ R1a @ R2a @ X5 @ X5 ) ) ) ).
% fo_fmla.rel_refl
thf(fact_145_fo__fmla_Orel__cong,axiom,
! [X5: fo_fmla_nat_nat,Ya: fo_fmla_nat_nat,Y: fo_fmla_nat_nat,Xa: fo_fmla_nat_nat,R1: nat > nat > $o,R1a: nat > nat > $o,R2: nat > nat > $o,R2a: nat > nat > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z1: nat,Y13: nat] :
( ( member_nat @ Z1 @ ( fo_set7155282118041507904at_nat @ Ya ) )
=> ( ( member_nat @ Y13 @ ( fo_set7155282118041507904at_nat @ Xa ) )
=> ( ( R1 @ Z1 @ Y13 )
= ( R1a @ Z1 @ Y13 ) ) ) )
=> ( ! [Z22: nat,Y22: nat] :
( ( member_nat @ Z22 @ ( fo_set4065224739847495105at_nat @ Ya ) )
=> ( ( member_nat @ Y22 @ ( fo_set4065224739847495105at_nat @ Xa ) )
=> ( ( R2 @ Z22 @ Y22 )
= ( R2a @ Z22 @ Y22 ) ) ) )
=> ( ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ X5 @ Y )
= ( fo_rel4206018205878794776at_nat @ R1a @ R2a @ Ya @ Xa ) ) ) ) ) ) ).
% fo_fmla.rel_cong
thf(fact_146_fo__fmla_Orel__cong,axiom,
! [X5: fo_fmla_nat_b,Ya: fo_fmla_nat_b,Y: fo_fmla_nat_b,Xa: fo_fmla_nat_b,R1: nat > nat > $o,R1a: nat > nat > $o,R2: b > b > $o,R2a: b > b > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z1: nat,Y13: nat] :
( ( member_nat @ Z1 @ ( fo_set3273380792390310543_nat_b @ Ya ) )
=> ( ( member_nat @ Y13 @ ( fo_set3273380792390310543_nat_b @ Xa ) )
=> ( ( R1 @ Z1 @ Y13 )
= ( R1a @ Z1 @ Y13 ) ) ) )
=> ( ! [Z22: b,Y22: b] :
( ( member_b @ Z22 @ ( fo_set7690413055645365070_nat_b @ Ya ) )
=> ( ( member_b @ Y22 @ ( fo_set7690413055645365070_nat_b @ Xa ) )
=> ( ( R2 @ Z22 @ Y22 )
= ( R2a @ Z22 @ Y22 ) ) ) )
=> ( ( fo_rel8447764700627778998at_b_b @ R1 @ R2 @ X5 @ Y )
= ( fo_rel8447764700627778998at_b_b @ R1a @ R2a @ Ya @ Xa ) ) ) ) ) ) ).
% fo_fmla.rel_cong
thf(fact_147_fo__fmla_Orel__cong,axiom,
! [X5: fo_fmla_nat_b,Ya: fo_fmla_nat_b,Y: fo_fmla_nat_a,Xa: fo_fmla_nat_a,R1: nat > nat > $o,R1a: nat > nat > $o,R2: b > a > $o,R2a: b > a > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z1: nat,Y13: nat] :
( ( member_nat @ Z1 @ ( fo_set3273380792390310543_nat_b @ Ya ) )
=> ( ( member_nat @ Y13 @ ( fo_set3273380792390310542_nat_a @ Xa ) )
=> ( ( R1 @ Z1 @ Y13 )
= ( R1a @ Z1 @ Y13 ) ) ) )
=> ( ! [Z22: b,Y22: a] :
( ( member_b @ Z22 @ ( fo_set7690413055645365070_nat_b @ Ya ) )
=> ( ( member_a @ Y22 @ ( fo_set7690413055645365069_nat_a @ Xa ) )
=> ( ( R2 @ Z22 @ Y22 )
= ( R2a @ Z22 @ Y22 ) ) ) )
=> ( ( fo_rel8447764700627778997at_b_a @ R1 @ R2 @ X5 @ Y )
= ( fo_rel8447764700627778997at_b_a @ R1a @ R2a @ Ya @ Xa ) ) ) ) ) ) ).
% fo_fmla.rel_cong
thf(fact_148_fo__fmla_Orel__cong,axiom,
! [X5: fo_fmla_nat_a,Ya: fo_fmla_nat_a,Y: fo_fmla_nat_b,Xa: fo_fmla_nat_b,R1: nat > nat > $o,R1a: nat > nat > $o,R2: a > b > $o,R2a: a > b > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z1: nat,Y13: nat] :
( ( member_nat @ Z1 @ ( fo_set3273380792390310542_nat_a @ Ya ) )
=> ( ( member_nat @ Y13 @ ( fo_set3273380792390310543_nat_b @ Xa ) )
=> ( ( R1 @ Z1 @ Y13 )
= ( R1a @ Z1 @ Y13 ) ) ) )
=> ( ! [Z22: a,Y22: b] :
( ( member_a @ Z22 @ ( fo_set7690413055645365069_nat_a @ Ya ) )
=> ( ( member_b @ Y22 @ ( fo_set7690413055645365070_nat_b @ Xa ) )
=> ( ( R2 @ Z22 @ Y22 )
= ( R2a @ Z22 @ Y22 ) ) ) )
=> ( ( fo_rel2011848245745679223at_a_b @ R1 @ R2 @ X5 @ Y )
= ( fo_rel2011848245745679223at_a_b @ R1a @ R2a @ Ya @ Xa ) ) ) ) ) ) ).
% fo_fmla.rel_cong
thf(fact_149_fo__fmla_Orel__cong,axiom,
! [X5: fo_fmla_nat_a,Ya: fo_fmla_nat_a,Y: fo_fmla_nat_a,Xa: fo_fmla_nat_a,R1: nat > nat > $o,R1a: nat > nat > $o,R2: a > a > $o,R2a: a > a > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z1: nat,Y13: nat] :
( ( member_nat @ Z1 @ ( fo_set3273380792390310542_nat_a @ Ya ) )
=> ( ( member_nat @ Y13 @ ( fo_set3273380792390310542_nat_a @ Xa ) )
=> ( ( R1 @ Z1 @ Y13 )
= ( R1a @ Z1 @ Y13 ) ) ) )
=> ( ! [Z22: a,Y22: a] :
( ( member_a @ Z22 @ ( fo_set7690413055645365069_nat_a @ Ya ) )
=> ( ( member_a @ Y22 @ ( fo_set7690413055645365069_nat_a @ Xa ) )
=> ( ( R2 @ Z22 @ Y22 )
= ( R2a @ Z22 @ Y22 ) ) ) )
=> ( ( fo_rel2011848245745679222at_a_a @ R1 @ R2 @ X5 @ Y )
= ( fo_rel2011848245745679222at_a_a @ R1a @ R2a @ Ya @ Xa ) ) ) ) ) ) ).
% fo_fmla.rel_cong
thf(fact_150_fo__fmla_Orel__cong,axiom,
! [X5: fo_fmla_nat_b,Ya: fo_fmla_nat_b,Y: fo_fmla_b_nat,Xa: fo_fmla_b_nat,R1: nat > b > $o,R1a: nat > b > $o,R2: b > nat > $o,R2a: b > nat > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z1: nat,Y13: b] :
( ( member_nat @ Z1 @ ( fo_set3273380792390310543_nat_b @ Ya ) )
=> ( ( member_b @ Y13 @ ( fo_set476893620168418609_b_nat @ Xa ) )
=> ( ( R1 @ Z1 @ Y13 )
= ( R1a @ Z1 @ Y13 ) ) ) )
=> ( ! [Z22: b,Y22: nat] :
( ( member_b @ Z22 @ ( fo_set7690413055645365070_nat_b @ Ya ) )
=> ( ( member_nat @ Y22 @ ( fo_set4893925883423473136_b_nat @ Xa ) )
=> ( ( R2 @ Z22 @ Y22 )
= ( R2a @ Z22 @ Y22 ) ) ) )
=> ( ( fo_rel155626513440320054_b_nat @ R1 @ R2 @ X5 @ Y )
= ( fo_rel155626513440320054_b_nat @ R1a @ R2a @ Ya @ Xa ) ) ) ) ) ) ).
% fo_fmla.rel_cong
thf(fact_151_fo__fmla_Orel__cong,axiom,
! [X5: fo_fmla_nat_b,Ya: fo_fmla_nat_b,Y: fo_fmla_b_b,Xa: fo_fmla_b_b,R1: nat > b > $o,R1a: nat > b > $o,R2: b > b > $o,R2a: b > b > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z1: nat,Y13: b] :
( ( member_nat @ Z1 @ ( fo_set3273380792390310543_nat_b @ Ya ) )
=> ( ( member_b @ Y13 @ ( fo_set1_fo_fmla_b_b @ Xa ) )
=> ( ( R1 @ Z1 @ Y13 )
= ( R1a @ Z1 @ Y13 ) ) ) )
=> ( ! [Z22: b,Y22: b] :
( ( member_b @ Z22 @ ( fo_set7690413055645365070_nat_b @ Ya ) )
=> ( ( member_b @ Y22 @ ( fo_set2_fo_fmla_b_b @ Xa ) )
=> ( ( R2 @ Z22 @ Y22 )
= ( R2a @ Z22 @ Y22 ) ) ) )
=> ( ( fo_rel7296952225388976665_b_b_b @ R1 @ R2 @ X5 @ Y )
= ( fo_rel7296952225388976665_b_b_b @ R1a @ R2a @ Ya @ Xa ) ) ) ) ) ) ).
% fo_fmla.rel_cong
thf(fact_152_fo__fmla_Orel__cong,axiom,
! [X5: fo_fmla_nat_b,Ya: fo_fmla_nat_b,Y: fo_fmla_b_a,Xa: fo_fmla_b_a,R1: nat > b > $o,R1a: nat > b > $o,R2: b > a > $o,R2a: b > a > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z1: nat,Y13: b] :
( ( member_nat @ Z1 @ ( fo_set3273380792390310543_nat_b @ Ya ) )
=> ( ( member_b @ Y13 @ ( fo_set1_fo_fmla_b_a @ Xa ) )
=> ( ( R1 @ Z1 @ Y13 )
= ( R1a @ Z1 @ Y13 ) ) ) )
=> ( ! [Z22: b,Y22: a] :
( ( member_b @ Z22 @ ( fo_set7690413055645365070_nat_b @ Ya ) )
=> ( ( member_a @ Y22 @ ( fo_set2_fo_fmla_b_a @ Xa ) )
=> ( ( R2 @ Z22 @ Y22 )
= ( R2a @ Z22 @ Y22 ) ) ) )
=> ( ( fo_rel7296952225388976664_b_b_a @ R1 @ R2 @ X5 @ Y )
= ( fo_rel7296952225388976664_b_b_a @ R1a @ R2a @ Ya @ Xa ) ) ) ) ) ) ).
% fo_fmla.rel_cong
thf(fact_153_fo__fmla_Orel__cong,axiom,
! [X5: fo_fmla_nat_a,Ya: fo_fmla_nat_a,Y: fo_fmla_b_nat,Xa: fo_fmla_b_nat,R1: nat > b > $o,R1a: nat > b > $o,R2: a > nat > $o,R2a: a > nat > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z1: nat,Y13: b] :
( ( member_nat @ Z1 @ ( fo_set3273380792390310542_nat_a @ Ya ) )
=> ( ( member_b @ Y13 @ ( fo_set476893620168418609_b_nat @ Xa ) )
=> ( ( R1 @ Z1 @ Y13 )
= ( R1a @ Z1 @ Y13 ) ) ) )
=> ( ! [Z22: a,Y22: nat] :
( ( member_a @ Z22 @ ( fo_set7690413055645365069_nat_a @ Ya ) )
=> ( ( member_nat @ Y22 @ ( fo_set4893925883423473136_b_nat @ Xa ) )
=> ( ( R2 @ Z22 @ Y22 )
= ( R2a @ Z22 @ Y22 ) ) ) )
=> ( ( fo_rel8143554221338701365_a_nat @ R1 @ R2 @ X5 @ Y )
= ( fo_rel8143554221338701365_a_nat @ R1a @ R2a @ Ya @ Xa ) ) ) ) ) ) ).
% fo_fmla.rel_cong
thf(fact_154_fo__fmla_Orel__cong,axiom,
! [X5: fo_fmla_nat_a,Ya: fo_fmla_nat_a,Y: fo_fmla_b_b,Xa: fo_fmla_b_b,R1: nat > b > $o,R1a: nat > b > $o,R2: a > b > $o,R2a: a > b > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z1: nat,Y13: b] :
( ( member_nat @ Z1 @ ( fo_set3273380792390310542_nat_a @ Ya ) )
=> ( ( member_b @ Y13 @ ( fo_set1_fo_fmla_b_b @ Xa ) )
=> ( ( R1 @ Z1 @ Y13 )
= ( R1a @ Z1 @ Y13 ) ) ) )
=> ( ! [Z22: a,Y22: b] :
( ( member_a @ Z22 @ ( fo_set7690413055645365069_nat_a @ Ya ) )
=> ( ( member_b @ Y22 @ ( fo_set2_fo_fmla_b_b @ Xa ) )
=> ( ( R2 @ Z22 @ Y22 )
= ( R2a @ Z22 @ Y22 ) ) ) )
=> ( ( fo_rel861035770506876890_b_a_b @ R1 @ R2 @ X5 @ Y )
= ( fo_rel861035770506876890_b_a_b @ R1a @ R2a @ Ya @ Xa ) ) ) ) ) ) ).
% fo_fmla.rel_cong
thf(fact_155_list__fo__term_Ocases,axiom,
! [X5: fo_term_a] :
( ! [C2: a] :
( X5
!= ( fo_Const_a @ C2 ) )
=> ~ ! [V: nat] :
( X5
!= ( fo_Var_a @ V ) ) ) ).
% list_fo_term.cases
thf(fact_156_list__fo__term_Ocases,axiom,
! [X5: fo_term_nat] :
( ! [C2: nat] :
( X5
!= ( fo_Const_nat @ C2 ) )
=> ~ ! [V: nat] :
( X5
!= ( fo_Var_nat @ V ) ) ) ).
% list_fo_term.cases
thf(fact_157_fo__term_Oset__intros,axiom,
! [X1: fo_term_a] : ( member_fo_term_a @ X1 @ ( fo_set9081760999098759482term_a @ ( fo_Const_fo_term_a @ X1 ) ) ) ).
% fo_term.set_intros
thf(fact_158_fo__term_Oset__intros,axiom,
! [X1: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X1 @ ( fo_set5182096037969159431_a_nat @ ( fo_Con7354235876602600827_a_nat @ X1 ) ) ) ).
% fo_term.set_intros
thf(fact_159_fo__term_Oset__intros,axiom,
! [X1: set_nat] : ( member_set_nat @ X1 @ ( fo_set3263689133970341252et_nat @ ( fo_Const_set_nat @ X1 ) ) ) ).
% fo_term.set_intros
thf(fact_160_fo__term_Oset__intros,axiom,
! [X1: b] : ( member_b @ X1 @ ( fo_set_fo_term_b @ ( fo_Const_b @ X1 ) ) ) ).
% fo_term.set_intros
thf(fact_161_fo__term_Oset__intros,axiom,
! [X1: a] : ( member_a @ X1 @ ( fo_set_fo_term_a @ ( fo_Const_a @ X1 ) ) ) ).
% fo_term.set_intros
thf(fact_162_fo__term_Oset__intros,axiom,
! [X1: nat] : ( member_nat @ X1 @ ( fo_set_fo_term_nat @ ( fo_Const_nat @ X1 ) ) ) ).
% fo_term.set_intros
thf(fact_163_fo__term_Orel__induct,axiom,
! [R3: a > nat > $o,X5: fo_term_a,Y: fo_term_nat,Q: fo_term_a > fo_term_nat > $o] :
( ( fo_rel_fo_term_a_nat @ R3 @ X5 @ Y )
=> ( ! [A1: a,B1: nat] :
( ( R3 @ A1 @ B1 )
=> ( Q @ ( fo_Const_a @ A1 ) @ ( fo_Const_nat @ B1 ) ) )
=> ( ! [A22: nat,B22: nat] :
( ( A22 = B22 )
=> ( Q @ ( fo_Var_a @ A22 ) @ ( fo_Var_nat @ B22 ) ) )
=> ( Q @ X5 @ Y ) ) ) ) ).
% fo_term.rel_induct
thf(fact_164_fo__term_Orel__induct,axiom,
! [R3: nat > a > $o,X5: fo_term_nat,Y: fo_term_a,Q: fo_term_nat > fo_term_a > $o] :
( ( fo_rel_fo_term_nat_a @ R3 @ X5 @ Y )
=> ( ! [A1: nat,B1: a] :
( ( R3 @ A1 @ B1 )
=> ( Q @ ( fo_Const_nat @ A1 ) @ ( fo_Const_a @ B1 ) ) )
=> ( ! [A22: nat,B22: nat] :
( ( A22 = B22 )
=> ( Q @ ( fo_Var_nat @ A22 ) @ ( fo_Var_a @ B22 ) ) )
=> ( Q @ X5 @ Y ) ) ) ) ).
% fo_term.rel_induct
thf(fact_165_fo__term_Orel__induct,axiom,
! [R3: a > a > $o,X5: fo_term_a,Y: fo_term_a,Q: fo_term_a > fo_term_a > $o] :
( ( fo_rel_fo_term_a_a @ R3 @ X5 @ Y )
=> ( ! [A1: a,B1: a] :
( ( R3 @ A1 @ B1 )
=> ( Q @ ( fo_Const_a @ A1 ) @ ( fo_Const_a @ B1 ) ) )
=> ( ! [A22: nat,B22: nat] :
( ( A22 = B22 )
=> ( Q @ ( fo_Var_a @ A22 ) @ ( fo_Var_a @ B22 ) ) )
=> ( Q @ X5 @ Y ) ) ) ) ).
% fo_term.rel_induct
thf(fact_166_fo__term_Orel__induct,axiom,
! [R3: nat > nat > $o,X5: fo_term_nat,Y: fo_term_nat,Q: fo_term_nat > fo_term_nat > $o] :
( ( fo_rel1533169934621585718at_nat @ R3 @ X5 @ Y )
=> ( ! [A1: nat,B1: nat] :
( ( R3 @ A1 @ B1 )
=> ( Q @ ( fo_Const_nat @ A1 ) @ ( fo_Const_nat @ B1 ) ) )
=> ( ! [A22: nat,B22: nat] :
( ( A22 = B22 )
=> ( Q @ ( fo_Var_nat @ A22 ) @ ( fo_Var_nat @ B22 ) ) )
=> ( Q @ X5 @ Y ) ) ) ) ).
% fo_term.rel_induct
thf(fact_167_fo__term_Oset__cases,axiom,
! [E: fo_term_a,A: fo_term_fo_term_a] :
( ( member_fo_term_a @ E @ ( fo_set9081760999098759482term_a @ A ) )
=> ( A
= ( fo_Const_fo_term_a @ E ) ) ) ).
% fo_term.set_cases
thf(fact_168_fo__term_Oset__cases,axiom,
! [E: list_Sum_sum_a_nat,A: fo_ter518426593632715199_a_nat] :
( ( member408289922725080238_a_nat @ E @ ( fo_set5182096037969159431_a_nat @ A ) )
=> ( A
= ( fo_Con7354235876602600827_a_nat @ E ) ) ) ).
% fo_term.set_cases
thf(fact_169_fo__term_Oset__cases,axiom,
! [E: set_nat,A: fo_term_set_nat] :
( ( member_set_nat @ E @ ( fo_set3263689133970341252et_nat @ A ) )
=> ( A
= ( fo_Const_set_nat @ E ) ) ) ).
% fo_term.set_cases
thf(fact_170_fo__term_Oset__cases,axiom,
! [E: b,A: fo_term_b] :
( ( member_b @ E @ ( fo_set_fo_term_b @ A ) )
=> ( A
= ( fo_Const_b @ E ) ) ) ).
% fo_term.set_cases
thf(fact_171_fo__term_Oset__cases,axiom,
! [E: a,A: fo_term_a] :
( ( member_a @ E @ ( fo_set_fo_term_a @ A ) )
=> ( A
= ( fo_Const_a @ E ) ) ) ).
% fo_term.set_cases
thf(fact_172_fo__term_Oset__cases,axiom,
! [E: nat,A: fo_term_nat] :
( ( member_nat @ E @ ( fo_set_fo_term_nat @ A ) )
=> ( A
= ( fo_Const_nat @ E ) ) ) ).
% fo_term.set_cases
thf(fact_173_fo__term_Orel__cases,axiom,
! [R3: a > nat > $o,A: fo_term_a,B: fo_term_nat] :
( ( fo_rel_fo_term_a_nat @ R3 @ A @ B )
=> ( ! [X: a] :
( ( A
= ( fo_Const_a @ X ) )
=> ! [Y5: nat] :
( ( B
= ( fo_Const_nat @ Y5 ) )
=> ~ ( R3 @ X @ Y5 ) ) )
=> ~ ! [Xa2: nat] :
( ( A
= ( fo_Var_a @ Xa2 ) )
=> ! [Ya2: nat] :
( ( B
= ( fo_Var_nat @ Ya2 ) )
=> ( Xa2 != Ya2 ) ) ) ) ) ).
% fo_term.rel_cases
thf(fact_174_fo__term_Orel__cases,axiom,
! [R3: nat > a > $o,A: fo_term_nat,B: fo_term_a] :
( ( fo_rel_fo_term_nat_a @ R3 @ A @ B )
=> ( ! [X: nat] :
( ( A
= ( fo_Const_nat @ X ) )
=> ! [Y5: a] :
( ( B
= ( fo_Const_a @ Y5 ) )
=> ~ ( R3 @ X @ Y5 ) ) )
=> ~ ! [Xa2: nat] :
( ( A
= ( fo_Var_nat @ Xa2 ) )
=> ! [Ya2: nat] :
( ( B
= ( fo_Var_a @ Ya2 ) )
=> ( Xa2 != Ya2 ) ) ) ) ) ).
% fo_term.rel_cases
thf(fact_175_fo__term_Orel__cases,axiom,
! [R3: a > a > $o,A: fo_term_a,B: fo_term_a] :
( ( fo_rel_fo_term_a_a @ R3 @ A @ B )
=> ( ! [X: a] :
( ( A
= ( fo_Const_a @ X ) )
=> ! [Y5: a] :
( ( B
= ( fo_Const_a @ Y5 ) )
=> ~ ( R3 @ X @ Y5 ) ) )
=> ~ ! [Xa2: nat] :
( ( A
= ( fo_Var_a @ Xa2 ) )
=> ! [Ya2: nat] :
( ( B
= ( fo_Var_a @ Ya2 ) )
=> ( Xa2 != Ya2 ) ) ) ) ) ).
% fo_term.rel_cases
thf(fact_176_fo__term_Orel__cases,axiom,
! [R3: nat > nat > $o,A: fo_term_nat,B: fo_term_nat] :
( ( fo_rel1533169934621585718at_nat @ R3 @ A @ B )
=> ( ! [X: nat] :
( ( A
= ( fo_Const_nat @ X ) )
=> ! [Y5: nat] :
( ( B
= ( fo_Const_nat @ Y5 ) )
=> ~ ( R3 @ X @ Y5 ) ) )
=> ~ ! [Xa2: nat] :
( ( A
= ( fo_Var_nat @ Xa2 ) )
=> ! [Ya2: nat] :
( ( B
= ( fo_Var_nat @ Ya2 ) )
=> ( Xa2 != Ya2 ) ) ) ) ) ).
% fo_term.rel_cases
thf(fact_177_fo__term_Orel__eq,axiom,
( ( fo_rel_fo_term_a_a
@ ^ [Y6: a,Z4: a] : ( Y6 = Z4 ) )
= ( ^ [Y6: fo_term_a,Z4: fo_term_a] : ( Y6 = Z4 ) ) ) ).
% fo_term.rel_eq
thf(fact_178_fo__term_Orel__eq,axiom,
( ( fo_rel1533169934621585718at_nat
@ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
= ( ^ [Y6: fo_term_nat,Z4: fo_term_nat] : ( Y6 = Z4 ) ) ) ).
% fo_term.rel_eq
thf(fact_179_fo__fmla_Orel__eq,axiom,
( ( fo_rel8103664758052585748_a_b_b
@ ^ [Y6: a,Z4: a] : ( Y6 = Z4 )
@ ^ [Y6: b,Z4: b] : ( Y6 = Z4 ) )
= ( ^ [Y6: fo_fmla_a_b,Z4: fo_fmla_a_b] : ( Y6 = Z4 ) ) ) ).
% fo_fmla.rel_eq
thf(fact_180_fo__fmla_Orel__eq,axiom,
( ( fo_rel4206018205878794776at_nat
@ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
@ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
= ( ^ [Y6: fo_fmla_nat_nat,Z4: fo_fmla_nat_nat] : ( Y6 = Z4 ) ) ) ).
% fo_fmla.rel_eq
thf(fact_181_fo__term_Oexhaust,axiom,
! [Y: fo_term_a] :
( ! [X13: a] :
( Y
!= ( fo_Const_a @ X13 ) )
=> ~ ! [X23: nat] :
( Y
!= ( fo_Var_a @ X23 ) ) ) ).
% fo_term.exhaust
thf(fact_182_fo__term_Oexhaust,axiom,
! [Y: fo_term_nat] :
( ! [X13: nat] :
( Y
!= ( fo_Const_nat @ X13 ) )
=> ~ ! [X23: nat] :
( Y
!= ( fo_Var_nat @ X23 ) ) ) ).
% fo_term.exhaust
thf(fact_183_fo__term_Orel__distinct_I1_J,axiom,
! [R3: nat > a > $o,X1: nat,Y2: nat] :
~ ( fo_rel_fo_term_nat_a @ R3 @ ( fo_Const_nat @ X1 ) @ ( fo_Var_a @ Y2 ) ) ).
% fo_term.rel_distinct(1)
thf(fact_184_fo__term_Orel__distinct_I1_J,axiom,
! [R3: a > a > $o,X1: a,Y2: nat] :
~ ( fo_rel_fo_term_a_a @ R3 @ ( fo_Const_a @ X1 ) @ ( fo_Var_a @ Y2 ) ) ).
% fo_term.rel_distinct(1)
thf(fact_185_fo__term_Orel__distinct_I1_J,axiom,
! [R3: nat > nat > $o,X1: nat,Y2: nat] :
~ ( fo_rel1533169934621585718at_nat @ R3 @ ( fo_Const_nat @ X1 ) @ ( fo_Var_nat @ Y2 ) ) ).
% fo_term.rel_distinct(1)
thf(fact_186_fo__term_Orel__distinct_I2_J,axiom,
! [R3: a > nat > $o,Y2: nat,X1: nat] :
~ ( fo_rel_fo_term_a_nat @ R3 @ ( fo_Var_a @ Y2 ) @ ( fo_Const_nat @ X1 ) ) ).
% fo_term.rel_distinct(2)
thf(fact_187_fo__term_Orel__distinct_I2_J,axiom,
! [R3: a > a > $o,Y2: nat,X1: a] :
~ ( fo_rel_fo_term_a_a @ R3 @ ( fo_Var_a @ Y2 ) @ ( fo_Const_a @ X1 ) ) ).
% fo_term.rel_distinct(2)
thf(fact_188_fo__term_Orel__distinct_I2_J,axiom,
! [R3: nat > nat > $o,Y2: nat,X1: nat] :
~ ( fo_rel1533169934621585718at_nat @ R3 @ ( fo_Var_nat @ Y2 ) @ ( fo_Const_nat @ X1 ) ) ).
% fo_term.rel_distinct(2)
thf(fact_189_fo__term_Orel__intros_I1_J,axiom,
! [R3: a > nat > $o,X1: a,Y1: nat] :
( ( R3 @ X1 @ Y1 )
=> ( fo_rel_fo_term_a_nat @ R3 @ ( fo_Const_a @ X1 ) @ ( fo_Const_nat @ Y1 ) ) ) ).
% fo_term.rel_intros(1)
thf(fact_190_fo__term_Orel__intros_I1_J,axiom,
! [R3: nat > a > $o,X1: nat,Y1: a] :
( ( R3 @ X1 @ Y1 )
=> ( fo_rel_fo_term_nat_a @ R3 @ ( fo_Const_nat @ X1 ) @ ( fo_Const_a @ Y1 ) ) ) ).
% fo_term.rel_intros(1)
thf(fact_191_fo__term_Orel__intros_I1_J,axiom,
! [R3: a > a > $o,X1: a,Y1: a] :
( ( R3 @ X1 @ Y1 )
=> ( fo_rel_fo_term_a_a @ R3 @ ( fo_Const_a @ X1 ) @ ( fo_Const_a @ Y1 ) ) ) ).
% fo_term.rel_intros(1)
thf(fact_192_fo__term_Orel__intros_I1_J,axiom,
! [R3: nat > nat > $o,X1: nat,Y1: nat] :
( ( R3 @ X1 @ Y1 )
=> ( fo_rel1533169934621585718at_nat @ R3 @ ( fo_Const_nat @ X1 ) @ ( fo_Const_nat @ Y1 ) ) ) ).
% fo_term.rel_intros(1)
thf(fact_193_fo__term_Orel__intros_I2_J,axiom,
! [X22: nat,Y2: nat,R3: a > a > $o] :
( ( X22 = Y2 )
=> ( fo_rel_fo_term_a_a @ R3 @ ( fo_Var_a @ X22 ) @ ( fo_Var_a @ Y2 ) ) ) ).
% fo_term.rel_intros(2)
thf(fact_194_fo__term_Orel__intros_I2_J,axiom,
! [X22: nat,Y2: nat,R3: nat > nat > $o] :
( ( X22 = Y2 )
=> ( fo_rel1533169934621585718at_nat @ R3 @ ( fo_Var_nat @ X22 ) @ ( fo_Var_nat @ Y2 ) ) ) ).
% fo_term.rel_intros(2)
thf(fact_195_fo__term_Orel__inject_I1_J,axiom,
! [R3: a > nat > $o,X1: a,Y1: nat] :
( ( fo_rel_fo_term_a_nat @ R3 @ ( fo_Const_a @ X1 ) @ ( fo_Const_nat @ Y1 ) )
= ( R3 @ X1 @ Y1 ) ) ).
% fo_term.rel_inject(1)
thf(fact_196_fo__term_Orel__inject_I1_J,axiom,
! [R3: nat > a > $o,X1: nat,Y1: a] :
( ( fo_rel_fo_term_nat_a @ R3 @ ( fo_Const_nat @ X1 ) @ ( fo_Const_a @ Y1 ) )
= ( R3 @ X1 @ Y1 ) ) ).
% fo_term.rel_inject(1)
thf(fact_197_fo__term_Orel__inject_I1_J,axiom,
! [R3: a > a > $o,X1: a,Y1: a] :
( ( fo_rel_fo_term_a_a @ R3 @ ( fo_Const_a @ X1 ) @ ( fo_Const_a @ Y1 ) )
= ( R3 @ X1 @ Y1 ) ) ).
% fo_term.rel_inject(1)
thf(fact_198_fo__term_Orel__inject_I1_J,axiom,
! [R3: nat > nat > $o,X1: nat,Y1: nat] :
( ( fo_rel1533169934621585718at_nat @ R3 @ ( fo_Const_nat @ X1 ) @ ( fo_Const_nat @ Y1 ) )
= ( R3 @ X1 @ Y1 ) ) ).
% fo_term.rel_inject(1)
thf(fact_199_fo__term_Orel__inject_I2_J,axiom,
! [R3: a > a > $o,X22: nat,Y2: nat] :
( ( fo_rel_fo_term_a_a @ R3 @ ( fo_Var_a @ X22 ) @ ( fo_Var_a @ Y2 ) )
= ( X22 = Y2 ) ) ).
% fo_term.rel_inject(2)
thf(fact_200_fo__term_Orel__inject_I2_J,axiom,
! [R3: nat > nat > $o,X22: nat,Y2: nat] :
( ( fo_rel1533169934621585718at_nat @ R3 @ ( fo_Var_nat @ X22 ) @ ( fo_Var_nat @ Y2 ) )
= ( X22 = Y2 ) ) ).
% fo_term.rel_inject(2)
thf(fact_201_fo__fmla_Oset__intros_I1_J,axiom,
! [Y: fo_term_a,X12: list_fo_term_a,Ya: a,X11: b] :
( ( member_fo_term_a @ Y @ ( set_fo_term_a2 @ X12 ) )
=> ( ( member_a @ Ya @ ( fo_set_fo_term_a @ Y ) )
=> ( member_a @ Ya @ ( fo_set1_fo_fmla_a_b @ ( fo_Pred_b_a @ X11 @ X12 ) ) ) ) ) ).
% fo_fmla.set_intros(1)
thf(fact_202_fo__fmla_Oset__intros_I1_J,axiom,
! [Y: fo_term_nat,X12: list_fo_term_nat,Ya: nat,X11: nat] :
( ( member_fo_term_nat @ Y @ ( set_fo_term_nat2 @ X12 ) )
=> ( ( member_nat @ Ya @ ( fo_set_fo_term_nat @ Y ) )
=> ( member_nat @ Ya @ ( fo_set7155282118041507904at_nat @ ( fo_Pred_nat_nat @ X11 @ X12 ) ) ) ) ) ).
% fo_fmla.set_intros(1)
thf(fact_203_fo__fmla_Orel__intros_I1_J,axiom,
! [R2: b > b > $o,X11: b,Y11: b,R1: a > a > $o,X12: list_fo_term_a,Y12: list_fo_term_a] :
( ( R2 @ X11 @ Y11 )
=> ( ( list_a2487961086254413575term_a @ ( fo_rel_fo_term_a_a @ R1 ) @ X12 @ Y12 )
=> ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Pred_b_a @ X11 @ X12 ) @ ( fo_Pred_b_a @ Y11 @ Y12 ) ) ) ) ).
% fo_fmla.rel_intros(1)
thf(fact_204_fo__fmla_Orel__intros_I1_J,axiom,
! [R2: nat > nat > $o,X11: nat,Y11: nat,R1: nat > nat > $o,X12: list_fo_term_nat,Y12: list_fo_term_nat] :
( ( R2 @ X11 @ Y11 )
=> ( ( list_a3012696415560123489rm_nat @ ( fo_rel1533169934621585718at_nat @ R1 ) @ X12 @ Y12 )
=> ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Pred_nat_nat @ X11 @ X12 ) @ ( fo_Pred_nat_nat @ Y11 @ Y12 ) ) ) ) ).
% fo_fmla.rel_intros(1)
thf(fact_205_fo__fmla_Orel__inject_I1_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X11: b,X12: list_fo_term_a,Y11: b,Y12: list_fo_term_a] :
( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Pred_b_a @ X11 @ X12 ) @ ( fo_Pred_b_a @ Y11 @ Y12 ) )
= ( ( R2 @ X11 @ Y11 )
& ( list_a2487961086254413575term_a @ ( fo_rel_fo_term_a_a @ R1 ) @ X12 @ Y12 ) ) ) ).
% fo_fmla.rel_inject(1)
thf(fact_206_fo__fmla_Orel__inject_I1_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X11: nat,X12: list_fo_term_nat,Y11: nat,Y12: list_fo_term_nat] :
( ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Pred_nat_nat @ X11 @ X12 ) @ ( fo_Pred_nat_nat @ Y11 @ Y12 ) )
= ( ( R2 @ X11 @ Y11 )
& ( list_a3012696415560123489rm_nat @ ( fo_rel1533169934621585718at_nat @ R1 ) @ X12 @ Y12 ) ) ) ).
% fo_fmla.rel_inject(1)
thf(fact_207_fo__term_Odistinct_I1_J,axiom,
! [X1: a,X22: nat] :
( ( fo_Const_a @ X1 )
!= ( fo_Var_a @ X22 ) ) ).
% fo_term.distinct(1)
thf(fact_208_fo__term_Odistinct_I1_J,axiom,
! [X1: nat,X22: nat] :
( ( fo_Const_nat @ X1 )
!= ( fo_Var_nat @ X22 ) ) ).
% fo_term.distinct(1)
thf(fact_209_fo__term_Osimps_I16_J,axiom,
! [X22: nat] :
( ( fo_set_fo_term_nat @ ( fo_Var_nat @ X22 ) )
= bot_bot_set_nat ) ).
% fo_term.simps(16)
thf(fact_210_fo__term_Osimps_I16_J,axiom,
! [X22: nat] :
( ( fo_set_fo_term_a @ ( fo_Var_a @ X22 ) )
= bot_bot_set_a ) ).
% fo_term.simps(16)
thf(fact_211_fo__term_Osimps_I16_J,axiom,
! [X22: nat] :
( ( fo_set_fo_term_b @ ( fo_Var_b @ X22 ) )
= bot_bot_set_b ) ).
% fo_term.simps(16)
thf(fact_212_sp__equiv__pair_Ocases,axiom,
! [X5: produc859450856879609959at_nat] :
~ ! [A5: nat,B3: nat,A6: nat,B5: nat] :
( X5
!= ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A5 @ B3 ) @ ( product_Pair_nat_nat @ A6 @ B5 ) ) ) ).
% sp_equiv_pair.cases
thf(fact_213_sp__equiv__pair_Ocases,axiom,
! [X5: produc7223814127285706151_a_nat] :
~ ! [A5: product_prod_b_nat > set_list_a,B3: produc4672180596006801056_a_nat,A6: product_prod_b_nat > set_list_a,B5: produc4672180596006801056_a_nat] :
( X5
!= ( produc5415104756258485911_a_nat @ ( produc6651248262528101210_a_nat @ A5 @ B3 ) @ ( produc6651248262528101210_a_nat @ A6 @ B5 ) ) ) ).
% sp_equiv_pair.cases
thf(fact_214_sp__equiv__pair_Ocases,axiom,
! [X5: produc1259525717903798759_nat_a] :
~ ! [A5: product_prod_b_nat > set_list_a,B3: nat > a,A6: product_prod_b_nat > set_list_a,B5: nat > a] :
( X5
!= ( produc4949270592693755223_nat_a @ ( produc2895298938842563487_nat_a @ A5 @ B3 ) @ ( produc2895298938842563487_nat_a @ A6 @ B5 ) ) ) ).
% sp_equiv_pair.cases
thf(fact_215_sp__equiv__pair_Ocases,axiom,
! [X5: produc3551143267141641511_a_nat] :
~ ! [A5: nat > sum_sum_a_nat,B3: set_Sum_sum_a_nat,A6: nat > sum_sum_a_nat,B5: set_Sum_sum_a_nat] :
( X5
!= ( produc4653759813812500759_a_nat @ ( produc3720304352952013712_a_nat @ A5 @ B3 ) @ ( produc3720304352952013712_a_nat @ A6 @ B5 ) ) ) ).
% sp_equiv_pair.cases
thf(fact_216_sp__equiv__pair_Ocases,axiom,
! [X5: produc613052087796603485_b_nat] :
~ ! [A5: b,B3: nat,A6: b,B5: nat] :
( X5
!= ( produc2793586353817733269_b_nat @ ( product_Pair_b_nat @ A5 @ B3 ) @ ( product_Pair_b_nat @ A6 @ B5 ) ) ) ).
% sp_equiv_pair.cases
thf(fact_217_fo__fmla_Orel__intros_I3_J,axiom,
! [R1: a > a > $o,X31: fo_term_a,Y31: fo_term_a,X32: fo_term_a,Y32: fo_term_a,R2: b > b > $o] :
( ( fo_rel_fo_term_a_a @ R1 @ X31 @ Y31 )
=> ( ( fo_rel_fo_term_a_a @ R1 @ X32 @ Y32 )
=> ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Eqa_a_b @ X31 @ X32 ) @ ( fo_Eqa_a_b @ Y31 @ Y32 ) ) ) ) ).
% fo_fmla.rel_intros(3)
thf(fact_218_fo__fmla_Orel__intros_I3_J,axiom,
! [R1: nat > nat > $o,X31: fo_term_nat,Y31: fo_term_nat,X32: fo_term_nat,Y32: fo_term_nat,R2: nat > nat > $o] :
( ( fo_rel1533169934621585718at_nat @ R1 @ X31 @ Y31 )
=> ( ( fo_rel1533169934621585718at_nat @ R1 @ X32 @ Y32 )
=> ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Eqa_nat_nat @ X31 @ X32 ) @ ( fo_Eqa_nat_nat @ Y31 @ Y32 ) ) ) ) ).
% fo_fmla.rel_intros(3)
thf(fact_219_fo__fmla_Orel__inject_I3_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X31: fo_term_a,X32: fo_term_a,Y31: fo_term_a,Y32: fo_term_a] :
( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Eqa_a_b @ X31 @ X32 ) @ ( fo_Eqa_a_b @ Y31 @ Y32 ) )
= ( ( fo_rel_fo_term_a_a @ R1 @ X31 @ Y31 )
& ( fo_rel_fo_term_a_a @ R1 @ X32 @ Y32 ) ) ) ).
% fo_fmla.rel_inject(3)
thf(fact_220_fo__fmla_Orel__inject_I3_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X31: fo_term_nat,X32: fo_term_nat,Y31: fo_term_nat,Y32: fo_term_nat] :
( ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Eqa_nat_nat @ X31 @ X32 ) @ ( fo_Eqa_nat_nat @ Y31 @ Y32 ) )
= ( ( fo_rel1533169934621585718at_nat @ R1 @ X31 @ Y31 )
& ( fo_rel1533169934621585718at_nat @ R1 @ X32 @ Y32 ) ) ) ).
% fo_fmla.rel_inject(3)
thf(fact_221_fo__fmla_Osimps_I163_J,axiom,
! [X31: fo_term_nat,X32: fo_term_nat] :
( ( fo_set4065224739847495105at_nat @ ( fo_Eqa_nat_nat @ X31 @ X32 ) )
= bot_bot_set_nat ) ).
% fo_fmla.simps(163)
thf(fact_222_fo__fmla_Osimps_I163_J,axiom,
! [X31: fo_term_a,X32: fo_term_a] :
( ( fo_set2_fo_fmla_a_b @ ( fo_Eqa_a_b @ X31 @ X32 ) )
= bot_bot_set_b ) ).
% fo_fmla.simps(163)
thf(fact_223_fo__fmla_Osimps_I154_J,axiom,
! [X22: $o] :
( ( fo_set7155282118041507904at_nat @ ( fo_Bool_nat_nat @ X22 ) )
= bot_bot_set_nat ) ).
% fo_fmla.simps(154)
thf(fact_224_fo__fmla_Osimps_I154_J,axiom,
! [X22: $o] :
( ( fo_set1_fo_fmla_a_b @ ( fo_Bool_a_b @ X22 ) )
= bot_bot_set_a ) ).
% fo_fmla.simps(154)
thf(fact_225_fo__fmla_Oset__intros_I2_J,axiom,
! [Yb2: a,X31: fo_term_a,X32: fo_term_a] :
( ( member_a @ Yb2 @ ( fo_set_fo_term_a @ X31 ) )
=> ( member_a @ Yb2 @ ( fo_set1_fo_fmla_a_b @ ( fo_Eqa_a_b @ X31 @ X32 ) ) ) ) ).
% fo_fmla.set_intros(2)
thf(fact_226_fo__fmla_Oset__intros_I2_J,axiom,
! [Yb2: nat,X31: fo_term_nat,X32: fo_term_nat] :
( ( member_nat @ Yb2 @ ( fo_set_fo_term_nat @ X31 ) )
=> ( member_nat @ Yb2 @ ( fo_set7155282118041507904at_nat @ ( fo_Eqa_nat_nat @ X31 @ X32 ) ) ) ) ).
% fo_fmla.set_intros(2)
thf(fact_227_fo__fmla_Oset__intros_I3_J,axiom,
! [Yc: a,X32: fo_term_a,X31: fo_term_a] :
( ( member_a @ Yc @ ( fo_set_fo_term_a @ X32 ) )
=> ( member_a @ Yc @ ( fo_set1_fo_fmla_a_b @ ( fo_Eqa_a_b @ X31 @ X32 ) ) ) ) ).
% fo_fmla.set_intros(3)
thf(fact_228_fo__fmla_Oset__intros_I3_J,axiom,
! [Yc: nat,X32: fo_term_nat,X31: fo_term_nat] :
( ( member_nat @ Yc @ ( fo_set_fo_term_nat @ X32 ) )
=> ( member_nat @ Yc @ ( fo_set7155282118041507904at_nat @ ( fo_Eqa_nat_nat @ X31 @ X32 ) ) ) ) ).
% fo_fmla.set_intros(3)
thf(fact_229_fo__fmla_Osimps_I162_J,axiom,
! [X22: $o] :
( ( fo_set4065224739847495105at_nat @ ( fo_Bool_nat_nat @ X22 ) )
= bot_bot_set_nat ) ).
% fo_fmla.simps(162)
thf(fact_230_fo__fmla_Osimps_I162_J,axiom,
! [X22: $o] :
( ( fo_set2_fo_fmla_a_b @ ( fo_Bool_a_b @ X22 ) )
= bot_bot_set_b ) ).
% fo_fmla.simps(162)
thf(fact_231_fo__fmla_Orel__inject_I5_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X51: fo_fmla_nat_nat,X52: fo_fmla_nat_nat,Y51: fo_fmla_nat_nat,Y52: fo_fmla_nat_nat] :
( ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Conj_nat_nat @ X51 @ X52 ) @ ( fo_Conj_nat_nat @ Y51 @ Y52 ) )
= ( ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ X51 @ Y51 )
& ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ X52 @ Y52 ) ) ) ).
% fo_fmla.rel_inject(5)
thf(fact_232_fo__fmla_Orel__inject_I5_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X51: fo_fmla_a_b,X52: fo_fmla_a_b,Y51: fo_fmla_a_b,Y52: fo_fmla_a_b] :
( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Conj_a_b @ X51 @ X52 ) @ ( fo_Conj_a_b @ Y51 @ Y52 ) )
= ( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X51 @ Y51 )
& ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X52 @ Y52 ) ) ) ).
% fo_fmla.rel_inject(5)
thf(fact_233_fo__fmla_Orel__intros_I5_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X51: fo_fmla_nat_nat,Y51: fo_fmla_nat_nat,X52: fo_fmla_nat_nat,Y52: fo_fmla_nat_nat] :
( ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ X51 @ Y51 )
=> ( ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ X52 @ Y52 )
=> ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Conj_nat_nat @ X51 @ X52 ) @ ( fo_Conj_nat_nat @ Y51 @ Y52 ) ) ) ) ).
% fo_fmla.rel_intros(5)
thf(fact_234_fo__fmla_Orel__intros_I5_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X51: fo_fmla_a_b,Y51: fo_fmla_a_b,X52: fo_fmla_a_b,Y52: fo_fmla_a_b] :
( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X51 @ Y51 )
=> ( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X52 @ Y52 )
=> ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Conj_a_b @ X51 @ X52 ) @ ( fo_Conj_a_b @ Y51 @ Y52 ) ) ) ) ).
% fo_fmla.rel_intros(5)
thf(fact_235_fo__fmla_Orel__inject_I6_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X61: fo_fmla_nat_nat,X62: fo_fmla_nat_nat,Y61: fo_fmla_nat_nat,Y62: fo_fmla_nat_nat] :
( ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Disj_nat_nat @ X61 @ X62 ) @ ( fo_Disj_nat_nat @ Y61 @ Y62 ) )
= ( ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ X61 @ Y61 )
& ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ X62 @ Y62 ) ) ) ).
% fo_fmla.rel_inject(6)
thf(fact_236_fo__fmla_Orel__inject_I6_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X61: fo_fmla_a_b,X62: fo_fmla_a_b,Y61: fo_fmla_a_b,Y62: fo_fmla_a_b] :
( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Disj_a_b @ X61 @ X62 ) @ ( fo_Disj_a_b @ Y61 @ Y62 ) )
= ( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X61 @ Y61 )
& ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X62 @ Y62 ) ) ) ).
% fo_fmla.rel_inject(6)
thf(fact_237_fo__fmla_Orel__intros_I6_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X61: fo_fmla_nat_nat,Y61: fo_fmla_nat_nat,X62: fo_fmla_nat_nat,Y62: fo_fmla_nat_nat] :
( ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ X61 @ Y61 )
=> ( ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ X62 @ Y62 )
=> ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Disj_nat_nat @ X61 @ X62 ) @ ( fo_Disj_nat_nat @ Y61 @ Y62 ) ) ) ) ).
% fo_fmla.rel_intros(6)
thf(fact_238_fo__fmla_Orel__intros_I6_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X61: fo_fmla_a_b,Y61: fo_fmla_a_b,X62: fo_fmla_a_b,Y62: fo_fmla_a_b] :
( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X61 @ Y61 )
=> ( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X62 @ Y62 )
=> ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Disj_a_b @ X61 @ X62 ) @ ( fo_Disj_a_b @ Y61 @ Y62 ) ) ) ) ).
% fo_fmla.rel_intros(6)
thf(fact_239_fo__fmla_Orel__inject_I4_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X4: fo_fmla_nat_nat,Y4: fo_fmla_nat_nat] :
( ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Neg_nat_nat @ X4 ) @ ( fo_Neg_nat_nat @ Y4 ) )
= ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ X4 @ Y4 ) ) ).
% fo_fmla.rel_inject(4)
thf(fact_240_fo__fmla_Orel__inject_I4_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X4: fo_fmla_a_b,Y4: fo_fmla_a_b] :
( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Neg_a_b @ X4 ) @ ( fo_Neg_a_b @ Y4 ) )
= ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X4 @ Y4 ) ) ).
% fo_fmla.rel_inject(4)
thf(fact_241_fo__fmla_Orel__intros_I4_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X4: fo_fmla_nat_nat,Y4: fo_fmla_nat_nat] :
( ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ X4 @ Y4 )
=> ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Neg_nat_nat @ X4 ) @ ( fo_Neg_nat_nat @ Y4 ) ) ) ).
% fo_fmla.rel_intros(4)
thf(fact_242_fo__fmla_Orel__intros_I4_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X4: fo_fmla_a_b,Y4: fo_fmla_a_b] :
( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X4 @ Y4 )
=> ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Neg_a_b @ X4 ) @ ( fo_Neg_a_b @ Y4 ) ) ) ).
% fo_fmla.rel_intros(4)
thf(fact_243_fo__fmla_Orel__inject_I7_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X71: nat,X72: fo_fmla_nat_nat,Y71: nat,Y72: fo_fmla_nat_nat] :
( ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Exists_nat_nat @ X71 @ X72 ) @ ( fo_Exists_nat_nat @ Y71 @ Y72 ) )
= ( ( X71 = Y71 )
& ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ X72 @ Y72 ) ) ) ).
% fo_fmla.rel_inject(7)
thf(fact_244_fo__fmla_Orel__inject_I7_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X71: nat,X72: fo_fmla_a_b,Y71: nat,Y72: fo_fmla_a_b] :
( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Exists_a_b @ X71 @ X72 ) @ ( fo_Exists_a_b @ Y71 @ Y72 ) )
= ( ( X71 = Y71 )
& ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X72 @ Y72 ) ) ) ).
% fo_fmla.rel_inject(7)
thf(fact_245_fo__fmla_Orel__intros_I7_J,axiom,
! [X71: nat,Y71: nat,R1: nat > nat > $o,R2: nat > nat > $o,X72: fo_fmla_nat_nat,Y72: fo_fmla_nat_nat] :
( ( X71 = Y71 )
=> ( ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ X72 @ Y72 )
=> ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Exists_nat_nat @ X71 @ X72 ) @ ( fo_Exists_nat_nat @ Y71 @ Y72 ) ) ) ) ).
% fo_fmla.rel_intros(7)
thf(fact_246_fo__fmla_Orel__intros_I7_J,axiom,
! [X71: nat,Y71: nat,R1: a > a > $o,R2: b > b > $o,X72: fo_fmla_a_b,Y72: fo_fmla_a_b] :
( ( X71 = Y71 )
=> ( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X72 @ Y72 )
=> ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Exists_a_b @ X71 @ X72 ) @ ( fo_Exists_a_b @ Y71 @ Y72 ) ) ) ) ).
% fo_fmla.rel_intros(7)
thf(fact_247_fo__fmla_Orel__inject_I8_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X81: nat,X82: fo_fmla_a_b,Y81: nat,Y82: fo_fmla_a_b] :
( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Forall_a_b @ X81 @ X82 ) @ ( fo_Forall_a_b @ Y81 @ Y82 ) )
= ( ( X81 = Y81 )
& ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X82 @ Y82 ) ) ) ).
% fo_fmla.rel_inject(8)
thf(fact_248_fo__fmla_Orel__inject_I8_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X81: nat,X82: fo_fmla_nat_nat,Y81: nat,Y82: fo_fmla_nat_nat] :
( ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Forall_nat_nat @ X81 @ X82 ) @ ( fo_Forall_nat_nat @ Y81 @ Y82 ) )
= ( ( X81 = Y81 )
& ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ X82 @ Y82 ) ) ) ).
% fo_fmla.rel_inject(8)
thf(fact_249_fo__fmla_Orel__intros_I8_J,axiom,
! [X81: nat,Y81: nat,R1: a > a > $o,R2: b > b > $o,X82: fo_fmla_a_b,Y82: fo_fmla_a_b] :
( ( X81 = Y81 )
=> ( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X82 @ Y82 )
=> ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Forall_a_b @ X81 @ X82 ) @ ( fo_Forall_a_b @ Y81 @ Y82 ) ) ) ) ).
% fo_fmla.rel_intros(8)
thf(fact_250_fo__fmla_Orel__intros_I8_J,axiom,
! [X81: nat,Y81: nat,R1: nat > nat > $o,R2: nat > nat > $o,X82: fo_fmla_nat_nat,Y82: fo_fmla_nat_nat] :
( ( X81 = Y81 )
=> ( ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ X82 @ Y82 )
=> ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Forall_nat_nat @ X81 @ X82 ) @ ( fo_Forall_nat_nat @ Y81 @ Y82 ) ) ) ) ).
% fo_fmla.rel_intros(8)
thf(fact_251_fo__fmla_Orel__inject_I2_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X22: $o,Y2: $o] :
( ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Bool_nat_nat @ X22 ) @ ( fo_Bool_nat_nat @ Y2 ) )
= ( X22 = Y2 ) ) ).
% fo_fmla.rel_inject(2)
thf(fact_252_fo__fmla_Orel__inject_I2_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X22: $o,Y2: $o] :
( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Bool_a_b @ X22 ) @ ( fo_Bool_a_b @ Y2 ) )
= ( X22 = Y2 ) ) ).
% fo_fmla.rel_inject(2)
thf(fact_253_fo__fmla_Orel__intros_I2_J,axiom,
! [X22: $o,Y2: $o,R1: nat > nat > $o,R2: nat > nat > $o] :
( ( X22 = Y2 )
=> ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Bool_nat_nat @ X22 ) @ ( fo_Bool_nat_nat @ Y2 ) ) ) ).
% fo_fmla.rel_intros(2)
thf(fact_254_fo__fmla_Orel__intros_I2_J,axiom,
! [X22: $o,Y2: $o,R1: a > a > $o,R2: b > b > $o] :
( ( X22 = Y2 )
=> ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Bool_a_b @ X22 ) @ ( fo_Bool_a_b @ Y2 ) ) ) ).
% fo_fmla.rel_intros(2)
thf(fact_255_eval__eterm_Osimps_I2_J,axiom,
! [Sigma: nat > sum_sum_a_nat,N2: nat] :
( ( eval_eterm_a_nat @ Sigma @ ( fo_Var_a @ N2 ) )
= ( Sigma @ N2 ) ) ).
% eval_eterm.simps(2)
thf(fact_256_fo__fmla_Oset__intros_I6_J,axiom,
! [Yh: nat,X52: fo_fmla_nat_nat,X51: fo_fmla_nat_nat] :
( ( member_nat @ Yh @ ( fo_set7155282118041507904at_nat @ X52 ) )
=> ( member_nat @ Yh @ ( fo_set7155282118041507904at_nat @ ( fo_Conj_nat_nat @ X51 @ X52 ) ) ) ) ).
% fo_fmla.set_intros(6)
thf(fact_257_fo__fmla_Oset__intros_I6_J,axiom,
! [Yh: a,X52: fo_fmla_a_b,X51: fo_fmla_a_b] :
( ( member_a @ Yh @ ( fo_set1_fo_fmla_a_b @ X52 ) )
=> ( member_a @ Yh @ ( fo_set1_fo_fmla_a_b @ ( fo_Conj_a_b @ X51 @ X52 ) ) ) ) ).
% fo_fmla.set_intros(6)
thf(fact_258_fo__fmla_Oset__intros_I5_J,axiom,
! [Yf: nat,X51: fo_fmla_nat_nat,X52: fo_fmla_nat_nat] :
( ( member_nat @ Yf @ ( fo_set7155282118041507904at_nat @ X51 ) )
=> ( member_nat @ Yf @ ( fo_set7155282118041507904at_nat @ ( fo_Conj_nat_nat @ X51 @ X52 ) ) ) ) ).
% fo_fmla.set_intros(5)
thf(fact_259_fo__fmla_Oset__intros_I5_J,axiom,
! [Yf: a,X51: fo_fmla_a_b,X52: fo_fmla_a_b] :
( ( member_a @ Yf @ ( fo_set1_fo_fmla_a_b @ X51 ) )
=> ( member_a @ Yf @ ( fo_set1_fo_fmla_a_b @ ( fo_Conj_a_b @ X51 @ X52 ) ) ) ) ).
% fo_fmla.set_intros(5)
thf(fact_260_fo__fmla_Oset__intros_I8_J,axiom,
! [Yl: nat,X62: fo_fmla_nat_nat,X61: fo_fmla_nat_nat] :
( ( member_nat @ Yl @ ( fo_set7155282118041507904at_nat @ X62 ) )
=> ( member_nat @ Yl @ ( fo_set7155282118041507904at_nat @ ( fo_Disj_nat_nat @ X61 @ X62 ) ) ) ) ).
% fo_fmla.set_intros(8)
thf(fact_261_fo__fmla_Oset__intros_I8_J,axiom,
! [Yl: a,X62: fo_fmla_a_b,X61: fo_fmla_a_b] :
( ( member_a @ Yl @ ( fo_set1_fo_fmla_a_b @ X62 ) )
=> ( member_a @ Yl @ ( fo_set1_fo_fmla_a_b @ ( fo_Disj_a_b @ X61 @ X62 ) ) ) ) ).
% fo_fmla.set_intros(8)
thf(fact_262_fo__fmla_Oset__intros_I7_J,axiom,
! [Yj: nat,X61: fo_fmla_nat_nat,X62: fo_fmla_nat_nat] :
( ( member_nat @ Yj @ ( fo_set7155282118041507904at_nat @ X61 ) )
=> ( member_nat @ Yj @ ( fo_set7155282118041507904at_nat @ ( fo_Disj_nat_nat @ X61 @ X62 ) ) ) ) ).
% fo_fmla.set_intros(7)
thf(fact_263_fo__fmla_Oset__intros_I7_J,axiom,
! [Yj: a,X61: fo_fmla_a_b,X62: fo_fmla_a_b] :
( ( member_a @ Yj @ ( fo_set1_fo_fmla_a_b @ X61 ) )
=> ( member_a @ Yj @ ( fo_set1_fo_fmla_a_b @ ( fo_Disj_a_b @ X61 @ X62 ) ) ) ) ).
% fo_fmla.set_intros(7)
thf(fact_264_fo__fmla_Osimps_I156_J,axiom,
! [X4: fo_fmla_nat_nat] :
( ( fo_set7155282118041507904at_nat @ ( fo_Neg_nat_nat @ X4 ) )
= ( fo_set7155282118041507904at_nat @ X4 ) ) ).
% fo_fmla.simps(156)
thf(fact_265_fo__fmla_Osimps_I156_J,axiom,
! [X4: fo_fmla_a_b] :
( ( fo_set1_fo_fmla_a_b @ ( fo_Neg_a_b @ X4 ) )
= ( fo_set1_fo_fmla_a_b @ X4 ) ) ).
% fo_fmla.simps(156)
thf(fact_266_fo__fmla_Oset__intros_I4_J,axiom,
! [Yd: nat,X4: fo_fmla_nat_nat] :
( ( member_nat @ Yd @ ( fo_set7155282118041507904at_nat @ X4 ) )
=> ( member_nat @ Yd @ ( fo_set7155282118041507904at_nat @ ( fo_Neg_nat_nat @ X4 ) ) ) ) ).
% fo_fmla.set_intros(4)
thf(fact_267_fo__fmla_Oset__intros_I4_J,axiom,
! [Yd: a,X4: fo_fmla_a_b] :
( ( member_a @ Yd @ ( fo_set1_fo_fmla_a_b @ X4 ) )
=> ( member_a @ Yd @ ( fo_set1_fo_fmla_a_b @ ( fo_Neg_a_b @ X4 ) ) ) ) ).
% fo_fmla.set_intros(4)
thf(fact_268_fo__fmla_Osimps_I159_J,axiom,
! [X71: nat,X72: fo_fmla_nat_nat] :
( ( fo_set7155282118041507904at_nat @ ( fo_Exists_nat_nat @ X71 @ X72 ) )
= ( fo_set7155282118041507904at_nat @ X72 ) ) ).
% fo_fmla.simps(159)
thf(fact_269_fo__fmla_Osimps_I159_J,axiom,
! [X71: nat,X72: fo_fmla_a_b] :
( ( fo_set1_fo_fmla_a_b @ ( fo_Exists_a_b @ X71 @ X72 ) )
= ( fo_set1_fo_fmla_a_b @ X72 ) ) ).
% fo_fmla.simps(159)
thf(fact_270_fo__fmla_Oset__intros_I9_J,axiom,
! [Yn: nat,X72: fo_fmla_nat_nat,X71: nat] :
( ( member_nat @ Yn @ ( fo_set7155282118041507904at_nat @ X72 ) )
=> ( member_nat @ Yn @ ( fo_set7155282118041507904at_nat @ ( fo_Exists_nat_nat @ X71 @ X72 ) ) ) ) ).
% fo_fmla.set_intros(9)
thf(fact_271_fo__fmla_Oset__intros_I9_J,axiom,
! [Yn: a,X72: fo_fmla_a_b,X71: nat] :
( ( member_a @ Yn @ ( fo_set1_fo_fmla_a_b @ X72 ) )
=> ( member_a @ Yn @ ( fo_set1_fo_fmla_a_b @ ( fo_Exists_a_b @ X71 @ X72 ) ) ) ) ).
% fo_fmla.set_intros(9)
thf(fact_272_fo__fmla_Osimps_I160_J,axiom,
! [X81: nat,X82: fo_fmla_a_b] :
( ( fo_set1_fo_fmla_a_b @ ( fo_Forall_a_b @ X81 @ X82 ) )
= ( fo_set1_fo_fmla_a_b @ X82 ) ) ).
% fo_fmla.simps(160)
thf(fact_273_fo__fmla_Osimps_I160_J,axiom,
! [X81: nat,X82: fo_fmla_nat_nat] :
( ( fo_set7155282118041507904at_nat @ ( fo_Forall_nat_nat @ X81 @ X82 ) )
= ( fo_set7155282118041507904at_nat @ X82 ) ) ).
% fo_fmla.simps(160)
thf(fact_274_fo__fmla_Oset__intros_I10_J,axiom,
! [Yp: a,X82: fo_fmla_a_b,X81: nat] :
( ( member_a @ Yp @ ( fo_set1_fo_fmla_a_b @ X82 ) )
=> ( member_a @ Yp @ ( fo_set1_fo_fmla_a_b @ ( fo_Forall_a_b @ X81 @ X82 ) ) ) ) ).
% fo_fmla.set_intros(10)
thf(fact_275_fo__fmla_Oset__intros_I10_J,axiom,
! [Yp: nat,X82: fo_fmla_nat_nat,X81: nat] :
( ( member_nat @ Yp @ ( fo_set7155282118041507904at_nat @ X82 ) )
=> ( member_nat @ Yp @ ( fo_set7155282118041507904at_nat @ ( fo_Forall_nat_nat @ X81 @ X82 ) ) ) ) ).
% fo_fmla.set_intros(10)
thf(fact_276_fo__fmla_Oset__intros_I14_J,axiom,
! [Za: nat,X52: fo_fmla_nat_nat,X51: fo_fmla_nat_nat] :
( ( member_nat @ Za @ ( fo_set4065224739847495105at_nat @ X52 ) )
=> ( member_nat @ Za @ ( fo_set4065224739847495105at_nat @ ( fo_Conj_nat_nat @ X51 @ X52 ) ) ) ) ).
% fo_fmla.set_intros(14)
thf(fact_277_fo__fmla_Oset__intros_I14_J,axiom,
! [Za: b,X52: fo_fmla_a_b,X51: fo_fmla_a_b] :
( ( member_b @ Za @ ( fo_set2_fo_fmla_a_b @ X52 ) )
=> ( member_b @ Za @ ( fo_set2_fo_fmla_a_b @ ( fo_Conj_a_b @ X51 @ X52 ) ) ) ) ).
% fo_fmla.set_intros(14)
thf(fact_278_fo__fmla_Oset__intros_I13_J,axiom,
! [Yy: nat,X51: fo_fmla_nat_nat,X52: fo_fmla_nat_nat] :
( ( member_nat @ Yy @ ( fo_set4065224739847495105at_nat @ X51 ) )
=> ( member_nat @ Yy @ ( fo_set4065224739847495105at_nat @ ( fo_Conj_nat_nat @ X51 @ X52 ) ) ) ) ).
% fo_fmla.set_intros(13)
thf(fact_279_fo__fmla_Oset__intros_I13_J,axiom,
! [Yy: b,X51: fo_fmla_a_b,X52: fo_fmla_a_b] :
( ( member_b @ Yy @ ( fo_set2_fo_fmla_a_b @ X51 ) )
=> ( member_b @ Yy @ ( fo_set2_fo_fmla_a_b @ ( fo_Conj_a_b @ X51 @ X52 ) ) ) ) ).
% fo_fmla.set_intros(13)
thf(fact_280_fo__fmla_Oset__intros_I16_J,axiom,
! [Ze: nat,X62: fo_fmla_nat_nat,X61: fo_fmla_nat_nat] :
( ( member_nat @ Ze @ ( fo_set4065224739847495105at_nat @ X62 ) )
=> ( member_nat @ Ze @ ( fo_set4065224739847495105at_nat @ ( fo_Disj_nat_nat @ X61 @ X62 ) ) ) ) ).
% fo_fmla.set_intros(16)
thf(fact_281_fo__fmla_Oset__intros_I16_J,axiom,
! [Ze: b,X62: fo_fmla_a_b,X61: fo_fmla_a_b] :
( ( member_b @ Ze @ ( fo_set2_fo_fmla_a_b @ X62 ) )
=> ( member_b @ Ze @ ( fo_set2_fo_fmla_a_b @ ( fo_Disj_a_b @ X61 @ X62 ) ) ) ) ).
% fo_fmla.set_intros(16)
thf(fact_282_fo__fmla_Oset__intros_I15_J,axiom,
! [Zc: nat,X61: fo_fmla_nat_nat,X62: fo_fmla_nat_nat] :
( ( member_nat @ Zc @ ( fo_set4065224739847495105at_nat @ X61 ) )
=> ( member_nat @ Zc @ ( fo_set4065224739847495105at_nat @ ( fo_Disj_nat_nat @ X61 @ X62 ) ) ) ) ).
% fo_fmla.set_intros(15)
thf(fact_283_fo__fmla_Oset__intros_I15_J,axiom,
! [Zc: b,X61: fo_fmla_a_b,X62: fo_fmla_a_b] :
( ( member_b @ Zc @ ( fo_set2_fo_fmla_a_b @ X61 ) )
=> ( member_b @ Zc @ ( fo_set2_fo_fmla_a_b @ ( fo_Disj_a_b @ X61 @ X62 ) ) ) ) ).
% fo_fmla.set_intros(15)
thf(fact_284_fo__fmla_Osimps_I164_J,axiom,
! [X4: fo_fmla_nat_nat] :
( ( fo_set4065224739847495105at_nat @ ( fo_Neg_nat_nat @ X4 ) )
= ( fo_set4065224739847495105at_nat @ X4 ) ) ).
% fo_fmla.simps(164)
thf(fact_285_fo__fmla_Osimps_I164_J,axiom,
! [X4: fo_fmla_a_b] :
( ( fo_set2_fo_fmla_a_b @ ( fo_Neg_a_b @ X4 ) )
= ( fo_set2_fo_fmla_a_b @ X4 ) ) ).
% fo_fmla.simps(164)
thf(fact_286_fo__fmla_Oset__intros_I12_J,axiom,
! [Yw: nat,X4: fo_fmla_nat_nat] :
( ( member_nat @ Yw @ ( fo_set4065224739847495105at_nat @ X4 ) )
=> ( member_nat @ Yw @ ( fo_set4065224739847495105at_nat @ ( fo_Neg_nat_nat @ X4 ) ) ) ) ).
% fo_fmla.set_intros(12)
thf(fact_287_fo__fmla_Oset__intros_I12_J,axiom,
! [Yw: b,X4: fo_fmla_a_b] :
( ( member_b @ Yw @ ( fo_set2_fo_fmla_a_b @ X4 ) )
=> ( member_b @ Yw @ ( fo_set2_fo_fmla_a_b @ ( fo_Neg_a_b @ X4 ) ) ) ) ).
% fo_fmla.set_intros(12)
thf(fact_288_fo__fmla_Osimps_I167_J,axiom,
! [X71: nat,X72: fo_fmla_nat_nat] :
( ( fo_set4065224739847495105at_nat @ ( fo_Exists_nat_nat @ X71 @ X72 ) )
= ( fo_set4065224739847495105at_nat @ X72 ) ) ).
% fo_fmla.simps(167)
thf(fact_289_fo__fmla_Osimps_I167_J,axiom,
! [X71: nat,X72: fo_fmla_a_b] :
( ( fo_set2_fo_fmla_a_b @ ( fo_Exists_a_b @ X71 @ X72 ) )
= ( fo_set2_fo_fmla_a_b @ X72 ) ) ).
% fo_fmla.simps(167)
thf(fact_290_fo__fmla_Oset__intros_I17_J,axiom,
! [Zg: nat,X72: fo_fmla_nat_nat,X71: nat] :
( ( member_nat @ Zg @ ( fo_set4065224739847495105at_nat @ X72 ) )
=> ( member_nat @ Zg @ ( fo_set4065224739847495105at_nat @ ( fo_Exists_nat_nat @ X71 @ X72 ) ) ) ) ).
% fo_fmla.set_intros(17)
thf(fact_291_fo__fmla_Oset__intros_I17_J,axiom,
! [Zg: b,X72: fo_fmla_a_b,X71: nat] :
( ( member_b @ Zg @ ( fo_set2_fo_fmla_a_b @ X72 ) )
=> ( member_b @ Zg @ ( fo_set2_fo_fmla_a_b @ ( fo_Exists_a_b @ X71 @ X72 ) ) ) ) ).
% fo_fmla.set_intros(17)
thf(fact_292_fo__fmla_Osimps_I168_J,axiom,
! [X81: nat,X82: fo_fmla_a_b] :
( ( fo_set2_fo_fmla_a_b @ ( fo_Forall_a_b @ X81 @ X82 ) )
= ( fo_set2_fo_fmla_a_b @ X82 ) ) ).
% fo_fmla.simps(168)
thf(fact_293_fo__fmla_Osimps_I168_J,axiom,
! [X81: nat,X82: fo_fmla_nat_nat] :
( ( fo_set4065224739847495105at_nat @ ( fo_Forall_nat_nat @ X81 @ X82 ) )
= ( fo_set4065224739847495105at_nat @ X82 ) ) ).
% fo_fmla.simps(168)
thf(fact_294_fo__fmla_Oset__intros_I18_J,axiom,
! [Zi: b,X82: fo_fmla_a_b,X81: nat] :
( ( member_b @ Zi @ ( fo_set2_fo_fmla_a_b @ X82 ) )
=> ( member_b @ Zi @ ( fo_set2_fo_fmla_a_b @ ( fo_Forall_a_b @ X81 @ X82 ) ) ) ) ).
% fo_fmla.set_intros(18)
thf(fact_295_fo__fmla_Oset__intros_I18_J,axiom,
! [Zi: nat,X82: fo_fmla_nat_nat,X81: nat] :
( ( member_nat @ Zi @ ( fo_set4065224739847495105at_nat @ X82 ) )
=> ( member_nat @ Zi @ ( fo_set4065224739847495105at_nat @ ( fo_Forall_nat_nat @ X81 @ X82 ) ) ) ) ).
% fo_fmla.set_intros(18)
thf(fact_296_fo__fmla_Oset__intros_I11_J,axiom,
! [X11: b,X12: list_fo_term_a] : ( member_b @ X11 @ ( fo_set2_fo_fmla_a_b @ ( fo_Pred_b_a @ X11 @ X12 ) ) ) ).
% fo_fmla.set_intros(11)
thf(fact_297_fo__fmla_Oset__intros_I11_J,axiom,
! [X11: nat,X12: list_fo_term_nat] : ( member_nat @ X11 @ ( fo_set4065224739847495105at_nat @ ( fo_Pred_nat_nat @ X11 @ X12 ) ) ) ).
% fo_fmla.set_intros(11)
thf(fact_298_sp__equiv__pair_Oelims_I3_J,axiom,
! [X5: product_prod_nat_nat,Xa: product_prod_nat_nat] :
( ~ ( sp_equ885134019442699404at_nat @ X5 @ Xa )
=> ~ ! [A5: nat,B3: nat] :
( ( X5
= ( product_Pair_nat_nat @ A5 @ B3 ) )
=> ! [A6: nat,B5: nat] :
( ( Xa
= ( product_Pair_nat_nat @ A6 @ B5 ) )
=> ( ( A5 = A6 )
= ( B3 = B5 ) ) ) ) ) ).
% sp_equiv_pair.elims(3)
thf(fact_299_sp__equiv__pair_Oelims_I3_J,axiom,
! [X5: produc5433867242478491114_a_nat,Xa: produc5433867242478491114_a_nat] :
( ~ ( sp_equ4301142204835971757_a_nat @ X5 @ Xa )
=> ~ ! [A5: product_prod_b_nat > set_list_a,B3: produc4672180596006801056_a_nat] :
( ( X5
= ( produc6651248262528101210_a_nat @ A5 @ B3 ) )
=> ! [A6: product_prod_b_nat > set_list_a,B5: produc4672180596006801056_a_nat] :
( ( Xa
= ( produc6651248262528101210_a_nat @ A6 @ B5 ) )
=> ( ( A5 = A6 )
= ( B3 = B5 ) ) ) ) ) ).
% sp_equiv_pair.elims(3)
thf(fact_300_sp__equiv__pair_Oelims_I3_J,axiom,
! [X5: produc5835360497134304175_nat_a,Xa: produc5835360497134304175_nat_a] :
( ~ ( sp_equ547821432705481458_nat_a @ X5 @ Xa )
=> ~ ! [A5: product_prod_b_nat > set_list_a,B3: nat > a] :
( ( X5
= ( produc2895298938842563487_nat_a @ A5 @ B3 ) )
=> ! [A6: product_prod_b_nat > set_list_a,B5: nat > a] :
( ( Xa
= ( produc2895298938842563487_nat_a @ A6 @ B5 ) )
=> ( ( A5 = A6 )
= ( B3 = B5 ) ) ) ) ) ).
% sp_equiv_pair.elims(3)
thf(fact_301_sp__equiv__pair_Oelims_I3_J,axiom,
! [X5: produc4672180596006801056_a_nat,Xa: produc4672180596006801056_a_nat] :
( ~ ( sp_equ1372826846814931683_a_nat @ X5 @ Xa )
=> ~ ! [A5: nat > sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( ( X5
= ( produc3720304352952013712_a_nat @ A5 @ B3 ) )
=> ! [A6: nat > sum_sum_a_nat,B5: set_Sum_sum_a_nat] :
( ( Xa
= ( produc3720304352952013712_a_nat @ A6 @ B5 ) )
=> ( ( A5 = A6 )
= ( B3 = B5 ) ) ) ) ) ).
% sp_equiv_pair.elims(3)
thf(fact_302_sp__equiv__pair_Oelims_I3_J,axiom,
! [X5: product_prod_b_nat,Xa: product_prod_b_nat] :
( ~ ( sp_equiv_pair_b_nat @ X5 @ Xa )
=> ~ ! [A5: b,B3: nat] :
( ( X5
= ( product_Pair_b_nat @ A5 @ B3 ) )
=> ! [A6: b,B5: nat] :
( ( Xa
= ( product_Pair_b_nat @ A6 @ B5 ) )
=> ( ( A5 = A6 )
= ( B3 = B5 ) ) ) ) ) ).
% sp_equiv_pair.elims(3)
thf(fact_303_sp__equiv__pair_Oelims_I2_J,axiom,
! [X5: product_prod_nat_nat,Xa: product_prod_nat_nat] :
( ( sp_equ885134019442699404at_nat @ X5 @ Xa )
=> ~ ! [A5: nat,B3: nat] :
( ( X5
= ( product_Pair_nat_nat @ A5 @ B3 ) )
=> ! [A6: nat,B5: nat] :
( ( Xa
= ( product_Pair_nat_nat @ A6 @ B5 ) )
=> ( ( A5 = A6 )
= ( B3 != B5 ) ) ) ) ) ).
% sp_equiv_pair.elims(2)
thf(fact_304_sp__equiv__pair_Oelims_I2_J,axiom,
! [X5: produc5433867242478491114_a_nat,Xa: produc5433867242478491114_a_nat] :
( ( sp_equ4301142204835971757_a_nat @ X5 @ Xa )
=> ~ ! [A5: product_prod_b_nat > set_list_a,B3: produc4672180596006801056_a_nat] :
( ( X5
= ( produc6651248262528101210_a_nat @ A5 @ B3 ) )
=> ! [A6: product_prod_b_nat > set_list_a,B5: produc4672180596006801056_a_nat] :
( ( Xa
= ( produc6651248262528101210_a_nat @ A6 @ B5 ) )
=> ( ( A5 = A6 )
= ( B3 != B5 ) ) ) ) ) ).
% sp_equiv_pair.elims(2)
thf(fact_305_sp__equiv__pair_Oelims_I2_J,axiom,
! [X5: produc5835360497134304175_nat_a,Xa: produc5835360497134304175_nat_a] :
( ( sp_equ547821432705481458_nat_a @ X5 @ Xa )
=> ~ ! [A5: product_prod_b_nat > set_list_a,B3: nat > a] :
( ( X5
= ( produc2895298938842563487_nat_a @ A5 @ B3 ) )
=> ! [A6: product_prod_b_nat > set_list_a,B5: nat > a] :
( ( Xa
= ( produc2895298938842563487_nat_a @ A6 @ B5 ) )
=> ( ( A5 = A6 )
= ( B3 != B5 ) ) ) ) ) ).
% sp_equiv_pair.elims(2)
thf(fact_306_sp__equiv__pair_Oelims_I2_J,axiom,
! [X5: produc4672180596006801056_a_nat,Xa: produc4672180596006801056_a_nat] :
( ( sp_equ1372826846814931683_a_nat @ X5 @ Xa )
=> ~ ! [A5: nat > sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( ( X5
= ( produc3720304352952013712_a_nat @ A5 @ B3 ) )
=> ! [A6: nat > sum_sum_a_nat,B5: set_Sum_sum_a_nat] :
( ( Xa
= ( produc3720304352952013712_a_nat @ A6 @ B5 ) )
=> ( ( A5 = A6 )
= ( B3 != B5 ) ) ) ) ) ).
% sp_equiv_pair.elims(2)
thf(fact_307_sp__equiv__pair_Oelims_I2_J,axiom,
! [X5: product_prod_b_nat,Xa: product_prod_b_nat] :
( ( sp_equiv_pair_b_nat @ X5 @ Xa )
=> ~ ! [A5: b,B3: nat] :
( ( X5
= ( product_Pair_b_nat @ A5 @ B3 ) )
=> ! [A6: b,B5: nat] :
( ( Xa
= ( product_Pair_b_nat @ A6 @ B5 ) )
=> ( ( A5 = A6 )
= ( B3 != B5 ) ) ) ) ) ).
% sp_equiv_pair.elims(2)
thf(fact_308_sp__equiv__pair_Oelims_I1_J,axiom,
! [X5: product_prod_nat_nat,Xa: product_prod_nat_nat,Y: $o] :
( ( ( sp_equ885134019442699404at_nat @ X5 @ Xa )
= Y )
=> ~ ! [A5: nat,B3: nat] :
( ( X5
= ( product_Pair_nat_nat @ A5 @ B3 ) )
=> ! [A6: nat,B5: nat] :
( ( Xa
= ( product_Pair_nat_nat @ A6 @ B5 ) )
=> ( Y
= ( ( A5 = A6 )
= ( B3 != B5 ) ) ) ) ) ) ).
% sp_equiv_pair.elims(1)
thf(fact_309_sp__equiv__pair_Oelims_I1_J,axiom,
! [X5: produc5433867242478491114_a_nat,Xa: produc5433867242478491114_a_nat,Y: $o] :
( ( ( sp_equ4301142204835971757_a_nat @ X5 @ Xa )
= Y )
=> ~ ! [A5: product_prod_b_nat > set_list_a,B3: produc4672180596006801056_a_nat] :
( ( X5
= ( produc6651248262528101210_a_nat @ A5 @ B3 ) )
=> ! [A6: product_prod_b_nat > set_list_a,B5: produc4672180596006801056_a_nat] :
( ( Xa
= ( produc6651248262528101210_a_nat @ A6 @ B5 ) )
=> ( Y
= ( ( A5 = A6 )
= ( B3 != B5 ) ) ) ) ) ) ).
% sp_equiv_pair.elims(1)
thf(fact_310_sp__equiv__pair_Oelims_I1_J,axiom,
! [X5: produc5835360497134304175_nat_a,Xa: produc5835360497134304175_nat_a,Y: $o] :
( ( ( sp_equ547821432705481458_nat_a @ X5 @ Xa )
= Y )
=> ~ ! [A5: product_prod_b_nat > set_list_a,B3: nat > a] :
( ( X5
= ( produc2895298938842563487_nat_a @ A5 @ B3 ) )
=> ! [A6: product_prod_b_nat > set_list_a,B5: nat > a] :
( ( Xa
= ( produc2895298938842563487_nat_a @ A6 @ B5 ) )
=> ( Y
= ( ( A5 = A6 )
= ( B3 != B5 ) ) ) ) ) ) ).
% sp_equiv_pair.elims(1)
thf(fact_311_sp__equiv__pair_Oelims_I1_J,axiom,
! [X5: produc4672180596006801056_a_nat,Xa: produc4672180596006801056_a_nat,Y: $o] :
( ( ( sp_equ1372826846814931683_a_nat @ X5 @ Xa )
= Y )
=> ~ ! [A5: nat > sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( ( X5
= ( produc3720304352952013712_a_nat @ A5 @ B3 ) )
=> ! [A6: nat > sum_sum_a_nat,B5: set_Sum_sum_a_nat] :
( ( Xa
= ( produc3720304352952013712_a_nat @ A6 @ B5 ) )
=> ( Y
= ( ( A5 = A6 )
= ( B3 != B5 ) ) ) ) ) ) ).
% sp_equiv_pair.elims(1)
thf(fact_312_sp__equiv__pair_Oelims_I1_J,axiom,
! [X5: product_prod_b_nat,Xa: product_prod_b_nat,Y: $o] :
( ( ( sp_equiv_pair_b_nat @ X5 @ Xa )
= Y )
=> ~ ! [A5: b,B3: nat] :
( ( X5
= ( product_Pair_b_nat @ A5 @ B3 ) )
=> ! [A6: b,B5: nat] :
( ( Xa
= ( product_Pair_b_nat @ A6 @ B5 ) )
=> ( Y
= ( ( A5 = A6 )
= ( B3 != B5 ) ) ) ) ) ) ).
% sp_equiv_pair.elims(1)
thf(fact_313_sp__equiv__pair_Osimps,axiom,
! [A: nat,B: nat,A7: nat,B6: nat] :
( ( sp_equ885134019442699404at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ A7 @ B6 ) )
= ( ( A = A7 )
= ( B = B6 ) ) ) ).
% sp_equiv_pair.simps
thf(fact_314_sp__equiv__pair_Osimps,axiom,
! [A: product_prod_b_nat > set_list_a,B: produc4672180596006801056_a_nat,A7: product_prod_b_nat > set_list_a,B6: produc4672180596006801056_a_nat] :
( ( sp_equ4301142204835971757_a_nat @ ( produc6651248262528101210_a_nat @ A @ B ) @ ( produc6651248262528101210_a_nat @ A7 @ B6 ) )
= ( ( A = A7 )
= ( B = B6 ) ) ) ).
% sp_equiv_pair.simps
thf(fact_315_sp__equiv__pair_Osimps,axiom,
! [A: product_prod_b_nat > set_list_a,B: nat > a,A7: product_prod_b_nat > set_list_a,B6: nat > a] :
( ( sp_equ547821432705481458_nat_a @ ( produc2895298938842563487_nat_a @ A @ B ) @ ( produc2895298938842563487_nat_a @ A7 @ B6 ) )
= ( ( A = A7 )
= ( B = B6 ) ) ) ).
% sp_equiv_pair.simps
thf(fact_316_sp__equiv__pair_Osimps,axiom,
! [A: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat,A7: nat > sum_sum_a_nat,B6: set_Sum_sum_a_nat] :
( ( sp_equ1372826846814931683_a_nat @ ( produc3720304352952013712_a_nat @ A @ B ) @ ( produc3720304352952013712_a_nat @ A7 @ B6 ) )
= ( ( A = A7 )
= ( B = B6 ) ) ) ).
% sp_equiv_pair.simps
thf(fact_317_sp__equiv__pair_Osimps,axiom,
! [A: b,B: nat,A7: b,B6: nat] :
( ( sp_equiv_pair_b_nat @ ( product_Pair_b_nat @ A @ B ) @ ( product_Pair_b_nat @ A7 @ B6 ) )
= ( ( A = A7 )
= ( B = B6 ) ) ) ).
% sp_equiv_pair.simps
thf(fact_318_fo__fmla_Orel__distinct_I30_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y51: fo_fmla_a_b,Y52: fo_fmla_a_b,X31: fo_term_a,X32: fo_term_a] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Conj_a_b @ Y51 @ Y52 ) @ ( fo_Eqa_a_b @ X31 @ X32 ) ) ).
% fo_fmla.rel_distinct(30)
thf(fact_319_fo__fmla_Orel__distinct_I30_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y51: fo_fmla_nat_nat,Y52: fo_fmla_nat_nat,X31: fo_term_nat,X32: fo_term_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Conj_nat_nat @ Y51 @ Y52 ) @ ( fo_Eqa_nat_nat @ X31 @ X32 ) ) ).
% fo_fmla.rel_distinct(30)
thf(fact_320_fo__fmla_Orel__distinct_I29_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X31: fo_term_a,X32: fo_term_a,Y51: fo_fmla_a_b,Y52: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Eqa_a_b @ X31 @ X32 ) @ ( fo_Conj_a_b @ Y51 @ Y52 ) ) ).
% fo_fmla.rel_distinct(29)
thf(fact_321_fo__fmla_Orel__distinct_I29_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X31: fo_term_nat,X32: fo_term_nat,Y51: fo_fmla_nat_nat,Y52: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Eqa_nat_nat @ X31 @ X32 ) @ ( fo_Conj_nat_nat @ Y51 @ Y52 ) ) ).
% fo_fmla.rel_distinct(29)
thf(fact_322_fo__fmla_Orel__distinct_I32_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y61: fo_fmla_a_b,Y62: fo_fmla_a_b,X31: fo_term_a,X32: fo_term_a] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Disj_a_b @ Y61 @ Y62 ) @ ( fo_Eqa_a_b @ X31 @ X32 ) ) ).
% fo_fmla.rel_distinct(32)
thf(fact_323_fo__fmla_Orel__distinct_I32_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y61: fo_fmla_nat_nat,Y62: fo_fmla_nat_nat,X31: fo_term_nat,X32: fo_term_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Disj_nat_nat @ Y61 @ Y62 ) @ ( fo_Eqa_nat_nat @ X31 @ X32 ) ) ).
% fo_fmla.rel_distinct(32)
thf(fact_324_fo__fmla_Orel__distinct_I31_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X31: fo_term_a,X32: fo_term_a,Y61: fo_fmla_a_b,Y62: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Eqa_a_b @ X31 @ X32 ) @ ( fo_Disj_a_b @ Y61 @ Y62 ) ) ).
% fo_fmla.rel_distinct(31)
thf(fact_325_fo__fmla_Orel__distinct_I31_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X31: fo_term_nat,X32: fo_term_nat,Y61: fo_fmla_nat_nat,Y62: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Eqa_nat_nat @ X31 @ X32 ) @ ( fo_Disj_nat_nat @ Y61 @ Y62 ) ) ).
% fo_fmla.rel_distinct(31)
thf(fact_326_fo__fmla_Orel__distinct_I28_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y4: fo_fmla_a_b,X31: fo_term_a,X32: fo_term_a] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Neg_a_b @ Y4 ) @ ( fo_Eqa_a_b @ X31 @ X32 ) ) ).
% fo_fmla.rel_distinct(28)
thf(fact_327_fo__fmla_Orel__distinct_I28_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y4: fo_fmla_nat_nat,X31: fo_term_nat,X32: fo_term_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Neg_nat_nat @ Y4 ) @ ( fo_Eqa_nat_nat @ X31 @ X32 ) ) ).
% fo_fmla.rel_distinct(28)
thf(fact_328_fo__fmla_Orel__distinct_I27_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X31: fo_term_a,X32: fo_term_a,Y4: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Eqa_a_b @ X31 @ X32 ) @ ( fo_Neg_a_b @ Y4 ) ) ).
% fo_fmla.rel_distinct(27)
thf(fact_329_fo__fmla_Orel__distinct_I27_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X31: fo_term_nat,X32: fo_term_nat,Y4: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Eqa_nat_nat @ X31 @ X32 ) @ ( fo_Neg_nat_nat @ Y4 ) ) ).
% fo_fmla.rel_distinct(27)
thf(fact_330_fo__fmla_Orel__distinct_I34_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y71: nat,Y72: fo_fmla_a_b,X31: fo_term_a,X32: fo_term_a] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Exists_a_b @ Y71 @ Y72 ) @ ( fo_Eqa_a_b @ X31 @ X32 ) ) ).
% fo_fmla.rel_distinct(34)
thf(fact_331_fo__fmla_Orel__distinct_I34_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y71: nat,Y72: fo_fmla_nat_nat,X31: fo_term_nat,X32: fo_term_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Exists_nat_nat @ Y71 @ Y72 ) @ ( fo_Eqa_nat_nat @ X31 @ X32 ) ) ).
% fo_fmla.rel_distinct(34)
thf(fact_332_fo__fmla_Orel__distinct_I33_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X31: fo_term_a,X32: fo_term_a,Y71: nat,Y72: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Eqa_a_b @ X31 @ X32 ) @ ( fo_Exists_a_b @ Y71 @ Y72 ) ) ).
% fo_fmla.rel_distinct(33)
thf(fact_333_fo__fmla_Orel__distinct_I33_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X31: fo_term_nat,X32: fo_term_nat,Y71: nat,Y72: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Eqa_nat_nat @ X31 @ X32 ) @ ( fo_Exists_nat_nat @ Y71 @ Y72 ) ) ).
% fo_fmla.rel_distinct(33)
thf(fact_334_fo__fmla_Orel__distinct_I36_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y81: nat,Y82: fo_fmla_a_b,X31: fo_term_a,X32: fo_term_a] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Forall_a_b @ Y81 @ Y82 ) @ ( fo_Eqa_a_b @ X31 @ X32 ) ) ).
% fo_fmla.rel_distinct(36)
thf(fact_335_fo__fmla_Orel__distinct_I36_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y81: nat,Y82: fo_fmla_nat_nat,X31: fo_term_nat,X32: fo_term_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Forall_nat_nat @ Y81 @ Y82 ) @ ( fo_Eqa_nat_nat @ X31 @ X32 ) ) ).
% fo_fmla.rel_distinct(36)
thf(fact_336_fo__fmla_Orel__distinct_I35_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X31: fo_term_a,X32: fo_term_a,Y81: nat,Y82: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Eqa_a_b @ X31 @ X32 ) @ ( fo_Forall_a_b @ Y81 @ Y82 ) ) ).
% fo_fmla.rel_distinct(35)
thf(fact_337_fo__fmla_Orel__distinct_I35_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X31: fo_term_nat,X32: fo_term_nat,Y81: nat,Y82: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Eqa_nat_nat @ X31 @ X32 ) @ ( fo_Forall_nat_nat @ Y81 @ Y82 ) ) ).
% fo_fmla.rel_distinct(35)
thf(fact_338_fo__fmla_Orel__distinct_I3_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X11: b,X12: list_fo_term_a,Y31: fo_term_a,Y32: fo_term_a] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Pred_b_a @ X11 @ X12 ) @ ( fo_Eqa_a_b @ Y31 @ Y32 ) ) ).
% fo_fmla.rel_distinct(3)
thf(fact_339_fo__fmla_Orel__distinct_I3_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X11: nat,X12: list_fo_term_nat,Y31: fo_term_nat,Y32: fo_term_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Pred_nat_nat @ X11 @ X12 ) @ ( fo_Eqa_nat_nat @ Y31 @ Y32 ) ) ).
% fo_fmla.rel_distinct(3)
thf(fact_340_fo__fmla_Orel__distinct_I4_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y31: fo_term_a,Y32: fo_term_a,X11: b,X12: list_fo_term_a] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Eqa_a_b @ Y31 @ Y32 ) @ ( fo_Pred_b_a @ X11 @ X12 ) ) ).
% fo_fmla.rel_distinct(4)
thf(fact_341_fo__fmla_Orel__distinct_I4_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y31: fo_term_nat,Y32: fo_term_nat,X11: nat,X12: list_fo_term_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Eqa_nat_nat @ Y31 @ Y32 ) @ ( fo_Pred_nat_nat @ X11 @ X12 ) ) ).
% fo_fmla.rel_distinct(4)
thf(fact_342_fo__fmla_Orel__distinct_I46_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y61: fo_fmla_nat_nat,Y62: fo_fmla_nat_nat,X51: fo_fmla_nat_nat,X52: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Disj_nat_nat @ Y61 @ Y62 ) @ ( fo_Conj_nat_nat @ X51 @ X52 ) ) ).
% fo_fmla.rel_distinct(46)
thf(fact_343_fo__fmla_Orel__distinct_I46_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y61: fo_fmla_a_b,Y62: fo_fmla_a_b,X51: fo_fmla_a_b,X52: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Disj_a_b @ Y61 @ Y62 ) @ ( fo_Conj_a_b @ X51 @ X52 ) ) ).
% fo_fmla.rel_distinct(46)
thf(fact_344_fo__fmla_Orel__distinct_I45_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X51: fo_fmla_nat_nat,X52: fo_fmla_nat_nat,Y61: fo_fmla_nat_nat,Y62: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Conj_nat_nat @ X51 @ X52 ) @ ( fo_Disj_nat_nat @ Y61 @ Y62 ) ) ).
% fo_fmla.rel_distinct(45)
thf(fact_345_fo__fmla_Orel__distinct_I45_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X51: fo_fmla_a_b,X52: fo_fmla_a_b,Y61: fo_fmla_a_b,Y62: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Conj_a_b @ X51 @ X52 ) @ ( fo_Disj_a_b @ Y61 @ Y62 ) ) ).
% fo_fmla.rel_distinct(45)
thf(fact_346_fo__fmla_Orel__distinct_I38_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y51: fo_fmla_nat_nat,Y52: fo_fmla_nat_nat,X4: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Conj_nat_nat @ Y51 @ Y52 ) @ ( fo_Neg_nat_nat @ X4 ) ) ).
% fo_fmla.rel_distinct(38)
thf(fact_347_fo__fmla_Orel__distinct_I38_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y51: fo_fmla_a_b,Y52: fo_fmla_a_b,X4: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Conj_a_b @ Y51 @ Y52 ) @ ( fo_Neg_a_b @ X4 ) ) ).
% fo_fmla.rel_distinct(38)
thf(fact_348_fo__fmla_Orel__distinct_I37_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X4: fo_fmla_nat_nat,Y51: fo_fmla_nat_nat,Y52: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Neg_nat_nat @ X4 ) @ ( fo_Conj_nat_nat @ Y51 @ Y52 ) ) ).
% fo_fmla.rel_distinct(37)
thf(fact_349_fo__fmla_Orel__distinct_I37_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X4: fo_fmla_a_b,Y51: fo_fmla_a_b,Y52: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Neg_a_b @ X4 ) @ ( fo_Conj_a_b @ Y51 @ Y52 ) ) ).
% fo_fmla.rel_distinct(37)
thf(fact_350_fo__fmla_Orel__distinct_I16_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y31: fo_term_a,Y32: fo_term_a,X22: $o] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Eqa_a_b @ Y31 @ Y32 ) @ ( fo_Bool_a_b @ X22 ) ) ).
% fo_fmla.rel_distinct(16)
thf(fact_351_fo__fmla_Orel__distinct_I16_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y31: fo_term_nat,Y32: fo_term_nat,X22: $o] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Eqa_nat_nat @ Y31 @ Y32 ) @ ( fo_Bool_nat_nat @ X22 ) ) ).
% fo_fmla.rel_distinct(16)
thf(fact_352_fo__fmla_Orel__distinct_I15_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X22: $o,Y31: fo_term_a,Y32: fo_term_a] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Bool_a_b @ X22 ) @ ( fo_Eqa_a_b @ Y31 @ Y32 ) ) ).
% fo_fmla.rel_distinct(15)
thf(fact_353_fo__fmla_Orel__distinct_I15_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X22: $o,Y31: fo_term_nat,Y32: fo_term_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Bool_nat_nat @ X22 ) @ ( fo_Eqa_nat_nat @ Y31 @ Y32 ) ) ).
% fo_fmla.rel_distinct(15)
thf(fact_354_fo__fmla_Orel__distinct_I48_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y71: nat,Y72: fo_fmla_nat_nat,X51: fo_fmla_nat_nat,X52: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Exists_nat_nat @ Y71 @ Y72 ) @ ( fo_Conj_nat_nat @ X51 @ X52 ) ) ).
% fo_fmla.rel_distinct(48)
thf(fact_355_fo__fmla_Orel__distinct_I48_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y71: nat,Y72: fo_fmla_a_b,X51: fo_fmla_a_b,X52: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Exists_a_b @ Y71 @ Y72 ) @ ( fo_Conj_a_b @ X51 @ X52 ) ) ).
% fo_fmla.rel_distinct(48)
thf(fact_356_fo__fmla_Orel__distinct_I47_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X51: fo_fmla_nat_nat,X52: fo_fmla_nat_nat,Y71: nat,Y72: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Conj_nat_nat @ X51 @ X52 ) @ ( fo_Exists_nat_nat @ Y71 @ Y72 ) ) ).
% fo_fmla.rel_distinct(47)
thf(fact_357_fo__fmla_Orel__distinct_I47_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X51: fo_fmla_a_b,X52: fo_fmla_a_b,Y71: nat,Y72: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Conj_a_b @ X51 @ X52 ) @ ( fo_Exists_a_b @ Y71 @ Y72 ) ) ).
% fo_fmla.rel_distinct(47)
thf(fact_358_fo__fmla_Orel__distinct_I50_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y81: nat,Y82: fo_fmla_a_b,X51: fo_fmla_a_b,X52: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Forall_a_b @ Y81 @ Y82 ) @ ( fo_Conj_a_b @ X51 @ X52 ) ) ).
% fo_fmla.rel_distinct(50)
thf(fact_359_fo__fmla_Orel__distinct_I50_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y81: nat,Y82: fo_fmla_nat_nat,X51: fo_fmla_nat_nat,X52: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Forall_nat_nat @ Y81 @ Y82 ) @ ( fo_Conj_nat_nat @ X51 @ X52 ) ) ).
% fo_fmla.rel_distinct(50)
thf(fact_360_fo__fmla_Orel__distinct_I49_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X51: fo_fmla_a_b,X52: fo_fmla_a_b,Y81: nat,Y82: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Conj_a_b @ X51 @ X52 ) @ ( fo_Forall_a_b @ Y81 @ Y82 ) ) ).
% fo_fmla.rel_distinct(49)
thf(fact_361_fo__fmla_Orel__distinct_I49_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X51: fo_fmla_nat_nat,X52: fo_fmla_nat_nat,Y81: nat,Y82: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Conj_nat_nat @ X51 @ X52 ) @ ( fo_Forall_nat_nat @ Y81 @ Y82 ) ) ).
% fo_fmla.rel_distinct(49)
thf(fact_362_fo__fmla_Orel__distinct_I7_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X11: b,X12: list_fo_term_a,Y51: fo_fmla_a_b,Y52: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Pred_b_a @ X11 @ X12 ) @ ( fo_Conj_a_b @ Y51 @ Y52 ) ) ).
% fo_fmla.rel_distinct(7)
thf(fact_363_fo__fmla_Orel__distinct_I7_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X11: nat,X12: list_fo_term_nat,Y51: fo_fmla_nat_nat,Y52: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Pred_nat_nat @ X11 @ X12 ) @ ( fo_Conj_nat_nat @ Y51 @ Y52 ) ) ).
% fo_fmla.rel_distinct(7)
thf(fact_364_fo__fmla_Orel__distinct_I8_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y51: fo_fmla_a_b,Y52: fo_fmla_a_b,X11: b,X12: list_fo_term_a] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Conj_a_b @ Y51 @ Y52 ) @ ( fo_Pred_b_a @ X11 @ X12 ) ) ).
% fo_fmla.rel_distinct(8)
thf(fact_365_fo__fmla_Orel__distinct_I8_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y51: fo_fmla_nat_nat,Y52: fo_fmla_nat_nat,X11: nat,X12: list_fo_term_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Conj_nat_nat @ Y51 @ Y52 ) @ ( fo_Pred_nat_nat @ X11 @ X12 ) ) ).
% fo_fmla.rel_distinct(8)
thf(fact_366_fo__fmla_Orel__distinct_I40_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y61: fo_fmla_nat_nat,Y62: fo_fmla_nat_nat,X4: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Disj_nat_nat @ Y61 @ Y62 ) @ ( fo_Neg_nat_nat @ X4 ) ) ).
% fo_fmla.rel_distinct(40)
thf(fact_367_fo__fmla_Orel__distinct_I40_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y61: fo_fmla_a_b,Y62: fo_fmla_a_b,X4: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Disj_a_b @ Y61 @ Y62 ) @ ( fo_Neg_a_b @ X4 ) ) ).
% fo_fmla.rel_distinct(40)
thf(fact_368_fo__fmla_Orel__distinct_I39_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X4: fo_fmla_nat_nat,Y61: fo_fmla_nat_nat,Y62: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Neg_nat_nat @ X4 ) @ ( fo_Disj_nat_nat @ Y61 @ Y62 ) ) ).
% fo_fmla.rel_distinct(39)
thf(fact_369_fo__fmla_Orel__distinct_I39_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X4: fo_fmla_a_b,Y61: fo_fmla_a_b,Y62: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Neg_a_b @ X4 ) @ ( fo_Disj_a_b @ Y61 @ Y62 ) ) ).
% fo_fmla.rel_distinct(39)
thf(fact_370_fo__fmla_Orel__distinct_I52_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y71: nat,Y72: fo_fmla_nat_nat,X61: fo_fmla_nat_nat,X62: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Exists_nat_nat @ Y71 @ Y72 ) @ ( fo_Disj_nat_nat @ X61 @ X62 ) ) ).
% fo_fmla.rel_distinct(52)
thf(fact_371_fo__fmla_Orel__distinct_I52_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y71: nat,Y72: fo_fmla_a_b,X61: fo_fmla_a_b,X62: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Exists_a_b @ Y71 @ Y72 ) @ ( fo_Disj_a_b @ X61 @ X62 ) ) ).
% fo_fmla.rel_distinct(52)
thf(fact_372_fo__fmla_Orel__distinct_I51_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X61: fo_fmla_nat_nat,X62: fo_fmla_nat_nat,Y71: nat,Y72: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Disj_nat_nat @ X61 @ X62 ) @ ( fo_Exists_nat_nat @ Y71 @ Y72 ) ) ).
% fo_fmla.rel_distinct(51)
thf(fact_373_fo__fmla_Orel__distinct_I51_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X61: fo_fmla_a_b,X62: fo_fmla_a_b,Y71: nat,Y72: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Disj_a_b @ X61 @ X62 ) @ ( fo_Exists_a_b @ Y71 @ Y72 ) ) ).
% fo_fmla.rel_distinct(51)
thf(fact_374_fo__fmla_Orel__distinct_I54_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y81: nat,Y82: fo_fmla_a_b,X61: fo_fmla_a_b,X62: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Forall_a_b @ Y81 @ Y82 ) @ ( fo_Disj_a_b @ X61 @ X62 ) ) ).
% fo_fmla.rel_distinct(54)
thf(fact_375_fo__fmla_Orel__distinct_I54_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y81: nat,Y82: fo_fmla_nat_nat,X61: fo_fmla_nat_nat,X62: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Forall_nat_nat @ Y81 @ Y82 ) @ ( fo_Disj_nat_nat @ X61 @ X62 ) ) ).
% fo_fmla.rel_distinct(54)
thf(fact_376_fo__fmla_Orel__distinct_I53_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X61: fo_fmla_a_b,X62: fo_fmla_a_b,Y81: nat,Y82: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Disj_a_b @ X61 @ X62 ) @ ( fo_Forall_a_b @ Y81 @ Y82 ) ) ).
% fo_fmla.rel_distinct(53)
thf(fact_377_fo__fmla_Orel__distinct_I53_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X61: fo_fmla_nat_nat,X62: fo_fmla_nat_nat,Y81: nat,Y82: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Disj_nat_nat @ X61 @ X62 ) @ ( fo_Forall_nat_nat @ Y81 @ Y82 ) ) ).
% fo_fmla.rel_distinct(53)
thf(fact_378_fo__fmla_Orel__distinct_I9_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X11: b,X12: list_fo_term_a,Y61: fo_fmla_a_b,Y62: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Pred_b_a @ X11 @ X12 ) @ ( fo_Disj_a_b @ Y61 @ Y62 ) ) ).
% fo_fmla.rel_distinct(9)
thf(fact_379_fo__fmla_Orel__distinct_I9_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X11: nat,X12: list_fo_term_nat,Y61: fo_fmla_nat_nat,Y62: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Pred_nat_nat @ X11 @ X12 ) @ ( fo_Disj_nat_nat @ Y61 @ Y62 ) ) ).
% fo_fmla.rel_distinct(9)
thf(fact_380_fo__fmla_Orel__distinct_I10_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y61: fo_fmla_a_b,Y62: fo_fmla_a_b,X11: b,X12: list_fo_term_a] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Disj_a_b @ Y61 @ Y62 ) @ ( fo_Pred_b_a @ X11 @ X12 ) ) ).
% fo_fmla.rel_distinct(10)
thf(fact_381_fo__fmla_Orel__distinct_I10_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y61: fo_fmla_nat_nat,Y62: fo_fmla_nat_nat,X11: nat,X12: list_fo_term_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Disj_nat_nat @ Y61 @ Y62 ) @ ( fo_Pred_nat_nat @ X11 @ X12 ) ) ).
% fo_fmla.rel_distinct(10)
thf(fact_382_fo__fmla_Orel__distinct_I42_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y71: nat,Y72: fo_fmla_nat_nat,X4: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Exists_nat_nat @ Y71 @ Y72 ) @ ( fo_Neg_nat_nat @ X4 ) ) ).
% fo_fmla.rel_distinct(42)
thf(fact_383_fo__fmla_Orel__distinct_I42_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y71: nat,Y72: fo_fmla_a_b,X4: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Exists_a_b @ Y71 @ Y72 ) @ ( fo_Neg_a_b @ X4 ) ) ).
% fo_fmla.rel_distinct(42)
thf(fact_384_fo__fmla_Orel__distinct_I41_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X4: fo_fmla_nat_nat,Y71: nat,Y72: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Neg_nat_nat @ X4 ) @ ( fo_Exists_nat_nat @ Y71 @ Y72 ) ) ).
% fo_fmla.rel_distinct(41)
thf(fact_385_fo__fmla_Orel__distinct_I41_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X4: fo_fmla_a_b,Y71: nat,Y72: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Neg_a_b @ X4 ) @ ( fo_Exists_a_b @ Y71 @ Y72 ) ) ).
% fo_fmla.rel_distinct(41)
thf(fact_386_fo__fmla_Orel__distinct_I20_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y51: fo_fmla_nat_nat,Y52: fo_fmla_nat_nat,X22: $o] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Conj_nat_nat @ Y51 @ Y52 ) @ ( fo_Bool_nat_nat @ X22 ) ) ).
% fo_fmla.rel_distinct(20)
thf(fact_387_fo__fmla_Orel__distinct_I20_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y51: fo_fmla_a_b,Y52: fo_fmla_a_b,X22: $o] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Conj_a_b @ Y51 @ Y52 ) @ ( fo_Bool_a_b @ X22 ) ) ).
% fo_fmla.rel_distinct(20)
thf(fact_388_fo__fmla_Orel__distinct_I19_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X22: $o,Y51: fo_fmla_nat_nat,Y52: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Bool_nat_nat @ X22 ) @ ( fo_Conj_nat_nat @ Y51 @ Y52 ) ) ).
% fo_fmla.rel_distinct(19)
thf(fact_389_fo__fmla_Orel__distinct_I19_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X22: $o,Y51: fo_fmla_a_b,Y52: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Bool_a_b @ X22 ) @ ( fo_Conj_a_b @ Y51 @ Y52 ) ) ).
% fo_fmla.rel_distinct(19)
thf(fact_390_fo__fmla_Orel__distinct_I44_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y81: nat,Y82: fo_fmla_a_b,X4: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Forall_a_b @ Y81 @ Y82 ) @ ( fo_Neg_a_b @ X4 ) ) ).
% fo_fmla.rel_distinct(44)
thf(fact_391_fo__fmla_Orel__distinct_I44_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y81: nat,Y82: fo_fmla_nat_nat,X4: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Forall_nat_nat @ Y81 @ Y82 ) @ ( fo_Neg_nat_nat @ X4 ) ) ).
% fo_fmla.rel_distinct(44)
thf(fact_392_fo__fmla_Orel__distinct_I43_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X4: fo_fmla_a_b,Y81: nat,Y82: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Neg_a_b @ X4 ) @ ( fo_Forall_a_b @ Y81 @ Y82 ) ) ).
% fo_fmla.rel_distinct(43)
thf(fact_393_fo__fmla_Orel__distinct_I43_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X4: fo_fmla_nat_nat,Y81: nat,Y82: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Neg_nat_nat @ X4 ) @ ( fo_Forall_nat_nat @ Y81 @ Y82 ) ) ).
% fo_fmla.rel_distinct(43)
thf(fact_394_fo__fmla_Orel__distinct_I5_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X11: b,X12: list_fo_term_a,Y4: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Pred_b_a @ X11 @ X12 ) @ ( fo_Neg_a_b @ Y4 ) ) ).
% fo_fmla.rel_distinct(5)
thf(fact_395_fo__fmla_Orel__distinct_I5_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X11: nat,X12: list_fo_term_nat,Y4: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Pred_nat_nat @ X11 @ X12 ) @ ( fo_Neg_nat_nat @ Y4 ) ) ).
% fo_fmla.rel_distinct(5)
thf(fact_396_fo__fmla_Orel__distinct_I6_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y4: fo_fmla_a_b,X11: b,X12: list_fo_term_a] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Neg_a_b @ Y4 ) @ ( fo_Pred_b_a @ X11 @ X12 ) ) ).
% fo_fmla.rel_distinct(6)
thf(fact_397_fo__fmla_Orel__distinct_I6_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y4: fo_fmla_nat_nat,X11: nat,X12: list_fo_term_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Neg_nat_nat @ Y4 ) @ ( fo_Pred_nat_nat @ X11 @ X12 ) ) ).
% fo_fmla.rel_distinct(6)
thf(fact_398_fo__fmla_Orel__distinct_I56_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y81: nat,Y82: fo_fmla_a_b,X71: nat,X72: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Forall_a_b @ Y81 @ Y82 ) @ ( fo_Exists_a_b @ X71 @ X72 ) ) ).
% fo_fmla.rel_distinct(56)
thf(fact_399_fo__fmla_Orel__distinct_I56_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y81: nat,Y82: fo_fmla_nat_nat,X71: nat,X72: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Forall_nat_nat @ Y81 @ Y82 ) @ ( fo_Exists_nat_nat @ X71 @ X72 ) ) ).
% fo_fmla.rel_distinct(56)
thf(fact_400_fo__fmla_Orel__distinct_I55_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X71: nat,X72: fo_fmla_a_b,Y81: nat,Y82: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Exists_a_b @ X71 @ X72 ) @ ( fo_Forall_a_b @ Y81 @ Y82 ) ) ).
% fo_fmla.rel_distinct(55)
thf(fact_401_fo__fmla_Orel__distinct_I55_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X71: nat,X72: fo_fmla_nat_nat,Y81: nat,Y82: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Exists_nat_nat @ X71 @ X72 ) @ ( fo_Forall_nat_nat @ Y81 @ Y82 ) ) ).
% fo_fmla.rel_distinct(55)
thf(fact_402_fo__fmla_Orel__distinct_I11_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X11: b,X12: list_fo_term_a,Y71: nat,Y72: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Pred_b_a @ X11 @ X12 ) @ ( fo_Exists_a_b @ Y71 @ Y72 ) ) ).
% fo_fmla.rel_distinct(11)
thf(fact_403_fo__fmla_Orel__distinct_I11_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X11: nat,X12: list_fo_term_nat,Y71: nat,Y72: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Pred_nat_nat @ X11 @ X12 ) @ ( fo_Exists_nat_nat @ Y71 @ Y72 ) ) ).
% fo_fmla.rel_distinct(11)
thf(fact_404_fo__fmla_Orel__distinct_I12_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y71: nat,Y72: fo_fmla_a_b,X11: b,X12: list_fo_term_a] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Exists_a_b @ Y71 @ Y72 ) @ ( fo_Pred_b_a @ X11 @ X12 ) ) ).
% fo_fmla.rel_distinct(12)
thf(fact_405_fo__fmla_Orel__distinct_I12_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y71: nat,Y72: fo_fmla_nat_nat,X11: nat,X12: list_fo_term_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Exists_nat_nat @ Y71 @ Y72 ) @ ( fo_Pred_nat_nat @ X11 @ X12 ) ) ).
% fo_fmla.rel_distinct(12)
thf(fact_406_fo__fmla_Orel__distinct_I13_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X11: b,X12: list_fo_term_a,Y81: nat,Y82: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Pred_b_a @ X11 @ X12 ) @ ( fo_Forall_a_b @ Y81 @ Y82 ) ) ).
% fo_fmla.rel_distinct(13)
thf(fact_407_fo__fmla_Orel__distinct_I13_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X11: nat,X12: list_fo_term_nat,Y81: nat,Y82: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Pred_nat_nat @ X11 @ X12 ) @ ( fo_Forall_nat_nat @ Y81 @ Y82 ) ) ).
% fo_fmla.rel_distinct(13)
thf(fact_408_fo__fmla_Orel__distinct_I14_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y81: nat,Y82: fo_fmla_a_b,X11: b,X12: list_fo_term_a] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Forall_a_b @ Y81 @ Y82 ) @ ( fo_Pred_b_a @ X11 @ X12 ) ) ).
% fo_fmla.rel_distinct(14)
thf(fact_409_fo__fmla_Orel__distinct_I14_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y81: nat,Y82: fo_fmla_nat_nat,X11: nat,X12: list_fo_term_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Forall_nat_nat @ Y81 @ Y82 ) @ ( fo_Pred_nat_nat @ X11 @ X12 ) ) ).
% fo_fmla.rel_distinct(14)
thf(fact_410_fo__fmla_Orel__distinct_I22_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y61: fo_fmla_nat_nat,Y62: fo_fmla_nat_nat,X22: $o] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Disj_nat_nat @ Y61 @ Y62 ) @ ( fo_Bool_nat_nat @ X22 ) ) ).
% fo_fmla.rel_distinct(22)
thf(fact_411_fo__fmla_Orel__distinct_I22_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y61: fo_fmla_a_b,Y62: fo_fmla_a_b,X22: $o] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Disj_a_b @ Y61 @ Y62 ) @ ( fo_Bool_a_b @ X22 ) ) ).
% fo_fmla.rel_distinct(22)
thf(fact_412_fo__fmla_Orel__distinct_I21_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X22: $o,Y61: fo_fmla_nat_nat,Y62: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Bool_nat_nat @ X22 ) @ ( fo_Disj_nat_nat @ Y61 @ Y62 ) ) ).
% fo_fmla.rel_distinct(21)
thf(fact_413_fo__fmla_Orel__distinct_I21_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X22: $o,Y61: fo_fmla_a_b,Y62: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Bool_a_b @ X22 ) @ ( fo_Disj_a_b @ Y61 @ Y62 ) ) ).
% fo_fmla.rel_distinct(21)
thf(fact_414_fo__fmla_Orel__distinct_I18_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y4: fo_fmla_nat_nat,X22: $o] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Neg_nat_nat @ Y4 ) @ ( fo_Bool_nat_nat @ X22 ) ) ).
% fo_fmla.rel_distinct(18)
thf(fact_415_fo__fmla_Orel__distinct_I18_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y4: fo_fmla_a_b,X22: $o] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Neg_a_b @ Y4 ) @ ( fo_Bool_a_b @ X22 ) ) ).
% fo_fmla.rel_distinct(18)
thf(fact_416_fo__fmla_Orel__distinct_I17_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X22: $o,Y4: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Bool_nat_nat @ X22 ) @ ( fo_Neg_nat_nat @ Y4 ) ) ).
% fo_fmla.rel_distinct(17)
thf(fact_417_fo__fmla_Orel__distinct_I17_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X22: $o,Y4: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Bool_a_b @ X22 ) @ ( fo_Neg_a_b @ Y4 ) ) ).
% fo_fmla.rel_distinct(17)
thf(fact_418_fo__fmla_Orel__distinct_I24_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y71: nat,Y72: fo_fmla_nat_nat,X22: $o] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Exists_nat_nat @ Y71 @ Y72 ) @ ( fo_Bool_nat_nat @ X22 ) ) ).
% fo_fmla.rel_distinct(24)
thf(fact_419_fo__fmla_Orel__distinct_I24_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y71: nat,Y72: fo_fmla_a_b,X22: $o] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Exists_a_b @ Y71 @ Y72 ) @ ( fo_Bool_a_b @ X22 ) ) ).
% fo_fmla.rel_distinct(24)
thf(fact_420_fo__fmla_Orel__distinct_I23_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X22: $o,Y71: nat,Y72: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Bool_nat_nat @ X22 ) @ ( fo_Exists_nat_nat @ Y71 @ Y72 ) ) ).
% fo_fmla.rel_distinct(23)
thf(fact_421_fo__fmla_Orel__distinct_I23_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X22: $o,Y71: nat,Y72: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Bool_a_b @ X22 ) @ ( fo_Exists_a_b @ Y71 @ Y72 ) ) ).
% fo_fmla.rel_distinct(23)
thf(fact_422_fo__fmla_Orel__distinct_I26_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y81: nat,Y82: fo_fmla_a_b,X22: $o] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Forall_a_b @ Y81 @ Y82 ) @ ( fo_Bool_a_b @ X22 ) ) ).
% fo_fmla.rel_distinct(26)
thf(fact_423_fo__fmla_Orel__distinct_I26_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y81: nat,Y82: fo_fmla_nat_nat,X22: $o] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Forall_nat_nat @ Y81 @ Y82 ) @ ( fo_Bool_nat_nat @ X22 ) ) ).
% fo_fmla.rel_distinct(26)
thf(fact_424_fo__fmla_Orel__distinct_I25_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X22: $o,Y81: nat,Y82: fo_fmla_a_b] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Bool_a_b @ X22 ) @ ( fo_Forall_a_b @ Y81 @ Y82 ) ) ).
% fo_fmla.rel_distinct(25)
thf(fact_425_fo__fmla_Orel__distinct_I25_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X22: $o,Y81: nat,Y82: fo_fmla_nat_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Bool_nat_nat @ X22 ) @ ( fo_Forall_nat_nat @ Y81 @ Y82 ) ) ).
% fo_fmla.rel_distinct(25)
thf(fact_426_fo__fmla_Orel__distinct_I1_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,X11: b,X12: list_fo_term_a,Y2: $o] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Pred_b_a @ X11 @ X12 ) @ ( fo_Bool_a_b @ Y2 ) ) ).
% fo_fmla.rel_distinct(1)
thf(fact_427_fo__fmla_Orel__distinct_I1_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,X11: nat,X12: list_fo_term_nat,Y2: $o] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Pred_nat_nat @ X11 @ X12 ) @ ( fo_Bool_nat_nat @ Y2 ) ) ).
% fo_fmla.rel_distinct(1)
thf(fact_428_fo__fmla_Orel__distinct_I2_J,axiom,
! [R1: a > a > $o,R2: b > b > $o,Y2: $o,X11: b,X12: list_fo_term_a] :
~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ ( fo_Bool_a_b @ Y2 ) @ ( fo_Pred_b_a @ X11 @ X12 ) ) ).
% fo_fmla.rel_distinct(2)
thf(fact_429_fo__fmla_Orel__distinct_I2_J,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,Y2: $o,X11: nat,X12: list_fo_term_nat] :
~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ ( fo_Bool_nat_nat @ Y2 ) @ ( fo_Pred_nat_nat @ X11 @ X12 ) ) ).
% fo_fmla.rel_distinct(2)
thf(fact_430_fo__fmla_Orel__cases,axiom,
! [R1: nat > nat > $o,R2: nat > nat > $o,A: fo_fmla_nat_nat,B: fo_fmla_nat_nat] :
( ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ A @ B )
=> ( ! [X13: nat,X2a: list_fo_term_nat] :
( ( A
= ( fo_Pred_nat_nat @ X13 @ X2a ) )
=> ! [Y13: nat,Y2a: list_fo_term_nat] :
( ( B
= ( fo_Pred_nat_nat @ Y13 @ Y2a ) )
=> ( ( R2 @ X13 @ Y13 )
=> ~ ( list_a3012696415560123489rm_nat @ ( fo_rel1533169934621585718at_nat @ R1 ) @ X2a @ Y2a ) ) ) )
=> ( ! [X: $o] :
( ( A
= ( fo_Bool_nat_nat @ X ) )
=> ! [Y5: $o] :
( ( B
= ( fo_Bool_nat_nat @ Y5 ) )
=> ( X = (~ Y5) ) ) )
=> ( ! [X1a: fo_term_nat,X2b: fo_term_nat] :
( ( A
= ( fo_Eqa_nat_nat @ X1a @ X2b ) )
=> ! [Y1a: fo_term_nat,Y2b: fo_term_nat] :
( ( B
= ( fo_Eqa_nat_nat @ Y1a @ Y2b ) )
=> ( ( fo_rel1533169934621585718at_nat @ R1 @ X1a @ Y1a )
=> ~ ( fo_rel1533169934621585718at_nat @ R1 @ X2b @ Y2b ) ) ) )
=> ( ! [Xa2: fo_fmla_nat_nat] :
( ( A
= ( fo_Neg_nat_nat @ Xa2 ) )
=> ! [Ya2: fo_fmla_nat_nat] :
( ( B
= ( fo_Neg_nat_nat @ Ya2 ) )
=> ~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ Xa2 @ Ya2 ) ) )
=> ( ! [X1b: fo_fmla_nat_nat,X2c: fo_fmla_nat_nat] :
( ( A
= ( fo_Conj_nat_nat @ X1b @ X2c ) )
=> ! [Y1b: fo_fmla_nat_nat,Y2c: fo_fmla_nat_nat] :
( ( B
= ( fo_Conj_nat_nat @ Y1b @ Y2c ) )
=> ( ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ X1b @ Y1b )
=> ~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ X2c @ Y2c ) ) ) )
=> ( ! [X1c: fo_fmla_nat_nat,X2d: fo_fmla_nat_nat] :
( ( A
= ( fo_Disj_nat_nat @ X1c @ X2d ) )
=> ! [Y1c: fo_fmla_nat_nat,Y2d: fo_fmla_nat_nat] :
( ( B
= ( fo_Disj_nat_nat @ Y1c @ Y2d ) )
=> ( ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ X1c @ Y1c )
=> ~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ X2d @ Y2d ) ) ) )
=> ( ! [X1d: nat,X2e: fo_fmla_nat_nat] :
( ( A
= ( fo_Exists_nat_nat @ X1d @ X2e ) )
=> ! [Y1d: nat,Y2e: fo_fmla_nat_nat] :
( ( B
= ( fo_Exists_nat_nat @ Y1d @ Y2e ) )
=> ( ( X1d = Y1d )
=> ~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ X2e @ Y2e ) ) ) )
=> ~ ! [X1e: nat,X2f: fo_fmla_nat_nat] :
( ( A
= ( fo_Forall_nat_nat @ X1e @ X2f ) )
=> ! [Y1e: nat,Y2f: fo_fmla_nat_nat] :
( ( B
= ( fo_Forall_nat_nat @ Y1e @ Y2f ) )
=> ( ( X1e = Y1e )
=> ~ ( fo_rel4206018205878794776at_nat @ R1 @ R2 @ X2f @ Y2f ) ) ) ) ) ) ) ) ) ) ) ) ).
% fo_fmla.rel_cases
thf(fact_431_fo__fmla_Orel__cases,axiom,
! [R1: a > a > $o,R2: b > b > $o,A: fo_fmla_a_b,B: fo_fmla_a_b] :
( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ A @ B )
=> ( ! [X13: b,X2a: list_fo_term_a] :
( ( A
= ( fo_Pred_b_a @ X13 @ X2a ) )
=> ! [Y13: b,Y2a: list_fo_term_a] :
( ( B
= ( fo_Pred_b_a @ Y13 @ Y2a ) )
=> ( ( R2 @ X13 @ Y13 )
=> ~ ( list_a2487961086254413575term_a @ ( fo_rel_fo_term_a_a @ R1 ) @ X2a @ Y2a ) ) ) )
=> ( ! [X: $o] :
( ( A
= ( fo_Bool_a_b @ X ) )
=> ! [Y5: $o] :
( ( B
= ( fo_Bool_a_b @ Y5 ) )
=> ( X = (~ Y5) ) ) )
=> ( ! [X1a: fo_term_a,X2b: fo_term_a] :
( ( A
= ( fo_Eqa_a_b @ X1a @ X2b ) )
=> ! [Y1a: fo_term_a,Y2b: fo_term_a] :
( ( B
= ( fo_Eqa_a_b @ Y1a @ Y2b ) )
=> ( ( fo_rel_fo_term_a_a @ R1 @ X1a @ Y1a )
=> ~ ( fo_rel_fo_term_a_a @ R1 @ X2b @ Y2b ) ) ) )
=> ( ! [Xa2: fo_fmla_a_b] :
( ( A
= ( fo_Neg_a_b @ Xa2 ) )
=> ! [Ya2: fo_fmla_a_b] :
( ( B
= ( fo_Neg_a_b @ Ya2 ) )
=> ~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ Xa2 @ Ya2 ) ) )
=> ( ! [X1b: fo_fmla_a_b,X2c: fo_fmla_a_b] :
( ( A
= ( fo_Conj_a_b @ X1b @ X2c ) )
=> ! [Y1b: fo_fmla_a_b,Y2c: fo_fmla_a_b] :
( ( B
= ( fo_Conj_a_b @ Y1b @ Y2c ) )
=> ( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X1b @ Y1b )
=> ~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X2c @ Y2c ) ) ) )
=> ( ! [X1c: fo_fmla_a_b,X2d: fo_fmla_a_b] :
( ( A
= ( fo_Disj_a_b @ X1c @ X2d ) )
=> ! [Y1c: fo_fmla_a_b,Y2d: fo_fmla_a_b] :
( ( B
= ( fo_Disj_a_b @ Y1c @ Y2d ) )
=> ( ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X1c @ Y1c )
=> ~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X2d @ Y2d ) ) ) )
=> ( ! [X1d: nat,X2e: fo_fmla_a_b] :
( ( A
= ( fo_Exists_a_b @ X1d @ X2e ) )
=> ! [Y1d: nat,Y2e: fo_fmla_a_b] :
( ( B
= ( fo_Exists_a_b @ Y1d @ Y2e ) )
=> ( ( X1d = Y1d )
=> ~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X2e @ Y2e ) ) ) )
=> ~ ! [X1e: nat,X2f: fo_fmla_a_b] :
( ( A
= ( fo_Forall_a_b @ X1e @ X2f ) )
=> ! [Y1e: nat,Y2f: fo_fmla_a_b] :
( ( B
= ( fo_Forall_a_b @ Y1e @ Y2f ) )
=> ( ( X1e = Y1e )
=> ~ ( fo_rel8103664758052585748_a_b_b @ R1 @ R2 @ X2f @ Y2f ) ) ) ) ) ) ) ) ) ) ) ) ).
% fo_fmla.rel_cases
thf(fact_432_empty__iff,axiom,
! [C: fo_term_a] :
~ ( member_fo_term_a @ C @ bot_bo4735268219511357444term_a ) ).
% empty_iff
thf(fact_433_empty__iff,axiom,
! [C: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ C @ bot_bo1033123847703346641_a_nat ) ).
% empty_iff
thf(fact_434_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_435_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_436_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_437_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_438_all__not__in__conv,axiom,
! [A3: set_fo_term_a] :
( ( ! [X3: fo_term_a] :
~ ( member_fo_term_a @ X3 @ A3 ) )
= ( A3 = bot_bo4735268219511357444term_a ) ) ).
% all_not_in_conv
thf(fact_439_all__not__in__conv,axiom,
! [A3: set_li6526943997496501093_a_nat] :
( ( ! [X3: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ X3 @ A3 ) )
= ( A3 = bot_bo1033123847703346641_a_nat ) ) ).
% all_not_in_conv
thf(fact_440_all__not__in__conv,axiom,
! [A3: set_set_nat] :
( ( ! [X3: set_nat] :
~ ( member_set_nat @ X3 @ A3 ) )
= ( A3 = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_441_all__not__in__conv,axiom,
! [A3: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_442_all__not__in__conv,axiom,
! [A3: set_b] :
( ( ! [X3: b] :
~ ( member_b @ X3 @ A3 ) )
= ( A3 = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_443_all__not__in__conv,axiom,
! [A3: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A3 ) )
= ( A3 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_444_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A7: nat,B6: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A7 @ B6 ) )
= ( ( A = A7 )
& ( B = B6 ) ) ) ).
% old.prod.inject
thf(fact_445_old_Oprod_Oinject,axiom,
! [A: product_prod_b_nat > set_list_a,B: produc4672180596006801056_a_nat,A7: product_prod_b_nat > set_list_a,B6: produc4672180596006801056_a_nat] :
( ( ( produc6651248262528101210_a_nat @ A @ B )
= ( produc6651248262528101210_a_nat @ A7 @ B6 ) )
= ( ( A = A7 )
& ( B = B6 ) ) ) ).
% old.prod.inject
thf(fact_446_old_Oprod_Oinject,axiom,
! [A: product_prod_b_nat > set_list_a,B: nat > a,A7: product_prod_b_nat > set_list_a,B6: nat > a] :
( ( ( produc2895298938842563487_nat_a @ A @ B )
= ( produc2895298938842563487_nat_a @ A7 @ B6 ) )
= ( ( A = A7 )
& ( B = B6 ) ) ) ).
% old.prod.inject
thf(fact_447_old_Oprod_Oinject,axiom,
! [A: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat,A7: nat > sum_sum_a_nat,B6: set_Sum_sum_a_nat] :
( ( ( produc3720304352952013712_a_nat @ A @ B )
= ( produc3720304352952013712_a_nat @ A7 @ B6 ) )
= ( ( A = A7 )
& ( B = B6 ) ) ) ).
% old.prod.inject
thf(fact_448_old_Oprod_Oinject,axiom,
! [A: b,B: nat,A7: b,B6: nat] :
( ( ( product_Pair_b_nat @ A @ B )
= ( product_Pair_b_nat @ A7 @ B6 ) )
= ( ( A = A7 )
& ( B = B6 ) ) ) ).
% old.prod.inject
thf(fact_449_prod_Oinject,axiom,
! [X1: nat,X22: nat,Y1: nat,Y2: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X22 )
= ( product_Pair_nat_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_450_prod_Oinject,axiom,
! [X1: product_prod_b_nat > set_list_a,X22: produc4672180596006801056_a_nat,Y1: product_prod_b_nat > set_list_a,Y2: produc4672180596006801056_a_nat] :
( ( ( produc6651248262528101210_a_nat @ X1 @ X22 )
= ( produc6651248262528101210_a_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_451_prod_Oinject,axiom,
! [X1: product_prod_b_nat > set_list_a,X22: nat > a,Y1: product_prod_b_nat > set_list_a,Y2: nat > a] :
( ( ( produc2895298938842563487_nat_a @ X1 @ X22 )
= ( produc2895298938842563487_nat_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_452_prod_Oinject,axiom,
! [X1: nat > sum_sum_a_nat,X22: set_Sum_sum_a_nat,Y1: nat > sum_sum_a_nat,Y2: set_Sum_sum_a_nat] :
( ( ( produc3720304352952013712_a_nat @ X1 @ X22 )
= ( produc3720304352952013712_a_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_453_prod_Oinject,axiom,
! [X1: b,X22: nat,Y1: b,Y2: nat] :
( ( ( product_Pair_b_nat @ X1 @ X22 )
= ( product_Pair_b_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_454_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_455_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_456_empty__Collect__eq,axiom,
! [P: b > $o] :
( ( bot_bot_set_b
= ( collect_b @ P ) )
= ( ! [X3: b] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_457_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_458_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_459_Collect__empty__eq,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( ! [X3: b] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_460_eval__term_Ocases,axiom,
! [X5: produc7543280176765104058term_a] :
( ! [Sigma3: nat > a,C2: a] :
( X5
!= ( produc5936395021994102698term_a @ Sigma3 @ ( fo_Const_a @ C2 ) ) )
=> ~ ! [Sigma3: nat > a,N: nat] :
( X5
!= ( produc5936395021994102698term_a @ Sigma3 @ ( fo_Var_a @ N ) ) ) ) ).
% eval_term.cases
thf(fact_461_eval__term_Ocases,axiom,
! [X5: produc5895652031813375564rm_nat] :
( ! [Sigma3: nat > nat,C2: nat] :
( X5
!= ( produc3609431976081568388rm_nat @ Sigma3 @ ( fo_Const_nat @ C2 ) ) )
=> ~ ! [Sigma3: nat > nat,N: nat] :
( X5
!= ( produc3609431976081568388rm_nat @ Sigma3 @ ( fo_Var_nat @ N ) ) ) ) ).
% eval_term.cases
thf(fact_462_Pair__inject,axiom,
! [A: nat,B: nat,A7: nat,B6: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A7 @ B6 ) )
=> ~ ( ( A = A7 )
=> ( B != B6 ) ) ) ).
% Pair_inject
thf(fact_463_Pair__inject,axiom,
! [A: product_prod_b_nat > set_list_a,B: produc4672180596006801056_a_nat,A7: product_prod_b_nat > set_list_a,B6: produc4672180596006801056_a_nat] :
( ( ( produc6651248262528101210_a_nat @ A @ B )
= ( produc6651248262528101210_a_nat @ A7 @ B6 ) )
=> ~ ( ( A = A7 )
=> ( B != B6 ) ) ) ).
% Pair_inject
thf(fact_464_Pair__inject,axiom,
! [A: product_prod_b_nat > set_list_a,B: nat > a,A7: product_prod_b_nat > set_list_a,B6: nat > a] :
( ( ( produc2895298938842563487_nat_a @ A @ B )
= ( produc2895298938842563487_nat_a @ A7 @ B6 ) )
=> ~ ( ( A = A7 )
=> ( B != B6 ) ) ) ).
% Pair_inject
thf(fact_465_Pair__inject,axiom,
! [A: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat,A7: nat > sum_sum_a_nat,B6: set_Sum_sum_a_nat] :
( ( ( produc3720304352952013712_a_nat @ A @ B )
= ( produc3720304352952013712_a_nat @ A7 @ B6 ) )
=> ~ ( ( A = A7 )
=> ( B != B6 ) ) ) ).
% Pair_inject
thf(fact_466_Pair__inject,axiom,
! [A: b,B: nat,A7: b,B6: nat] :
( ( ( product_Pair_b_nat @ A @ B )
= ( product_Pair_b_nat @ A7 @ B6 ) )
=> ~ ( ( A = A7 )
=> ( B != B6 ) ) ) ).
% Pair_inject
thf(fact_467_prod__cases,axiom,
! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
( ! [A5: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A5 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_468_prod__cases,axiom,
! [P: produc5433867242478491114_a_nat > $o,P2: produc5433867242478491114_a_nat] :
( ! [A5: product_prod_b_nat > set_list_a,B3: produc4672180596006801056_a_nat] : ( P @ ( produc6651248262528101210_a_nat @ A5 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_469_prod__cases,axiom,
! [P: produc5835360497134304175_nat_a > $o,P2: produc5835360497134304175_nat_a] :
( ! [A5: product_prod_b_nat > set_list_a,B3: nat > a] : ( P @ ( produc2895298938842563487_nat_a @ A5 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_470_prod__cases,axiom,
! [P: produc4672180596006801056_a_nat > $o,P2: produc4672180596006801056_a_nat] :
( ! [A5: nat > sum_sum_a_nat,B3: set_Sum_sum_a_nat] : ( P @ ( produc3720304352952013712_a_nat @ A5 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_471_prod__cases,axiom,
! [P: product_prod_b_nat > $o,P2: product_prod_b_nat] :
( ! [A5: b,B3: nat] : ( P @ ( product_Pair_b_nat @ A5 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_472_surj__pair,axiom,
! [P2: product_prod_nat_nat] :
? [X: nat,Y5: nat] :
( P2
= ( product_Pair_nat_nat @ X @ Y5 ) ) ).
% surj_pair
thf(fact_473_surj__pair,axiom,
! [P2: produc5433867242478491114_a_nat] :
? [X: product_prod_b_nat > set_list_a,Y5: produc4672180596006801056_a_nat] :
( P2
= ( produc6651248262528101210_a_nat @ X @ Y5 ) ) ).
% surj_pair
thf(fact_474_surj__pair,axiom,
! [P2: produc5835360497134304175_nat_a] :
? [X: product_prod_b_nat > set_list_a,Y5: nat > a] :
( P2
= ( produc2895298938842563487_nat_a @ X @ Y5 ) ) ).
% surj_pair
thf(fact_475_surj__pair,axiom,
! [P2: produc4672180596006801056_a_nat] :
? [X: nat > sum_sum_a_nat,Y5: set_Sum_sum_a_nat] :
( P2
= ( produc3720304352952013712_a_nat @ X @ Y5 ) ) ).
% surj_pair
thf(fact_476_surj__pair,axiom,
! [P2: product_prod_b_nat] :
? [X: b,Y5: nat] :
( P2
= ( product_Pair_b_nat @ X @ Y5 ) ) ).
% surj_pair
thf(fact_477_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_nat_nat] :
~ ! [A5: nat,B3: nat] :
( Y
!= ( product_Pair_nat_nat @ A5 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_478_old_Oprod_Oexhaust,axiom,
! [Y: produc5433867242478491114_a_nat] :
~ ! [A5: product_prod_b_nat > set_list_a,B3: produc4672180596006801056_a_nat] :
( Y
!= ( produc6651248262528101210_a_nat @ A5 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_479_old_Oprod_Oexhaust,axiom,
! [Y: produc5835360497134304175_nat_a] :
~ ! [A5: product_prod_b_nat > set_list_a,B3: nat > a] :
( Y
!= ( produc2895298938842563487_nat_a @ A5 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_480_old_Oprod_Oexhaust,axiom,
! [Y: produc4672180596006801056_a_nat] :
~ ! [A5: nat > sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( Y
!= ( produc3720304352952013712_a_nat @ A5 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_481_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_b_nat] :
~ ! [A5: b,B3: nat] :
( Y
!= ( product_Pair_b_nat @ A5 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_482_ex__in__conv,axiom,
! [A3: set_fo_term_a] :
( ( ? [X3: fo_term_a] : ( member_fo_term_a @ X3 @ A3 ) )
= ( A3 != bot_bo4735268219511357444term_a ) ) ).
% ex_in_conv
thf(fact_483_ex__in__conv,axiom,
! [A3: set_li6526943997496501093_a_nat] :
( ( ? [X3: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X3 @ A3 ) )
= ( A3 != bot_bo1033123847703346641_a_nat ) ) ).
% ex_in_conv
thf(fact_484_ex__in__conv,axiom,
! [A3: set_set_nat] :
( ( ? [X3: set_nat] : ( member_set_nat @ X3 @ A3 ) )
= ( A3 != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_485_ex__in__conv,axiom,
! [A3: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A3 ) )
= ( A3 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_486_ex__in__conv,axiom,
! [A3: set_b] :
( ( ? [X3: b] : ( member_b @ X3 @ A3 ) )
= ( A3 != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_487_ex__in__conv,axiom,
! [A3: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A3 ) )
= ( A3 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_488_equals0I,axiom,
! [A3: set_fo_term_a] :
( ! [Y5: fo_term_a] :
~ ( member_fo_term_a @ Y5 @ A3 )
=> ( A3 = bot_bo4735268219511357444term_a ) ) ).
% equals0I
thf(fact_489_equals0I,axiom,
! [A3: set_li6526943997496501093_a_nat] :
( ! [Y5: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ Y5 @ A3 )
=> ( A3 = bot_bo1033123847703346641_a_nat ) ) ).
% equals0I
thf(fact_490_equals0I,axiom,
! [A3: set_set_nat] :
( ! [Y5: set_nat] :
~ ( member_set_nat @ Y5 @ A3 )
=> ( A3 = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_491_equals0I,axiom,
! [A3: set_a] :
( ! [Y5: a] :
~ ( member_a @ Y5 @ A3 )
=> ( A3 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_492_equals0I,axiom,
! [A3: set_b] :
( ! [Y5: b] :
~ ( member_b @ Y5 @ A3 )
=> ( A3 = bot_bot_set_b ) ) ).
% equals0I
thf(fact_493_equals0I,axiom,
! [A3: set_nat] :
( ! [Y5: nat] :
~ ( member_nat @ Y5 @ A3 )
=> ( A3 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_494_equals0D,axiom,
! [A3: set_fo_term_a,A: fo_term_a] :
( ( A3 = bot_bo4735268219511357444term_a )
=> ~ ( member_fo_term_a @ A @ A3 ) ) ).
% equals0D
thf(fact_495_equals0D,axiom,
! [A3: set_li6526943997496501093_a_nat,A: list_Sum_sum_a_nat] :
( ( A3 = bot_bo1033123847703346641_a_nat )
=> ~ ( member408289922725080238_a_nat @ A @ A3 ) ) ).
% equals0D
thf(fact_496_equals0D,axiom,
! [A3: set_set_nat,A: set_nat] :
( ( A3 = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A @ A3 ) ) ).
% equals0D
thf(fact_497_equals0D,axiom,
! [A3: set_a,A: a] :
( ( A3 = bot_bot_set_a )
=> ~ ( member_a @ A @ A3 ) ) ).
% equals0D
thf(fact_498_equals0D,axiom,
! [A3: set_b,A: b] :
( ( A3 = bot_bot_set_b )
=> ~ ( member_b @ A @ A3 ) ) ).
% equals0D
thf(fact_499_equals0D,axiom,
! [A3: set_nat,A: nat] :
( ( A3 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A3 ) ) ).
% equals0D
thf(fact_500_emptyE,axiom,
! [A: fo_term_a] :
~ ( member_fo_term_a @ A @ bot_bo4735268219511357444term_a ) ).
% emptyE
thf(fact_501_emptyE,axiom,
! [A: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ).
% emptyE
thf(fact_502_emptyE,axiom,
! [A: set_nat] :
~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_503_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_504_emptyE,axiom,
! [A: b] :
~ ( member_b @ A @ bot_bot_set_b ) ).
% emptyE
thf(fact_505_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_506_prod__induct3,axiom,
! [P: produc5433867242478491114_a_nat > $o,X5: produc5433867242478491114_a_nat] :
( ! [A5: product_prod_b_nat > set_list_a,B3: nat > sum_sum_a_nat,C2: set_Sum_sum_a_nat] : ( P @ ( produc6651248262528101210_a_nat @ A5 @ ( produc3720304352952013712_a_nat @ B3 @ C2 ) ) )
=> ( P @ X5 ) ) ).
% prod_induct3
thf(fact_507_prod__cases3,axiom,
! [Y: produc5433867242478491114_a_nat] :
~ ! [A5: product_prod_b_nat > set_list_a,B3: nat > sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( Y
!= ( produc6651248262528101210_a_nat @ A5 @ ( produc3720304352952013712_a_nat @ B3 @ C2 ) ) ) ).
% prod_cases3
thf(fact_508_bot__apply,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : bot_bot_o ) ) ).
% bot_apply
thf(fact_509_list__all2__same,axiom,
! [P: a > a > $o,Xs: list_a] :
( ( list_all2_a_a @ P @ Xs @ Xs )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( P @ X3 @ X3 ) ) ) ) ).
% list_all2_same
thf(fact_510_list__all2__same,axiom,
! [P: fo_term_a > fo_term_a > $o,Xs: list_fo_term_a] :
( ( list_a2487961086254413575term_a @ P @ Xs @ Xs )
= ( ! [X3: fo_term_a] :
( ( member_fo_term_a @ X3 @ ( set_fo_term_a2 @ Xs ) )
=> ( P @ X3 @ X3 ) ) ) ) ).
% list_all2_same
thf(fact_511_list__all2__same,axiom,
! [P: nat > nat > $o,Xs: list_nat] :
( ( list_all2_nat_nat @ P @ Xs @ Xs )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 @ X3 ) ) ) ) ).
% list_all2_same
thf(fact_512_list_Orel__refl__strong,axiom,
! [X5: list_l4703314356710769291_a_nat,Ra: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
( ! [Z3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Z3 @ ( set_li2392974972034027290_a_nat @ X5 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( list_a5617550422536820423_a_nat @ Ra @ X5 @ X5 ) ) ).
% list.rel_refl_strong
thf(fact_513_list_Orel__refl__strong,axiom,
! [X5: list_set_nat,Ra: set_nat > set_nat > $o] :
( ! [Z3: set_nat] :
( ( member_set_nat @ Z3 @ ( set_set_nat2 @ X5 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( list_a8687986634888186261et_nat @ Ra @ X5 @ X5 ) ) ).
% list.rel_refl_strong
thf(fact_514_list_Orel__refl__strong,axiom,
! [X5: list_b,Ra: b > b > $o] :
( ! [Z3: b] :
( ( member_b @ Z3 @ ( set_b2 @ X5 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( list_all2_b_b @ Ra @ X5 @ X5 ) ) ).
% list.rel_refl_strong
thf(fact_515_list_Orel__refl__strong,axiom,
! [X5: list_a,Ra: a > a > $o] :
( ! [Z3: a] :
( ( member_a @ Z3 @ ( set_a2 @ X5 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( list_all2_a_a @ Ra @ X5 @ X5 ) ) ).
% list.rel_refl_strong
thf(fact_516_list_Orel__refl__strong,axiom,
! [X5: list_fo_term_a,Ra: fo_term_a > fo_term_a > $o] :
( ! [Z3: fo_term_a] :
( ( member_fo_term_a @ Z3 @ ( set_fo_term_a2 @ X5 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( list_a2487961086254413575term_a @ Ra @ X5 @ X5 ) ) ).
% list.rel_refl_strong
thf(fact_517_list_Orel__refl__strong,axiom,
! [X5: list_nat,Ra: nat > nat > $o] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ X5 ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( list_all2_nat_nat @ Ra @ X5 @ X5 ) ) ).
% list.rel_refl_strong
thf(fact_518_list_Orel__mono__strong,axiom,
! [R3: nat > nat > $o,X5: list_nat,Y: list_nat,Ra: nat > nat > $o] :
( ( list_all2_nat_nat @ R3 @ X5 @ Y )
=> ( ! [Z3: nat,Yb: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ X5 ) )
=> ( ( member_nat @ Yb @ ( set_nat2 @ Y ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( list_all2_nat_nat @ Ra @ X5 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_519_list_Orel__mono__strong,axiom,
! [R3: b > b > $o,X5: list_b,Y: list_b,Ra: b > b > $o] :
( ( list_all2_b_b @ R3 @ X5 @ Y )
=> ( ! [Z3: b,Yb: b] :
( ( member_b @ Z3 @ ( set_b2 @ X5 ) )
=> ( ( member_b @ Yb @ ( set_b2 @ Y ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( list_all2_b_b @ Ra @ X5 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_520_list_Orel__mono__strong,axiom,
! [R3: b > nat > $o,X5: list_b,Y: list_nat,Ra: b > nat > $o] :
( ( list_all2_b_nat @ R3 @ X5 @ Y )
=> ( ! [Z3: b,Yb: nat] :
( ( member_b @ Z3 @ ( set_b2 @ X5 ) )
=> ( ( member_nat @ Yb @ ( set_nat2 @ Y ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( list_all2_b_nat @ Ra @ X5 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_521_list_Orel__mono__strong,axiom,
! [R3: b > a > $o,X5: list_b,Y: list_a,Ra: b > a > $o] :
( ( list_all2_b_a @ R3 @ X5 @ Y )
=> ( ! [Z3: b,Yb: a] :
( ( member_b @ Z3 @ ( set_b2 @ X5 ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Y ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( list_all2_b_a @ Ra @ X5 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_522_list_Orel__mono__strong,axiom,
! [R3: nat > b > $o,X5: list_nat,Y: list_b,Ra: nat > b > $o] :
( ( list_all2_nat_b @ R3 @ X5 @ Y )
=> ( ! [Z3: nat,Yb: b] :
( ( member_nat @ Z3 @ ( set_nat2 @ X5 ) )
=> ( ( member_b @ Yb @ ( set_b2 @ Y ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( list_all2_nat_b @ Ra @ X5 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_523_list_Orel__mono__strong,axiom,
! [R3: nat > a > $o,X5: list_nat,Y: list_a,Ra: nat > a > $o] :
( ( list_all2_nat_a @ R3 @ X5 @ Y )
=> ( ! [Z3: nat,Yb: a] :
( ( member_nat @ Z3 @ ( set_nat2 @ X5 ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Y ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( list_all2_nat_a @ Ra @ X5 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_524_list_Orel__mono__strong,axiom,
! [R3: a > b > $o,X5: list_a,Y: list_b,Ra: a > b > $o] :
( ( list_all2_a_b @ R3 @ X5 @ Y )
=> ( ! [Z3: a,Yb: b] :
( ( member_a @ Z3 @ ( set_a2 @ X5 ) )
=> ( ( member_b @ Yb @ ( set_b2 @ Y ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( list_all2_a_b @ Ra @ X5 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_525_list_Orel__mono__strong,axiom,
! [R3: a > nat > $o,X5: list_a,Y: list_nat,Ra: a > nat > $o] :
( ( list_all2_a_nat @ R3 @ X5 @ Y )
=> ( ! [Z3: a,Yb: nat] :
( ( member_a @ Z3 @ ( set_a2 @ X5 ) )
=> ( ( member_nat @ Yb @ ( set_nat2 @ Y ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( list_all2_a_nat @ Ra @ X5 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_526_list_Orel__mono__strong,axiom,
! [R3: a > a > $o,X5: list_a,Y: list_a,Ra: a > a > $o] :
( ( list_all2_a_a @ R3 @ X5 @ Y )
=> ( ! [Z3: a,Yb: a] :
( ( member_a @ Z3 @ ( set_a2 @ X5 ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Y ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( list_all2_a_a @ Ra @ X5 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_527_list_Orel__mono__strong,axiom,
! [R3: set_nat > b > $o,X5: list_set_nat,Y: list_b,Ra: set_nat > b > $o] :
( ( list_all2_set_nat_b @ R3 @ X5 @ Y )
=> ( ! [Z3: set_nat,Yb: b] :
( ( member_set_nat @ Z3 @ ( set_set_nat2 @ X5 ) )
=> ( ( member_b @ Yb @ ( set_b2 @ Y ) )
=> ( ( R3 @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( list_all2_set_nat_b @ Ra @ X5 @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_528_list_Orel__cong,axiom,
! [X5: list_nat,Ya: list_nat,Y: list_nat,Xa: list_nat,R3: nat > nat > $o,Ra: nat > nat > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: nat,Yb: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
=> ( ( member_nat @ Yb @ ( set_nat2 @ Xa ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( list_all2_nat_nat @ R3 @ X5 @ Y )
= ( list_all2_nat_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).
% list.rel_cong
thf(fact_529_list_Orel__cong,axiom,
! [X5: list_b,Ya: list_b,Y: list_b,Xa: list_b,R3: b > b > $o,Ra: b > b > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: b,Yb: b] :
( ( member_b @ Z3 @ ( set_b2 @ Ya ) )
=> ( ( member_b @ Yb @ ( set_b2 @ Xa ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( list_all2_b_b @ R3 @ X5 @ Y )
= ( list_all2_b_b @ Ra @ Ya @ Xa ) ) ) ) ) ).
% list.rel_cong
thf(fact_530_list_Orel__cong,axiom,
! [X5: list_b,Ya: list_b,Y: list_nat,Xa: list_nat,R3: b > nat > $o,Ra: b > nat > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: b,Yb: nat] :
( ( member_b @ Z3 @ ( set_b2 @ Ya ) )
=> ( ( member_nat @ Yb @ ( set_nat2 @ Xa ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( list_all2_b_nat @ R3 @ X5 @ Y )
= ( list_all2_b_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).
% list.rel_cong
thf(fact_531_list_Orel__cong,axiom,
! [X5: list_b,Ya: list_b,Y: list_a,Xa: list_a,R3: b > a > $o,Ra: b > a > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: b,Yb: a] :
( ( member_b @ Z3 @ ( set_b2 @ Ya ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Xa ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( list_all2_b_a @ R3 @ X5 @ Y )
= ( list_all2_b_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% list.rel_cong
thf(fact_532_list_Orel__cong,axiom,
! [X5: list_nat,Ya: list_nat,Y: list_b,Xa: list_b,R3: nat > b > $o,Ra: nat > b > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: nat,Yb: b] :
( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
=> ( ( member_b @ Yb @ ( set_b2 @ Xa ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( list_all2_nat_b @ R3 @ X5 @ Y )
= ( list_all2_nat_b @ Ra @ Ya @ Xa ) ) ) ) ) ).
% list.rel_cong
thf(fact_533_list_Orel__cong,axiom,
! [X5: list_nat,Ya: list_nat,Y: list_a,Xa: list_a,R3: nat > a > $o,Ra: nat > a > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: nat,Yb: a] :
( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Xa ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( list_all2_nat_a @ R3 @ X5 @ Y )
= ( list_all2_nat_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% list.rel_cong
thf(fact_534_list_Orel__cong,axiom,
! [X5: list_a,Ya: list_a,Y: list_b,Xa: list_b,R3: a > b > $o,Ra: a > b > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: a,Yb: b] :
( ( member_a @ Z3 @ ( set_a2 @ Ya ) )
=> ( ( member_b @ Yb @ ( set_b2 @ Xa ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( list_all2_a_b @ R3 @ X5 @ Y )
= ( list_all2_a_b @ Ra @ Ya @ Xa ) ) ) ) ) ).
% list.rel_cong
thf(fact_535_list_Orel__cong,axiom,
! [X5: list_a,Ya: list_a,Y: list_nat,Xa: list_nat,R3: a > nat > $o,Ra: a > nat > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: a,Yb: nat] :
( ( member_a @ Z3 @ ( set_a2 @ Ya ) )
=> ( ( member_nat @ Yb @ ( set_nat2 @ Xa ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( list_all2_a_nat @ R3 @ X5 @ Y )
= ( list_all2_a_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).
% list.rel_cong
thf(fact_536_list_Orel__cong,axiom,
! [X5: list_a,Ya: list_a,Y: list_a,Xa: list_a,R3: a > a > $o,Ra: a > a > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: a,Yb: a] :
( ( member_a @ Z3 @ ( set_a2 @ Ya ) )
=> ( ( member_a @ Yb @ ( set_a2 @ Xa ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( list_all2_a_a @ R3 @ X5 @ Y )
= ( list_all2_a_a @ Ra @ Ya @ Xa ) ) ) ) ) ).
% list.rel_cong
thf(fact_537_list_Orel__cong,axiom,
! [X5: list_set_nat,Ya: list_set_nat,Y: list_b,Xa: list_b,R3: set_nat > b > $o,Ra: set_nat > b > $o] :
( ( X5 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: set_nat,Yb: b] :
( ( member_set_nat @ Z3 @ ( set_set_nat2 @ Ya ) )
=> ( ( member_b @ Yb @ ( set_b2 @ Xa ) )
=> ( ( R3 @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( list_all2_set_nat_b @ R3 @ X5 @ Y )
= ( list_all2_set_nat_b @ Ra @ Ya @ Xa ) ) ) ) ) ).
% list.rel_cong
thf(fact_538_list__fo__term__set,axiom,
! [T: fo_term_fo_term_a] :
( ( set_fo_term_a2 @ ( list_f7871435398800439599term_a @ T ) )
= ( fo_set9081760999098759482term_a @ T ) ) ).
% list_fo_term_set
thf(fact_539_list__fo__term__set,axiom,
! [T: fo_term_a] :
( ( set_a2 @ ( list_fo_term_a2 @ T ) )
= ( fo_set_fo_term_a @ T ) ) ).
% list_fo_term_set
thf(fact_540_list__fo__term__set,axiom,
! [T: fo_term_nat] :
( ( set_nat2 @ ( list_fo_term_nat2 @ T ) )
= ( fo_set_fo_term_nat @ T ) ) ).
% list_fo_term_set
thf(fact_541_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_542_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_543_bot__set__def,axiom,
( bot_bot_set_b
= ( collect_b @ bot_bot_b_o ) ) ).
% bot_set_def
thf(fact_544_bot__fun__def,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_545_list_Orel__eq,axiom,
( ( list_a2487961086254413575term_a
@ ^ [Y6: fo_term_a,Z4: fo_term_a] : ( Y6 = Z4 ) )
= ( ^ [Y6: list_fo_term_a,Z4: list_fo_term_a] : ( Y6 = Z4 ) ) ) ).
% list.rel_eq
thf(fact_546_list_Orel__eq,axiom,
( ( list_all2_nat_nat
@ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
= ( ^ [Y6: list_nat,Z4: list_nat] : ( Y6 = Z4 ) ) ) ).
% list.rel_eq
thf(fact_547_list_Orel__refl,axiom,
! [Ra: fo_term_a > fo_term_a > $o,X5: list_fo_term_a] :
( ! [X: fo_term_a] : ( Ra @ X @ X )
=> ( list_a2487961086254413575term_a @ Ra @ X5 @ X5 ) ) ).
% list.rel_refl
thf(fact_548_list_Orel__refl,axiom,
! [Ra: nat > nat > $o,X5: list_nat] :
( ! [X: nat] : ( Ra @ X @ X )
=> ( list_all2_nat_nat @ Ra @ X5 @ X5 ) ) ).
% list.rel_refl
thf(fact_549_list__all2__eq,axiom,
( ( ^ [Y6: list_fo_term_a,Z4: list_fo_term_a] : ( Y6 = Z4 ) )
= ( list_a2487961086254413575term_a
@ ^ [Y6: fo_term_a,Z4: fo_term_a] : ( Y6 = Z4 ) ) ) ).
% list_all2_eq
thf(fact_550_list__all2__eq,axiom,
( ( ^ [Y6: list_nat,Z4: list_nat] : ( Y6 = Z4 ) )
= ( list_all2_nat_nat
@ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) ) ) ).
% list_all2_eq
thf(fact_551_list__all2__mono,axiom,
! [P: fo_term_a > fo_term_a > $o,Xs: list_fo_term_a,Ys: list_fo_term_a,Q: fo_term_a > fo_term_a > $o] :
( ( list_a2487961086254413575term_a @ P @ Xs @ Ys )
=> ( ! [Xs2: fo_term_a,Ys2: fo_term_a] :
( ( P @ Xs2 @ Ys2 )
=> ( Q @ Xs2 @ Ys2 ) )
=> ( list_a2487961086254413575term_a @ Q @ Xs @ Ys ) ) ) ).
% list_all2_mono
thf(fact_552_list__all2__mono,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat,Q: nat > nat > $o] :
( ( list_all2_nat_nat @ P @ Xs @ Ys )
=> ( ! [Xs2: nat,Ys2: nat] :
( ( P @ Xs2 @ Ys2 )
=> ( Q @ Xs2 @ Ys2 ) )
=> ( list_all2_nat_nat @ Q @ Xs @ Ys ) ) ) ).
% list_all2_mono
thf(fact_553_list__all2__refl,axiom,
! [P: fo_term_a > fo_term_a > $o,Xs: list_fo_term_a] :
( ! [X: fo_term_a] : ( P @ X @ X )
=> ( list_a2487961086254413575term_a @ P @ Xs @ Xs ) ) ).
% list_all2_refl
thf(fact_554_list__all2__refl,axiom,
! [P: nat > nat > $o,Xs: list_nat] :
( ! [X: nat] : ( P @ X @ X )
=> ( list_all2_nat_nat @ P @ Xs @ Xs ) ) ).
% list_all2_refl
thf(fact_555_list__all2__trans,axiom,
! [P1: fo_term_a > fo_term_a > $o,P22: fo_term_a > fo_term_a > $o,P3: fo_term_a > fo_term_a > $o,As: list_fo_term_a,Bs: list_fo_term_a,Cs: list_fo_term_a] :
( ! [A5: fo_term_a,B3: fo_term_a,C2: fo_term_a] :
( ( P1 @ A5 @ B3 )
=> ( ( P22 @ B3 @ C2 )
=> ( P3 @ A5 @ C2 ) ) )
=> ( ( list_a2487961086254413575term_a @ P1 @ As @ Bs )
=> ( ( list_a2487961086254413575term_a @ P22 @ Bs @ Cs )
=> ( list_a2487961086254413575term_a @ P3 @ As @ Cs ) ) ) ) ).
% list_all2_trans
thf(fact_556_list__all2__trans,axiom,
! [P1: nat > nat > $o,P22: nat > nat > $o,P3: nat > nat > $o,As: list_nat,Bs: list_nat,Cs: list_nat] :
( ! [A5: nat,B3: nat,C2: nat] :
( ( P1 @ A5 @ B3 )
=> ( ( P22 @ B3 @ C2 )
=> ( P3 @ A5 @ C2 ) ) )
=> ( ( list_all2_nat_nat @ P1 @ As @ Bs )
=> ( ( list_all2_nat_nat @ P22 @ Bs @ Cs )
=> ( list_all2_nat_nat @ P3 @ As @ Cs ) ) ) ) ).
% list_all2_trans
thf(fact_557_list__all2__antisym,axiom,
! [P: fo_term_a > fo_term_a > $o,Q: fo_term_a > fo_term_a > $o,Xs: list_fo_term_a,Ys: list_fo_term_a] :
( ! [X: fo_term_a,Y5: fo_term_a] :
( ( P @ X @ Y5 )
=> ( ( Q @ Y5 @ X )
=> ( X = Y5 ) ) )
=> ( ( list_a2487961086254413575term_a @ P @ Xs @ Ys )
=> ( ( list_a2487961086254413575term_a @ Q @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ) ).
% list_all2_antisym
thf(fact_558_list__all2__antisym,axiom,
! [P: nat > nat > $o,Q: nat > nat > $o,Xs: list_nat,Ys: list_nat] :
( ! [X: nat,Y5: nat] :
( ( P @ X @ Y5 )
=> ( ( Q @ Y5 @ X )
=> ( X = Y5 ) ) )
=> ( ( list_all2_nat_nat @ P @ Xs @ Ys )
=> ( ( list_all2_nat_nat @ Q @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ) ).
% list_all2_antisym
thf(fact_559_same__fstI,axiom,
! [P: ( product_prod_b_nat > set_list_a ) > $o,X5: product_prod_b_nat > set_list_a,Y7: produc4672180596006801056_a_nat,Y: produc4672180596006801056_a_nat,R3: ( product_prod_b_nat > set_list_a ) > set_Pr4466880552907744775_a_nat] :
( ( P @ X5 )
=> ( ( member3492506527488239696_a_nat @ ( produc4653759813812500759_a_nat @ Y7 @ Y ) @ ( R3 @ X5 ) )
=> ( member2098227167877521616_a_nat @ ( produc5415104756258485911_a_nat @ ( produc6651248262528101210_a_nat @ X5 @ Y7 ) @ ( produc6651248262528101210_a_nat @ X5 @ Y ) ) @ ( same_f6297344053069427631_a_nat @ P @ R3 ) ) ) ) ).
% same_fstI
thf(fact_560_same__fstI,axiom,
! [P: ( product_prod_b_nat > set_list_a ) > $o,X5: product_prod_b_nat > set_list_a,Y7: nat > a,Y: nat > a,R3: ( product_prod_b_nat > set_list_a ) > set_Pr2991377731101678151_nat_a] :
( ( P @ X5 )
=> ( ( member3167612539812461712_nat_a @ ( produc6854975703358627799_nat_a @ Y7 @ Y ) @ ( R3 @ X5 ) )
=> ( member1200888978250396944_nat_a @ ( produc4949270592693755223_nat_a @ ( produc2895298938842563487_nat_a @ X5 @ Y7 ) @ ( produc2895298938842563487_nat_a @ X5 @ Y ) ) @ ( same_f1314808272193629172_nat_a @ P @ R3 ) ) ) ) ).
% same_fstI
thf(fact_561_same__fstI,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,X5: nat > sum_sum_a_nat,Y7: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,R3: ( nat > sum_sum_a_nat ) > set_Pr3952102297599872711_a_nat] :
( ( P @ X5 )
=> ( ( member983090737300173072_a_nat @ ( produc5825311282902856023_a_nat @ Y7 @ Y ) @ ( R3 @ X5 ) )
=> ( member3492506527488239696_a_nat @ ( produc4653759813812500759_a_nat @ ( produc3720304352952013712_a_nat @ X5 @ Y7 ) @ ( produc3720304352952013712_a_nat @ X5 @ Y ) ) @ ( same_f2139813686303079397_a_nat @ P @ R3 ) ) ) ) ).
% same_fstI
thf(fact_562_same__fstI,axiom,
! [P: nat > $o,X5: nat,Y7: nat,Y: nat,R3: nat > set_Pr1261947904930325089at_nat] :
( ( P @ X5 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y7 @ Y ) @ ( R3 @ X5 ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X5 @ Y7 ) @ ( product_Pair_nat_nat @ X5 @ Y ) ) @ ( same_fst_nat_nat @ P @ R3 ) ) ) ) ).
% same_fstI
thf(fact_563_same__fstI,axiom,
! [P: b > $o,X5: b,Y7: nat,Y: nat,R3: b > set_Pr1261947904930325089at_nat] :
( ( P @ X5 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y7 @ Y ) @ ( R3 @ X5 ) )
=> ( member4966892972527255302_b_nat @ ( produc2793586353817733269_b_nat @ ( product_Pair_b_nat @ X5 @ Y7 ) @ ( product_Pair_b_nat @ X5 @ Y ) ) @ ( same_fst_b_nat @ P @ R3 ) ) ) ) ).
% same_fstI
thf(fact_564_SP_Osimps_I9_J,axiom,
! [Vb2: a,Va2: fo_term_a] :
( ( sP_a_b @ ( fo_Eqa_a_b @ ( fo_Const_a @ Vb2 ) @ Va2 ) )
= bot_bot_set_nat ) ).
% SP.simps(9)
thf(fact_565_SP_Osimps_I10_J,axiom,
! [V2: fo_term_a,Vb2: a] :
( ( sP_a_b @ ( fo_Eqa_a_b @ V2 @ ( fo_Const_a @ Vb2 ) ) )
= bot_bot_set_nat ) ).
% SP.simps(10)
thf(fact_566_fo__fmla_Osimps_I155_J,axiom,
! [X31: fo_term_a,X32: fo_term_a] :
( ( fo_set1_fo_fmla_a_b @ ( fo_Eqa_a_b @ X31 @ X32 ) )
= ( sup_sup_set_a @ ( fo_set_fo_term_a @ X31 ) @ ( fo_set_fo_term_a @ X32 ) ) ) ).
% fo_fmla.simps(155)
thf(fact_567_fo__fmla_Osimps_I155_J,axiom,
! [X31: fo_term_nat,X32: fo_term_nat] :
( ( fo_set7155282118041507904at_nat @ ( fo_Eqa_nat_nat @ X31 @ X32 ) )
= ( sup_sup_set_nat @ ( fo_set_fo_term_nat @ X31 ) @ ( fo_set_fo_term_nat @ X32 ) ) ) ).
% fo_fmla.simps(155)
thf(fact_568_fv__fo__term__set_Osimps_I2_J,axiom,
! [V2: nat] :
( ( fv_fo_term_set_nat @ ( fo_Const_nat @ V2 ) )
= bot_bot_set_nat ) ).
% fv_fo_term_set.simps(2)
thf(fact_569_fv__fo__term__set_Osimps_I2_J,axiom,
! [V2: a] :
( ( fv_fo_term_set_a @ ( fo_Const_a @ V2 ) )
= bot_bot_set_nat ) ).
% fv_fo_term_set.simps(2)
thf(fact_570_UnCI,axiom,
! [C: fo_term_a,B7: set_fo_term_a,A3: set_fo_term_a] :
( ( ~ ( member_fo_term_a @ C @ B7 )
=> ( member_fo_term_a @ C @ A3 ) )
=> ( member_fo_term_a @ C @ ( sup_su8271228240639168364term_a @ A3 @ B7 ) ) ) ).
% UnCI
thf(fact_571_UnCI,axiom,
! [C: list_Sum_sum_a_nat,B7: set_li6526943997496501093_a_nat,A3: set_li6526943997496501093_a_nat] :
( ( ~ ( member408289922725080238_a_nat @ C @ B7 )
=> ( member408289922725080238_a_nat @ C @ A3 ) )
=> ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A3 @ B7 ) ) ) ).
% UnCI
thf(fact_572_UnCI,axiom,
! [C: set_nat,B7: set_set_nat,A3: set_set_nat] :
( ( ~ ( member_set_nat @ C @ B7 )
=> ( member_set_nat @ C @ A3 ) )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B7 ) ) ) ).
% UnCI
thf(fact_573_UnCI,axiom,
! [C: a,B7: set_a,A3: set_a] :
( ( ~ ( member_a @ C @ B7 )
=> ( member_a @ C @ A3 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B7 ) ) ) ).
% UnCI
thf(fact_574_UnCI,axiom,
! [C: b,B7: set_b,A3: set_b] :
( ( ~ ( member_b @ C @ B7 )
=> ( member_b @ C @ A3 ) )
=> ( member_b @ C @ ( sup_sup_set_b @ A3 @ B7 ) ) ) ).
% UnCI
thf(fact_575_UnCI,axiom,
! [C: nat,B7: set_nat,A3: set_nat] :
( ( ~ ( member_nat @ C @ B7 )
=> ( member_nat @ C @ A3 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B7 ) ) ) ).
% UnCI
thf(fact_576_Un__iff,axiom,
! [C: fo_term_a,A3: set_fo_term_a,B7: set_fo_term_a] :
( ( member_fo_term_a @ C @ ( sup_su8271228240639168364term_a @ A3 @ B7 ) )
= ( ( member_fo_term_a @ C @ A3 )
| ( member_fo_term_a @ C @ B7 ) ) ) ).
% Un_iff
thf(fact_577_Un__iff,axiom,
! [C: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat,B7: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A3 @ B7 ) )
= ( ( member408289922725080238_a_nat @ C @ A3 )
| ( member408289922725080238_a_nat @ C @ B7 ) ) ) ).
% Un_iff
thf(fact_578_Un__iff,axiom,
! [C: set_nat,A3: set_set_nat,B7: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B7 ) )
= ( ( member_set_nat @ C @ A3 )
| ( member_set_nat @ C @ B7 ) ) ) ).
% Un_iff
thf(fact_579_Un__iff,axiom,
! [C: a,A3: set_a,B7: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B7 ) )
= ( ( member_a @ C @ A3 )
| ( member_a @ C @ B7 ) ) ) ).
% Un_iff
thf(fact_580_Un__iff,axiom,
! [C: b,A3: set_b,B7: set_b] :
( ( member_b @ C @ ( sup_sup_set_b @ A3 @ B7 ) )
= ( ( member_b @ C @ A3 )
| ( member_b @ C @ B7 ) ) ) ).
% Un_iff
thf(fact_581_Un__iff,axiom,
! [C: nat,A3: set_nat,B7: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B7 ) )
= ( ( member_nat @ C @ A3 )
| ( member_nat @ C @ B7 ) ) ) ).
% Un_iff
thf(fact_582_Un__empty,axiom,
! [A3: set_nat,B7: set_nat] :
( ( ( sup_sup_set_nat @ A3 @ B7 )
= bot_bot_set_nat )
= ( ( A3 = bot_bot_set_nat )
& ( B7 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_583_Un__empty,axiom,
! [A3: set_a,B7: set_a] :
( ( ( sup_sup_set_a @ A3 @ B7 )
= bot_bot_set_a )
= ( ( A3 = bot_bot_set_a )
& ( B7 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_584_Un__empty,axiom,
! [A3: set_b,B7: set_b] :
( ( ( sup_sup_set_b @ A3 @ B7 )
= bot_bot_set_b )
= ( ( A3 = bot_bot_set_b )
& ( B7 = bot_bot_set_b ) ) ) ).
% Un_empty
thf(fact_585_SP_Osimps_I4_J,axiom,
! [Phi: fo_fmla_a_b,Psi: fo_fmla_a_b] :
( ( sP_a_b @ ( fo_Disj_a_b @ Phi @ Psi ) )
= ( sup_sup_set_nat @ ( sP_a_b @ Phi ) @ ( sP_a_b @ Psi ) ) ) ).
% SP.simps(4)
thf(fact_586_SP_Osimps_I3_J,axiom,
! [Phi: fo_fmla_a_b,Psi: fo_fmla_a_b] :
( ( sP_a_b @ ( fo_Conj_a_b @ Phi @ Psi ) )
= ( sup_sup_set_nat @ ( sP_a_b @ Phi ) @ ( sP_a_b @ Psi ) ) ) ).
% SP.simps(3)
thf(fact_587_UnE,axiom,
! [C: fo_term_a,A3: set_fo_term_a,B7: set_fo_term_a] :
( ( member_fo_term_a @ C @ ( sup_su8271228240639168364term_a @ A3 @ B7 ) )
=> ( ~ ( member_fo_term_a @ C @ A3 )
=> ( member_fo_term_a @ C @ B7 ) ) ) ).
% UnE
thf(fact_588_UnE,axiom,
! [C: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat,B7: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A3 @ B7 ) )
=> ( ~ ( member408289922725080238_a_nat @ C @ A3 )
=> ( member408289922725080238_a_nat @ C @ B7 ) ) ) ).
% UnE
thf(fact_589_UnE,axiom,
! [C: set_nat,A3: set_set_nat,B7: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B7 ) )
=> ( ~ ( member_set_nat @ C @ A3 )
=> ( member_set_nat @ C @ B7 ) ) ) ).
% UnE
thf(fact_590_UnE,axiom,
! [C: a,A3: set_a,B7: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B7 ) )
=> ( ~ ( member_a @ C @ A3 )
=> ( member_a @ C @ B7 ) ) ) ).
% UnE
thf(fact_591_UnE,axiom,
! [C: b,A3: set_b,B7: set_b] :
( ( member_b @ C @ ( sup_sup_set_b @ A3 @ B7 ) )
=> ( ~ ( member_b @ C @ A3 )
=> ( member_b @ C @ B7 ) ) ) ).
% UnE
thf(fact_592_UnE,axiom,
! [C: nat,A3: set_nat,B7: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B7 ) )
=> ( ~ ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B7 ) ) ) ).
% UnE
thf(fact_593_UnI1,axiom,
! [C: fo_term_a,A3: set_fo_term_a,B7: set_fo_term_a] :
( ( member_fo_term_a @ C @ A3 )
=> ( member_fo_term_a @ C @ ( sup_su8271228240639168364term_a @ A3 @ B7 ) ) ) ).
% UnI1
thf(fact_594_UnI1,axiom,
! [C: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat,B7: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ A3 )
=> ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A3 @ B7 ) ) ) ).
% UnI1
thf(fact_595_UnI1,axiom,
! [C: set_nat,A3: set_set_nat,B7: set_set_nat] :
( ( member_set_nat @ C @ A3 )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B7 ) ) ) ).
% UnI1
thf(fact_596_UnI1,axiom,
! [C: a,A3: set_a,B7: set_a] :
( ( member_a @ C @ A3 )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B7 ) ) ) ).
% UnI1
thf(fact_597_UnI1,axiom,
! [C: b,A3: set_b,B7: set_b] :
( ( member_b @ C @ A3 )
=> ( member_b @ C @ ( sup_sup_set_b @ A3 @ B7 ) ) ) ).
% UnI1
thf(fact_598_UnI1,axiom,
! [C: nat,A3: set_nat,B7: set_nat] :
( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B7 ) ) ) ).
% UnI1
thf(fact_599_UnI2,axiom,
! [C: fo_term_a,B7: set_fo_term_a,A3: set_fo_term_a] :
( ( member_fo_term_a @ C @ B7 )
=> ( member_fo_term_a @ C @ ( sup_su8271228240639168364term_a @ A3 @ B7 ) ) ) ).
% UnI2
thf(fact_600_UnI2,axiom,
! [C: list_Sum_sum_a_nat,B7: set_li6526943997496501093_a_nat,A3: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ B7 )
=> ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A3 @ B7 ) ) ) ).
% UnI2
thf(fact_601_UnI2,axiom,
! [C: set_nat,B7: set_set_nat,A3: set_set_nat] :
( ( member_set_nat @ C @ B7 )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B7 ) ) ) ).
% UnI2
thf(fact_602_UnI2,axiom,
! [C: a,B7: set_a,A3: set_a] :
( ( member_a @ C @ B7 )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B7 ) ) ) ).
% UnI2
thf(fact_603_UnI2,axiom,
! [C: b,B7: set_b,A3: set_b] :
( ( member_b @ C @ B7 )
=> ( member_b @ C @ ( sup_sup_set_b @ A3 @ B7 ) ) ) ).
% UnI2
thf(fact_604_UnI2,axiom,
! [C: nat,B7: set_nat,A3: set_nat] :
( ( member_nat @ C @ B7 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B7 ) ) ) ).
% UnI2
thf(fact_605_bex__Un,axiom,
! [A3: set_nat,B7: set_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A3 @ B7 ) )
& ( P @ X3 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( P @ X3 ) )
| ? [X3: nat] :
( ( member_nat @ X3 @ B7 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_606_bex__Un,axiom,
! [A3: set_a,B7: set_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A3 @ B7 ) )
& ( P @ X3 ) ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A3 )
& ( P @ X3 ) )
| ? [X3: a] :
( ( member_a @ X3 @ B7 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_607_bex__Un,axiom,
! [A3: set_b,B7: set_b,P: b > $o] :
( ( ? [X3: b] :
( ( member_b @ X3 @ ( sup_sup_set_b @ A3 @ B7 ) )
& ( P @ X3 ) ) )
= ( ? [X3: b] :
( ( member_b @ X3 @ A3 )
& ( P @ X3 ) )
| ? [X3: b] :
( ( member_b @ X3 @ B7 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_608_ball__Un,axiom,
! [A3: set_nat,B7: set_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A3 @ B7 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( P @ X3 ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B7 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_609_ball__Un,axiom,
! [A3: set_a,B7: set_a,P: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A3 @ B7 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( P @ X3 ) )
& ! [X3: a] :
( ( member_a @ X3 @ B7 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_610_ball__Un,axiom,
! [A3: set_b,B7: set_b,P: b > $o] :
( ( ! [X3: b] :
( ( member_b @ X3 @ ( sup_sup_set_b @ A3 @ B7 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: b] :
( ( member_b @ X3 @ A3 )
=> ( P @ X3 ) )
& ! [X3: b] :
( ( member_b @ X3 @ B7 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_611_Un__assoc,axiom,
! [A3: set_nat,B7: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A3 @ B7 ) @ C3 )
= ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ B7 @ C3 ) ) ) ).
% Un_assoc
thf(fact_612_Un__assoc,axiom,
! [A3: set_a,B7: set_a,C3: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A3 @ B7 ) @ C3 )
= ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B7 @ C3 ) ) ) ).
% Un_assoc
thf(fact_613_Un__assoc,axiom,
! [A3: set_b,B7: set_b,C3: set_b] :
( ( sup_sup_set_b @ ( sup_sup_set_b @ A3 @ B7 ) @ C3 )
= ( sup_sup_set_b @ A3 @ ( sup_sup_set_b @ B7 @ C3 ) ) ) ).
% Un_assoc
thf(fact_614_Un__absorb,axiom,
! [A3: set_nat] :
( ( sup_sup_set_nat @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_615_Un__absorb,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_616_Un__absorb,axiom,
! [A3: set_b] :
( ( sup_sup_set_b @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_617_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A8: set_nat,B8: set_nat] : ( sup_sup_set_nat @ B8 @ A8 ) ) ) ).
% Un_commute
thf(fact_618_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A8: set_a,B8: set_a] : ( sup_sup_set_a @ B8 @ A8 ) ) ) ).
% Un_commute
thf(fact_619_Un__commute,axiom,
( sup_sup_set_b
= ( ^ [A8: set_b,B8: set_b] : ( sup_sup_set_b @ B8 @ A8 ) ) ) ).
% Un_commute
thf(fact_620_Un__left__absorb,axiom,
! [A3: set_nat,B7: set_nat] :
( ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ A3 @ B7 ) )
= ( sup_sup_set_nat @ A3 @ B7 ) ) ).
% Un_left_absorb
thf(fact_621_Un__left__absorb,axiom,
! [A3: set_a,B7: set_a] :
( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ A3 @ B7 ) )
= ( sup_sup_set_a @ A3 @ B7 ) ) ).
% Un_left_absorb
thf(fact_622_Un__left__absorb,axiom,
! [A3: set_b,B7: set_b] :
( ( sup_sup_set_b @ A3 @ ( sup_sup_set_b @ A3 @ B7 ) )
= ( sup_sup_set_b @ A3 @ B7 ) ) ).
% Un_left_absorb
thf(fact_623_Un__left__commute,axiom,
! [A3: set_nat,B7: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ B7 @ C3 ) )
= ( sup_sup_set_nat @ B7 @ ( sup_sup_set_nat @ A3 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_624_Un__left__commute,axiom,
! [A3: set_a,B7: set_a,C3: set_a] :
( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B7 @ C3 ) )
= ( sup_sup_set_a @ B7 @ ( sup_sup_set_a @ A3 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_625_Un__left__commute,axiom,
! [A3: set_b,B7: set_b,C3: set_b] :
( ( sup_sup_set_b @ A3 @ ( sup_sup_set_b @ B7 @ C3 ) )
= ( sup_sup_set_b @ B7 @ ( sup_sup_set_b @ A3 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_626_fv__fo__fmla__list__Conj,axiom,
! [Phi: fo_fmla_a_b,Psi: fo_fmla_a_b] :
( ( fv_fo_fmla_list_a_b @ ( fo_Conj_a_b @ Phi @ Psi ) )
= ( fv_fo_fmla_list_a_b @ ( fo_Conj_a_b @ Psi @ Phi ) ) ) ).
% fv_fo_fmla_list_Conj
thf(fact_627_Un__empty__right,axiom,
! [A3: set_nat] :
( ( sup_sup_set_nat @ A3 @ bot_bot_set_nat )
= A3 ) ).
% Un_empty_right
thf(fact_628_Un__empty__right,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ A3 @ bot_bot_set_a )
= A3 ) ).
% Un_empty_right
thf(fact_629_Un__empty__right,axiom,
! [A3: set_b] :
( ( sup_sup_set_b @ A3 @ bot_bot_set_b )
= A3 ) ).
% Un_empty_right
thf(fact_630_Un__empty__left,axiom,
! [B7: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B7 )
= B7 ) ).
% Un_empty_left
thf(fact_631_Un__empty__left,axiom,
! [B7: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B7 )
= B7 ) ).
% Un_empty_left
thf(fact_632_Un__empty__left,axiom,
! [B7: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ B7 )
= B7 ) ).
% Un_empty_left
thf(fact_633_fv__fo__fmla__list__eq,axiom,
! [Phi: fo_fmla_a_b,Psi: fo_fmla_a_b] :
( ( ( fv_fo_fmla_a_b @ Phi )
= ( fv_fo_fmla_a_b @ Psi ) )
=> ( ( fv_fo_fmla_list_a_b @ Phi )
= ( fv_fo_fmla_list_a_b @ Psi ) ) ) ).
% fv_fo_fmla_list_eq
thf(fact_634_fv__fo__fmla__list__set,axiom,
! [Phi: fo_fmla_a_b] :
( ( set_nat2 @ ( fv_fo_fmla_list_a_b @ Phi ) )
= ( fv_fo_fmla_a_b @ Phi ) ) ).
% fv_fo_fmla_list_set
thf(fact_635_fv__fo__fmla_Osimps_I3_J,axiom,
! [T: fo_term_a,T2: fo_term_a] :
( ( fv_fo_fmla_a_b @ ( fo_Eqa_a_b @ T @ T2 ) )
= ( sup_sup_set_nat @ ( fv_fo_term_set_a @ T ) @ ( fv_fo_term_set_a @ T2 ) ) ) ).
% fv_fo_fmla.simps(3)
thf(fact_636_SP_Osimps_I2_J,axiom,
! [Phi: fo_fmla_a_b] :
( ( sP_a_b @ ( fo_Neg_a_b @ Phi ) )
= ( sP_a_b @ Phi ) ) ).
% SP.simps(2)
thf(fact_637_fv__fo__term__setD,axiom,
! [N2: nat,T: fo_term_a] :
( ( member_nat @ N2 @ ( fv_fo_term_set_a @ T ) )
=> ( T
= ( fo_Var_a @ N2 ) ) ) ).
% fv_fo_term_setD
thf(fact_638_eval__eterm__cong,axiom,
! [T: fo_term_a,Sigma: nat > sum_sum_a_nat,Sigma2: nat > sum_sum_a_nat] :
( ! [N: nat] :
( ( member_nat @ N @ ( fv_fo_term_set_a @ T ) )
=> ( ( Sigma @ N )
= ( Sigma2 @ N ) ) )
=> ( ( eval_eterm_a_nat @ Sigma @ T )
= ( eval_eterm_a_nat @ Sigma2 @ T ) ) ) ).
% eval_eterm_cong
thf(fact_639_fo__fmla_Osimps_I157_J,axiom,
! [X51: fo_fmla_nat_nat,X52: fo_fmla_nat_nat] :
( ( fo_set7155282118041507904at_nat @ ( fo_Conj_nat_nat @ X51 @ X52 ) )
= ( sup_sup_set_nat @ ( fo_set7155282118041507904at_nat @ X51 ) @ ( fo_set7155282118041507904at_nat @ X52 ) ) ) ).
% fo_fmla.simps(157)
thf(fact_640_fo__fmla_Osimps_I157_J,axiom,
! [X51: fo_fmla_a_b,X52: fo_fmla_a_b] :
( ( fo_set1_fo_fmla_a_b @ ( fo_Conj_a_b @ X51 @ X52 ) )
= ( sup_sup_set_a @ ( fo_set1_fo_fmla_a_b @ X51 ) @ ( fo_set1_fo_fmla_a_b @ X52 ) ) ) ).
% fo_fmla.simps(157)
thf(fact_641_fo__fmla_Osimps_I158_J,axiom,
! [X61: fo_fmla_nat_nat,X62: fo_fmla_nat_nat] :
( ( fo_set7155282118041507904at_nat @ ( fo_Disj_nat_nat @ X61 @ X62 ) )
= ( sup_sup_set_nat @ ( fo_set7155282118041507904at_nat @ X61 ) @ ( fo_set7155282118041507904at_nat @ X62 ) ) ) ).
% fo_fmla.simps(158)
thf(fact_642_fo__fmla_Osimps_I158_J,axiom,
! [X61: fo_fmla_a_b,X62: fo_fmla_a_b] :
( ( fo_set1_fo_fmla_a_b @ ( fo_Disj_a_b @ X61 @ X62 ) )
= ( sup_sup_set_a @ ( fo_set1_fo_fmla_a_b @ X61 ) @ ( fo_set1_fo_fmla_a_b @ X62 ) ) ) ).
% fo_fmla.simps(158)
thf(fact_643_fo__fmla_Osimps_I165_J,axiom,
! [X51: fo_fmla_nat_nat,X52: fo_fmla_nat_nat] :
( ( fo_set4065224739847495105at_nat @ ( fo_Conj_nat_nat @ X51 @ X52 ) )
= ( sup_sup_set_nat @ ( fo_set4065224739847495105at_nat @ X51 ) @ ( fo_set4065224739847495105at_nat @ X52 ) ) ) ).
% fo_fmla.simps(165)
thf(fact_644_fo__fmla_Osimps_I165_J,axiom,
! [X51: fo_fmla_a_b,X52: fo_fmla_a_b] :
( ( fo_set2_fo_fmla_a_b @ ( fo_Conj_a_b @ X51 @ X52 ) )
= ( sup_sup_set_b @ ( fo_set2_fo_fmla_a_b @ X51 ) @ ( fo_set2_fo_fmla_a_b @ X52 ) ) ) ).
% fo_fmla.simps(165)
thf(fact_645_fo__fmla_Osimps_I166_J,axiom,
! [X61: fo_fmla_nat_nat,X62: fo_fmla_nat_nat] :
( ( fo_set4065224739847495105at_nat @ ( fo_Disj_nat_nat @ X61 @ X62 ) )
= ( sup_sup_set_nat @ ( fo_set4065224739847495105at_nat @ X61 ) @ ( fo_set4065224739847495105at_nat @ X62 ) ) ) ).
% fo_fmla.simps(166)
thf(fact_646_fo__fmla_Osimps_I166_J,axiom,
! [X61: fo_fmla_a_b,X62: fo_fmla_a_b] :
( ( fo_set2_fo_fmla_a_b @ ( fo_Disj_a_b @ X61 @ X62 ) )
= ( sup_sup_set_b @ ( fo_set2_fo_fmla_a_b @ X61 ) @ ( fo_set2_fo_fmla_a_b @ X62 ) ) ) ).
% fo_fmla.simps(166)
thf(fact_647_fv__fo__fmla_Osimps_I5_J,axiom,
! [Phi: fo_fmla_a_b,Psi: fo_fmla_a_b] :
( ( fv_fo_fmla_a_b @ ( fo_Conj_a_b @ Phi @ Psi ) )
= ( sup_sup_set_nat @ ( fv_fo_fmla_a_b @ Phi ) @ ( fv_fo_fmla_a_b @ Psi ) ) ) ).
% fv_fo_fmla.simps(5)
thf(fact_648_fv__fo__fmla_Osimps_I6_J,axiom,
! [Phi: fo_fmla_a_b,Psi: fo_fmla_a_b] :
( ( fv_fo_fmla_a_b @ ( fo_Disj_a_b @ Phi @ Psi ) )
= ( sup_sup_set_nat @ ( fv_fo_fmla_a_b @ Phi ) @ ( fv_fo_fmla_a_b @ Psi ) ) ) ).
% fv_fo_fmla.simps(6)
thf(fact_649_SP_Osimps_I7_J,axiom,
! [V2: b,Va2: list_fo_term_a] :
( ( sP_a_b @ ( fo_Pred_b_a @ V2 @ Va2 ) )
= bot_bot_set_nat ) ).
% SP.simps(7)
thf(fact_650_SP_Osimps_I8_J,axiom,
! [V2: $o] :
( ( sP_a_b @ ( fo_Bool_a_b @ V2 ) )
= bot_bot_set_nat ) ).
% SP.simps(8)
thf(fact_651_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_652_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_653_sup__bot_Oright__neutral,axiom,
! [A: set_b] :
( ( sup_sup_set_b @ A @ bot_bot_set_b )
= A ) ).
% sup_bot.right_neutral
thf(fact_654_sup__bot_Oright__neutral,axiom,
! [A: nat > $o] :
( ( sup_sup_nat_o @ A @ bot_bot_nat_o )
= A ) ).
% sup_bot.right_neutral
thf(fact_655_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A @ B ) )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_656_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A @ B ) )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_657_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_b,B: set_b] :
( ( bot_bot_set_b
= ( sup_sup_set_b @ A @ B ) )
= ( ( A = bot_bot_set_b )
& ( B = bot_bot_set_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_658_sup__bot_Oneutr__eq__iff,axiom,
! [A: nat > $o,B: nat > $o] :
( ( bot_bot_nat_o
= ( sup_sup_nat_o @ A @ B ) )
= ( ( A = bot_bot_nat_o )
& ( B = bot_bot_nat_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_659_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_660_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_661_sup__bot_Oleft__neutral,axiom,
! [A: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_662_sup__bot_Oleft__neutral,axiom,
! [A: nat > $o] :
( ( sup_sup_nat_o @ bot_bot_nat_o @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_663_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_664_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A @ B )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_665_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_b,B: set_b] :
( ( ( sup_sup_set_b @ A @ B )
= bot_bot_set_b )
= ( ( A = bot_bot_set_b )
& ( B = bot_bot_set_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_666_sup__bot_Oeq__neutr__iff,axiom,
! [A: nat > $o,B: nat > $o] :
( ( ( sup_sup_nat_o @ A @ B )
= bot_bot_nat_o )
= ( ( A = bot_bot_nat_o )
& ( B = bot_bot_nat_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_667_sup__eq__bot__iff,axiom,
! [X5: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X5 @ Y )
= bot_bot_set_nat )
= ( ( X5 = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_668_sup__eq__bot__iff,axiom,
! [X5: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X5 @ Y )
= bot_bot_set_a )
= ( ( X5 = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_669_sup__eq__bot__iff,axiom,
! [X5: set_b,Y: set_b] :
( ( ( sup_sup_set_b @ X5 @ Y )
= bot_bot_set_b )
= ( ( X5 = bot_bot_set_b )
& ( Y = bot_bot_set_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_670_sup__eq__bot__iff,axiom,
! [X5: nat > $o,Y: nat > $o] :
( ( ( sup_sup_nat_o @ X5 @ Y )
= bot_bot_nat_o )
= ( ( X5 = bot_bot_nat_o )
& ( Y = bot_bot_nat_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_671_bot__eq__sup__iff,axiom,
! [X5: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X5 @ Y ) )
= ( ( X5 = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_672_bot__eq__sup__iff,axiom,
! [X5: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X5 @ Y ) )
= ( ( X5 = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_673_bot__eq__sup__iff,axiom,
! [X5: set_b,Y: set_b] :
( ( bot_bot_set_b
= ( sup_sup_set_b @ X5 @ Y ) )
= ( ( X5 = bot_bot_set_b )
& ( Y = bot_bot_set_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_674_bot__eq__sup__iff,axiom,
! [X5: nat > $o,Y: nat > $o] :
( ( bot_bot_nat_o
= ( sup_sup_nat_o @ X5 @ Y ) )
= ( ( X5 = bot_bot_nat_o )
& ( Y = bot_bot_nat_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_675_sup__bot__right,axiom,
! [X5: set_nat] :
( ( sup_sup_set_nat @ X5 @ bot_bot_set_nat )
= X5 ) ).
% sup_bot_right
thf(fact_676_sup__bot__right,axiom,
! [X5: set_a] :
( ( sup_sup_set_a @ X5 @ bot_bot_set_a )
= X5 ) ).
% sup_bot_right
thf(fact_677_sup__bot__right,axiom,
! [X5: set_b] :
( ( sup_sup_set_b @ X5 @ bot_bot_set_b )
= X5 ) ).
% sup_bot_right
thf(fact_678_sup__bot__right,axiom,
! [X5: nat > $o] :
( ( sup_sup_nat_o @ X5 @ bot_bot_nat_o )
= X5 ) ).
% sup_bot_right
thf(fact_679_sup__bot__left,axiom,
! [X5: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X5 )
= X5 ) ).
% sup_bot_left
thf(fact_680_sup__bot__left,axiom,
! [X5: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X5 )
= X5 ) ).
% sup_bot_left
thf(fact_681_sup__bot__left,axiom,
! [X5: set_b] :
( ( sup_sup_set_b @ bot_bot_set_b @ X5 )
= X5 ) ).
% sup_bot_left
thf(fact_682_sup__bot__left,axiom,
! [X5: nat > $o] :
( ( sup_sup_nat_o @ bot_bot_nat_o @ X5 )
= X5 ) ).
% sup_bot_left
thf(fact_683_set__union,axiom,
! [Xs: list_fo_term_a,Ys: list_fo_term_a] :
( ( set_fo_term_a2 @ ( union_fo_term_a @ Xs @ Ys ) )
= ( sup_su8271228240639168364term_a @ ( set_fo_term_a2 @ Xs ) @ ( set_fo_term_a2 @ Ys ) ) ) ).
% set_union
thf(fact_684_set__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( union_nat @ Xs @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_685_set__union,axiom,
! [Xs: list_a,Ys: list_a] :
( ( set_a2 @ ( union_a @ Xs @ Ys ) )
= ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ).
% set_union
thf(fact_686_set__union,axiom,
! [Xs: list_b,Ys: list_b] :
( ( set_b2 @ ( union_b @ Xs @ Ys ) )
= ( sup_sup_set_b @ ( set_b2 @ Xs ) @ ( set_b2 @ Ys ) ) ) ).
% set_union
thf(fact_687_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_688_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_689_Collect__empty__eq__bot,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( P = bot_bot_b_o ) ) ).
% Collect_empty_eq_bot
thf(fact_690_bot__empty__eq,axiom,
( bot_bot_fo_term_a_o
= ( ^ [X3: fo_term_a] : ( member_fo_term_a @ X3 @ bot_bo4735268219511357444term_a ) ) ) ).
% bot_empty_eq
thf(fact_691_bot__empty__eq,axiom,
( bot_bo9042073657639083596_nat_o
= ( ^ [X3: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X3 @ bot_bo1033123847703346641_a_nat ) ) ) ).
% bot_empty_eq
thf(fact_692_bot__empty__eq,axiom,
( bot_bot_set_nat_o
= ( ^ [X3: set_nat] : ( member_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_693_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_694_bot__empty__eq,axiom,
( bot_bot_b_o
= ( ^ [X3: b] : ( member_b @ X3 @ bot_bot_set_b ) ) ) ).
% bot_empty_eq
thf(fact_695_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_696_fv__fo__fmla_Oelims,axiom,
! [X5: fo_fmla_a_b,Y: set_nat] :
( ( ( fv_fo_fmla_a_b @ X5 )
= Y )
=> ( ! [Uu: b,Ts: list_fo_term_a] :
( ( X5
= ( fo_Pred_b_a @ Uu @ Ts ) )
=> ( Y
!= ( fv_fo_terms_set_a @ Ts ) ) )
=> ( ( ? [B3: $o] :
( X5
= ( fo_Bool_a_b @ B3 ) )
=> ( Y != bot_bot_set_nat ) )
=> ( ! [T3: fo_term_a,T4: fo_term_a] :
( ( X5
= ( fo_Eqa_a_b @ T3 @ T4 ) )
=> ( Y
!= ( sup_sup_set_nat @ ( fv_fo_term_set_a @ T3 ) @ ( fv_fo_term_set_a @ T4 ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Neg_a_b @ Phi2 ) )
=> ( Y
!= ( fv_fo_fmla_a_b @ Phi2 ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Conj_a_b @ Phi2 @ Psi2 ) )
=> ( Y
!= ( sup_sup_set_nat @ ( fv_fo_fmla_a_b @ Phi2 ) @ ( fv_fo_fmla_a_b @ Psi2 ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Disj_a_b @ Phi2 @ Psi2 ) )
=> ( Y
!= ( sup_sup_set_nat @ ( fv_fo_fmla_a_b @ Phi2 ) @ ( fv_fo_fmla_a_b @ Psi2 ) ) ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Exists_a_b @ N @ Phi2 ) )
=> ( Y
!= ( minus_minus_set_nat @ ( fv_fo_fmla_a_b @ Phi2 ) @ ( insert_nat2 @ N @ bot_bot_set_nat ) ) ) )
=> ~ ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Forall_a_b @ N @ Phi2 ) )
=> ( Y
!= ( minus_minus_set_nat @ ( fv_fo_fmla_a_b @ Phi2 ) @ ( insert_nat2 @ N @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% fv_fo_fmla.elims
thf(fact_697_SP_Oelims,axiom,
! [X5: fo_fmla_a_b,Y: set_nat] :
( ( ( sP_a_b @ X5 )
= Y )
=> ( ! [N: nat,N3: nat] :
( ( X5
= ( fo_Eqa_a_b @ ( fo_Var_a @ N ) @ ( fo_Var_a @ N3 ) ) )
=> ~ ( ( ( N != N3 )
=> ( Y
= ( insert_nat2 @ N @ ( insert_nat2 @ N3 @ bot_bot_set_nat ) ) ) )
& ( ( N = N3 )
=> ( Y = bot_bot_set_nat ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Neg_a_b @ Phi2 ) )
=> ( Y
!= ( sP_a_b @ Phi2 ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Conj_a_b @ Phi2 @ Psi2 ) )
=> ( Y
!= ( sup_sup_set_nat @ ( sP_a_b @ Phi2 ) @ ( sP_a_b @ Psi2 ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Disj_a_b @ Phi2 @ Psi2 ) )
=> ( Y
!= ( sup_sup_set_nat @ ( sP_a_b @ Phi2 ) @ ( sP_a_b @ Psi2 ) ) ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Exists_a_b @ N @ Phi2 ) )
=> ( Y
!= ( minus_minus_set_nat @ ( sP_a_b @ Phi2 ) @ ( insert_nat2 @ N @ bot_bot_set_nat ) ) ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Forall_a_b @ N @ Phi2 ) )
=> ( Y
!= ( minus_minus_set_nat @ ( sP_a_b @ Phi2 ) @ ( insert_nat2 @ N @ bot_bot_set_nat ) ) ) )
=> ( ( ? [V: b,Va: list_fo_term_a] :
( X5
= ( fo_Pred_b_a @ V @ Va ) )
=> ( Y != bot_bot_set_nat ) )
=> ( ( ? [V: $o] :
( X5
= ( fo_Bool_a_b @ V ) )
=> ( Y != bot_bot_set_nat ) )
=> ( ( ? [Vb: a,Va: fo_term_a] :
( X5
= ( fo_Eqa_a_b @ ( fo_Const_a @ Vb ) @ Va ) )
=> ( Y != bot_bot_set_nat ) )
=> ~ ( ? [V: fo_term_a,Vb: a] :
( X5
= ( fo_Eqa_a_b @ V @ ( fo_Const_a @ Vb ) ) )
=> ( Y != bot_bot_set_nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% SP.elims
thf(fact_698_boolean__algebra_Odisj__zero__right,axiom,
! [X5: set_nat] :
( ( sup_sup_set_nat @ X5 @ bot_bot_set_nat )
= X5 ) ).
% boolean_algebra.disj_zero_right
thf(fact_699_boolean__algebra_Odisj__zero__right,axiom,
! [X5: set_a] :
( ( sup_sup_set_a @ X5 @ bot_bot_set_a )
= X5 ) ).
% boolean_algebra.disj_zero_right
thf(fact_700_boolean__algebra_Odisj__zero__right,axiom,
! [X5: set_b] :
( ( sup_sup_set_b @ X5 @ bot_bot_set_b )
= X5 ) ).
% boolean_algebra.disj_zero_right
thf(fact_701_boolean__algebra_Odisj__zero__right,axiom,
! [X5: nat > $o] :
( ( sup_sup_nat_o @ X5 @ bot_bot_nat_o )
= X5 ) ).
% boolean_algebra.disj_zero_right
thf(fact_702_insertCI,axiom,
! [A: sum_sum_a_nat,B7: set_Sum_sum_a_nat,B: sum_sum_a_nat] :
( ( ~ ( member_Sum_sum_a_nat @ A @ B7 )
=> ( A = B ) )
=> ( member_Sum_sum_a_nat @ A @ ( insert_Sum_sum_a_nat2 @ B @ B7 ) ) ) ).
% insertCI
thf(fact_703_insertCI,axiom,
! [A: fo_term_a,B7: set_fo_term_a,B: fo_term_a] :
( ( ~ ( member_fo_term_a @ A @ B7 )
=> ( A = B ) )
=> ( member_fo_term_a @ A @ ( insert_fo_term_a2 @ B @ B7 ) ) ) ).
% insertCI
thf(fact_704_insertCI,axiom,
! [A: list_Sum_sum_a_nat,B7: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat] :
( ( ~ ( member408289922725080238_a_nat @ A @ B7 )
=> ( A = B ) )
=> ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ B @ B7 ) ) ) ).
% insertCI
thf(fact_705_insertCI,axiom,
! [A: set_nat,B7: set_set_nat,B: set_nat] :
( ( ~ ( member_set_nat @ A @ B7 )
=> ( A = B ) )
=> ( member_set_nat @ A @ ( insert_set_nat2 @ B @ B7 ) ) ) ).
% insertCI
thf(fact_706_insertCI,axiom,
! [A: b,B7: set_b,B: b] :
( ( ~ ( member_b @ A @ B7 )
=> ( A = B ) )
=> ( member_b @ A @ ( insert_b2 @ B @ B7 ) ) ) ).
% insertCI
thf(fact_707_insertCI,axiom,
! [A: a,B7: set_a,B: a] :
( ( ~ ( member_a @ A @ B7 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a2 @ B @ B7 ) ) ) ).
% insertCI
thf(fact_708_insertCI,axiom,
! [A: nat,B7: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B7 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat2 @ B @ B7 ) ) ) ).
% insertCI
thf(fact_709_insert__iff,axiom,
! [A: sum_sum_a_nat,B: sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A @ ( insert_Sum_sum_a_nat2 @ B @ A3 ) )
= ( ( A = B )
| ( member_Sum_sum_a_nat @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_710_insert__iff,axiom,
! [A: fo_term_a,B: fo_term_a,A3: set_fo_term_a] :
( ( member_fo_term_a @ A @ ( insert_fo_term_a2 @ B @ A3 ) )
= ( ( A = B )
| ( member_fo_term_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_711_insert__iff,axiom,
! [A: list_Sum_sum_a_nat,B: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ B @ A3 ) )
= ( ( A = B )
| ( member408289922725080238_a_nat @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_712_insert__iff,axiom,
! [A: set_nat,B: set_nat,A3: set_set_nat] :
( ( member_set_nat @ A @ ( insert_set_nat2 @ B @ A3 ) )
= ( ( A = B )
| ( member_set_nat @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_713_insert__iff,axiom,
! [A: b,B: b,A3: set_b] :
( ( member_b @ A @ ( insert_b2 @ B @ A3 ) )
= ( ( A = B )
| ( member_b @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_714_insert__iff,axiom,
! [A: a,B: a,A3: set_a] :
( ( member_a @ A @ ( insert_a2 @ B @ A3 ) )
= ( ( A = B )
| ( member_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_715_insert__iff,axiom,
! [A: nat,B: nat,A3: set_nat] :
( ( member_nat @ A @ ( insert_nat2 @ B @ A3 ) )
= ( ( A = B )
| ( member_nat @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_716_insert__absorb2,axiom,
! [X5: nat,A3: set_nat] :
( ( insert_nat2 @ X5 @ ( insert_nat2 @ X5 @ A3 ) )
= ( insert_nat2 @ X5 @ A3 ) ) ).
% insert_absorb2
thf(fact_717_insert__absorb2,axiom,
! [X5: sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( insert_Sum_sum_a_nat2 @ X5 @ ( insert_Sum_sum_a_nat2 @ X5 @ A3 ) )
= ( insert_Sum_sum_a_nat2 @ X5 @ A3 ) ) ).
% insert_absorb2
thf(fact_718_DiffI,axiom,
! [C: fo_term_a,A3: set_fo_term_a,B7: set_fo_term_a] :
( ( member_fo_term_a @ C @ A3 )
=> ( ~ ( member_fo_term_a @ C @ B7 )
=> ( member_fo_term_a @ C @ ( minus_6854963972745519743term_a @ A3 @ B7 ) ) ) ) ).
% DiffI
thf(fact_719_DiffI,axiom,
! [C: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat,B7: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ A3 )
=> ( ~ ( member408289922725080238_a_nat @ C @ B7 )
=> ( member408289922725080238_a_nat @ C @ ( minus_7395159227704179404_a_nat @ A3 @ B7 ) ) ) ) ).
% DiffI
thf(fact_720_DiffI,axiom,
! [C: set_nat,A3: set_set_nat,B7: set_set_nat] :
( ( member_set_nat @ C @ A3 )
=> ( ~ ( member_set_nat @ C @ B7 )
=> ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B7 ) ) ) ) ).
% DiffI
thf(fact_721_DiffI,axiom,
! [C: b,A3: set_b,B7: set_b] :
( ( member_b @ C @ A3 )
=> ( ~ ( member_b @ C @ B7 )
=> ( member_b @ C @ ( minus_minus_set_b @ A3 @ B7 ) ) ) ) ).
% DiffI
thf(fact_722_DiffI,axiom,
! [C: a,A3: set_a,B7: set_a] :
( ( member_a @ C @ A3 )
=> ( ~ ( member_a @ C @ B7 )
=> ( member_a @ C @ ( minus_minus_set_a @ A3 @ B7 ) ) ) ) ).
% DiffI
thf(fact_723_DiffI,axiom,
! [C: nat,A3: set_nat,B7: set_nat] :
( ( member_nat @ C @ A3 )
=> ( ~ ( member_nat @ C @ B7 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B7 ) ) ) ) ).
% DiffI
thf(fact_724_Diff__iff,axiom,
! [C: fo_term_a,A3: set_fo_term_a,B7: set_fo_term_a] :
( ( member_fo_term_a @ C @ ( minus_6854963972745519743term_a @ A3 @ B7 ) )
= ( ( member_fo_term_a @ C @ A3 )
& ~ ( member_fo_term_a @ C @ B7 ) ) ) ).
% Diff_iff
thf(fact_725_Diff__iff,axiom,
! [C: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat,B7: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( minus_7395159227704179404_a_nat @ A3 @ B7 ) )
= ( ( member408289922725080238_a_nat @ C @ A3 )
& ~ ( member408289922725080238_a_nat @ C @ B7 ) ) ) ).
% Diff_iff
thf(fact_726_Diff__iff,axiom,
! [C: set_nat,A3: set_set_nat,B7: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B7 ) )
= ( ( member_set_nat @ C @ A3 )
& ~ ( member_set_nat @ C @ B7 ) ) ) ).
% Diff_iff
thf(fact_727_Diff__iff,axiom,
! [C: b,A3: set_b,B7: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A3 @ B7 ) )
= ( ( member_b @ C @ A3 )
& ~ ( member_b @ C @ B7 ) ) ) ).
% Diff_iff
thf(fact_728_Diff__iff,axiom,
! [C: a,A3: set_a,B7: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B7 ) )
= ( ( member_a @ C @ A3 )
& ~ ( member_a @ C @ B7 ) ) ) ).
% Diff_iff
thf(fact_729_Diff__iff,axiom,
! [C: nat,A3: set_nat,B7: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B7 ) )
= ( ( member_nat @ C @ A3 )
& ~ ( member_nat @ C @ B7 ) ) ) ).
% Diff_iff
thf(fact_730_Diff__idemp,axiom,
! [A3: set_nat,B7: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ B7 ) @ B7 )
= ( minus_minus_set_nat @ A3 @ B7 ) ) ).
% Diff_idemp
thf(fact_731_singletonI,axiom,
! [A: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ A @ ( insert_Sum_sum_a_nat2 @ A @ bot_bo3438331934148233675_a_nat ) ) ).
% singletonI
thf(fact_732_singletonI,axiom,
! [A: fo_term_a] : ( member_fo_term_a @ A @ ( insert_fo_term_a2 @ A @ bot_bo4735268219511357444term_a ) ) ).
% singletonI
thf(fact_733_singletonI,axiom,
! [A: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ) ).
% singletonI
thf(fact_734_singletonI,axiom,
! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat2 @ A @ bot_bot_set_set_nat ) ) ).
% singletonI
thf(fact_735_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a2 @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_736_singletonI,axiom,
! [A: b] : ( member_b @ A @ ( insert_b2 @ A @ bot_bot_set_b ) ) ).
% singletonI
thf(fact_737_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_738_Diff__cancel,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ A3 @ A3 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_739_Diff__cancel,axiom,
! [A3: set_b] :
( ( minus_minus_set_b @ A3 @ A3 )
= bot_bot_set_b ) ).
% Diff_cancel
thf(fact_740_Diff__cancel,axiom,
! [A3: set_nat] :
( ( minus_minus_set_nat @ A3 @ A3 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_741_empty__Diff,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A3 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_742_empty__Diff,axiom,
! [A3: set_b] :
( ( minus_minus_set_b @ bot_bot_set_b @ A3 )
= bot_bot_set_b ) ).
% empty_Diff
thf(fact_743_empty__Diff,axiom,
! [A3: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A3 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_744_Diff__empty,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ A3 @ bot_bot_set_a )
= A3 ) ).
% Diff_empty
thf(fact_745_Diff__empty,axiom,
! [A3: set_b] :
( ( minus_minus_set_b @ A3 @ bot_bot_set_b )
= A3 ) ).
% Diff_empty
thf(fact_746_Diff__empty,axiom,
! [A3: set_nat] :
( ( minus_minus_set_nat @ A3 @ bot_bot_set_nat )
= A3 ) ).
% Diff_empty
thf(fact_747_Un__insert__left,axiom,
! [A: sum_sum_a_nat,B7: set_Sum_sum_a_nat,C3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( insert_Sum_sum_a_nat2 @ A @ B7 ) @ C3 )
= ( insert_Sum_sum_a_nat2 @ A @ ( sup_su6804446743777130803_a_nat @ B7 @ C3 ) ) ) ).
% Un_insert_left
thf(fact_748_Un__insert__left,axiom,
! [A: nat,B7: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat2 @ A @ B7 ) @ C3 )
= ( insert_nat2 @ A @ ( sup_sup_set_nat @ B7 @ C3 ) ) ) ).
% Un_insert_left
thf(fact_749_Un__insert__left,axiom,
! [A: a,B7: set_a,C3: set_a] :
( ( sup_sup_set_a @ ( insert_a2 @ A @ B7 ) @ C3 )
= ( insert_a2 @ A @ ( sup_sup_set_a @ B7 @ C3 ) ) ) ).
% Un_insert_left
thf(fact_750_Un__insert__left,axiom,
! [A: b,B7: set_b,C3: set_b] :
( ( sup_sup_set_b @ ( insert_b2 @ A @ B7 ) @ C3 )
= ( insert_b2 @ A @ ( sup_sup_set_b @ B7 @ C3 ) ) ) ).
% Un_insert_left
thf(fact_751_Un__insert__right,axiom,
! [A3: set_Sum_sum_a_nat,A: sum_sum_a_nat,B7: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A3 @ ( insert_Sum_sum_a_nat2 @ A @ B7 ) )
= ( insert_Sum_sum_a_nat2 @ A @ ( sup_su6804446743777130803_a_nat @ A3 @ B7 ) ) ) ).
% Un_insert_right
thf(fact_752_Un__insert__right,axiom,
! [A3: set_nat,A: nat,B7: set_nat] :
( ( sup_sup_set_nat @ A3 @ ( insert_nat2 @ A @ B7 ) )
= ( insert_nat2 @ A @ ( sup_sup_set_nat @ A3 @ B7 ) ) ) ).
% Un_insert_right
thf(fact_753_Un__insert__right,axiom,
! [A3: set_a,A: a,B7: set_a] :
( ( sup_sup_set_a @ A3 @ ( insert_a2 @ A @ B7 ) )
= ( insert_a2 @ A @ ( sup_sup_set_a @ A3 @ B7 ) ) ) ).
% Un_insert_right
thf(fact_754_Un__insert__right,axiom,
! [A3: set_b,A: b,B7: set_b] :
( ( sup_sup_set_b @ A3 @ ( insert_b2 @ A @ B7 ) )
= ( insert_b2 @ A @ ( sup_sup_set_b @ A3 @ B7 ) ) ) ).
% Un_insert_right
thf(fact_755_Diff__insert0,axiom,
! [X5: sum_sum_a_nat,A3: set_Sum_sum_a_nat,B7: set_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ X5 @ A3 )
=> ( ( minus_1134630996077396038_a_nat @ A3 @ ( insert_Sum_sum_a_nat2 @ X5 @ B7 ) )
= ( minus_1134630996077396038_a_nat @ A3 @ B7 ) ) ) ).
% Diff_insert0
thf(fact_756_Diff__insert0,axiom,
! [X5: fo_term_a,A3: set_fo_term_a,B7: set_fo_term_a] :
( ~ ( member_fo_term_a @ X5 @ A3 )
=> ( ( minus_6854963972745519743term_a @ A3 @ ( insert_fo_term_a2 @ X5 @ B7 ) )
= ( minus_6854963972745519743term_a @ A3 @ B7 ) ) ) ).
% Diff_insert0
thf(fact_757_Diff__insert0,axiom,
! [X5: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat,B7: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X5 @ A3 )
=> ( ( minus_7395159227704179404_a_nat @ A3 @ ( insert2950094090816004437_a_nat @ X5 @ B7 ) )
= ( minus_7395159227704179404_a_nat @ A3 @ B7 ) ) ) ).
% Diff_insert0
thf(fact_758_Diff__insert0,axiom,
! [X5: set_nat,A3: set_set_nat,B7: set_set_nat] :
( ~ ( member_set_nat @ X5 @ A3 )
=> ( ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat2 @ X5 @ B7 ) )
= ( minus_2163939370556025621et_nat @ A3 @ B7 ) ) ) ).
% Diff_insert0
thf(fact_759_Diff__insert0,axiom,
! [X5: b,A3: set_b,B7: set_b] :
( ~ ( member_b @ X5 @ A3 )
=> ( ( minus_minus_set_b @ A3 @ ( insert_b2 @ X5 @ B7 ) )
= ( minus_minus_set_b @ A3 @ B7 ) ) ) ).
% Diff_insert0
thf(fact_760_Diff__insert0,axiom,
! [X5: a,A3: set_a,B7: set_a] :
( ~ ( member_a @ X5 @ A3 )
=> ( ( minus_minus_set_a @ A3 @ ( insert_a2 @ X5 @ B7 ) )
= ( minus_minus_set_a @ A3 @ B7 ) ) ) ).
% Diff_insert0
thf(fact_761_Diff__insert0,axiom,
! [X5: nat,A3: set_nat,B7: set_nat] :
( ~ ( member_nat @ X5 @ A3 )
=> ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X5 @ B7 ) )
= ( minus_minus_set_nat @ A3 @ B7 ) ) ) ).
% Diff_insert0
thf(fact_762_insert__Diff1,axiom,
! [X5: sum_sum_a_nat,B7: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X5 @ B7 )
=> ( ( minus_1134630996077396038_a_nat @ ( insert_Sum_sum_a_nat2 @ X5 @ A3 ) @ B7 )
= ( minus_1134630996077396038_a_nat @ A3 @ B7 ) ) ) ).
% insert_Diff1
thf(fact_763_insert__Diff1,axiom,
! [X5: fo_term_a,B7: set_fo_term_a,A3: set_fo_term_a] :
( ( member_fo_term_a @ X5 @ B7 )
=> ( ( minus_6854963972745519743term_a @ ( insert_fo_term_a2 @ X5 @ A3 ) @ B7 )
= ( minus_6854963972745519743term_a @ A3 @ B7 ) ) ) ).
% insert_Diff1
thf(fact_764_insert__Diff1,axiom,
! [X5: list_Sum_sum_a_nat,B7: set_li6526943997496501093_a_nat,A3: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ B7 )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X5 @ A3 ) @ B7 )
= ( minus_7395159227704179404_a_nat @ A3 @ B7 ) ) ) ).
% insert_Diff1
thf(fact_765_insert__Diff1,axiom,
! [X5: set_nat,B7: set_set_nat,A3: set_set_nat] :
( ( member_set_nat @ X5 @ B7 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat2 @ X5 @ A3 ) @ B7 )
= ( minus_2163939370556025621et_nat @ A3 @ B7 ) ) ) ).
% insert_Diff1
thf(fact_766_insert__Diff1,axiom,
! [X5: b,B7: set_b,A3: set_b] :
( ( member_b @ X5 @ B7 )
=> ( ( minus_minus_set_b @ ( insert_b2 @ X5 @ A3 ) @ B7 )
= ( minus_minus_set_b @ A3 @ B7 ) ) ) ).
% insert_Diff1
thf(fact_767_insert__Diff1,axiom,
! [X5: a,B7: set_a,A3: set_a] :
( ( member_a @ X5 @ B7 )
=> ( ( minus_minus_set_a @ ( insert_a2 @ X5 @ A3 ) @ B7 )
= ( minus_minus_set_a @ A3 @ B7 ) ) ) ).
% insert_Diff1
thf(fact_768_insert__Diff1,axiom,
! [X5: nat,B7: set_nat,A3: set_nat] :
( ( member_nat @ X5 @ B7 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X5 @ A3 ) @ B7 )
= ( minus_minus_set_nat @ A3 @ B7 ) ) ) ).
% insert_Diff1
thf(fact_769_Un__Diff__cancel,axiom,
! [A3: set_a,B7: set_a] :
( ( sup_sup_set_a @ A3 @ ( minus_minus_set_a @ B7 @ A3 ) )
= ( sup_sup_set_a @ A3 @ B7 ) ) ).
% Un_Diff_cancel
thf(fact_770_Un__Diff__cancel,axiom,
! [A3: set_b,B7: set_b] :
( ( sup_sup_set_b @ A3 @ ( minus_minus_set_b @ B7 @ A3 ) )
= ( sup_sup_set_b @ A3 @ B7 ) ) ).
% Un_Diff_cancel
thf(fact_771_Un__Diff__cancel,axiom,
! [A3: set_nat,B7: set_nat] :
( ( sup_sup_set_nat @ A3 @ ( minus_minus_set_nat @ B7 @ A3 ) )
= ( sup_sup_set_nat @ A3 @ B7 ) ) ).
% Un_Diff_cancel
thf(fact_772_Un__Diff__cancel2,axiom,
! [B7: set_a,A3: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B7 @ A3 ) @ A3 )
= ( sup_sup_set_a @ B7 @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_773_Un__Diff__cancel2,axiom,
! [B7: set_b,A3: set_b] :
( ( sup_sup_set_b @ ( minus_minus_set_b @ B7 @ A3 ) @ A3 )
= ( sup_sup_set_b @ B7 @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_774_Un__Diff__cancel2,axiom,
! [B7: set_nat,A3: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B7 @ A3 ) @ A3 )
= ( sup_sup_set_nat @ B7 @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_775_insert__Diff__single,axiom,
! [A: sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( insert_Sum_sum_a_nat2 @ A @ ( minus_1134630996077396038_a_nat @ A3 @ ( insert_Sum_sum_a_nat2 @ A @ bot_bo3438331934148233675_a_nat ) ) )
= ( insert_Sum_sum_a_nat2 @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_776_insert__Diff__single,axiom,
! [A: a,A3: set_a] :
( ( insert_a2 @ A @ ( minus_minus_set_a @ A3 @ ( insert_a2 @ A @ bot_bot_set_a ) ) )
= ( insert_a2 @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_777_insert__Diff__single,axiom,
! [A: b,A3: set_b] :
( ( insert_b2 @ A @ ( minus_minus_set_b @ A3 @ ( insert_b2 @ A @ bot_bot_set_b ) ) )
= ( insert_b2 @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_778_insert__Diff__single,axiom,
! [A: nat,A3: set_nat] :
( ( insert_nat2 @ A @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
= ( insert_nat2 @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_779_DiffE,axiom,
! [C: fo_term_a,A3: set_fo_term_a,B7: set_fo_term_a] :
( ( member_fo_term_a @ C @ ( minus_6854963972745519743term_a @ A3 @ B7 ) )
=> ~ ( ( member_fo_term_a @ C @ A3 )
=> ( member_fo_term_a @ C @ B7 ) ) ) ).
% DiffE
thf(fact_780_DiffE,axiom,
! [C: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat,B7: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( minus_7395159227704179404_a_nat @ A3 @ B7 ) )
=> ~ ( ( member408289922725080238_a_nat @ C @ A3 )
=> ( member408289922725080238_a_nat @ C @ B7 ) ) ) ).
% DiffE
thf(fact_781_DiffE,axiom,
! [C: set_nat,A3: set_set_nat,B7: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B7 ) )
=> ~ ( ( member_set_nat @ C @ A3 )
=> ( member_set_nat @ C @ B7 ) ) ) ).
% DiffE
thf(fact_782_DiffE,axiom,
! [C: b,A3: set_b,B7: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A3 @ B7 ) )
=> ~ ( ( member_b @ C @ A3 )
=> ( member_b @ C @ B7 ) ) ) ).
% DiffE
thf(fact_783_DiffE,axiom,
! [C: a,A3: set_a,B7: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B7 ) )
=> ~ ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B7 ) ) ) ).
% DiffE
thf(fact_784_DiffE,axiom,
! [C: nat,A3: set_nat,B7: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B7 ) )
=> ~ ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B7 ) ) ) ).
% DiffE
thf(fact_785_DiffD1,axiom,
! [C: fo_term_a,A3: set_fo_term_a,B7: set_fo_term_a] :
( ( member_fo_term_a @ C @ ( minus_6854963972745519743term_a @ A3 @ B7 ) )
=> ( member_fo_term_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_786_DiffD1,axiom,
! [C: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat,B7: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( minus_7395159227704179404_a_nat @ A3 @ B7 ) )
=> ( member408289922725080238_a_nat @ C @ A3 ) ) ).
% DiffD1
thf(fact_787_DiffD1,axiom,
! [C: set_nat,A3: set_set_nat,B7: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B7 ) )
=> ( member_set_nat @ C @ A3 ) ) ).
% DiffD1
thf(fact_788_DiffD1,axiom,
! [C: b,A3: set_b,B7: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A3 @ B7 ) )
=> ( member_b @ C @ A3 ) ) ).
% DiffD1
thf(fact_789_DiffD1,axiom,
! [C: a,A3: set_a,B7: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B7 ) )
=> ( member_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_790_DiffD1,axiom,
! [C: nat,A3: set_nat,B7: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B7 ) )
=> ( member_nat @ C @ A3 ) ) ).
% DiffD1
thf(fact_791_DiffD2,axiom,
! [C: fo_term_a,A3: set_fo_term_a,B7: set_fo_term_a] :
( ( member_fo_term_a @ C @ ( minus_6854963972745519743term_a @ A3 @ B7 ) )
=> ~ ( member_fo_term_a @ C @ B7 ) ) ).
% DiffD2
thf(fact_792_DiffD2,axiom,
! [C: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat,B7: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( minus_7395159227704179404_a_nat @ A3 @ B7 ) )
=> ~ ( member408289922725080238_a_nat @ C @ B7 ) ) ).
% DiffD2
thf(fact_793_DiffD2,axiom,
! [C: set_nat,A3: set_set_nat,B7: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B7 ) )
=> ~ ( member_set_nat @ C @ B7 ) ) ).
% DiffD2
thf(fact_794_DiffD2,axiom,
! [C: b,A3: set_b,B7: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A3 @ B7 ) )
=> ~ ( member_b @ C @ B7 ) ) ).
% DiffD2
thf(fact_795_DiffD2,axiom,
! [C: a,A3: set_a,B7: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B7 ) )
=> ~ ( member_a @ C @ B7 ) ) ).
% DiffD2
thf(fact_796_DiffD2,axiom,
! [C: nat,A3: set_nat,B7: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B7 ) )
=> ~ ( member_nat @ C @ B7 ) ) ).
% DiffD2
thf(fact_797_insertE,axiom,
! [A: sum_sum_a_nat,B: sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A @ ( insert_Sum_sum_a_nat2 @ B @ A3 ) )
=> ( ( A != B )
=> ( member_Sum_sum_a_nat @ A @ A3 ) ) ) ).
% insertE
thf(fact_798_insertE,axiom,
! [A: fo_term_a,B: fo_term_a,A3: set_fo_term_a] :
( ( member_fo_term_a @ A @ ( insert_fo_term_a2 @ B @ A3 ) )
=> ( ( A != B )
=> ( member_fo_term_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_799_insertE,axiom,
! [A: list_Sum_sum_a_nat,B: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ B @ A3 ) )
=> ( ( A != B )
=> ( member408289922725080238_a_nat @ A @ A3 ) ) ) ).
% insertE
thf(fact_800_insertE,axiom,
! [A: set_nat,B: set_nat,A3: set_set_nat] :
( ( member_set_nat @ A @ ( insert_set_nat2 @ B @ A3 ) )
=> ( ( A != B )
=> ( member_set_nat @ A @ A3 ) ) ) ).
% insertE
thf(fact_801_insertE,axiom,
! [A: b,B: b,A3: set_b] :
( ( member_b @ A @ ( insert_b2 @ B @ A3 ) )
=> ( ( A != B )
=> ( member_b @ A @ A3 ) ) ) ).
% insertE
thf(fact_802_insertE,axiom,
! [A: a,B: a,A3: set_a] :
( ( member_a @ A @ ( insert_a2 @ B @ A3 ) )
=> ( ( A != B )
=> ( member_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_803_insertE,axiom,
! [A: nat,B: nat,A3: set_nat] :
( ( member_nat @ A @ ( insert_nat2 @ B @ A3 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A3 ) ) ) ).
% insertE
thf(fact_804_insertI1,axiom,
! [A: sum_sum_a_nat,B7: set_Sum_sum_a_nat] : ( member_Sum_sum_a_nat @ A @ ( insert_Sum_sum_a_nat2 @ A @ B7 ) ) ).
% insertI1
thf(fact_805_insertI1,axiom,
! [A: fo_term_a,B7: set_fo_term_a] : ( member_fo_term_a @ A @ ( insert_fo_term_a2 @ A @ B7 ) ) ).
% insertI1
thf(fact_806_insertI1,axiom,
! [A: list_Sum_sum_a_nat,B7: set_li6526943997496501093_a_nat] : ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ A @ B7 ) ) ).
% insertI1
thf(fact_807_insertI1,axiom,
! [A: set_nat,B7: set_set_nat] : ( member_set_nat @ A @ ( insert_set_nat2 @ A @ B7 ) ) ).
% insertI1
thf(fact_808_insertI1,axiom,
! [A: b,B7: set_b] : ( member_b @ A @ ( insert_b2 @ A @ B7 ) ) ).
% insertI1
thf(fact_809_insertI1,axiom,
! [A: a,B7: set_a] : ( member_a @ A @ ( insert_a2 @ A @ B7 ) ) ).
% insertI1
thf(fact_810_insertI1,axiom,
! [A: nat,B7: set_nat] : ( member_nat @ A @ ( insert_nat2 @ A @ B7 ) ) ).
% insertI1
thf(fact_811_insertI2,axiom,
! [A: sum_sum_a_nat,B7: set_Sum_sum_a_nat,B: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A @ B7 )
=> ( member_Sum_sum_a_nat @ A @ ( insert_Sum_sum_a_nat2 @ B @ B7 ) ) ) ).
% insertI2
thf(fact_812_insertI2,axiom,
! [A: fo_term_a,B7: set_fo_term_a,B: fo_term_a] :
( ( member_fo_term_a @ A @ B7 )
=> ( member_fo_term_a @ A @ ( insert_fo_term_a2 @ B @ B7 ) ) ) ).
% insertI2
thf(fact_813_insertI2,axiom,
! [A: list_Sum_sum_a_nat,B7: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ A @ B7 )
=> ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ B @ B7 ) ) ) ).
% insertI2
thf(fact_814_insertI2,axiom,
! [A: set_nat,B7: set_set_nat,B: set_nat] :
( ( member_set_nat @ A @ B7 )
=> ( member_set_nat @ A @ ( insert_set_nat2 @ B @ B7 ) ) ) ).
% insertI2
thf(fact_815_insertI2,axiom,
! [A: b,B7: set_b,B: b] :
( ( member_b @ A @ B7 )
=> ( member_b @ A @ ( insert_b2 @ B @ B7 ) ) ) ).
% insertI2
thf(fact_816_insertI2,axiom,
! [A: a,B7: set_a,B: a] :
( ( member_a @ A @ B7 )
=> ( member_a @ A @ ( insert_a2 @ B @ B7 ) ) ) ).
% insertI2
thf(fact_817_insertI2,axiom,
! [A: nat,B7: set_nat,B: nat] :
( ( member_nat @ A @ B7 )
=> ( member_nat @ A @ ( insert_nat2 @ B @ B7 ) ) ) ).
% insertI2
thf(fact_818_Set_Oset__insert,axiom,
! [X5: sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X5 @ A3 )
=> ~ ! [B9: set_Sum_sum_a_nat] :
( ( A3
= ( insert_Sum_sum_a_nat2 @ X5 @ B9 ) )
=> ( member_Sum_sum_a_nat @ X5 @ B9 ) ) ) ).
% Set.set_insert
thf(fact_819_Set_Oset__insert,axiom,
! [X5: fo_term_a,A3: set_fo_term_a] :
( ( member_fo_term_a @ X5 @ A3 )
=> ~ ! [B9: set_fo_term_a] :
( ( A3
= ( insert_fo_term_a2 @ X5 @ B9 ) )
=> ( member_fo_term_a @ X5 @ B9 ) ) ) ).
% Set.set_insert
thf(fact_820_Set_Oset__insert,axiom,
! [X5: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ A3 )
=> ~ ! [B9: set_li6526943997496501093_a_nat] :
( ( A3
= ( insert2950094090816004437_a_nat @ X5 @ B9 ) )
=> ( member408289922725080238_a_nat @ X5 @ B9 ) ) ) ).
% Set.set_insert
thf(fact_821_Set_Oset__insert,axiom,
! [X5: set_nat,A3: set_set_nat] :
( ( member_set_nat @ X5 @ A3 )
=> ~ ! [B9: set_set_nat] :
( ( A3
= ( insert_set_nat2 @ X5 @ B9 ) )
=> ( member_set_nat @ X5 @ B9 ) ) ) ).
% Set.set_insert
thf(fact_822_Set_Oset__insert,axiom,
! [X5: b,A3: set_b] :
( ( member_b @ X5 @ A3 )
=> ~ ! [B9: set_b] :
( ( A3
= ( insert_b2 @ X5 @ B9 ) )
=> ( member_b @ X5 @ B9 ) ) ) ).
% Set.set_insert
thf(fact_823_Set_Oset__insert,axiom,
! [X5: a,A3: set_a] :
( ( member_a @ X5 @ A3 )
=> ~ ! [B9: set_a] :
( ( A3
= ( insert_a2 @ X5 @ B9 ) )
=> ( member_a @ X5 @ B9 ) ) ) ).
% Set.set_insert
thf(fact_824_Set_Oset__insert,axiom,
! [X5: nat,A3: set_nat] :
( ( member_nat @ X5 @ A3 )
=> ~ ! [B9: set_nat] :
( ( A3
= ( insert_nat2 @ X5 @ B9 ) )
=> ( member_nat @ X5 @ B9 ) ) ) ).
% Set.set_insert
thf(fact_825_insert__ident,axiom,
! [X5: sum_sum_a_nat,A3: set_Sum_sum_a_nat,B7: set_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ X5 @ A3 )
=> ( ~ ( member_Sum_sum_a_nat @ X5 @ B7 )
=> ( ( ( insert_Sum_sum_a_nat2 @ X5 @ A3 )
= ( insert_Sum_sum_a_nat2 @ X5 @ B7 ) )
= ( A3 = B7 ) ) ) ) ).
% insert_ident
thf(fact_826_insert__ident,axiom,
! [X5: fo_term_a,A3: set_fo_term_a,B7: set_fo_term_a] :
( ~ ( member_fo_term_a @ X5 @ A3 )
=> ( ~ ( member_fo_term_a @ X5 @ B7 )
=> ( ( ( insert_fo_term_a2 @ X5 @ A3 )
= ( insert_fo_term_a2 @ X5 @ B7 ) )
= ( A3 = B7 ) ) ) ) ).
% insert_ident
thf(fact_827_insert__ident,axiom,
! [X5: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat,B7: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X5 @ A3 )
=> ( ~ ( member408289922725080238_a_nat @ X5 @ B7 )
=> ( ( ( insert2950094090816004437_a_nat @ X5 @ A3 )
= ( insert2950094090816004437_a_nat @ X5 @ B7 ) )
= ( A3 = B7 ) ) ) ) ).
% insert_ident
thf(fact_828_insert__ident,axiom,
! [X5: set_nat,A3: set_set_nat,B7: set_set_nat] :
( ~ ( member_set_nat @ X5 @ A3 )
=> ( ~ ( member_set_nat @ X5 @ B7 )
=> ( ( ( insert_set_nat2 @ X5 @ A3 )
= ( insert_set_nat2 @ X5 @ B7 ) )
= ( A3 = B7 ) ) ) ) ).
% insert_ident
thf(fact_829_insert__ident,axiom,
! [X5: b,A3: set_b,B7: set_b] :
( ~ ( member_b @ X5 @ A3 )
=> ( ~ ( member_b @ X5 @ B7 )
=> ( ( ( insert_b2 @ X5 @ A3 )
= ( insert_b2 @ X5 @ B7 ) )
= ( A3 = B7 ) ) ) ) ).
% insert_ident
thf(fact_830_insert__ident,axiom,
! [X5: a,A3: set_a,B7: set_a] :
( ~ ( member_a @ X5 @ A3 )
=> ( ~ ( member_a @ X5 @ B7 )
=> ( ( ( insert_a2 @ X5 @ A3 )
= ( insert_a2 @ X5 @ B7 ) )
= ( A3 = B7 ) ) ) ) ).
% insert_ident
thf(fact_831_insert__ident,axiom,
! [X5: nat,A3: set_nat,B7: set_nat] :
( ~ ( member_nat @ X5 @ A3 )
=> ( ~ ( member_nat @ X5 @ B7 )
=> ( ( ( insert_nat2 @ X5 @ A3 )
= ( insert_nat2 @ X5 @ B7 ) )
= ( A3 = B7 ) ) ) ) ).
% insert_ident
thf(fact_832_insert__absorb,axiom,
! [A: sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A @ A3 )
=> ( ( insert_Sum_sum_a_nat2 @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_833_insert__absorb,axiom,
! [A: fo_term_a,A3: set_fo_term_a] :
( ( member_fo_term_a @ A @ A3 )
=> ( ( insert_fo_term_a2 @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_834_insert__absorb,axiom,
! [A: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ A3 )
=> ( ( insert2950094090816004437_a_nat @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_835_insert__absorb,axiom,
! [A: set_nat,A3: set_set_nat] :
( ( member_set_nat @ A @ A3 )
=> ( ( insert_set_nat2 @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_836_insert__absorb,axiom,
! [A: b,A3: set_b] :
( ( member_b @ A @ A3 )
=> ( ( insert_b2 @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_837_insert__absorb,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ( ( insert_a2 @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_838_insert__absorb,axiom,
! [A: nat,A3: set_nat] :
( ( member_nat @ A @ A3 )
=> ( ( insert_nat2 @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_839_insert__eq__iff,axiom,
! [A: sum_sum_a_nat,A3: set_Sum_sum_a_nat,B: sum_sum_a_nat,B7: set_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ A @ A3 )
=> ( ~ ( member_Sum_sum_a_nat @ B @ B7 )
=> ( ( ( insert_Sum_sum_a_nat2 @ A @ A3 )
= ( insert_Sum_sum_a_nat2 @ B @ B7 ) )
= ( ( ( A = B )
=> ( A3 = B7 ) )
& ( ( A != B )
=> ? [C4: set_Sum_sum_a_nat] :
( ( A3
= ( insert_Sum_sum_a_nat2 @ B @ C4 ) )
& ~ ( member_Sum_sum_a_nat @ B @ C4 )
& ( B7
= ( insert_Sum_sum_a_nat2 @ A @ C4 ) )
& ~ ( member_Sum_sum_a_nat @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_840_insert__eq__iff,axiom,
! [A: fo_term_a,A3: set_fo_term_a,B: fo_term_a,B7: set_fo_term_a] :
( ~ ( member_fo_term_a @ A @ A3 )
=> ( ~ ( member_fo_term_a @ B @ B7 )
=> ( ( ( insert_fo_term_a2 @ A @ A3 )
= ( insert_fo_term_a2 @ B @ B7 ) )
= ( ( ( A = B )
=> ( A3 = B7 ) )
& ( ( A != B )
=> ? [C4: set_fo_term_a] :
( ( A3
= ( insert_fo_term_a2 @ B @ C4 ) )
& ~ ( member_fo_term_a @ B @ C4 )
& ( B7
= ( insert_fo_term_a2 @ A @ C4 ) )
& ~ ( member_fo_term_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_841_insert__eq__iff,axiom,
! [A: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat,B7: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ A @ A3 )
=> ( ~ ( member408289922725080238_a_nat @ B @ B7 )
=> ( ( ( insert2950094090816004437_a_nat @ A @ A3 )
= ( insert2950094090816004437_a_nat @ B @ B7 ) )
= ( ( ( A = B )
=> ( A3 = B7 ) )
& ( ( A != B )
=> ? [C4: set_li6526943997496501093_a_nat] :
( ( A3
= ( insert2950094090816004437_a_nat @ B @ C4 ) )
& ~ ( member408289922725080238_a_nat @ B @ C4 )
& ( B7
= ( insert2950094090816004437_a_nat @ A @ C4 ) )
& ~ ( member408289922725080238_a_nat @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_842_insert__eq__iff,axiom,
! [A: set_nat,A3: set_set_nat,B: set_nat,B7: set_set_nat] :
( ~ ( member_set_nat @ A @ A3 )
=> ( ~ ( member_set_nat @ B @ B7 )
=> ( ( ( insert_set_nat2 @ A @ A3 )
= ( insert_set_nat2 @ B @ B7 ) )
= ( ( ( A = B )
=> ( A3 = B7 ) )
& ( ( A != B )
=> ? [C4: set_set_nat] :
( ( A3
= ( insert_set_nat2 @ B @ C4 ) )
& ~ ( member_set_nat @ B @ C4 )
& ( B7
= ( insert_set_nat2 @ A @ C4 ) )
& ~ ( member_set_nat @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_843_insert__eq__iff,axiom,
! [A: b,A3: set_b,B: b,B7: set_b] :
( ~ ( member_b @ A @ A3 )
=> ( ~ ( member_b @ B @ B7 )
=> ( ( ( insert_b2 @ A @ A3 )
= ( insert_b2 @ B @ B7 ) )
= ( ( ( A = B )
=> ( A3 = B7 ) )
& ( ( A != B )
=> ? [C4: set_b] :
( ( A3
= ( insert_b2 @ B @ C4 ) )
& ~ ( member_b @ B @ C4 )
& ( B7
= ( insert_b2 @ A @ C4 ) )
& ~ ( member_b @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_844_insert__eq__iff,axiom,
! [A: a,A3: set_a,B: a,B7: set_a] :
( ~ ( member_a @ A @ A3 )
=> ( ~ ( member_a @ B @ B7 )
=> ( ( ( insert_a2 @ A @ A3 )
= ( insert_a2 @ B @ B7 ) )
= ( ( ( A = B )
=> ( A3 = B7 ) )
& ( ( A != B )
=> ? [C4: set_a] :
( ( A3
= ( insert_a2 @ B @ C4 ) )
& ~ ( member_a @ B @ C4 )
& ( B7
= ( insert_a2 @ A @ C4 ) )
& ~ ( member_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_845_insert__eq__iff,axiom,
! [A: nat,A3: set_nat,B: nat,B7: set_nat] :
( ~ ( member_nat @ A @ A3 )
=> ( ~ ( member_nat @ B @ B7 )
=> ( ( ( insert_nat2 @ A @ A3 )
= ( insert_nat2 @ B @ B7 ) )
= ( ( ( A = B )
=> ( A3 = B7 ) )
& ( ( A != B )
=> ? [C4: set_nat] :
( ( A3
= ( insert_nat2 @ B @ C4 ) )
& ~ ( member_nat @ B @ C4 )
& ( B7
= ( insert_nat2 @ A @ C4 ) )
& ~ ( member_nat @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_846_insert__Diff__if,axiom,
! [X5: sum_sum_a_nat,B7: set_Sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( ( member_Sum_sum_a_nat @ X5 @ B7 )
=> ( ( minus_1134630996077396038_a_nat @ ( insert_Sum_sum_a_nat2 @ X5 @ A3 ) @ B7 )
= ( minus_1134630996077396038_a_nat @ A3 @ B7 ) ) )
& ( ~ ( member_Sum_sum_a_nat @ X5 @ B7 )
=> ( ( minus_1134630996077396038_a_nat @ ( insert_Sum_sum_a_nat2 @ X5 @ A3 ) @ B7 )
= ( insert_Sum_sum_a_nat2 @ X5 @ ( minus_1134630996077396038_a_nat @ A3 @ B7 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_847_insert__Diff__if,axiom,
! [X5: fo_term_a,B7: set_fo_term_a,A3: set_fo_term_a] :
( ( ( member_fo_term_a @ X5 @ B7 )
=> ( ( minus_6854963972745519743term_a @ ( insert_fo_term_a2 @ X5 @ A3 ) @ B7 )
= ( minus_6854963972745519743term_a @ A3 @ B7 ) ) )
& ( ~ ( member_fo_term_a @ X5 @ B7 )
=> ( ( minus_6854963972745519743term_a @ ( insert_fo_term_a2 @ X5 @ A3 ) @ B7 )
= ( insert_fo_term_a2 @ X5 @ ( minus_6854963972745519743term_a @ A3 @ B7 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_848_insert__Diff__if,axiom,
! [X5: list_Sum_sum_a_nat,B7: set_li6526943997496501093_a_nat,A3: set_li6526943997496501093_a_nat] :
( ( ( member408289922725080238_a_nat @ X5 @ B7 )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X5 @ A3 ) @ B7 )
= ( minus_7395159227704179404_a_nat @ A3 @ B7 ) ) )
& ( ~ ( member408289922725080238_a_nat @ X5 @ B7 )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X5 @ A3 ) @ B7 )
= ( insert2950094090816004437_a_nat @ X5 @ ( minus_7395159227704179404_a_nat @ A3 @ B7 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_849_insert__Diff__if,axiom,
! [X5: set_nat,B7: set_set_nat,A3: set_set_nat] :
( ( ( member_set_nat @ X5 @ B7 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat2 @ X5 @ A3 ) @ B7 )
= ( minus_2163939370556025621et_nat @ A3 @ B7 ) ) )
& ( ~ ( member_set_nat @ X5 @ B7 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat2 @ X5 @ A3 ) @ B7 )
= ( insert_set_nat2 @ X5 @ ( minus_2163939370556025621et_nat @ A3 @ B7 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_850_insert__Diff__if,axiom,
! [X5: b,B7: set_b,A3: set_b] :
( ( ( member_b @ X5 @ B7 )
=> ( ( minus_minus_set_b @ ( insert_b2 @ X5 @ A3 ) @ B7 )
= ( minus_minus_set_b @ A3 @ B7 ) ) )
& ( ~ ( member_b @ X5 @ B7 )
=> ( ( minus_minus_set_b @ ( insert_b2 @ X5 @ A3 ) @ B7 )
= ( insert_b2 @ X5 @ ( minus_minus_set_b @ A3 @ B7 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_851_insert__Diff__if,axiom,
! [X5: a,B7: set_a,A3: set_a] :
( ( ( member_a @ X5 @ B7 )
=> ( ( minus_minus_set_a @ ( insert_a2 @ X5 @ A3 ) @ B7 )
= ( minus_minus_set_a @ A3 @ B7 ) ) )
& ( ~ ( member_a @ X5 @ B7 )
=> ( ( minus_minus_set_a @ ( insert_a2 @ X5 @ A3 ) @ B7 )
= ( insert_a2 @ X5 @ ( minus_minus_set_a @ A3 @ B7 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_852_insert__Diff__if,axiom,
! [X5: nat,B7: set_nat,A3: set_nat] :
( ( ( member_nat @ X5 @ B7 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X5 @ A3 ) @ B7 )
= ( minus_minus_set_nat @ A3 @ B7 ) ) )
& ( ~ ( member_nat @ X5 @ B7 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X5 @ A3 ) @ B7 )
= ( insert_nat2 @ X5 @ ( minus_minus_set_nat @ A3 @ B7 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_853_insert__commute,axiom,
! [X5: nat,Y: nat,A3: set_nat] :
( ( insert_nat2 @ X5 @ ( insert_nat2 @ Y @ A3 ) )
= ( insert_nat2 @ Y @ ( insert_nat2 @ X5 @ A3 ) ) ) ).
% insert_commute
thf(fact_854_insert__commute,axiom,
! [X5: sum_sum_a_nat,Y: sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( insert_Sum_sum_a_nat2 @ X5 @ ( insert_Sum_sum_a_nat2 @ Y @ A3 ) )
= ( insert_Sum_sum_a_nat2 @ Y @ ( insert_Sum_sum_a_nat2 @ X5 @ A3 ) ) ) ).
% insert_commute
thf(fact_855_mk__disjoint__insert,axiom,
! [A: sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A @ A3 )
=> ? [B9: set_Sum_sum_a_nat] :
( ( A3
= ( insert_Sum_sum_a_nat2 @ A @ B9 ) )
& ~ ( member_Sum_sum_a_nat @ A @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_856_mk__disjoint__insert,axiom,
! [A: fo_term_a,A3: set_fo_term_a] :
( ( member_fo_term_a @ A @ A3 )
=> ? [B9: set_fo_term_a] :
( ( A3
= ( insert_fo_term_a2 @ A @ B9 ) )
& ~ ( member_fo_term_a @ A @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_857_mk__disjoint__insert,axiom,
! [A: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ A3 )
=> ? [B9: set_li6526943997496501093_a_nat] :
( ( A3
= ( insert2950094090816004437_a_nat @ A @ B9 ) )
& ~ ( member408289922725080238_a_nat @ A @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_858_mk__disjoint__insert,axiom,
! [A: set_nat,A3: set_set_nat] :
( ( member_set_nat @ A @ A3 )
=> ? [B9: set_set_nat] :
( ( A3
= ( insert_set_nat2 @ A @ B9 ) )
& ~ ( member_set_nat @ A @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_859_mk__disjoint__insert,axiom,
! [A: b,A3: set_b] :
( ( member_b @ A @ A3 )
=> ? [B9: set_b] :
( ( A3
= ( insert_b2 @ A @ B9 ) )
& ~ ( member_b @ A @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_860_mk__disjoint__insert,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ? [B9: set_a] :
( ( A3
= ( insert_a2 @ A @ B9 ) )
& ~ ( member_a @ A @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_861_mk__disjoint__insert,axiom,
! [A: nat,A3: set_nat] :
( ( member_nat @ A @ A3 )
=> ? [B9: set_nat] :
( ( A3
= ( insert_nat2 @ A @ B9 ) )
& ~ ( member_nat @ A @ B9 ) ) ) ).
% mk_disjoint_insert
thf(fact_862_Diff__insert__absorb,axiom,
! [X5: sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ X5 @ A3 )
=> ( ( minus_1134630996077396038_a_nat @ ( insert_Sum_sum_a_nat2 @ X5 @ A3 ) @ ( insert_Sum_sum_a_nat2 @ X5 @ bot_bo3438331934148233675_a_nat ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_863_Diff__insert__absorb,axiom,
! [X5: fo_term_a,A3: set_fo_term_a] :
( ~ ( member_fo_term_a @ X5 @ A3 )
=> ( ( minus_6854963972745519743term_a @ ( insert_fo_term_a2 @ X5 @ A3 ) @ ( insert_fo_term_a2 @ X5 @ bot_bo4735268219511357444term_a ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_864_Diff__insert__absorb,axiom,
! [X5: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X5 @ A3 )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X5 @ A3 ) @ ( insert2950094090816004437_a_nat @ X5 @ bot_bo1033123847703346641_a_nat ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_865_Diff__insert__absorb,axiom,
! [X5: set_nat,A3: set_set_nat] :
( ~ ( member_set_nat @ X5 @ A3 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat2 @ X5 @ A3 ) @ ( insert_set_nat2 @ X5 @ bot_bot_set_set_nat ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_866_Diff__insert__absorb,axiom,
! [X5: a,A3: set_a] :
( ~ ( member_a @ X5 @ A3 )
=> ( ( minus_minus_set_a @ ( insert_a2 @ X5 @ A3 ) @ ( insert_a2 @ X5 @ bot_bot_set_a ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_867_Diff__insert__absorb,axiom,
! [X5: b,A3: set_b] :
( ~ ( member_b @ X5 @ A3 )
=> ( ( minus_minus_set_b @ ( insert_b2 @ X5 @ A3 ) @ ( insert_b2 @ X5 @ bot_bot_set_b ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_868_Diff__insert__absorb,axiom,
! [X5: nat,A3: set_nat] :
( ~ ( member_nat @ X5 @ A3 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X5 @ A3 ) @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_869_Diff__insert2,axiom,
! [A3: set_Sum_sum_a_nat,A: sum_sum_a_nat,B7: set_Sum_sum_a_nat] :
( ( minus_1134630996077396038_a_nat @ A3 @ ( insert_Sum_sum_a_nat2 @ A @ B7 ) )
= ( minus_1134630996077396038_a_nat @ ( minus_1134630996077396038_a_nat @ A3 @ ( insert_Sum_sum_a_nat2 @ A @ bot_bo3438331934148233675_a_nat ) ) @ B7 ) ) ).
% Diff_insert2
thf(fact_870_Diff__insert2,axiom,
! [A3: set_a,A: a,B7: set_a] :
( ( minus_minus_set_a @ A3 @ ( insert_a2 @ A @ B7 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a2 @ A @ bot_bot_set_a ) ) @ B7 ) ) ).
% Diff_insert2
thf(fact_871_Diff__insert2,axiom,
! [A3: set_b,A: b,B7: set_b] :
( ( minus_minus_set_b @ A3 @ ( insert_b2 @ A @ B7 ) )
= ( minus_minus_set_b @ ( minus_minus_set_b @ A3 @ ( insert_b2 @ A @ bot_bot_set_b ) ) @ B7 ) ) ).
% Diff_insert2
thf(fact_872_Diff__insert2,axiom,
! [A3: set_nat,A: nat,B7: set_nat] :
( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ A @ B7 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) @ B7 ) ) ).
% Diff_insert2
thf(fact_873_insert__Diff,axiom,
! [A: sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A @ A3 )
=> ( ( insert_Sum_sum_a_nat2 @ A @ ( minus_1134630996077396038_a_nat @ A3 @ ( insert_Sum_sum_a_nat2 @ A @ bot_bo3438331934148233675_a_nat ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_874_insert__Diff,axiom,
! [A: fo_term_a,A3: set_fo_term_a] :
( ( member_fo_term_a @ A @ A3 )
=> ( ( insert_fo_term_a2 @ A @ ( minus_6854963972745519743term_a @ A3 @ ( insert_fo_term_a2 @ A @ bot_bo4735268219511357444term_a ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_875_insert__Diff,axiom,
! [A: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ A3 )
=> ( ( insert2950094090816004437_a_nat @ A @ ( minus_7395159227704179404_a_nat @ A3 @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_876_insert__Diff,axiom,
! [A: set_nat,A3: set_set_nat] :
( ( member_set_nat @ A @ A3 )
=> ( ( insert_set_nat2 @ A @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat2 @ A @ bot_bot_set_set_nat ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_877_insert__Diff,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ( ( insert_a2 @ A @ ( minus_minus_set_a @ A3 @ ( insert_a2 @ A @ bot_bot_set_a ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_878_insert__Diff,axiom,
! [A: b,A3: set_b] :
( ( member_b @ A @ A3 )
=> ( ( insert_b2 @ A @ ( minus_minus_set_b @ A3 @ ( insert_b2 @ A @ bot_bot_set_b ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_879_insert__Diff,axiom,
! [A: nat,A3: set_nat] :
( ( member_nat @ A @ A3 )
=> ( ( insert_nat2 @ A @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_880_Diff__insert,axiom,
! [A3: set_Sum_sum_a_nat,A: sum_sum_a_nat,B7: set_Sum_sum_a_nat] :
( ( minus_1134630996077396038_a_nat @ A3 @ ( insert_Sum_sum_a_nat2 @ A @ B7 ) )
= ( minus_1134630996077396038_a_nat @ ( minus_1134630996077396038_a_nat @ A3 @ B7 ) @ ( insert_Sum_sum_a_nat2 @ A @ bot_bo3438331934148233675_a_nat ) ) ) ).
% Diff_insert
thf(fact_881_Diff__insert,axiom,
! [A3: set_a,A: a,B7: set_a] :
( ( minus_minus_set_a @ A3 @ ( insert_a2 @ A @ B7 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ B7 ) @ ( insert_a2 @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_882_Diff__insert,axiom,
! [A3: set_b,A: b,B7: set_b] :
( ( minus_minus_set_b @ A3 @ ( insert_b2 @ A @ B7 ) )
= ( minus_minus_set_b @ ( minus_minus_set_b @ A3 @ B7 ) @ ( insert_b2 @ A @ bot_bot_set_b ) ) ) ).
% Diff_insert
thf(fact_883_Diff__insert,axiom,
! [A3: set_nat,A: nat,B7: set_nat] :
( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ A @ B7 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ B7 ) @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_884_singleton__inject,axiom,
! [A: sum_sum_a_nat,B: sum_sum_a_nat] :
( ( ( insert_Sum_sum_a_nat2 @ A @ bot_bo3438331934148233675_a_nat )
= ( insert_Sum_sum_a_nat2 @ B @ bot_bo3438331934148233675_a_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_885_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat2 @ A @ bot_bot_set_nat )
= ( insert_nat2 @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_886_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a2 @ A @ bot_bot_set_a )
= ( insert_a2 @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_887_singleton__inject,axiom,
! [A: b,B: b] :
( ( ( insert_b2 @ A @ bot_bot_set_b )
= ( insert_b2 @ B @ bot_bot_set_b ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_888_insert__not__empty,axiom,
! [A: sum_sum_a_nat,A3: set_Sum_sum_a_nat] :
( ( insert_Sum_sum_a_nat2 @ A @ A3 )
!= bot_bo3438331934148233675_a_nat ) ).
% insert_not_empty
thf(fact_889_insert__not__empty,axiom,
! [A: nat,A3: set_nat] :
( ( insert_nat2 @ A @ A3 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_890_insert__not__empty,axiom,
! [A: a,A3: set_a] :
( ( insert_a2 @ A @ A3 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_891_insert__not__empty,axiom,
! [A: b,A3: set_b] :
( ( insert_b2 @ A @ A3 )
!= bot_bot_set_b ) ).
% insert_not_empty
thf(fact_892_doubleton__eq__iff,axiom,
! [A: sum_sum_a_nat,B: sum_sum_a_nat,C: sum_sum_a_nat,D: sum_sum_a_nat] :
( ( ( insert_Sum_sum_a_nat2 @ A @ ( insert_Sum_sum_a_nat2 @ B @ bot_bo3438331934148233675_a_nat ) )
= ( insert_Sum_sum_a_nat2 @ C @ ( insert_Sum_sum_a_nat2 @ D @ bot_bo3438331934148233675_a_nat ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_893_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( insert_nat2 @ A @ ( insert_nat2 @ B @ bot_bot_set_nat ) )
= ( insert_nat2 @ C @ ( insert_nat2 @ D @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_894_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( insert_a2 @ A @ ( insert_a2 @ B @ bot_bot_set_a ) )
= ( insert_a2 @ C @ ( insert_a2 @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_895_doubleton__eq__iff,axiom,
! [A: b,B: b,C: b,D: b] :
( ( ( insert_b2 @ A @ ( insert_b2 @ B @ bot_bot_set_b ) )
= ( insert_b2 @ C @ ( insert_b2 @ D @ bot_bot_set_b ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_896_singleton__iff,axiom,
! [B: sum_sum_a_nat,A: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ B @ ( insert_Sum_sum_a_nat2 @ A @ bot_bo3438331934148233675_a_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_897_singleton__iff,axiom,
! [B: fo_term_a,A: fo_term_a] :
( ( member_fo_term_a @ B @ ( insert_fo_term_a2 @ A @ bot_bo4735268219511357444term_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_898_singleton__iff,axiom,
! [B: list_Sum_sum_a_nat,A: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ B @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_899_singleton__iff,axiom,
! [B: set_nat,A: set_nat] :
( ( member_set_nat @ B @ ( insert_set_nat2 @ A @ bot_bot_set_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_900_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a2 @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_901_singleton__iff,axiom,
! [B: b,A: b] :
( ( member_b @ B @ ( insert_b2 @ A @ bot_bot_set_b ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_902_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_903_singletonD,axiom,
! [B: sum_sum_a_nat,A: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ B @ ( insert_Sum_sum_a_nat2 @ A @ bot_bo3438331934148233675_a_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_904_singletonD,axiom,
! [B: fo_term_a,A: fo_term_a] :
( ( member_fo_term_a @ B @ ( insert_fo_term_a2 @ A @ bot_bo4735268219511357444term_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_905_singletonD,axiom,
! [B: list_Sum_sum_a_nat,A: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ B @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_906_singletonD,axiom,
! [B: set_nat,A: set_nat] :
( ( member_set_nat @ B @ ( insert_set_nat2 @ A @ bot_bot_set_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_907_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a2 @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_908_singletonD,axiom,
! [B: b,A: b] :
( ( member_b @ B @ ( insert_b2 @ A @ bot_bot_set_b ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_909_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_910_Un__Diff,axiom,
! [A3: set_a,B7: set_a,C3: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A3 @ B7 ) @ C3 )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A3 @ C3 ) @ ( minus_minus_set_a @ B7 @ C3 ) ) ) ).
% Un_Diff
thf(fact_911_Un__Diff,axiom,
! [A3: set_b,B7: set_b,C3: set_b] :
( ( minus_minus_set_b @ ( sup_sup_set_b @ A3 @ B7 ) @ C3 )
= ( sup_sup_set_b @ ( minus_minus_set_b @ A3 @ C3 ) @ ( minus_minus_set_b @ B7 @ C3 ) ) ) ).
% Un_Diff
thf(fact_912_Un__Diff,axiom,
! [A3: set_nat,B7: set_nat,C3: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A3 @ B7 ) @ C3 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ C3 ) @ ( minus_minus_set_nat @ B7 @ C3 ) ) ) ).
% Un_Diff
thf(fact_913_SP_Osimps_I5_J,axiom,
! [N2: nat,Phi: fo_fmla_a_b] :
( ( sP_a_b @ ( fo_Exists_a_b @ N2 @ Phi ) )
= ( minus_minus_set_nat @ ( sP_a_b @ Phi ) @ ( insert_nat2 @ N2 @ bot_bot_set_nat ) ) ) ).
% SP.simps(5)
thf(fact_914_fv__fo__fmla_Osimps_I7_J,axiom,
! [N2: nat,Phi: fo_fmla_a_b] :
( ( fv_fo_fmla_a_b @ ( fo_Exists_a_b @ N2 @ Phi ) )
= ( minus_minus_set_nat @ ( fv_fo_fmla_a_b @ Phi ) @ ( insert_nat2 @ N2 @ bot_bot_set_nat ) ) ) ).
% fv_fo_fmla.simps(7)
thf(fact_915_SP_Osimps_I6_J,axiom,
! [N2: nat,Phi: fo_fmla_a_b] :
( ( sP_a_b @ ( fo_Forall_a_b @ N2 @ Phi ) )
= ( minus_minus_set_nat @ ( sP_a_b @ Phi ) @ ( insert_nat2 @ N2 @ bot_bot_set_nat ) ) ) ).
% SP.simps(6)
thf(fact_916_fv__fo__fmla_Osimps_I8_J,axiom,
! [N2: nat,Phi: fo_fmla_a_b] :
( ( fv_fo_fmla_a_b @ ( fo_Forall_a_b @ N2 @ Phi ) )
= ( minus_minus_set_nat @ ( fv_fo_fmla_a_b @ Phi ) @ ( insert_nat2 @ N2 @ bot_bot_set_nat ) ) ) ).
% fv_fo_fmla.simps(8)
thf(fact_917_insert__is__Un,axiom,
( insert_Sum_sum_a_nat2
= ( ^ [A2: sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ ( insert_Sum_sum_a_nat2 @ A2 @ bot_bo3438331934148233675_a_nat ) ) ) ) ).
% insert_is_Un
thf(fact_918_insert__is__Un,axiom,
( insert_nat2
= ( ^ [A2: nat] : ( sup_sup_set_nat @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_919_insert__is__Un,axiom,
( insert_a2
= ( ^ [A2: a] : ( sup_sup_set_a @ ( insert_a2 @ A2 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_920_insert__is__Un,axiom,
( insert_b2
= ( ^ [A2: b] : ( sup_sup_set_b @ ( insert_b2 @ A2 @ bot_bot_set_b ) ) ) ) ).
% insert_is_Un
thf(fact_921_Un__singleton__iff,axiom,
! [A3: set_Sum_sum_a_nat,B7: set_Sum_sum_a_nat,X5: sum_sum_a_nat] :
( ( ( sup_su6804446743777130803_a_nat @ A3 @ B7 )
= ( insert_Sum_sum_a_nat2 @ X5 @ bot_bo3438331934148233675_a_nat ) )
= ( ( ( A3 = bot_bo3438331934148233675_a_nat )
& ( B7
= ( insert_Sum_sum_a_nat2 @ X5 @ bot_bo3438331934148233675_a_nat ) ) )
| ( ( A3
= ( insert_Sum_sum_a_nat2 @ X5 @ bot_bo3438331934148233675_a_nat ) )
& ( B7 = bot_bo3438331934148233675_a_nat ) )
| ( ( A3
= ( insert_Sum_sum_a_nat2 @ X5 @ bot_bo3438331934148233675_a_nat ) )
& ( B7
= ( insert_Sum_sum_a_nat2 @ X5 @ bot_bo3438331934148233675_a_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_922_Un__singleton__iff,axiom,
! [A3: set_nat,B7: set_nat,X5: nat] :
( ( ( sup_sup_set_nat @ A3 @ B7 )
= ( insert_nat2 @ X5 @ bot_bot_set_nat ) )
= ( ( ( A3 = bot_bot_set_nat )
& ( B7
= ( insert_nat2 @ X5 @ bot_bot_set_nat ) ) )
| ( ( A3
= ( insert_nat2 @ X5 @ bot_bot_set_nat ) )
& ( B7 = bot_bot_set_nat ) )
| ( ( A3
= ( insert_nat2 @ X5 @ bot_bot_set_nat ) )
& ( B7
= ( insert_nat2 @ X5 @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_923_Un__singleton__iff,axiom,
! [A3: set_a,B7: set_a,X5: a] :
( ( ( sup_sup_set_a @ A3 @ B7 )
= ( insert_a2 @ X5 @ bot_bot_set_a ) )
= ( ( ( A3 = bot_bot_set_a )
& ( B7
= ( insert_a2 @ X5 @ bot_bot_set_a ) ) )
| ( ( A3
= ( insert_a2 @ X5 @ bot_bot_set_a ) )
& ( B7 = bot_bot_set_a ) )
| ( ( A3
= ( insert_a2 @ X5 @ bot_bot_set_a ) )
& ( B7
= ( insert_a2 @ X5 @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_924_Un__singleton__iff,axiom,
! [A3: set_b,B7: set_b,X5: b] :
( ( ( sup_sup_set_b @ A3 @ B7 )
= ( insert_b2 @ X5 @ bot_bot_set_b ) )
= ( ( ( A3 = bot_bot_set_b )
& ( B7
= ( insert_b2 @ X5 @ bot_bot_set_b ) ) )
| ( ( A3
= ( insert_b2 @ X5 @ bot_bot_set_b ) )
& ( B7 = bot_bot_set_b ) )
| ( ( A3
= ( insert_b2 @ X5 @ bot_bot_set_b ) )
& ( B7
= ( insert_b2 @ X5 @ bot_bot_set_b ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_925_singleton__Un__iff,axiom,
! [X5: sum_sum_a_nat,A3: set_Sum_sum_a_nat,B7: set_Sum_sum_a_nat] :
( ( ( insert_Sum_sum_a_nat2 @ X5 @ bot_bo3438331934148233675_a_nat )
= ( sup_su6804446743777130803_a_nat @ A3 @ B7 ) )
= ( ( ( A3 = bot_bo3438331934148233675_a_nat )
& ( B7
= ( insert_Sum_sum_a_nat2 @ X5 @ bot_bo3438331934148233675_a_nat ) ) )
| ( ( A3
= ( insert_Sum_sum_a_nat2 @ X5 @ bot_bo3438331934148233675_a_nat ) )
& ( B7 = bot_bo3438331934148233675_a_nat ) )
| ( ( A3
= ( insert_Sum_sum_a_nat2 @ X5 @ bot_bo3438331934148233675_a_nat ) )
& ( B7
= ( insert_Sum_sum_a_nat2 @ X5 @ bot_bo3438331934148233675_a_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_926_singleton__Un__iff,axiom,
! [X5: nat,A3: set_nat,B7: set_nat] :
( ( ( insert_nat2 @ X5 @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A3 @ B7 ) )
= ( ( ( A3 = bot_bot_set_nat )
& ( B7
= ( insert_nat2 @ X5 @ bot_bot_set_nat ) ) )
| ( ( A3
= ( insert_nat2 @ X5 @ bot_bot_set_nat ) )
& ( B7 = bot_bot_set_nat ) )
| ( ( A3
= ( insert_nat2 @ X5 @ bot_bot_set_nat ) )
& ( B7
= ( insert_nat2 @ X5 @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_927_singleton__Un__iff,axiom,
! [X5: a,A3: set_a,B7: set_a] :
( ( ( insert_a2 @ X5 @ bot_bot_set_a )
= ( sup_sup_set_a @ A3 @ B7 ) )
= ( ( ( A3 = bot_bot_set_a )
& ( B7
= ( insert_a2 @ X5 @ bot_bot_set_a ) ) )
| ( ( A3
= ( insert_a2 @ X5 @ bot_bot_set_a ) )
& ( B7 = bot_bot_set_a ) )
| ( ( A3
= ( insert_a2 @ X5 @ bot_bot_set_a ) )
& ( B7
= ( insert_a2 @ X5 @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_928_singleton__Un__iff,axiom,
! [X5: b,A3: set_b,B7: set_b] :
( ( ( insert_b2 @ X5 @ bot_bot_set_b )
= ( sup_sup_set_b @ A3 @ B7 ) )
= ( ( ( A3 = bot_bot_set_b )
& ( B7
= ( insert_b2 @ X5 @ bot_bot_set_b ) ) )
| ( ( A3
= ( insert_b2 @ X5 @ bot_bot_set_b ) )
& ( B7 = bot_bot_set_b ) )
| ( ( A3
= ( insert_b2 @ X5 @ bot_bot_set_b ) )
& ( B7
= ( insert_b2 @ X5 @ bot_bot_set_b ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_929_fo__fmla_Osimps_I161_J,axiom,
! [X11: nat,X12: list_fo_term_nat] :
( ( fo_set4065224739847495105at_nat @ ( fo_Pred_nat_nat @ X11 @ X12 ) )
= ( insert_nat2 @ X11 @ bot_bot_set_nat ) ) ).
% fo_fmla.simps(161)
thf(fact_930_fo__fmla_Osimps_I161_J,axiom,
! [X11: b,X12: list_fo_term_a] :
( ( fo_set2_fo_fmla_a_b @ ( fo_Pred_b_a @ X11 @ X12 ) )
= ( insert_b2 @ X11 @ bot_bot_set_b ) ) ).
% fo_fmla.simps(161)
thf(fact_931_fo__term_Osimps_I15_J,axiom,
! [X1: sum_sum_a_nat] :
( ( fo_set5789020685998436993_a_nat @ ( fo_Con399356768421508341_a_nat @ X1 ) )
= ( insert_Sum_sum_a_nat2 @ X1 @ bot_bo3438331934148233675_a_nat ) ) ).
% fo_term.simps(15)
thf(fact_932_fo__term_Osimps_I15_J,axiom,
! [X1: nat] :
( ( fo_set_fo_term_nat @ ( fo_Const_nat @ X1 ) )
= ( insert_nat2 @ X1 @ bot_bot_set_nat ) ) ).
% fo_term.simps(15)
thf(fact_933_fo__term_Osimps_I15_J,axiom,
! [X1: a] :
( ( fo_set_fo_term_a @ ( fo_Const_a @ X1 ) )
= ( insert_a2 @ X1 @ bot_bot_set_a ) ) ).
% fo_term.simps(15)
thf(fact_934_fo__term_Osimps_I15_J,axiom,
! [X1: b] :
( ( fo_set_fo_term_b @ ( fo_Const_b @ X1 ) )
= ( insert_b2 @ X1 @ bot_bot_set_b ) ) ).
% fo_term.simps(15)
thf(fact_935_fv__fo__term__set_Osimps_I1_J,axiom,
! [N2: nat] :
( ( fv_fo_term_set_a @ ( fo_Var_a @ N2 ) )
= ( insert_nat2 @ N2 @ bot_bot_set_nat ) ) ).
% fv_fo_term_set.simps(1)
thf(fact_936_SP_Osimps_I1_J,axiom,
! [N2: nat,N4: nat] :
( ( ( N2 != N4 )
=> ( ( sP_a_b @ ( fo_Eqa_a_b @ ( fo_Var_a @ N2 ) @ ( fo_Var_a @ N4 ) ) )
= ( insert_nat2 @ N2 @ ( insert_nat2 @ N4 @ bot_bot_set_nat ) ) ) )
& ( ( N2 = N4 )
=> ( ( sP_a_b @ ( fo_Eqa_a_b @ ( fo_Var_a @ N2 ) @ ( fo_Var_a @ N4 ) ) )
= bot_bot_set_nat ) ) ) ).
% SP.simps(1)
thf(fact_937_fv__fo__term__set_Oelims,axiom,
! [X5: fo_term_nat,Y: set_nat] :
( ( ( fv_fo_term_set_nat @ X5 )
= Y )
=> ( ! [N: nat] :
( ( X5
= ( fo_Var_nat @ N ) )
=> ( Y
!= ( insert_nat2 @ N @ bot_bot_set_nat ) ) )
=> ~ ( ? [V: nat] :
( X5
= ( fo_Const_nat @ V ) )
=> ( Y != bot_bot_set_nat ) ) ) ) ).
% fv_fo_term_set.elims
thf(fact_938_fv__fo__term__set_Oelims,axiom,
! [X5: fo_term_a,Y: set_nat] :
( ( ( fv_fo_term_set_a @ X5 )
= Y )
=> ( ! [N: nat] :
( ( X5
= ( fo_Var_a @ N ) )
=> ( Y
!= ( insert_nat2 @ N @ bot_bot_set_nat ) ) )
=> ~ ( ? [V: a] :
( X5
= ( fo_Const_a @ V ) )
=> ( Y != bot_bot_set_nat ) ) ) ) ).
% fv_fo_term_set.elims
thf(fact_939_fv__fo__fmla_Opelims,axiom,
! [X5: fo_fmla_a_b,Y: set_nat] :
( ( ( fv_fo_fmla_a_b @ X5 )
= Y )
=> ( ( accp_fo_fmla_a_b @ fv_fo_fmla_rel_a_b @ X5 )
=> ( ! [Uu: b,Ts: list_fo_term_a] :
( ( X5
= ( fo_Pred_b_a @ Uu @ Ts ) )
=> ( ( Y
= ( fv_fo_terms_set_a @ Ts ) )
=> ~ ( accp_fo_fmla_a_b @ fv_fo_fmla_rel_a_b @ ( fo_Pred_b_a @ Uu @ Ts ) ) ) )
=> ( ! [B3: $o] :
( ( X5
= ( fo_Bool_a_b @ B3 ) )
=> ( ( Y = bot_bot_set_nat )
=> ~ ( accp_fo_fmla_a_b @ fv_fo_fmla_rel_a_b @ ( fo_Bool_a_b @ B3 ) ) ) )
=> ( ! [T3: fo_term_a,T4: fo_term_a] :
( ( X5
= ( fo_Eqa_a_b @ T3 @ T4 ) )
=> ( ( Y
= ( sup_sup_set_nat @ ( fv_fo_term_set_a @ T3 ) @ ( fv_fo_term_set_a @ T4 ) ) )
=> ~ ( accp_fo_fmla_a_b @ fv_fo_fmla_rel_a_b @ ( fo_Eqa_a_b @ T3 @ T4 ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Neg_a_b @ Phi2 ) )
=> ( ( Y
= ( fv_fo_fmla_a_b @ Phi2 ) )
=> ~ ( accp_fo_fmla_a_b @ fv_fo_fmla_rel_a_b @ ( fo_Neg_a_b @ Phi2 ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Conj_a_b @ Phi2 @ Psi2 ) )
=> ( ( Y
= ( sup_sup_set_nat @ ( fv_fo_fmla_a_b @ Phi2 ) @ ( fv_fo_fmla_a_b @ Psi2 ) ) )
=> ~ ( accp_fo_fmla_a_b @ fv_fo_fmla_rel_a_b @ ( fo_Conj_a_b @ Phi2 @ Psi2 ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Disj_a_b @ Phi2 @ Psi2 ) )
=> ( ( Y
= ( sup_sup_set_nat @ ( fv_fo_fmla_a_b @ Phi2 ) @ ( fv_fo_fmla_a_b @ Psi2 ) ) )
=> ~ ( accp_fo_fmla_a_b @ fv_fo_fmla_rel_a_b @ ( fo_Disj_a_b @ Phi2 @ Psi2 ) ) ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Exists_a_b @ N @ Phi2 ) )
=> ( ( Y
= ( minus_minus_set_nat @ ( fv_fo_fmla_a_b @ Phi2 ) @ ( insert_nat2 @ N @ bot_bot_set_nat ) ) )
=> ~ ( accp_fo_fmla_a_b @ fv_fo_fmla_rel_a_b @ ( fo_Exists_a_b @ N @ Phi2 ) ) ) )
=> ~ ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Forall_a_b @ N @ Phi2 ) )
=> ( ( Y
= ( minus_minus_set_nat @ ( fv_fo_fmla_a_b @ Phi2 ) @ ( insert_nat2 @ N @ bot_bot_set_nat ) ) )
=> ~ ( accp_fo_fmla_a_b @ fv_fo_fmla_rel_a_b @ ( fo_Forall_a_b @ N @ Phi2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% fv_fo_fmla.pelims
thf(fact_940_SP_Opelims,axiom,
! [X5: fo_fmla_a_b,Y: set_nat] :
( ( ( sP_a_b @ X5 )
= Y )
=> ( ( accp_fo_fmla_a_b @ sP_rel_a_b @ X5 )
=> ( ! [N: nat,N3: nat] :
( ( X5
= ( fo_Eqa_a_b @ ( fo_Var_a @ N ) @ ( fo_Var_a @ N3 ) ) )
=> ( ( ( ( N != N3 )
=> ( Y
= ( insert_nat2 @ N @ ( insert_nat2 @ N3 @ bot_bot_set_nat ) ) ) )
& ( ( N = N3 )
=> ( Y = bot_bot_set_nat ) ) )
=> ~ ( accp_fo_fmla_a_b @ sP_rel_a_b @ ( fo_Eqa_a_b @ ( fo_Var_a @ N ) @ ( fo_Var_a @ N3 ) ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Neg_a_b @ Phi2 ) )
=> ( ( Y
= ( sP_a_b @ Phi2 ) )
=> ~ ( accp_fo_fmla_a_b @ sP_rel_a_b @ ( fo_Neg_a_b @ Phi2 ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Conj_a_b @ Phi2 @ Psi2 ) )
=> ( ( Y
= ( sup_sup_set_nat @ ( sP_a_b @ Phi2 ) @ ( sP_a_b @ Psi2 ) ) )
=> ~ ( accp_fo_fmla_a_b @ sP_rel_a_b @ ( fo_Conj_a_b @ Phi2 @ Psi2 ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Disj_a_b @ Phi2 @ Psi2 ) )
=> ( ( Y
= ( sup_sup_set_nat @ ( sP_a_b @ Phi2 ) @ ( sP_a_b @ Psi2 ) ) )
=> ~ ( accp_fo_fmla_a_b @ sP_rel_a_b @ ( fo_Disj_a_b @ Phi2 @ Psi2 ) ) ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Exists_a_b @ N @ Phi2 ) )
=> ( ( Y
= ( minus_minus_set_nat @ ( sP_a_b @ Phi2 ) @ ( insert_nat2 @ N @ bot_bot_set_nat ) ) )
=> ~ ( accp_fo_fmla_a_b @ sP_rel_a_b @ ( fo_Exists_a_b @ N @ Phi2 ) ) ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Forall_a_b @ N @ Phi2 ) )
=> ( ( Y
= ( minus_minus_set_nat @ ( sP_a_b @ Phi2 ) @ ( insert_nat2 @ N @ bot_bot_set_nat ) ) )
=> ~ ( accp_fo_fmla_a_b @ sP_rel_a_b @ ( fo_Forall_a_b @ N @ Phi2 ) ) ) )
=> ( ! [V: b,Va: list_fo_term_a] :
( ( X5
= ( fo_Pred_b_a @ V @ Va ) )
=> ( ( Y = bot_bot_set_nat )
=> ~ ( accp_fo_fmla_a_b @ sP_rel_a_b @ ( fo_Pred_b_a @ V @ Va ) ) ) )
=> ( ! [V: $o] :
( ( X5
= ( fo_Bool_a_b @ V ) )
=> ( ( Y = bot_bot_set_nat )
=> ~ ( accp_fo_fmla_a_b @ sP_rel_a_b @ ( fo_Bool_a_b @ V ) ) ) )
=> ( ! [Vb: a,Va: fo_term_a] :
( ( X5
= ( fo_Eqa_a_b @ ( fo_Const_a @ Vb ) @ Va ) )
=> ( ( Y = bot_bot_set_nat )
=> ~ ( accp_fo_fmla_a_b @ sP_rel_a_b @ ( fo_Eqa_a_b @ ( fo_Const_a @ Vb ) @ Va ) ) ) )
=> ~ ! [V: fo_term_a,Vb: a] :
( ( X5
= ( fo_Eqa_a_b @ V @ ( fo_Const_a @ Vb ) ) )
=> ( ( Y = bot_bot_set_nat )
=> ~ ( accp_fo_fmla_a_b @ sP_rel_a_b @ ( fo_Eqa_a_b @ V @ ( fo_Const_a @ Vb ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% SP.pelims
thf(fact_941_is__singletonI,axiom,
! [X5: sum_sum_a_nat] : ( is_sin5176708635568246003_a_nat @ ( insert_Sum_sum_a_nat2 @ X5 @ bot_bo3438331934148233675_a_nat ) ) ).
% is_singletonI
thf(fact_942_is__singletonI,axiom,
! [X5: nat] : ( is_singleton_nat @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_943_is__singletonI,axiom,
! [X5: a] : ( is_singleton_a @ ( insert_a2 @ X5 @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_944_is__singletonI,axiom,
! [X5: b] : ( is_singleton_b @ ( insert_b2 @ X5 @ bot_bot_set_b ) ) ).
% is_singletonI
thf(fact_945_set__removeAll,axiom,
! [X5: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( remove3909449470355376139_a_nat @ X5 @ Xs ) )
= ( minus_1134630996077396038_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( insert_Sum_sum_a_nat2 @ X5 @ bot_bo3438331934148233675_a_nat ) ) ) ).
% set_removeAll
thf(fact_946_set__removeAll,axiom,
! [X5: fo_term_a,Xs: list_fo_term_a] :
( ( set_fo_term_a2 @ ( removeAll_fo_term_a @ X5 @ Xs ) )
= ( minus_6854963972745519743term_a @ ( set_fo_term_a2 @ Xs ) @ ( insert_fo_term_a2 @ X5 @ bot_bo4735268219511357444term_a ) ) ) ).
% set_removeAll
thf(fact_947_set__removeAll,axiom,
! [X5: a,Xs: list_a] :
( ( set_a2 @ ( removeAll_a @ X5 @ Xs ) )
= ( minus_minus_set_a @ ( set_a2 @ Xs ) @ ( insert_a2 @ X5 @ bot_bot_set_a ) ) ) ).
% set_removeAll
thf(fact_948_set__removeAll,axiom,
! [X5: b,Xs: list_b] :
( ( set_b2 @ ( removeAll_b @ X5 @ Xs ) )
= ( minus_minus_set_b @ ( set_b2 @ Xs ) @ ( insert_b2 @ X5 @ bot_bot_set_b ) ) ) ).
% set_removeAll
thf(fact_949_set__removeAll,axiom,
! [X5: nat,Xs: list_nat] :
( ( set_nat2 @ ( removeAll_nat @ X5 @ Xs ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) ) ) ).
% set_removeAll
thf(fact_950_SP__list__set,axiom,
! [Phi: fo_fmla_a_b] :
( ( set_nat2 @ ( sP_list_a_b @ Phi ) )
= ( sP_a_b @ Phi ) ) ).
% SP_list_set
thf(fact_951_Field__insert,axiom,
! [A: sum_sum_a_nat,B: sum_sum_a_nat,R4: set_Pr7343886759072863943_a_nat] :
( ( field_Sum_sum_a_nat @ ( insert900367560037198775_a_nat @ ( produc1212125651291703639_a_nat @ A @ B ) @ R4 ) )
= ( sup_su6804446743777130803_a_nat @ ( insert_Sum_sum_a_nat2 @ A @ ( insert_Sum_sum_a_nat2 @ B @ bot_bo3438331934148233675_a_nat ) ) @ ( field_Sum_sum_a_nat @ R4 ) ) ) ).
% Field_insert
thf(fact_952_Field__insert,axiom,
! [A: nat,B: nat,R4: set_Pr1261947904930325089at_nat] :
( ( field_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 ) )
= ( sup_sup_set_nat @ ( insert_nat2 @ A @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R4 ) ) ) ).
% Field_insert
thf(fact_953_Field__insert,axiom,
! [A: a,B: a,R4: set_Product_prod_a_a] :
( ( field_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A @ B ) @ R4 ) )
= ( sup_sup_set_a @ ( insert_a2 @ A @ ( insert_a2 @ B @ bot_bot_set_a ) ) @ ( field_a @ R4 ) ) ) ).
% Field_insert
thf(fact_954_Field__insert,axiom,
! [A: b,B: b,R4: set_Product_prod_b_b] :
( ( field_b @ ( insert1747480804371709111od_b_b @ ( product_Pair_b_b @ A @ B ) @ R4 ) )
= ( sup_sup_set_b @ ( insert_b2 @ A @ ( insert_b2 @ B @ bot_bot_set_b ) ) @ ( field_b @ R4 ) ) ) ).
% Field_insert
thf(fact_955_Field__empty,axiom,
( ( field_nat @ bot_bo2099793752762293965at_nat )
= bot_bot_set_nat ) ).
% Field_empty
thf(fact_956_Field__empty,axiom,
( ( field_a @ bot_bo3357376287454694259od_a_a )
= bot_bot_set_a ) ).
% Field_empty
thf(fact_957_Field__empty,axiom,
( ( field_b @ bot_bo2792761326896053555od_b_b )
= bot_bot_set_b ) ).
% Field_empty
thf(fact_958_removeAll__id,axiom,
! [X5: list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat] :
( ~ ( member408289922725080238_a_nat @ X5 @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( ( remove910890064017026449_a_nat @ X5 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_959_removeAll__id,axiom,
! [X5: set_nat,Xs: list_set_nat] :
( ~ ( member_set_nat @ X5 @ ( set_set_nat2 @ Xs ) )
=> ( ( removeAll_set_nat @ X5 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_960_removeAll__id,axiom,
! [X5: b,Xs: list_b] :
( ~ ( member_b @ X5 @ ( set_b2 @ Xs ) )
=> ( ( removeAll_b @ X5 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_961_removeAll__id,axiom,
! [X5: fo_term_a,Xs: list_fo_term_a] :
( ~ ( member_fo_term_a @ X5 @ ( set_fo_term_a2 @ Xs ) )
=> ( ( removeAll_fo_term_a @ X5 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_962_removeAll__id,axiom,
! [X5: a,Xs: list_a] :
( ~ ( member_a @ X5 @ ( set_a2 @ Xs ) )
=> ( ( removeAll_a @ X5 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_963_removeAll__id,axiom,
! [X5: nat,Xs: list_nat] :
( ~ ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
=> ( ( removeAll_nat @ X5 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_964_FieldI1,axiom,
! [I3: fo_term_a,J: fo_term_a,R3: set_Pr2098608728313050119term_a] :
( ( member3851078497659713104term_a @ ( produc255400929278251799term_a @ I3 @ J ) @ R3 )
=> ( member_fo_term_a @ I3 @ ( field_fo_term_a @ R3 ) ) ) ).
% FieldI1
thf(fact_965_FieldI1,axiom,
! [I3: list_Sum_sum_a_nat,J: list_Sum_sum_a_nat,R3: set_Pr4870381170404451655_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ I3 @ J ) @ R3 )
=> ( member408289922725080238_a_nat @ I3 @ ( field_8091184615201973458_a_nat @ R3 ) ) ) ).
% FieldI1
thf(fact_966_FieldI1,axiom,
! [I3: set_nat,J: set_nat,R3: set_Pr5488025237498180813et_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ I3 @ J ) @ R3 )
=> ( member_set_nat @ I3 @ ( field_set_nat @ R3 ) ) ) ).
% FieldI1
thf(fact_967_FieldI1,axiom,
! [I3: b,J: b,R3: set_Product_prod_b_b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ I3 @ J ) @ R3 )
=> ( member_b @ I3 @ ( field_b @ R3 ) ) ) ).
% FieldI1
thf(fact_968_FieldI1,axiom,
! [I3: a,J: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I3 @ J ) @ R3 )
=> ( member_a @ I3 @ ( field_a @ R3 ) ) ) ).
% FieldI1
thf(fact_969_FieldI1,axiom,
! [I3: nat,J: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I3 @ J ) @ R3 )
=> ( member_nat @ I3 @ ( field_nat @ R3 ) ) ) ).
% FieldI1
thf(fact_970_FieldI2,axiom,
! [I3: fo_term_a,J: fo_term_a,R3: set_Pr2098608728313050119term_a] :
( ( member3851078497659713104term_a @ ( produc255400929278251799term_a @ I3 @ J ) @ R3 )
=> ( member_fo_term_a @ J @ ( field_fo_term_a @ R3 ) ) ) ).
% FieldI2
thf(fact_971_FieldI2,axiom,
! [I3: list_Sum_sum_a_nat,J: list_Sum_sum_a_nat,R3: set_Pr4870381170404451655_a_nat] :
( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ I3 @ J ) @ R3 )
=> ( member408289922725080238_a_nat @ J @ ( field_8091184615201973458_a_nat @ R3 ) ) ) ).
% FieldI2
thf(fact_972_FieldI2,axiom,
! [I3: set_nat,J: set_nat,R3: set_Pr5488025237498180813et_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ I3 @ J ) @ R3 )
=> ( member_set_nat @ J @ ( field_set_nat @ R3 ) ) ) ).
% FieldI2
thf(fact_973_FieldI2,axiom,
! [I3: b,J: b,R3: set_Product_prod_b_b] :
( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ I3 @ J ) @ R3 )
=> ( member_b @ J @ ( field_b @ R3 ) ) ) ).
% FieldI2
thf(fact_974_FieldI2,axiom,
! [I3: a,J: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I3 @ J ) @ R3 )
=> ( member_a @ J @ ( field_a @ R3 ) ) ) ).
% FieldI2
thf(fact_975_FieldI2,axiom,
! [I3: nat,J: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I3 @ J ) @ R3 )
=> ( member_nat @ J @ ( field_nat @ R3 ) ) ) ).
% FieldI2
thf(fact_976_is__singletonI_H,axiom,
! [A3: set_fo_term_a] :
( ( A3 != bot_bo4735268219511357444term_a )
=> ( ! [X: fo_term_a,Y5: fo_term_a] :
( ( member_fo_term_a @ X @ A3 )
=> ( ( member_fo_term_a @ Y5 @ A3 )
=> ( X = Y5 ) ) )
=> ( is_sin1520682774489783212term_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_977_is__singletonI_H,axiom,
! [A3: set_li6526943997496501093_a_nat] :
( ( A3 != bot_bo1033123847703346641_a_nat )
=> ( ! [X: list_Sum_sum_a_nat,Y5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A3 )
=> ( ( member408289922725080238_a_nat @ Y5 @ A3 )
=> ( X = Y5 ) ) )
=> ( is_sin2231188923920309881_a_nat @ A3 ) ) ) ).
% is_singletonI'
thf(fact_978_is__singletonI_H,axiom,
! [A3: set_set_nat] :
( ( A3 != bot_bot_set_set_nat )
=> ( ! [X: set_nat,Y5: set_nat] :
( ( member_set_nat @ X @ A3 )
=> ( ( member_set_nat @ Y5 @ A3 )
=> ( X = Y5 ) ) )
=> ( is_singleton_set_nat @ A3 ) ) ) ).
% is_singletonI'
thf(fact_979_is__singletonI_H,axiom,
! [A3: set_a] :
( ( A3 != bot_bot_set_a )
=> ( ! [X: a,Y5: a] :
( ( member_a @ X @ A3 )
=> ( ( member_a @ Y5 @ A3 )
=> ( X = Y5 ) ) )
=> ( is_singleton_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_980_is__singletonI_H,axiom,
! [A3: set_b] :
( ( A3 != bot_bot_set_b )
=> ( ! [X: b,Y5: b] :
( ( member_b @ X @ A3 )
=> ( ( member_b @ Y5 @ A3 )
=> ( X = Y5 ) ) )
=> ( is_singleton_b @ A3 ) ) ) ).
% is_singletonI'
thf(fact_981_is__singletonI_H,axiom,
! [A3: set_nat] :
( ( A3 != bot_bot_set_nat )
=> ( ! [X: nat,Y5: nat] :
( ( member_nat @ X @ A3 )
=> ( ( member_nat @ Y5 @ A3 )
=> ( X = Y5 ) ) )
=> ( is_singleton_nat @ A3 ) ) ) ).
% is_singletonI'
thf(fact_982_is__singletonE,axiom,
! [A3: set_Sum_sum_a_nat] :
( ( is_sin5176708635568246003_a_nat @ A3 )
=> ~ ! [X: sum_sum_a_nat] :
( A3
!= ( insert_Sum_sum_a_nat2 @ X @ bot_bo3438331934148233675_a_nat ) ) ) ).
% is_singletonE
thf(fact_983_is__singletonE,axiom,
! [A3: set_nat] :
( ( is_singleton_nat @ A3 )
=> ~ ! [X: nat] :
( A3
!= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_984_is__singletonE,axiom,
! [A3: set_a] :
( ( is_singleton_a @ A3 )
=> ~ ! [X: a] :
( A3
!= ( insert_a2 @ X @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_985_is__singletonE,axiom,
! [A3: set_b] :
( ( is_singleton_b @ A3 )
=> ~ ! [X: b] :
( A3
!= ( insert_b2 @ X @ bot_bot_set_b ) ) ) ).
% is_singletonE
thf(fact_986_is__singleton__def,axiom,
( is_sin5176708635568246003_a_nat
= ( ^ [A8: set_Sum_sum_a_nat] :
? [X3: sum_sum_a_nat] :
( A8
= ( insert_Sum_sum_a_nat2 @ X3 @ bot_bo3438331934148233675_a_nat ) ) ) ) ).
% is_singleton_def
thf(fact_987_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A8: set_nat] :
? [X3: nat] :
( A8
= ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_988_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A8: set_a] :
? [X3: a] :
( A8
= ( insert_a2 @ X3 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_989_is__singleton__def,axiom,
( is_singleton_b
= ( ^ [A8: set_b] :
? [X3: b] :
( A8
= ( insert_b2 @ X3 @ bot_bot_set_b ) ) ) ) ).
% is_singleton_def
thf(fact_990_is__singleton__the__elem,axiom,
( is_sin5176708635568246003_a_nat
= ( ^ [A8: set_Sum_sum_a_nat] :
( A8
= ( insert_Sum_sum_a_nat2 @ ( the_el9213825397763150452_a_nat @ A8 ) @ bot_bo3438331934148233675_a_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_991_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A8: set_nat] :
( A8
= ( insert_nat2 @ ( the_elem_nat @ A8 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_992_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A8: set_a] :
( A8
= ( insert_a2 @ ( the_elem_a @ A8 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_993_is__singleton__the__elem,axiom,
( is_singleton_b
= ( ^ [A8: set_b] :
( A8
= ( insert_b2 @ ( the_elem_b @ A8 ) @ bot_bot_set_b ) ) ) ) ).
% is_singleton_the_elem
thf(fact_994_Set_Oremove__def,axiom,
( remove_Sum_sum_a_nat
= ( ^ [X3: sum_sum_a_nat,A8: set_Sum_sum_a_nat] : ( minus_1134630996077396038_a_nat @ A8 @ ( insert_Sum_sum_a_nat2 @ X3 @ bot_bo3438331934148233675_a_nat ) ) ) ) ).
% Set.remove_def
thf(fact_995_Set_Oremove__def,axiom,
( remove_a
= ( ^ [X3: a,A8: set_a] : ( minus_minus_set_a @ A8 @ ( insert_a2 @ X3 @ bot_bot_set_a ) ) ) ) ).
% Set.remove_def
thf(fact_996_Set_Oremove__def,axiom,
( remove_b
= ( ^ [X3: b,A8: set_b] : ( minus_minus_set_b @ A8 @ ( insert_b2 @ X3 @ bot_bot_set_b ) ) ) ) ).
% Set.remove_def
thf(fact_997_Set_Oremove__def,axiom,
( remove_nat
= ( ^ [X3: nat,A8: set_nat] : ( minus_minus_set_nat @ A8 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ).
% Set.remove_def
thf(fact_998_List_Oset__insert,axiom,
! [X5: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( insert_Sum_sum_a_nat @ X5 @ Xs ) )
= ( insert_Sum_sum_a_nat2 @ X5 @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_999_List_Oset__insert,axiom,
! [X5: fo_term_a,Xs: list_fo_term_a] :
( ( set_fo_term_a2 @ ( insert_fo_term_a @ X5 @ Xs ) )
= ( insert_fo_term_a2 @ X5 @ ( set_fo_term_a2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_1000_List_Oset__insert,axiom,
! [X5: nat,Xs: list_nat] :
( ( set_nat2 @ ( insert_nat @ X5 @ Xs ) )
= ( insert_nat2 @ X5 @ ( set_nat2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_1001_List_Oset__insert,axiom,
! [X5: a,Xs: list_a] :
( ( set_a2 @ ( insert_a @ X5 @ Xs ) )
= ( insert_a2 @ X5 @ ( set_a2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_1002_fv__fo__term__set_Opelims,axiom,
! [X5: fo_term_nat,Y: set_nat] :
( ( ( fv_fo_term_set_nat @ X5 )
= Y )
=> ( ( accp_fo_term_nat @ fv_fo_5930752286079999714el_nat @ X5 )
=> ( ! [N: nat] :
( ( X5
= ( fo_Var_nat @ N ) )
=> ( ( Y
= ( insert_nat2 @ N @ bot_bot_set_nat ) )
=> ~ ( accp_fo_term_nat @ fv_fo_5930752286079999714el_nat @ ( fo_Var_nat @ N ) ) ) )
=> ~ ! [V: nat] :
( ( X5
= ( fo_Const_nat @ V ) )
=> ( ( Y = bot_bot_set_nat )
=> ~ ( accp_fo_term_nat @ fv_fo_5930752286079999714el_nat @ ( fo_Const_nat @ V ) ) ) ) ) ) ) ).
% fv_fo_term_set.pelims
thf(fact_1003_fv__fo__term__set_Opelims,axiom,
! [X5: fo_term_a,Y: set_nat] :
( ( ( fv_fo_term_set_a @ X5 )
= Y )
=> ( ( accp_fo_term_a @ fv_fo_term_set_rel_a @ X5 )
=> ( ! [N: nat] :
( ( X5
= ( fo_Var_a @ N ) )
=> ( ( Y
= ( insert_nat2 @ N @ bot_bot_set_nat ) )
=> ~ ( accp_fo_term_a @ fv_fo_term_set_rel_a @ ( fo_Var_a @ N ) ) ) )
=> ~ ! [V: a] :
( ( X5
= ( fo_Const_a @ V ) )
=> ( ( Y = bot_bot_set_nat )
=> ~ ( accp_fo_term_a @ fv_fo_term_set_rel_a @ ( fo_Const_a @ V ) ) ) ) ) ) ) ).
% fv_fo_term_set.pelims
thf(fact_1004_fv__fo__terms__set__list,axiom,
! [Ts2: list_fo_term_a] :
( ( set_nat2 @ ( fv_fo_terms_list_a @ Ts2 ) )
= ( fv_fo_terms_set_a @ Ts2 ) ) ).
% fv_fo_terms_set_list
thf(fact_1005_Set_Omember__remove,axiom,
! [X5: fo_term_a,Y: fo_term_a,A3: set_fo_term_a] :
( ( member_fo_term_a @ X5 @ ( remove_fo_term_a @ Y @ A3 ) )
= ( ( member_fo_term_a @ X5 @ A3 )
& ( X5 != Y ) ) ) ).
% Set.member_remove
thf(fact_1006_Set_Omember__remove,axiom,
! [X5: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat,A3: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ ( remove5086202153292001386_a_nat @ Y @ A3 ) )
= ( ( member408289922725080238_a_nat @ X5 @ A3 )
& ( X5 != Y ) ) ) ).
% Set.member_remove
thf(fact_1007_Set_Omember__remove,axiom,
! [X5: set_nat,Y: set_nat,A3: set_set_nat] :
( ( member_set_nat @ X5 @ ( remove_set_nat @ Y @ A3 ) )
= ( ( member_set_nat @ X5 @ A3 )
& ( X5 != Y ) ) ) ).
% Set.member_remove
thf(fact_1008_Set_Omember__remove,axiom,
! [X5: b,Y: b,A3: set_b] :
( ( member_b @ X5 @ ( remove_b @ Y @ A3 ) )
= ( ( member_b @ X5 @ A3 )
& ( X5 != Y ) ) ) ).
% Set.member_remove
thf(fact_1009_Set_Omember__remove,axiom,
! [X5: a,Y: a,A3: set_a] :
( ( member_a @ X5 @ ( remove_a @ Y @ A3 ) )
= ( ( member_a @ X5 @ A3 )
& ( X5 != Y ) ) ) ).
% Set.member_remove
thf(fact_1010_Set_Omember__remove,axiom,
! [X5: nat,Y: nat,A3: set_nat] :
( ( member_nat @ X5 @ ( remove_nat @ Y @ A3 ) )
= ( ( member_nat @ X5 @ A3 )
& ( X5 != Y ) ) ) ).
% Set.member_remove
thf(fact_1011_in__set__insert,axiom,
! [X5: list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( ( insert3579561606878280865_a_nat @ X5 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_1012_in__set__insert,axiom,
! [X5: set_nat,Xs: list_set_nat] :
( ( member_set_nat @ X5 @ ( set_set_nat2 @ Xs ) )
=> ( ( insert_set_nat @ X5 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_1013_in__set__insert,axiom,
! [X5: b,Xs: list_b] :
( ( member_b @ X5 @ ( set_b2 @ Xs ) )
=> ( ( insert_b @ X5 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_1014_in__set__insert,axiom,
! [X5: fo_term_a,Xs: list_fo_term_a] :
( ( member_fo_term_a @ X5 @ ( set_fo_term_a2 @ Xs ) )
=> ( ( insert_fo_term_a @ X5 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_1015_in__set__insert,axiom,
! [X5: a,Xs: list_a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X5 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_1016_in__set__insert,axiom,
! [X5: nat,Xs: list_nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X5 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_1017_the__elem__eq,axiom,
! [X5: sum_sum_a_nat] :
( ( the_el9213825397763150452_a_nat @ ( insert_Sum_sum_a_nat2 @ X5 @ bot_bo3438331934148233675_a_nat ) )
= X5 ) ).
% the_elem_eq
thf(fact_1018_the__elem__eq,axiom,
! [X5: nat] :
( ( the_elem_nat @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) )
= X5 ) ).
% the_elem_eq
thf(fact_1019_the__elem__eq,axiom,
! [X5: a] :
( ( the_elem_a @ ( insert_a2 @ X5 @ bot_bot_set_a ) )
= X5 ) ).
% the_elem_eq
thf(fact_1020_the__elem__eq,axiom,
! [X5: b] :
( ( the_elem_b @ ( insert_b2 @ X5 @ bot_bot_set_b ) )
= X5 ) ).
% the_elem_eq
thf(fact_1021_remove__code_I1_J,axiom,
! [X5: fo_term_a,Xs: list_fo_term_a] :
( ( remove_fo_term_a @ X5 @ ( set_fo_term_a2 @ Xs ) )
= ( set_fo_term_a2 @ ( removeAll_fo_term_a @ X5 @ Xs ) ) ) ).
% remove_code(1)
thf(fact_1022_remove__code_I1_J,axiom,
! [X5: a,Xs: list_a] :
( ( remove_a @ X5 @ ( set_a2 @ Xs ) )
= ( set_a2 @ ( removeAll_a @ X5 @ Xs ) ) ) ).
% remove_code(1)
thf(fact_1023_remove__code_I1_J,axiom,
! [X5: nat,Xs: list_nat] :
( ( remove_nat @ X5 @ ( set_nat2 @ Xs ) )
= ( set_nat2 @ ( removeAll_nat @ X5 @ Xs ) ) ) ).
% remove_code(1)
thf(fact_1024_refl__on__singleton,axiom,
! [X5: sum_sum_a_nat] : ( refl_o828724780208292734_a_nat @ ( insert_Sum_sum_a_nat2 @ X5 @ bot_bo3438331934148233675_a_nat ) @ ( insert900367560037198775_a_nat @ ( produc1212125651291703639_a_nat @ X5 @ X5 ) @ bot_bo6795098209002113331_a_nat ) ) ).
% refl_on_singleton
thf(fact_1025_refl__on__singleton,axiom,
! [X5: nat] : ( refl_on_nat @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X5 @ X5 ) @ bot_bo2099793752762293965at_nat ) ) ).
% refl_on_singleton
thf(fact_1026_refl__on__singleton,axiom,
! [X5: a] : ( refl_on_a @ ( insert_a2 @ X5 @ bot_bot_set_a ) @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X5 @ X5 ) @ bot_bo3357376287454694259od_a_a ) ) ).
% refl_on_singleton
thf(fact_1027_refl__on__singleton,axiom,
! [X5: b] : ( refl_on_b @ ( insert_b2 @ X5 @ bot_bot_set_b ) @ ( insert1747480804371709111od_b_b @ ( product_Pair_b_b @ X5 @ X5 ) @ bot_bo2792761326896053555od_b_b ) ) ).
% refl_on_singleton
thf(fact_1028_remove__code_I2_J,axiom,
! [X5: nat,Xs: list_nat] :
( ( remove_nat @ X5 @ ( coset_nat @ Xs ) )
= ( coset_nat @ ( insert_nat @ X5 @ Xs ) ) ) ).
% remove_code(2)
thf(fact_1029_Range__insert,axiom,
! [A: nat,B: nat,R4: set_Pr1261947904930325089at_nat] :
( ( range_nat_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 ) )
= ( insert_nat2 @ B @ ( range_nat_nat @ R4 ) ) ) ).
% Range_insert
thf(fact_1030_Range__insert,axiom,
! [A: product_prod_b_nat > set_list_a,B: produc4672180596006801056_a_nat,R4: set_Pr1181062300739491274_a_nat] :
( ( range_4281096948368284472_a_nat @ ( insert8523128178420280250_a_nat @ ( produc6651248262528101210_a_nat @ A @ B ) @ R4 ) )
= ( insert2826030515666838640_a_nat @ B @ ( range_4281096948368284472_a_nat @ R4 ) ) ) ).
% Range_insert
thf(fact_1031_Range__insert,axiom,
! [A: product_prod_b_nat > set_list_a,B: nat > a,R4: set_Pr6389665502131816719_nat_a] :
( ( range_2790413621204530045_nat_a @ ( insert3989210416794341759_nat_a @ ( produc2895298938842563487_nat_a @ A @ B ) @ R4 ) )
= ( insert_nat_a @ B @ ( range_2790413621204530045_nat_a @ R4 ) ) ) ).
% Range_insert
thf(fact_1032_Range__insert,axiom,
! [A: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat,R4: set_Pr7813613929123410816_a_nat] :
( ( range_3615419035313980270_a_nat @ ( insert2826030515666838640_a_nat @ ( produc3720304352952013712_a_nat @ A @ B ) @ R4 ) )
= ( insert7996324640348233519_a_nat @ B @ ( range_3615419035313980270_a_nat @ R4 ) ) ) ).
% Range_insert
thf(fact_1033_Range__insert,axiom,
! [A: b,B: nat,R4: set_Pr1307281990691478580_b_nat] :
( ( range_b_nat @ ( insert66675715695368366_b_nat @ ( product_Pair_b_nat @ A @ B ) @ R4 ) )
= ( insert_nat2 @ B @ ( range_b_nat @ R4 ) ) ) ).
% Range_insert
thf(fact_1034_linear__order__on__singleton,axiom,
! [X5: sum_sum_a_nat] : ( order_3331697602794055092_a_nat @ ( insert_Sum_sum_a_nat2 @ X5 @ bot_bo3438331934148233675_a_nat ) @ ( insert900367560037198775_a_nat @ ( produc1212125651291703639_a_nat @ X5 @ X5 ) @ bot_bo6795098209002113331_a_nat ) ) ).
% linear_order_on_singleton
thf(fact_1035_linear__order__on__singleton,axiom,
! [X5: nat] : ( order_4473980167227706203on_nat @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X5 @ X5 ) @ bot_bo2099793752762293965at_nat ) ) ).
% linear_order_on_singleton
thf(fact_1036_linear__order__on__singleton,axiom,
! [X5: a] : ( order_8768733634509060147r_on_a @ ( insert_a2 @ X5 @ bot_bot_set_a ) @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X5 @ X5 ) @ bot_bo3357376287454694259od_a_a ) ) ).
% linear_order_on_singleton
thf(fact_1037_linear__order__on__singleton,axiom,
! [X5: b] : ( order_8768733634509060148r_on_b @ ( insert_b2 @ X5 @ bot_bot_set_b ) @ ( insert1747480804371709111od_b_b @ ( product_Pair_b_b @ X5 @ X5 ) @ bot_bo2792761326896053555od_b_b ) ) ).
% linear_order_on_singleton
thf(fact_1038_Domain__insert,axiom,
! [A: nat,B: nat,R4: set_Pr1261947904930325089at_nat] :
( ( domain_nat_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 ) )
= ( insert_nat2 @ A @ ( domain_nat_nat @ R4 ) ) ) ).
% Domain_insert
thf(fact_1039_Domain__insert,axiom,
! [A: product_prod_b_nat > set_list_a,B: produc4672180596006801056_a_nat,R4: set_Pr1181062300739491274_a_nat] :
( ( domain4253084160996794191_a_nat @ ( insert8523128178420280250_a_nat @ ( produc6651248262528101210_a_nat @ A @ B ) @ R4 ) )
= ( insert7903338688830906093list_a @ A @ ( domain4253084160996794191_a_nat @ R4 ) ) ) ).
% Domain_insert
thf(fact_1040_Domain__insert,axiom,
! [A: product_prod_b_nat > set_list_a,B: nat > a,R4: set_Pr6389665502131816719_nat_a] :
( ( domain2872409888413931412_nat_a @ ( insert3989210416794341759_nat_a @ ( produc2895298938842563487_nat_a @ A @ B ) @ R4 ) )
= ( insert7903338688830906093list_a @ A @ ( domain2872409888413931412_nat_a @ R4 ) ) ) ).
% Domain_insert
thf(fact_1041_Domain__insert,axiom,
! [A: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat,R4: set_Pr7813613929123410816_a_nat] :
( ( domain3697415302523381637_a_nat @ ( insert2826030515666838640_a_nat @ ( produc3720304352952013712_a_nat @ A @ B ) @ R4 ) )
= ( insert5265011953798106934_a_nat @ A @ ( domain3697415302523381637_a_nat @ R4 ) ) ) ).
% Domain_insert
thf(fact_1042_Domain__insert,axiom,
! [A: b,B: nat,R4: set_Pr1307281990691478580_b_nat] :
( ( domain_b_nat @ ( insert66675715695368366_b_nat @ ( product_Pair_b_nat @ A @ B ) @ R4 ) )
= ( insert_b2 @ A @ ( domain_b_nat @ R4 ) ) ) ).
% Domain_insert
thf(fact_1043_refl__onD2,axiom,
! [A3: set_fo_term_a,R4: set_Pr2098608728313050119term_a,X5: fo_term_a,Y: fo_term_a] :
( ( refl_on_fo_term_a @ A3 @ R4 )
=> ( ( member3851078497659713104term_a @ ( produc255400929278251799term_a @ X5 @ Y ) @ R4 )
=> ( member_fo_term_a @ Y @ A3 ) ) ) ).
% refl_onD2
thf(fact_1044_refl__onD2,axiom,
! [A3: set_li6526943997496501093_a_nat,R4: set_Pr4870381170404451655_a_nat,X5: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
( ( refl_o8238942462679651204_a_nat @ A3 @ R4 )
=> ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ X5 @ Y ) @ R4 )
=> ( member408289922725080238_a_nat @ Y @ A3 ) ) ) ).
% refl_onD2
thf(fact_1045_refl__onD2,axiom,
! [A3: set_set_nat,R4: set_Pr5488025237498180813et_nat,X5: set_nat,Y: set_nat] :
( ( refl_on_set_nat @ A3 @ R4 )
=> ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X5 @ Y ) @ R4 )
=> ( member_set_nat @ Y @ A3 ) ) ) ).
% refl_onD2
thf(fact_1046_refl__onD2,axiom,
! [A3: set_b,R4: set_Product_prod_b_b,X5: b,Y: b] :
( ( refl_on_b @ A3 @ R4 )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X5 @ Y ) @ R4 )
=> ( member_b @ Y @ A3 ) ) ) ).
% refl_onD2
thf(fact_1047_refl__onD2,axiom,
! [A3: set_a,R4: set_Product_prod_a_a,X5: a,Y: a] :
( ( refl_on_a @ A3 @ R4 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y ) @ R4 )
=> ( member_a @ Y @ A3 ) ) ) ).
% refl_onD2
thf(fact_1048_refl__onD2,axiom,
! [A3: set_nat,R4: set_Pr1261947904930325089at_nat,X5: nat,Y: nat] :
( ( refl_on_nat @ A3 @ R4 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y ) @ R4 )
=> ( member_nat @ Y @ A3 ) ) ) ).
% refl_onD2
thf(fact_1049_refl__onD1,axiom,
! [A3: set_fo_term_a,R4: set_Pr2098608728313050119term_a,X5: fo_term_a,Y: fo_term_a] :
( ( refl_on_fo_term_a @ A3 @ R4 )
=> ( ( member3851078497659713104term_a @ ( produc255400929278251799term_a @ X5 @ Y ) @ R4 )
=> ( member_fo_term_a @ X5 @ A3 ) ) ) ).
% refl_onD1
thf(fact_1050_refl__onD1,axiom,
! [A3: set_li6526943997496501093_a_nat,R4: set_Pr4870381170404451655_a_nat,X5: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
( ( refl_o8238942462679651204_a_nat @ A3 @ R4 )
=> ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ X5 @ Y ) @ R4 )
=> ( member408289922725080238_a_nat @ X5 @ A3 ) ) ) ).
% refl_onD1
thf(fact_1051_refl__onD1,axiom,
! [A3: set_set_nat,R4: set_Pr5488025237498180813et_nat,X5: set_nat,Y: set_nat] :
( ( refl_on_set_nat @ A3 @ R4 )
=> ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X5 @ Y ) @ R4 )
=> ( member_set_nat @ X5 @ A3 ) ) ) ).
% refl_onD1
thf(fact_1052_refl__onD1,axiom,
! [A3: set_b,R4: set_Product_prod_b_b,X5: b,Y: b] :
( ( refl_on_b @ A3 @ R4 )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ X5 @ Y ) @ R4 )
=> ( member_b @ X5 @ A3 ) ) ) ).
% refl_onD1
thf(fact_1053_refl__onD1,axiom,
! [A3: set_a,R4: set_Product_prod_a_a,X5: a,Y: a] :
( ( refl_on_a @ A3 @ R4 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y ) @ R4 )
=> ( member_a @ X5 @ A3 ) ) ) ).
% refl_onD1
thf(fact_1054_refl__onD1,axiom,
! [A3: set_nat,R4: set_Pr1261947904930325089at_nat,X5: nat,Y: nat] :
( ( refl_on_nat @ A3 @ R4 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y ) @ R4 )
=> ( member_nat @ X5 @ A3 ) ) ) ).
% refl_onD1
thf(fact_1055_refl__onD,axiom,
! [A3: set_fo_term_a,R4: set_Pr2098608728313050119term_a,A: fo_term_a] :
( ( refl_on_fo_term_a @ A3 @ R4 )
=> ( ( member_fo_term_a @ A @ A3 )
=> ( member3851078497659713104term_a @ ( produc255400929278251799term_a @ A @ A ) @ R4 ) ) ) ).
% refl_onD
thf(fact_1056_refl__onD,axiom,
! [A3: set_li6526943997496501093_a_nat,R4: set_Pr4870381170404451655_a_nat,A: list_Sum_sum_a_nat] :
( ( refl_o8238942462679651204_a_nat @ A3 @ R4 )
=> ( ( member408289922725080238_a_nat @ A @ A3 )
=> ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ A @ A ) @ R4 ) ) ) ).
% refl_onD
thf(fact_1057_refl__onD,axiom,
! [A3: set_set_nat,R4: set_Pr5488025237498180813et_nat,A: set_nat] :
( ( refl_on_set_nat @ A3 @ R4 )
=> ( ( member_set_nat @ A @ A3 )
=> ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A @ A ) @ R4 ) ) ) ).
% refl_onD
thf(fact_1058_refl__onD,axiom,
! [A3: set_b,R4: set_Product_prod_b_b,A: b] :
( ( refl_on_b @ A3 @ R4 )
=> ( ( member_b @ A @ A3 )
=> ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ A ) @ R4 ) ) ) ).
% refl_onD
thf(fact_1059_refl__onD,axiom,
! [A3: set_a,R4: set_Product_prod_a_a,A: a] :
( ( refl_on_a @ A3 @ R4 )
=> ( ( member_a @ A @ A3 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A ) @ R4 ) ) ) ).
% refl_onD
thf(fact_1060_refl__onD,axiom,
! [A3: set_nat,R4: set_Pr1261947904930325089at_nat,A: nat] :
( ( refl_on_nat @ A3 @ R4 )
=> ( ( member_nat @ A @ A3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R4 ) ) ) ).
% refl_onD
thf(fact_1061_refl__on__domain,axiom,
! [A3: set_fo_term_a,R4: set_Pr2098608728313050119term_a,A: fo_term_a,B: fo_term_a] :
( ( refl_on_fo_term_a @ A3 @ R4 )
=> ( ( member3851078497659713104term_a @ ( produc255400929278251799term_a @ A @ B ) @ R4 )
=> ( ( member_fo_term_a @ A @ A3 )
& ( member_fo_term_a @ B @ A3 ) ) ) ) ).
% refl_on_domain
thf(fact_1062_refl__on__domain,axiom,
! [A3: set_li6526943997496501093_a_nat,R4: set_Pr4870381170404451655_a_nat,A: list_Sum_sum_a_nat,B: list_Sum_sum_a_nat] :
( ( refl_o8238942462679651204_a_nat @ A3 @ R4 )
=> ( ( member7457213283480048528_a_nat @ ( produc7990843422341522135_a_nat @ A @ B ) @ R4 )
=> ( ( member408289922725080238_a_nat @ A @ A3 )
& ( member408289922725080238_a_nat @ B @ A3 ) ) ) ) ).
% refl_on_domain
thf(fact_1063_refl__on__domain,axiom,
! [A3: set_set_nat,R4: set_Pr5488025237498180813et_nat,A: set_nat,B: set_nat] :
( ( refl_on_set_nat @ A3 @ R4 )
=> ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A @ B ) @ R4 )
=> ( ( member_set_nat @ A @ A3 )
& ( member_set_nat @ B @ A3 ) ) ) ) ).
% refl_on_domain
thf(fact_1064_refl__on__domain,axiom,
! [A3: set_b,R4: set_Product_prod_b_b,A: b,B: b] :
( ( refl_on_b @ A3 @ R4 )
=> ( ( member7862447936710763792od_b_b @ ( product_Pair_b_b @ A @ B ) @ R4 )
=> ( ( member_b @ A @ A3 )
& ( member_b @ B @ A3 ) ) ) ) ).
% refl_on_domain
thf(fact_1065_refl__on__domain,axiom,
! [A3: set_a,R4: set_Product_prod_a_a,A: a,B: a] :
( ( refl_on_a @ A3 @ R4 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R4 )
=> ( ( member_a @ A @ A3 )
& ( member_a @ B @ A3 ) ) ) ) ).
% refl_on_domain
thf(fact_1066_refl__on__domain,axiom,
! [A3: set_nat,R4: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( refl_on_nat @ A3 @ R4 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 )
=> ( ( member_nat @ A @ A3 )
& ( member_nat @ B @ A3 ) ) ) ) ).
% refl_on_domain
thf(fact_1067_Domain_Ocases,axiom,
! [A: nat,R4: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A @ ( domain_nat_nat @ R4 ) )
=> ~ ! [B3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B3 ) @ R4 ) ) ).
% Domain.cases
thf(fact_1068_Domain_Ocases,axiom,
! [A: product_prod_b_nat > set_list_a,R4: set_Pr1181062300739491274_a_nat] :
( ( member8404886500659538246list_a @ A @ ( domain4253084160996794191_a_nat @ R4 ) )
=> ~ ! [B3: produc4672180596006801056_a_nat] :
~ ( member2219381696197041043_a_nat @ ( produc6651248262528101210_a_nat @ A @ B3 ) @ R4 ) ) ).
% Domain.cases
thf(fact_1069_Domain_Ocases,axiom,
! [A: product_prod_b_nat > set_list_a,R4: set_Pr6389665502131816719_nat_a] :
( ( member8404886500659538246list_a @ A @ ( domain2872409888413931412_nat_a @ R4 ) )
=> ~ ! [B3: nat > a] :
~ ( member9198066416134578520_nat_a @ ( produc2895298938842563487_nat_a @ A @ B3 ) @ R4 ) ) ).
% Domain.cases
thf(fact_1070_Domain_Ocases,axiom,
! [A: nat > sum_sum_a_nat,R4: set_Pr7813613929123410816_a_nat] :
( ( member8690443509505302927_a_nat @ A @ ( domain3697415302523381637_a_nat @ R4 ) )
=> ~ ! [B3: set_Sum_sum_a_nat] :
~ ( member8034886515007075401_a_nat @ ( produc3720304352952013712_a_nat @ A @ B3 ) @ R4 ) ) ).
% Domain.cases
thf(fact_1071_Domain_Ocases,axiom,
! [A: b,R4: set_Pr1307281990691478580_b_nat] :
( ( member_b @ A @ ( domain_b_nat @ R4 ) )
=> ~ ! [B3: nat] :
~ ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A @ B3 ) @ R4 ) ) ).
% Domain.cases
thf(fact_1072_Domain_Osimps,axiom,
! [A: nat,R4: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A @ ( domain_nat_nat @ R4 ) )
= ( ? [A2: nat,B2: nat] :
( ( A = A2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R4 ) ) ) ) ).
% Domain.simps
thf(fact_1073_Domain_Osimps,axiom,
! [A: product_prod_b_nat > set_list_a,R4: set_Pr1181062300739491274_a_nat] :
( ( member8404886500659538246list_a @ A @ ( domain4253084160996794191_a_nat @ R4 ) )
= ( ? [A2: product_prod_b_nat > set_list_a,B2: produc4672180596006801056_a_nat] :
( ( A = A2 )
& ( member2219381696197041043_a_nat @ ( produc6651248262528101210_a_nat @ A2 @ B2 ) @ R4 ) ) ) ) ).
% Domain.simps
thf(fact_1074_Domain_Osimps,axiom,
! [A: product_prod_b_nat > set_list_a,R4: set_Pr6389665502131816719_nat_a] :
( ( member8404886500659538246list_a @ A @ ( domain2872409888413931412_nat_a @ R4 ) )
= ( ? [A2: product_prod_b_nat > set_list_a,B2: nat > a] :
( ( A = A2 )
& ( member9198066416134578520_nat_a @ ( produc2895298938842563487_nat_a @ A2 @ B2 ) @ R4 ) ) ) ) ).
% Domain.simps
thf(fact_1075_Domain_Osimps,axiom,
! [A: nat > sum_sum_a_nat,R4: set_Pr7813613929123410816_a_nat] :
( ( member8690443509505302927_a_nat @ A @ ( domain3697415302523381637_a_nat @ R4 ) )
= ( ? [A2: nat > sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( A = A2 )
& ( member8034886515007075401_a_nat @ ( produc3720304352952013712_a_nat @ A2 @ B2 ) @ R4 ) ) ) ) ).
% Domain.simps
thf(fact_1076_Domain_Osimps,axiom,
! [A: b,R4: set_Pr1307281990691478580_b_nat] :
( ( member_b @ A @ ( domain_b_nat @ R4 ) )
= ( ? [A2: b,B2: nat] :
( ( A = A2 )
& ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A2 @ B2 ) @ R4 ) ) ) ) ).
% Domain.simps
thf(fact_1077_Domain_ODomainI,axiom,
! [A: nat,B: nat,R4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 )
=> ( member_nat @ A @ ( domain_nat_nat @ R4 ) ) ) ).
% Domain.DomainI
thf(fact_1078_Domain_ODomainI,axiom,
! [A: product_prod_b_nat > set_list_a,B: produc4672180596006801056_a_nat,R4: set_Pr1181062300739491274_a_nat] :
( ( member2219381696197041043_a_nat @ ( produc6651248262528101210_a_nat @ A @ B ) @ R4 )
=> ( member8404886500659538246list_a @ A @ ( domain4253084160996794191_a_nat @ R4 ) ) ) ).
% Domain.DomainI
thf(fact_1079_Domain_ODomainI,axiom,
! [A: product_prod_b_nat > set_list_a,B: nat > a,R4: set_Pr6389665502131816719_nat_a] :
( ( member9198066416134578520_nat_a @ ( produc2895298938842563487_nat_a @ A @ B ) @ R4 )
=> ( member8404886500659538246list_a @ A @ ( domain2872409888413931412_nat_a @ R4 ) ) ) ).
% Domain.DomainI
thf(fact_1080_Domain_ODomainI,axiom,
! [A: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat,R4: set_Pr7813613929123410816_a_nat] :
( ( member8034886515007075401_a_nat @ ( produc3720304352952013712_a_nat @ A @ B ) @ R4 )
=> ( member8690443509505302927_a_nat @ A @ ( domain3697415302523381637_a_nat @ R4 ) ) ) ).
% Domain.DomainI
thf(fact_1081_Domain_ODomainI,axiom,
! [A: b,B: nat,R4: set_Pr1307281990691478580_b_nat] :
( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A @ B ) @ R4 )
=> ( member_b @ A @ ( domain_b_nat @ R4 ) ) ) ).
% Domain.DomainI
thf(fact_1082_DomainE,axiom,
! [A: nat,R4: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A @ ( domain_nat_nat @ R4 ) )
=> ~ ! [B3: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B3 ) @ R4 ) ) ).
% DomainE
thf(fact_1083_DomainE,axiom,
! [A: product_prod_b_nat > set_list_a,R4: set_Pr1181062300739491274_a_nat] :
( ( member8404886500659538246list_a @ A @ ( domain4253084160996794191_a_nat @ R4 ) )
=> ~ ! [B3: produc4672180596006801056_a_nat] :
~ ( member2219381696197041043_a_nat @ ( produc6651248262528101210_a_nat @ A @ B3 ) @ R4 ) ) ).
% DomainE
thf(fact_1084_DomainE,axiom,
! [A: product_prod_b_nat > set_list_a,R4: set_Pr6389665502131816719_nat_a] :
( ( member8404886500659538246list_a @ A @ ( domain2872409888413931412_nat_a @ R4 ) )
=> ~ ! [B3: nat > a] :
~ ( member9198066416134578520_nat_a @ ( produc2895298938842563487_nat_a @ A @ B3 ) @ R4 ) ) ).
% DomainE
thf(fact_1085_DomainE,axiom,
! [A: nat > sum_sum_a_nat,R4: set_Pr7813613929123410816_a_nat] :
( ( member8690443509505302927_a_nat @ A @ ( domain3697415302523381637_a_nat @ R4 ) )
=> ~ ! [B3: set_Sum_sum_a_nat] :
~ ( member8034886515007075401_a_nat @ ( produc3720304352952013712_a_nat @ A @ B3 ) @ R4 ) ) ).
% DomainE
thf(fact_1086_DomainE,axiom,
! [A: b,R4: set_Pr1307281990691478580_b_nat] :
( ( member_b @ A @ ( domain_b_nat @ R4 ) )
=> ~ ! [B3: nat] :
~ ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A @ B3 ) @ R4 ) ) ).
% DomainE
thf(fact_1087_Domain__iff,axiom,
! [A: nat,R4: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A @ ( domain_nat_nat @ R4 ) )
= ( ? [Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ R4 ) ) ) ).
% Domain_iff
thf(fact_1088_Domain__iff,axiom,
! [A: product_prod_b_nat > set_list_a,R4: set_Pr1181062300739491274_a_nat] :
( ( member8404886500659538246list_a @ A @ ( domain4253084160996794191_a_nat @ R4 ) )
= ( ? [Y3: produc4672180596006801056_a_nat] : ( member2219381696197041043_a_nat @ ( produc6651248262528101210_a_nat @ A @ Y3 ) @ R4 ) ) ) ).
% Domain_iff
thf(fact_1089_Domain__iff,axiom,
! [A: product_prod_b_nat > set_list_a,R4: set_Pr6389665502131816719_nat_a] :
( ( member8404886500659538246list_a @ A @ ( domain2872409888413931412_nat_a @ R4 ) )
= ( ? [Y3: nat > a] : ( member9198066416134578520_nat_a @ ( produc2895298938842563487_nat_a @ A @ Y3 ) @ R4 ) ) ) ).
% Domain_iff
thf(fact_1090_Domain__iff,axiom,
! [A: nat > sum_sum_a_nat,R4: set_Pr7813613929123410816_a_nat] :
( ( member8690443509505302927_a_nat @ A @ ( domain3697415302523381637_a_nat @ R4 ) )
= ( ? [Y3: set_Sum_sum_a_nat] : ( member8034886515007075401_a_nat @ ( produc3720304352952013712_a_nat @ A @ Y3 ) @ R4 ) ) ) ).
% Domain_iff
thf(fact_1091_Domain__iff,axiom,
! [A: b,R4: set_Pr1307281990691478580_b_nat] :
( ( member_b @ A @ ( domain_b_nat @ R4 ) )
= ( ? [Y3: nat] : ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A @ Y3 ) @ R4 ) ) ) ).
% Domain_iff
thf(fact_1092_Range_Ocases,axiom,
! [A: nat,R4: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A @ ( range_nat_nat @ R4 ) )
=> ~ ! [A5: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ A ) @ R4 ) ) ).
% Range.cases
thf(fact_1093_Range_Ocases,axiom,
! [A: produc4672180596006801056_a_nat,R4: set_Pr1181062300739491274_a_nat] :
( ( member8034886515007075401_a_nat @ A @ ( range_4281096948368284472_a_nat @ R4 ) )
=> ~ ! [A5: product_prod_b_nat > set_list_a] :
~ ( member2219381696197041043_a_nat @ ( produc6651248262528101210_a_nat @ A5 @ A ) @ R4 ) ) ).
% Range.cases
thf(fact_1094_Range_Ocases,axiom,
! [A: nat > a,R4: set_Pr6389665502131816719_nat_a] :
( ( member_nat_a @ A @ ( range_2790413621204530045_nat_a @ R4 ) )
=> ~ ! [A5: product_prod_b_nat > set_list_a] :
~ ( member9198066416134578520_nat_a @ ( produc2895298938842563487_nat_a @ A5 @ A ) @ R4 ) ) ).
% Range.cases
thf(fact_1095_Range_Ocases,axiom,
! [A: set_Sum_sum_a_nat,R4: set_Pr7813613929123410816_a_nat] :
( ( member8098812455498974984_a_nat @ A @ ( range_3615419035313980270_a_nat @ R4 ) )
=> ~ ! [A5: nat > sum_sum_a_nat] :
~ ( member8034886515007075401_a_nat @ ( produc3720304352952013712_a_nat @ A5 @ A ) @ R4 ) ) ).
% Range.cases
thf(fact_1096_Range_Ocases,axiom,
! [A: nat,R4: set_Pr1307281990691478580_b_nat] :
( ( member_nat @ A @ ( range_b_nat @ R4 ) )
=> ~ ! [A5: b] :
~ ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A5 @ A ) @ R4 ) ) ).
% Range.cases
thf(fact_1097_Range_Osimps,axiom,
! [A: nat,R4: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A @ ( range_nat_nat @ R4 ) )
= ( ? [A2: nat,B2: nat] :
( ( A = B2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R4 ) ) ) ) ).
% Range.simps
thf(fact_1098_Range_Osimps,axiom,
! [A: produc4672180596006801056_a_nat,R4: set_Pr1181062300739491274_a_nat] :
( ( member8034886515007075401_a_nat @ A @ ( range_4281096948368284472_a_nat @ R4 ) )
= ( ? [A2: product_prod_b_nat > set_list_a,B2: produc4672180596006801056_a_nat] :
( ( A = B2 )
& ( member2219381696197041043_a_nat @ ( produc6651248262528101210_a_nat @ A2 @ B2 ) @ R4 ) ) ) ) ).
% Range.simps
thf(fact_1099_Range_Osimps,axiom,
! [A: nat > a,R4: set_Pr6389665502131816719_nat_a] :
( ( member_nat_a @ A @ ( range_2790413621204530045_nat_a @ R4 ) )
= ( ? [A2: product_prod_b_nat > set_list_a,B2: nat > a] :
( ( A = B2 )
& ( member9198066416134578520_nat_a @ ( produc2895298938842563487_nat_a @ A2 @ B2 ) @ R4 ) ) ) ) ).
% Range.simps
thf(fact_1100_Range_Osimps,axiom,
! [A: set_Sum_sum_a_nat,R4: set_Pr7813613929123410816_a_nat] :
( ( member8098812455498974984_a_nat @ A @ ( range_3615419035313980270_a_nat @ R4 ) )
= ( ? [A2: nat > sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( A = B2 )
& ( member8034886515007075401_a_nat @ ( produc3720304352952013712_a_nat @ A2 @ B2 ) @ R4 ) ) ) ) ).
% Range.simps
thf(fact_1101_Range_Osimps,axiom,
! [A: nat,R4: set_Pr1307281990691478580_b_nat] :
( ( member_nat @ A @ ( range_b_nat @ R4 ) )
= ( ? [A2: b,B2: nat] :
( ( A = B2 )
& ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A2 @ B2 ) @ R4 ) ) ) ) ).
% Range.simps
thf(fact_1102_Range_Ointros,axiom,
! [A: nat,B: nat,R4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 )
=> ( member_nat @ B @ ( range_nat_nat @ R4 ) ) ) ).
% Range.intros
thf(fact_1103_Range_Ointros,axiom,
! [A: product_prod_b_nat > set_list_a,B: produc4672180596006801056_a_nat,R4: set_Pr1181062300739491274_a_nat] :
( ( member2219381696197041043_a_nat @ ( produc6651248262528101210_a_nat @ A @ B ) @ R4 )
=> ( member8034886515007075401_a_nat @ B @ ( range_4281096948368284472_a_nat @ R4 ) ) ) ).
% Range.intros
thf(fact_1104_Range_Ointros,axiom,
! [A: product_prod_b_nat > set_list_a,B: nat > a,R4: set_Pr6389665502131816719_nat_a] :
( ( member9198066416134578520_nat_a @ ( produc2895298938842563487_nat_a @ A @ B ) @ R4 )
=> ( member_nat_a @ B @ ( range_2790413621204530045_nat_a @ R4 ) ) ) ).
% Range.intros
thf(fact_1105_Range_Ointros,axiom,
! [A: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat,R4: set_Pr7813613929123410816_a_nat] :
( ( member8034886515007075401_a_nat @ ( produc3720304352952013712_a_nat @ A @ B ) @ R4 )
=> ( member8098812455498974984_a_nat @ B @ ( range_3615419035313980270_a_nat @ R4 ) ) ) ).
% Range.intros
thf(fact_1106_Range_Ointros,axiom,
! [A: b,B: nat,R4: set_Pr1307281990691478580_b_nat] :
( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A @ B ) @ R4 )
=> ( member_nat @ B @ ( range_b_nat @ R4 ) ) ) ).
% Range.intros
thf(fact_1107_RangeE,axiom,
! [B: nat,R4: set_Pr1261947904930325089at_nat] :
( ( member_nat @ B @ ( range_nat_nat @ R4 ) )
=> ~ ! [A5: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B ) @ R4 ) ) ).
% RangeE
thf(fact_1108_RangeE,axiom,
! [B: produc4672180596006801056_a_nat,R4: set_Pr1181062300739491274_a_nat] :
( ( member8034886515007075401_a_nat @ B @ ( range_4281096948368284472_a_nat @ R4 ) )
=> ~ ! [A5: product_prod_b_nat > set_list_a] :
~ ( member2219381696197041043_a_nat @ ( produc6651248262528101210_a_nat @ A5 @ B ) @ R4 ) ) ).
% RangeE
thf(fact_1109_RangeE,axiom,
! [B: nat > a,R4: set_Pr6389665502131816719_nat_a] :
( ( member_nat_a @ B @ ( range_2790413621204530045_nat_a @ R4 ) )
=> ~ ! [A5: product_prod_b_nat > set_list_a] :
~ ( member9198066416134578520_nat_a @ ( produc2895298938842563487_nat_a @ A5 @ B ) @ R4 ) ) ).
% RangeE
thf(fact_1110_RangeE,axiom,
! [B: set_Sum_sum_a_nat,R4: set_Pr7813613929123410816_a_nat] :
( ( member8098812455498974984_a_nat @ B @ ( range_3615419035313980270_a_nat @ R4 ) )
=> ~ ! [A5: nat > sum_sum_a_nat] :
~ ( member8034886515007075401_a_nat @ ( produc3720304352952013712_a_nat @ A5 @ B ) @ R4 ) ) ).
% RangeE
thf(fact_1111_RangeE,axiom,
! [B: nat,R4: set_Pr1307281990691478580_b_nat] :
( ( member_nat @ B @ ( range_b_nat @ R4 ) )
=> ~ ! [A5: b] :
~ ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ A5 @ B ) @ R4 ) ) ).
% RangeE
thf(fact_1112_Range__iff,axiom,
! [A: nat,R4: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A @ ( range_nat_nat @ R4 ) )
= ( ? [Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ A ) @ R4 ) ) ) ).
% Range_iff
thf(fact_1113_Range__iff,axiom,
! [A: produc4672180596006801056_a_nat,R4: set_Pr1181062300739491274_a_nat] :
( ( member8034886515007075401_a_nat @ A @ ( range_4281096948368284472_a_nat @ R4 ) )
= ( ? [Y3: product_prod_b_nat > set_list_a] : ( member2219381696197041043_a_nat @ ( produc6651248262528101210_a_nat @ Y3 @ A ) @ R4 ) ) ) ).
% Range_iff
thf(fact_1114_Range__iff,axiom,
! [A: nat > a,R4: set_Pr6389665502131816719_nat_a] :
( ( member_nat_a @ A @ ( range_2790413621204530045_nat_a @ R4 ) )
= ( ? [Y3: product_prod_b_nat > set_list_a] : ( member9198066416134578520_nat_a @ ( produc2895298938842563487_nat_a @ Y3 @ A ) @ R4 ) ) ) ).
% Range_iff
thf(fact_1115_Range__iff,axiom,
! [A: set_Sum_sum_a_nat,R4: set_Pr7813613929123410816_a_nat] :
( ( member8098812455498974984_a_nat @ A @ ( range_3615419035313980270_a_nat @ R4 ) )
= ( ? [Y3: nat > sum_sum_a_nat] : ( member8034886515007075401_a_nat @ ( produc3720304352952013712_a_nat @ Y3 @ A ) @ R4 ) ) ) ).
% Range_iff
thf(fact_1116_Range__iff,axiom,
! [A: nat,R4: set_Pr1307281990691478580_b_nat] :
( ( member_nat @ A @ ( range_b_nat @ R4 ) )
= ( ? [Y3: b] : ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ Y3 @ A ) @ R4 ) ) ) ).
% Range_iff
thf(fact_1117_lnear__order__on__empty,axiom,
order_4473980167227706203on_nat @ bot_bot_set_nat @ bot_bo2099793752762293965at_nat ).
% lnear_order_on_empty
thf(fact_1118_lnear__order__on__empty,axiom,
order_8768733634509060147r_on_a @ bot_bot_set_a @ bot_bo3357376287454694259od_a_a ).
% lnear_order_on_empty
thf(fact_1119_lnear__order__on__empty,axiom,
order_8768733634509060148r_on_b @ bot_bot_set_b @ bot_bo2792761326896053555od_b_b ).
% lnear_order_on_empty
thf(fact_1120_refl__on__empty,axiom,
refl_on_nat @ bot_bot_set_nat @ bot_bo2099793752762293965at_nat ).
% refl_on_empty
thf(fact_1121_refl__on__empty,axiom,
refl_on_a @ bot_bot_set_a @ bot_bo3357376287454694259od_a_a ).
% refl_on_empty
thf(fact_1122_refl__on__empty,axiom,
refl_on_b @ bot_bot_set_b @ bot_bo2792761326896053555od_b_b ).
% refl_on_empty
thf(fact_1123_insert__code_I2_J,axiom,
! [X5: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( insert_Sum_sum_a_nat2 @ X5 @ ( coset_Sum_sum_a_nat @ Xs ) )
= ( coset_Sum_sum_a_nat @ ( remove3909449470355376139_a_nat @ X5 @ Xs ) ) ) ).
% insert_code(2)
thf(fact_1124_insert__code_I2_J,axiom,
! [X5: nat,Xs: list_nat] :
( ( insert_nat2 @ X5 @ ( coset_nat @ Xs ) )
= ( coset_nat @ ( removeAll_nat @ X5 @ Xs ) ) ) ).
% insert_code(2)
thf(fact_1125_pairwise__alt,axiom,
( pairwi7370142813935713258_a_nat
= ( ^ [R5: sum_sum_a_nat > sum_sum_a_nat > $o,S: set_Sum_sum_a_nat] :
! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ S )
=> ! [Y3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Y3 @ ( minus_1134630996077396038_a_nat @ S @ ( insert_Sum_sum_a_nat2 @ X3 @ bot_bo3438331934148233675_a_nat ) ) )
=> ( R5 @ X3 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_1126_pairwise__alt,axiom,
( pairwise_a
= ( ^ [R5: a > a > $o,S: set_a] :
! [X3: a] :
( ( member_a @ X3 @ S )
=> ! [Y3: a] :
( ( member_a @ Y3 @ ( minus_minus_set_a @ S @ ( insert_a2 @ X3 @ bot_bot_set_a ) ) )
=> ( R5 @ X3 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_1127_pairwise__alt,axiom,
( pairwise_b
= ( ^ [R5: b > b > $o,S: set_b] :
! [X3: b] :
( ( member_b @ X3 @ S )
=> ! [Y3: b] :
( ( member_b @ Y3 @ ( minus_minus_set_b @ S @ ( insert_b2 @ X3 @ bot_bot_set_b ) ) )
=> ( R5 @ X3 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_1128_pairwise__alt,axiom,
( pairwise_nat
= ( ^ [R5: nat > nat > $o,S: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( minus_minus_set_nat @ S @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) )
=> ( R5 @ X3 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_1129_override__on__insert,axiom,
! [F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,X5: nat,X2: set_nat] :
( ( overri2434899023996108783_a_nat @ F @ G @ ( insert_nat2 @ X5 @ X2 ) )
= ( fun_up180537416982607344_a_nat @ ( overri2434899023996108783_a_nat @ F @ G @ X2 ) @ X5 @ ( G @ X5 ) ) ) ).
% override_on_insert
thf(fact_1130_override__on__insert_H,axiom,
! [F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,X5: nat,X2: set_nat] :
( ( overri2434899023996108783_a_nat @ F @ G @ ( insert_nat2 @ X5 @ X2 ) )
= ( overri2434899023996108783_a_nat @ ( fun_up180537416982607344_a_nat @ F @ X5 @ ( G @ X5 ) ) @ G @ X2 ) ) ).
% override_on_insert'
thf(fact_1131_set__quicksort__part,axiom,
! [Ac: list_nat,X5: nat,Lts: list_nat,Eqs: list_nat,Gts: list_nat,Zs: list_nat] :
( ( set_nat2 @ ( set_or1804217446461887602rt_nat @ Ac @ X5 @ Lts @ Eqs @ Gts @ Zs ) )
= ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( set_nat2 @ Ac ) @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) ) @ ( set_nat2 @ Lts ) ) @ ( set_nat2 @ Eqs ) ) @ ( set_nat2 @ Gts ) ) @ ( set_nat2 @ Zs ) ) ) ).
% set_quicksort_part
thf(fact_1132_fun__upd__image,axiom,
! [X5: nat,A3: set_nat,F: nat > sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( ( member_nat @ X5 @ A3 )
=> ( ( image_7293268710728258664_a_nat @ ( fun_up180537416982607344_a_nat @ F @ X5 @ Y ) @ A3 )
= ( insert_Sum_sum_a_nat2 @ Y @ ( image_7293268710728258664_a_nat @ F @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat @ X5 @ A3 )
=> ( ( image_7293268710728258664_a_nat @ ( fun_up180537416982607344_a_nat @ F @ X5 @ Y ) @ A3 )
= ( image_7293268710728258664_a_nat @ F @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_1133_fun__upd__image,axiom,
! [X5: a,A3: set_a,F: a > nat,Y: nat] :
( ( ( member_a @ X5 @ A3 )
=> ( ( image_a_nat @ ( fun_upd_a_nat @ F @ X5 @ Y ) @ A3 )
= ( insert_nat2 @ Y @ ( image_a_nat @ F @ ( minus_minus_set_a @ A3 @ ( insert_a2 @ X5 @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ X5 @ A3 )
=> ( ( image_a_nat @ ( fun_upd_a_nat @ F @ X5 @ Y ) @ A3 )
= ( image_a_nat @ F @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_1134_fun__upd__image,axiom,
! [X5: b,A3: set_b,F: b > nat,Y: nat] :
( ( ( member_b @ X5 @ A3 )
=> ( ( image_b_nat @ ( fun_upd_b_nat @ F @ X5 @ Y ) @ A3 )
= ( insert_nat2 @ Y @ ( image_b_nat @ F @ ( minus_minus_set_b @ A3 @ ( insert_b2 @ X5 @ bot_bot_set_b ) ) ) ) ) )
& ( ~ ( member_b @ X5 @ A3 )
=> ( ( image_b_nat @ ( fun_upd_b_nat @ F @ X5 @ Y ) @ A3 )
= ( image_b_nat @ F @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_1135_fun__upd__image,axiom,
! [X5: nat,A3: set_nat,F: nat > nat,Y: nat] :
( ( ( member_nat @ X5 @ A3 )
=> ( ( image_nat_nat2 @ ( fun_upd_nat_nat @ F @ X5 @ Y ) @ A3 )
= ( insert_nat2 @ Y @ ( image_nat_nat2 @ F @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat @ X5 @ A3 )
=> ( ( image_nat_nat2 @ ( fun_upd_nat_nat @ F @ X5 @ Y ) @ A3 )
= ( image_nat_nat2 @ F @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_1136_fun__upd__image,axiom,
! [X5: fo_term_a,A3: set_fo_term_a,F: fo_term_a > nat,Y: nat] :
( ( ( member_fo_term_a @ X5 @ A3 )
=> ( ( image_fo_term_a_nat @ ( fun_up772808097823097291_a_nat @ F @ X5 @ Y ) @ A3 )
= ( insert_nat2 @ Y @ ( image_fo_term_a_nat @ F @ ( minus_6854963972745519743term_a @ A3 @ ( insert_fo_term_a2 @ X5 @ bot_bo4735268219511357444term_a ) ) ) ) ) )
& ( ~ ( member_fo_term_a @ X5 @ A3 )
=> ( ( image_fo_term_a_nat @ ( fun_up772808097823097291_a_nat @ F @ X5 @ Y ) @ A3 )
= ( image_fo_term_a_nat @ F @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_1137_fun__upd__image,axiom,
! [X5: set_nat,A3: set_set_nat,F: set_nat > nat,Y: nat] :
( ( ( member_set_nat @ X5 @ A3 )
=> ( ( image_set_nat_nat @ ( fun_upd_set_nat_nat @ F @ X5 @ Y ) @ A3 )
= ( insert_nat2 @ Y @ ( image_set_nat_nat @ F @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat2 @ X5 @ bot_bot_set_set_nat ) ) ) ) ) )
& ( ~ ( member_set_nat @ X5 @ A3 )
=> ( ( image_set_nat_nat @ ( fun_upd_set_nat_nat @ F @ X5 @ Y ) @ A3 )
= ( image_set_nat_nat @ F @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_1138_fun__upd__image,axiom,
! [X5: list_a,A3: set_list_a,F: list_a > set_a,Y: set_a] :
( ( ( member_list_a @ X5 @ A3 )
=> ( ( image_list_a_set_a @ ( fun_upd_list_a_set_a @ F @ X5 @ Y ) @ A3 )
= ( insert_set_a @ Y @ ( image_list_a_set_a @ F @ ( minus_646659088055828811list_a @ A3 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) ) ) ) )
& ( ~ ( member_list_a @ X5 @ A3 )
=> ( ( image_list_a_set_a @ ( fun_upd_list_a_set_a @ F @ X5 @ Y ) @ A3 )
= ( image_list_a_set_a @ F @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_1139_fun__upd__image,axiom,
! [X5: sum_sum_a_nat,A3: set_Sum_sum_a_nat,F: sum_sum_a_nat > nat,Y: nat] :
( ( ( member_Sum_sum_a_nat @ X5 @ A3 )
=> ( ( image_2473878607534554506at_nat @ ( fun_up4584519350643678994at_nat @ F @ X5 @ Y ) @ A3 )
= ( insert_nat2 @ Y @ ( image_2473878607534554506at_nat @ F @ ( minus_1134630996077396038_a_nat @ A3 @ ( insert_Sum_sum_a_nat2 @ X5 @ bot_bo3438331934148233675_a_nat ) ) ) ) ) )
& ( ~ ( member_Sum_sum_a_nat @ X5 @ A3 )
=> ( ( image_2473878607534554506at_nat @ ( fun_up4584519350643678994at_nat @ F @ X5 @ Y ) @ A3 )
= ( image_2473878607534554506at_nat @ F @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_1140_fun__upd__image,axiom,
! [X5: set_nat,A3: set_set_nat,F: set_nat > set_nat,Y: set_nat] :
( ( ( member_set_nat @ X5 @ A3 )
=> ( ( image_7916887816326733075et_nat @ ( fun_up2577977767889591691et_nat @ F @ X5 @ Y ) @ A3 )
= ( insert_set_nat2 @ Y @ ( image_7916887816326733075et_nat @ F @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat2 @ X5 @ bot_bot_set_set_nat ) ) ) ) ) )
& ( ~ ( member_set_nat @ X5 @ A3 )
=> ( ( image_7916887816326733075et_nat @ ( fun_up2577977767889591691et_nat @ F @ X5 @ Y ) @ A3 )
= ( image_7916887816326733075et_nat @ F @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_1141_fun__upd__image,axiom,
! [X5: a,A3: set_a,F: a > sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( ( member_a @ X5 @ A3 )
=> ( ( image_7873763678140191238_a_nat @ ( fun_up8976196915827720830_a_nat @ F @ X5 @ Y ) @ A3 )
= ( insert_Sum_sum_a_nat2 @ Y @ ( image_7873763678140191238_a_nat @ F @ ( minus_minus_set_a @ A3 @ ( insert_a2 @ X5 @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ X5 @ A3 )
=> ( ( image_7873763678140191238_a_nat @ ( fun_up8976196915827720830_a_nat @ F @ X5 @ Y ) @ A3 )
= ( image_7873763678140191238_a_nat @ F @ A3 ) ) ) ) ).
% fun_upd_image
thf(fact_1142_image__eqI,axiom,
! [B: nat,F: nat > nat,X5: nat,A3: set_nat] :
( ( B
= ( F @ X5 ) )
=> ( ( member_nat @ X5 @ A3 )
=> ( member_nat @ B @ ( image_nat_nat2 @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1143_image__eqI,axiom,
! [B: b,F: nat > b,X5: nat,A3: set_nat] :
( ( B
= ( F @ X5 ) )
=> ( ( member_nat @ X5 @ A3 )
=> ( member_b @ B @ ( image_nat_b @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1144_image__eqI,axiom,
! [B: a,F: nat > a,X5: nat,A3: set_nat] :
( ( B
= ( F @ X5 ) )
=> ( ( member_nat @ X5 @ A3 )
=> ( member_a @ B @ ( image_nat_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1145_image__eqI,axiom,
! [B: nat,F: b > nat,X5: b,A3: set_b] :
( ( B
= ( F @ X5 ) )
=> ( ( member_b @ X5 @ A3 )
=> ( member_nat @ B @ ( image_b_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1146_image__eqI,axiom,
! [B: b,F: b > b,X5: b,A3: set_b] :
( ( B
= ( F @ X5 ) )
=> ( ( member_b @ X5 @ A3 )
=> ( member_b @ B @ ( image_b_b @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1147_image__eqI,axiom,
! [B: a,F: b > a,X5: b,A3: set_b] :
( ( B
= ( F @ X5 ) )
=> ( ( member_b @ X5 @ A3 )
=> ( member_a @ B @ ( image_b_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1148_image__eqI,axiom,
! [B: nat,F: a > nat,X5: a,A3: set_a] :
( ( B
= ( F @ X5 ) )
=> ( ( member_a @ X5 @ A3 )
=> ( member_nat @ B @ ( image_a_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1149_image__eqI,axiom,
! [B: b,F: a > b,X5: a,A3: set_a] :
( ( B
= ( F @ X5 ) )
=> ( ( member_a @ X5 @ A3 )
=> ( member_b @ B @ ( image_a_b @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1150_image__eqI,axiom,
! [B: a,F: a > a,X5: a,A3: set_a] :
( ( B
= ( F @ X5 ) )
=> ( ( member_a @ X5 @ A3 )
=> ( member_a @ B @ ( image_a_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1151_image__eqI,axiom,
! [B: fo_term_a,F: nat > fo_term_a,X5: nat,A3: set_nat] :
( ( B
= ( F @ X5 ) )
=> ( ( member_nat @ X5 @ A3 )
=> ( member_fo_term_a @ B @ ( image_nat_fo_term_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1152_override__on__apply__in,axiom,
! [A: nat,A3: set_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( member_nat @ A @ A3 )
=> ( ( overri2434899023996108783_a_nat @ F @ G @ A3 @ A )
= ( G @ A ) ) ) ).
% override_on_apply_in
thf(fact_1153_override__on__apply__notin,axiom,
! [A: nat,A3: set_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ~ ( member_nat @ A @ A3 )
=> ( ( overri2434899023996108783_a_nat @ F @ G @ A3 @ A )
= ( F @ A ) ) ) ).
% override_on_apply_notin
thf(fact_1154_image__is__empty,axiom,
! [F: nat > nat,A3: set_nat] :
( ( ( image_nat_nat2 @ F @ A3 )
= bot_bot_set_nat )
= ( A3 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1155_image__is__empty,axiom,
! [F: a > nat,A3: set_a] :
( ( ( image_a_nat @ F @ A3 )
= bot_bot_set_nat )
= ( A3 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_1156_image__is__empty,axiom,
! [F: b > nat,A3: set_b] :
( ( ( image_b_nat @ F @ A3 )
= bot_bot_set_nat )
= ( A3 = bot_bot_set_b ) ) ).
% image_is_empty
thf(fact_1157_image__is__empty,axiom,
! [F: nat > a,A3: set_nat] :
( ( ( image_nat_a @ F @ A3 )
= bot_bot_set_a )
= ( A3 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1158_image__is__empty,axiom,
! [F: a > a,A3: set_a] :
( ( ( image_a_a @ F @ A3 )
= bot_bot_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_1159_image__is__empty,axiom,
! [F: b > a,A3: set_b] :
( ( ( image_b_a @ F @ A3 )
= bot_bot_set_a )
= ( A3 = bot_bot_set_b ) ) ).
% image_is_empty
thf(fact_1160_image__is__empty,axiom,
! [F: nat > b,A3: set_nat] :
( ( ( image_nat_b @ F @ A3 )
= bot_bot_set_b )
= ( A3 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1161_image__is__empty,axiom,
! [F: a > b,A3: set_a] :
( ( ( image_a_b @ F @ A3 )
= bot_bot_set_b )
= ( A3 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_1162_image__is__empty,axiom,
! [F: b > b,A3: set_b] :
( ( ( image_b_b @ F @ A3 )
= bot_bot_set_b )
= ( A3 = bot_bot_set_b ) ) ).
% image_is_empty
thf(fact_1163_image__is__empty,axiom,
! [F: set_nat > set_nat,A3: set_set_nat] :
( ( ( image_7916887816326733075et_nat @ F @ A3 )
= bot_bot_set_set_nat )
= ( A3 = bot_bot_set_set_nat ) ) ).
% image_is_empty
thf(fact_1164_empty__is__image,axiom,
! [F: b > b,A3: set_b] :
( ( bot_bot_set_b
= ( image_b_b @ F @ A3 ) )
= ( A3 = bot_bot_set_b ) ) ).
% empty_is_image
thf(fact_1165_imageI,axiom,
! [X5: nat,A3: set_nat,F: nat > nat] :
( ( member_nat @ X5 @ A3 )
=> ( member_nat @ ( F @ X5 ) @ ( image_nat_nat2 @ F @ A3 ) ) ) ).
% imageI
thf(fact_1166_pairwiseD,axiom,
! [R3: nat > nat > $o,S2: set_nat,X5: nat,Y: nat] :
( ( pairwise_nat @ R3 @ S2 )
=> ( ( member_nat @ X5 @ S2 )
=> ( ( member_nat @ Y @ S2 )
=> ( ( X5 != Y )
=> ( R3 @ X5 @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_1167_pairwiseI,axiom,
! [S2: set_nat,R3: nat > nat > $o] :
( ! [X: nat,Y5: nat] :
( ( member_nat @ X @ S2 )
=> ( ( member_nat @ Y5 @ S2 )
=> ( ( X != Y5 )
=> ( R3 @ X @ Y5 ) ) ) )
=> ( pairwise_nat @ R3 @ S2 ) ) ).
% pairwiseI
thf(fact_1168_rev__image__eqI,axiom,
! [X5: nat,A3: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X5 @ A3 )
=> ( ( B
= ( F @ X5 ) )
=> ( member_nat @ B @ ( image_nat_nat2 @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_1169_pairwise__insert,axiom,
! [R4: nat > nat > $o,X5: nat,S3: set_nat] :
( ( pairwise_nat @ R4 @ ( insert_nat2 @ X5 @ S3 ) )
= ( ! [Y3: nat] :
( ( ( member_nat @ Y3 @ S3 )
& ( Y3 != X5 ) )
=> ( ( R4 @ X5 @ Y3 )
& ( R4 @ Y3 @ X5 ) ) )
& ( pairwise_nat @ R4 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_1170_in__image__insert__iff,axiom,
! [B7: set_set_nat,X5: nat,A3: set_nat] :
( ! [C5: set_nat] :
( ( member_set_nat @ C5 @ B7 )
=> ~ ( member_nat @ X5 @ C5 ) )
=> ( ( member_set_nat @ A3 @ ( image_7916887816326733075et_nat @ ( insert_nat2 @ X5 ) @ B7 ) )
= ( ( member_nat @ X5 @ A3 )
& ( member_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) ) @ B7 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1171_inj__img__insertE,axiom,
! [F: nat > nat,A3: set_nat,X5: nat,B7: set_nat] :
( ( inj_on_nat_nat @ F @ A3 )
=> ( ~ ( member_nat @ X5 @ B7 )
=> ( ( ( insert_nat2 @ X5 @ B7 )
= ( image_nat_nat2 @ F @ A3 ) )
=> ~ ! [X7: nat,A9: set_nat] :
( ~ ( member_nat @ X7 @ A9 )
=> ( ( A3
= ( insert_nat2 @ X7 @ A9 ) )
=> ( ( X5
= ( F @ X7 ) )
=> ( B7
!= ( image_nat_nat2 @ F @ A9 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1172_inj__on__fun__updI,axiom,
! [F: nat > sum_sum_a_nat,A3: set_nat,Y: sum_sum_a_nat,X5: nat] :
( ( inj_on4348161877322679292_a_nat @ F @ A3 )
=> ( ~ ( member_Sum_sum_a_nat @ Y @ ( image_7293268710728258664_a_nat @ F @ A3 ) )
=> ( inj_on4348161877322679292_a_nat @ ( fun_up180537416982607344_a_nat @ F @ X5 @ Y ) @ A3 ) ) ) ).
% inj_on_fun_updI
thf(fact_1173_IntI,axiom,
! [C: nat,A3: set_nat,B7: set_nat] :
( ( member_nat @ C @ A3 )
=> ( ( member_nat @ C @ B7 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B7 ) ) ) ) ).
% IntI
thf(fact_1174_Int__iff,axiom,
! [C: nat,A3: set_nat,B7: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B7 ) )
= ( ( member_nat @ C @ A3 )
& ( member_nat @ C @ B7 ) ) ) ).
% Int_iff
thf(fact_1175_Int__insert__left__if0,axiom,
! [A: nat,C3: set_nat,B7: set_nat] :
( ~ ( member_nat @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B7 ) @ C3 )
= ( inf_inf_set_nat @ B7 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_1176_Int__insert__left__if1,axiom,
! [A: nat,C3: set_nat,B7: set_nat] :
( ( member_nat @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B7 ) @ C3 )
= ( insert_nat2 @ A @ ( inf_inf_set_nat @ B7 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1177_Int__insert__right__if0,axiom,
! [A: nat,A3: set_nat,B7: set_nat] :
( ~ ( member_nat @ A @ A3 )
=> ( ( inf_inf_set_nat @ A3 @ ( insert_nat2 @ A @ B7 ) )
= ( inf_inf_set_nat @ A3 @ B7 ) ) ) ).
% Int_insert_right_if0
thf(fact_1178_Int__insert__right__if1,axiom,
! [A: nat,A3: set_nat,B7: set_nat] :
( ( member_nat @ A @ A3 )
=> ( ( inf_inf_set_nat @ A3 @ ( insert_nat2 @ A @ B7 ) )
= ( insert_nat2 @ A @ ( inf_inf_set_nat @ A3 @ B7 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1179_insert__disjoint_I1_J,axiom,
! [A: nat,A3: set_nat,B7: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ A3 ) @ B7 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B7 )
& ( ( inf_inf_set_nat @ A3 @ B7 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_1180_insert__disjoint_I2_J,axiom,
! [A: nat,A3: set_nat,B7: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat2 @ A @ A3 ) @ B7 ) )
= ( ~ ( member_nat @ A @ B7 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A3 @ B7 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1181_disjoint__insert_I1_J,axiom,
! [B7: set_nat,A: nat,A3: set_nat] :
( ( ( inf_inf_set_nat @ B7 @ ( insert_nat2 @ A @ A3 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B7 )
& ( ( inf_inf_set_nat @ B7 @ A3 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_1182_disjoint__insert_I2_J,axiom,
! [A3: set_nat,B: nat,B7: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A3 @ ( insert_nat2 @ B @ B7 ) ) )
= ( ~ ( member_nat @ B @ A3 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A3 @ B7 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1183_map__fun__upd,axiom,
! [Y: nat,Xs: list_nat,F: nat > sum_sum_a_nat,V2: sum_sum_a_nat] :
( ~ ( member_nat @ Y @ ( set_nat2 @ Xs ) )
=> ( ( map_na823391071729141993_a_nat @ ( fun_up180537416982607344_a_nat @ F @ Y @ V2 ) @ Xs )
= ( map_na823391071729141993_a_nat @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_1184_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( F @ X )
= X ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_1185_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_1186_IntE,axiom,
! [C: nat,A3: set_nat,B7: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B7 ) )
=> ~ ( ( member_nat @ C @ A3 )
=> ~ ( member_nat @ C @ B7 ) ) ) ).
% IntE
thf(fact_1187_IntD1,axiom,
! [C: nat,A3: set_nat,B7: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B7 ) )
=> ( member_nat @ C @ A3 ) ) ).
% IntD1
thf(fact_1188_IntD2,axiom,
! [C: nat,A3: set_nat,B7: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B7 ) )
=> ( member_nat @ C @ B7 ) ) ).
% IntD2
thf(fact_1189_Int__insert__right,axiom,
! [A: nat,A3: set_nat,B7: set_nat] :
( ( ( member_nat @ A @ A3 )
=> ( ( inf_inf_set_nat @ A3 @ ( insert_nat2 @ A @ B7 ) )
= ( insert_nat2 @ A @ ( inf_inf_set_nat @ A3 @ B7 ) ) ) )
& ( ~ ( member_nat @ A @ A3 )
=> ( ( inf_inf_set_nat @ A3 @ ( insert_nat2 @ A @ B7 ) )
= ( inf_inf_set_nat @ A3 @ B7 ) ) ) ) ).
% Int_insert_right
thf(fact_1190_Int__insert__left,axiom,
! [A: nat,C3: set_nat,B7: set_nat] :
( ( ( member_nat @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B7 ) @ C3 )
= ( insert_nat2 @ A @ ( inf_inf_set_nat @ B7 @ C3 ) ) ) )
& ( ~ ( member_nat @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B7 ) @ C3 )
= ( inf_inf_set_nat @ B7 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_1191_disjoint__iff,axiom,
! [A3: set_nat,B7: set_nat] :
( ( ( inf_inf_set_nat @ A3 @ B7 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ~ ( member_nat @ X3 @ B7 ) ) ) ) ).
% disjoint_iff
thf(fact_1192_Int__emptyI,axiom,
! [A3: set_nat,B7: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ~ ( member_nat @ X @ B7 ) )
=> ( ( inf_inf_set_nat @ A3 @ B7 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_1193_esat_Oelims_I1_J,axiom,
! [X5: fo_fmla_a_b,Xa: product_prod_b_nat > set_list_a,Xb: nat > sum_sum_a_nat,Xc: set_Sum_sum_a_nat,Y: $o] :
( ( ( esat_a_b @ X5 @ Xa @ Xb @ Xc )
= Y )
=> ( ! [R: b,Ts: list_fo_term_a] :
( ( X5
= ( fo_Pred_b_a @ R @ Ts ) )
=> ( Y
= ( ~ ( member408289922725080238_a_nat @ ( eval_eterms_a_nat @ Xb @ Ts ) @ ( image_674313660629153798_a_nat @ ( map_a_Sum_sum_a_nat @ sum_Inl_a_nat ) @ ( Xa @ ( product_Pair_b_nat @ R @ ( size_s8359284885213242922term_a @ Ts ) ) ) ) ) ) ) )
=> ( ! [B3: $o] :
( ( X5
= ( fo_Bool_a_b @ B3 ) )
=> ( Y = (~ B3) ) )
=> ( ! [T3: fo_term_a,T4: fo_term_a] :
( ( X5
= ( fo_Eqa_a_b @ T3 @ T4 ) )
=> ( Y
= ( ( eval_eterm_a_nat @ Xb @ T3 )
!= ( eval_eterm_a_nat @ Xb @ T4 ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Neg_a_b @ Phi2 ) )
=> ( Y
= ( esat_a_b @ Phi2 @ Xa @ Xb @ Xc ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Conj_a_b @ Phi2 @ Psi2 ) )
=> ( Y
= ( ~ ( ( esat_a_b @ Phi2 @ Xa @ Xb @ Xc )
& ( esat_a_b @ Psi2 @ Xa @ Xb @ Xc ) ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Disj_a_b @ Phi2 @ Psi2 ) )
=> ( Y
= ( ~ ( ( esat_a_b @ Phi2 @ Xa @ Xb @ Xc )
| ( esat_a_b @ Psi2 @ Xa @ Xb @ Xc ) ) ) ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Exists_a_b @ N @ Phi2 ) )
=> ( Y
= ( ~ ? [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ Xc )
& ( esat_a_b @ Phi2 @ Xa @ ( fun_up180537416982607344_a_nat @ Xb @ N @ X3 ) @ Xc ) ) ) ) )
=> ~ ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Forall_a_b @ N @ Phi2 ) )
=> ( Y
= ( ~ ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ Xc )
=> ( esat_a_b @ Phi2 @ Xa @ ( fun_up180537416982607344_a_nat @ Xb @ N @ X3 ) @ Xc ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% esat.elims(1)
thf(fact_1194_esat_Oelims_I2_J,axiom,
! [X5: fo_fmla_a_b,Xa: product_prod_b_nat > set_list_a,Xb: nat > sum_sum_a_nat,Xc: set_Sum_sum_a_nat] :
( ( esat_a_b @ X5 @ Xa @ Xb @ Xc )
=> ( ! [R: b,Ts: list_fo_term_a] :
( ( X5
= ( fo_Pred_b_a @ R @ Ts ) )
=> ~ ( member408289922725080238_a_nat @ ( eval_eterms_a_nat @ Xb @ Ts ) @ ( image_674313660629153798_a_nat @ ( map_a_Sum_sum_a_nat @ sum_Inl_a_nat ) @ ( Xa @ ( product_Pair_b_nat @ R @ ( size_s8359284885213242922term_a @ Ts ) ) ) ) ) )
=> ( ! [B3: $o] :
( ( X5
= ( fo_Bool_a_b @ B3 ) )
=> ~ B3 )
=> ( ! [T3: fo_term_a,T4: fo_term_a] :
( ( X5
= ( fo_Eqa_a_b @ T3 @ T4 ) )
=> ( ( eval_eterm_a_nat @ Xb @ T3 )
!= ( eval_eterm_a_nat @ Xb @ T4 ) ) )
=> ( ! [Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Neg_a_b @ Phi2 ) )
=> ( esat_a_b @ Phi2 @ Xa @ Xb @ Xc ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Conj_a_b @ Phi2 @ Psi2 ) )
=> ~ ( ( esat_a_b @ Phi2 @ Xa @ Xb @ Xc )
& ( esat_a_b @ Psi2 @ Xa @ Xb @ Xc ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Disj_a_b @ Phi2 @ Psi2 ) )
=> ~ ( ( esat_a_b @ Phi2 @ Xa @ Xb @ Xc )
| ( esat_a_b @ Psi2 @ Xa @ Xb @ Xc ) ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Exists_a_b @ N @ Phi2 ) )
=> ~ ? [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ Xc )
& ( esat_a_b @ Phi2 @ Xa @ ( fun_up180537416982607344_a_nat @ Xb @ N @ X ) @ Xc ) ) )
=> ~ ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Forall_a_b @ N @ Phi2 ) )
=> ~ ! [X8: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X8 @ Xc )
=> ( esat_a_b @ Phi2 @ Xa @ ( fun_up180537416982607344_a_nat @ Xb @ N @ X8 ) @ Xc ) ) ) ) ) ) ) ) ) ) ) ).
% esat.elims(2)
thf(fact_1195_esat_Osimps_I1_J,axiom,
! [R4: b,Ts2: list_fo_term_a,I: product_prod_b_nat > set_list_a,Sigma: nat > sum_sum_a_nat,X2: set_Sum_sum_a_nat] :
( ( esat_a_b @ ( fo_Pred_b_a @ R4 @ Ts2 ) @ I @ Sigma @ X2 )
= ( member408289922725080238_a_nat @ ( eval_eterms_a_nat @ Sigma @ Ts2 ) @ ( image_674313660629153798_a_nat @ ( map_a_Sum_sum_a_nat @ sum_Inl_a_nat ) @ ( I @ ( product_Pair_b_nat @ R4 @ ( size_s8359284885213242922term_a @ Ts2 ) ) ) ) ) ) ).
% esat.simps(1)
thf(fact_1196_esat_Oelims_I3_J,axiom,
! [X5: fo_fmla_a_b,Xa: product_prod_b_nat > set_list_a,Xb: nat > sum_sum_a_nat,Xc: set_Sum_sum_a_nat] :
( ~ ( esat_a_b @ X5 @ Xa @ Xb @ Xc )
=> ( ! [R: b,Ts: list_fo_term_a] :
( ( X5
= ( fo_Pred_b_a @ R @ Ts ) )
=> ( member408289922725080238_a_nat @ ( eval_eterms_a_nat @ Xb @ Ts ) @ ( image_674313660629153798_a_nat @ ( map_a_Sum_sum_a_nat @ sum_Inl_a_nat ) @ ( Xa @ ( product_Pair_b_nat @ R @ ( size_s8359284885213242922term_a @ Ts ) ) ) ) ) )
=> ( ! [B3: $o] :
( ( X5
= ( fo_Bool_a_b @ B3 ) )
=> B3 )
=> ( ! [T3: fo_term_a,T4: fo_term_a] :
( ( X5
= ( fo_Eqa_a_b @ T3 @ T4 ) )
=> ( ( eval_eterm_a_nat @ Xb @ T3 )
= ( eval_eterm_a_nat @ Xb @ T4 ) ) )
=> ( ! [Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Neg_a_b @ Phi2 ) )
=> ~ ( esat_a_b @ Phi2 @ Xa @ Xb @ Xc ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Conj_a_b @ Phi2 @ Psi2 ) )
=> ( ( esat_a_b @ Phi2 @ Xa @ Xb @ Xc )
& ( esat_a_b @ Psi2 @ Xa @ Xb @ Xc ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Disj_a_b @ Phi2 @ Psi2 ) )
=> ( ( esat_a_b @ Phi2 @ Xa @ Xb @ Xc )
| ( esat_a_b @ Psi2 @ Xa @ Xb @ Xc ) ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Exists_a_b @ N @ Phi2 ) )
=> ? [X8: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X8 @ Xc )
& ( esat_a_b @ Phi2 @ Xa @ ( fun_up180537416982607344_a_nat @ Xb @ N @ X8 ) @ Xc ) ) )
=> ~ ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Forall_a_b @ N @ Phi2 ) )
=> ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ Xc )
=> ( esat_a_b @ Phi2 @ Xa @ ( fun_up180537416982607344_a_nat @ Xb @ N @ X ) @ Xc ) ) ) ) ) ) ) ) ) ) ) ).
% esat.elims(3)
thf(fact_1197_act__edom_Oelims,axiom,
! [X5: fo_fmla_a_b,Xa: product_prod_b_nat > set_list_a,Y: set_a] :
( ( ( act_edom_a_b @ X5 @ Xa )
= Y )
=> ( ! [R: b,Ts: list_fo_term_a] :
( ( X5
= ( fo_Pred_b_a @ R @ Ts ) )
=> ( Y
!= ( sup_sup_set_a @ ( ad_terms_a @ Ts ) @ ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ set_a2 @ ( Xa @ ( product_Pair_b_nat @ R @ ( size_s8359284885213242922term_a @ Ts ) ) ) ) ) ) ) )
=> ( ( ? [B3: $o] :
( X5
= ( fo_Bool_a_b @ B3 ) )
=> ( Y != bot_bot_set_a ) )
=> ( ! [T3: fo_term_a,T4: fo_term_a] :
( ( X5
= ( fo_Eqa_a_b @ T3 @ T4 ) )
=> ( Y
!= ( sup_sup_set_a @ ( fo_set_fo_term_a @ T3 ) @ ( fo_set_fo_term_a @ T4 ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Neg_a_b @ Phi2 ) )
=> ( Y
!= ( act_edom_a_b @ Phi2 @ Xa ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Conj_a_b @ Phi2 @ Psi2 ) )
=> ( Y
!= ( sup_sup_set_a @ ( act_edom_a_b @ Phi2 @ Xa ) @ ( act_edom_a_b @ Psi2 @ Xa ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Disj_a_b @ Phi2 @ Psi2 ) )
=> ( Y
!= ( sup_sup_set_a @ ( act_edom_a_b @ Phi2 @ Xa ) @ ( act_edom_a_b @ Psi2 @ Xa ) ) ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Exists_a_b @ N @ Phi2 ) )
=> ( Y
!= ( act_edom_a_b @ Phi2 @ Xa ) ) )
=> ~ ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Forall_a_b @ N @ Phi2 ) )
=> ( Y
!= ( act_edom_a_b @ Phi2 @ Xa ) ) ) ) ) ) ) ) ) ) ) ).
% act_edom.elims
thf(fact_1198_act__edom_Osimps_I7_J,axiom,
! [N2: nat,Phi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a] :
( ( act_edom_a_b @ ( fo_Exists_a_b @ N2 @ Phi ) @ I )
= ( act_edom_a_b @ Phi @ I ) ) ).
% act_edom.simps(7)
thf(fact_1199_act__edom_Osimps_I4_J,axiom,
! [Phi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a] :
( ( act_edom_a_b @ ( fo_Neg_a_b @ Phi ) @ I )
= ( act_edom_a_b @ Phi @ I ) ) ).
% act_edom.simps(4)
thf(fact_1200_act__edom_Osimps_I2_J,axiom,
! [B: $o,I: product_prod_b_nat > set_list_a] :
( ( act_edom_a_b @ ( fo_Bool_a_b @ B ) @ I )
= bot_bot_set_a ) ).
% act_edom.simps(2)
thf(fact_1201_act__edom_Osimps_I5_J,axiom,
! [Phi: fo_fmla_a_b,Psi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a] :
( ( act_edom_a_b @ ( fo_Conj_a_b @ Phi @ Psi ) @ I )
= ( sup_sup_set_a @ ( act_edom_a_b @ Phi @ I ) @ ( act_edom_a_b @ Psi @ I ) ) ) ).
% act_edom.simps(5)
thf(fact_1202_act__edom_Osimps_I6_J,axiom,
! [Phi: fo_fmla_a_b,Psi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a] :
( ( act_edom_a_b @ ( fo_Disj_a_b @ Phi @ Psi ) @ I )
= ( sup_sup_set_a @ ( act_edom_a_b @ Phi @ I ) @ ( act_edom_a_b @ Psi @ I ) ) ) ).
% act_edom.simps(6)
thf(fact_1203_subsetI,axiom,
! [A3: set_nat,B7: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B7 ) )
=> ( ord_less_eq_set_nat @ A3 @ B7 ) ) ).
% subsetI
thf(fact_1204_insert__subset,axiom,
! [X5: nat,A3: set_nat,B7: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat2 @ X5 @ A3 ) @ B7 )
= ( ( member_nat @ X5 @ B7 )
& ( ord_less_eq_set_nat @ A3 @ B7 ) ) ) ).
% insert_subset
thf(fact_1205_subset__insert,axiom,
! [X5: nat,A3: set_nat,B7: set_nat] :
( ~ ( member_nat @ X5 @ A3 )
=> ( ( ord_less_eq_set_nat @ A3 @ ( insert_nat2 @ X5 @ B7 ) )
= ( ord_less_eq_set_nat @ A3 @ B7 ) ) ) ).
% subset_insert
thf(fact_1206_image__subsetI,axiom,
! [A3: set_nat,F: nat > nat,B7: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( member_nat @ ( F @ X ) @ B7 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A3 ) @ B7 ) ) ).
% image_subsetI
thf(fact_1207_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A8: set_nat,B8: set_nat] :
! [T5: nat] :
( ( member_nat @ T5 @ A8 )
=> ( member_nat @ T5 @ B8 ) ) ) ) ).
% subset_iff
thf(fact_1208_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A8: set_nat,B8: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A8 )
=> ( member_nat @ X3 @ B8 ) ) ) ) ).
% subset_eq
thf(fact_1209_subsetD,axiom,
! [A3: set_nat,B7: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ B7 )
=> ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B7 ) ) ) ).
% subsetD
thf(fact_1210_in__mono,axiom,
! [A3: set_nat,B7: set_nat,X5: nat] :
( ( ord_less_eq_set_nat @ A3 @ B7 )
=> ( ( member_nat @ X5 @ A3 )
=> ( member_nat @ X5 @ B7 ) ) ) ).
% in_mono
thf(fact_1211_Int__Collect__mono,axiom,
! [A3: set_nat,B7: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A3 @ B7 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B7 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1212_subset__code_I1_J,axiom,
! [Xs: list_nat,B7: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B7 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X3 @ B7 ) ) ) ) ).
% subset_code(1)
thf(fact_1213_cSup__least,axiom,
! [X2: set_nat,Z: nat] :
( ( X2 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X2 )
=> ( ord_less_eq_nat @ X @ Z ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X2 ) @ Z ) ) ) ).
% cSup_least
thf(fact_1214_cSup__eq__non__empty,axiom,
! [X2: set_nat,A: nat] :
( ( X2 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat @ X @ X2 )
=> ( ord_less_eq_nat @ X @ A ) )
=> ( ! [Y5: nat] :
( ! [X8: nat] :
( ( member_nat @ X8 @ X2 )
=> ( ord_less_eq_nat @ X8 @ Y5 ) )
=> ( ord_less_eq_nat @ A @ Y5 ) )
=> ( ( complete_Sup_Sup_nat @ X2 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1215_inj__on__image__mem__iff,axiom,
! [F: nat > nat,B7: set_nat,A: nat,A3: set_nat] :
( ( inj_on_nat_nat @ F @ B7 )
=> ( ( member_nat @ A @ B7 )
=> ( ( ord_less_eq_set_nat @ A3 @ B7 )
=> ( ( member_nat @ ( F @ A ) @ ( image_nat_nat2 @ F @ A3 ) )
= ( member_nat @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1216_subset__Diff__insert,axiom,
! [A3: set_nat,B7: set_nat,X5: nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( minus_minus_set_nat @ B7 @ ( insert_nat2 @ X5 @ C3 ) ) )
= ( ( ord_less_eq_set_nat @ A3 @ ( minus_minus_set_nat @ B7 @ C3 ) )
& ~ ( member_nat @ X5 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1217_subset__code_I2_J,axiom,
! [A3: set_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( coset_nat @ Ys ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
=> ~ ( member_nat @ X3 @ A3 ) ) ) ) ).
% subset_code(2)
thf(fact_1218_SP__fv,axiom,
! [Phi: fo_fmla_a_b] : ( ord_less_eq_set_nat @ ( sP_a_b @ Phi ) @ ( fv_fo_fmla_a_b @ Phi ) ) ).
% SP_fv
thf(fact_1219_subset__insert__iff,axiom,
! [A3: set_nat,X5: nat,B7: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( insert_nat2 @ X5 @ B7 ) )
= ( ( ( member_nat @ X5 @ A3 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) ) @ B7 ) )
& ( ~ ( member_nat @ X5 @ A3 )
=> ( ord_less_eq_set_nat @ A3 @ B7 ) ) ) ) ).
% subset_insert_iff
thf(fact_1220_the__inv__into__into,axiom,
! [F: nat > nat,A3: set_nat,X5: nat,B7: set_nat] :
( ( inj_on_nat_nat @ F @ A3 )
=> ( ( member_nat @ X5 @ ( image_nat_nat2 @ F @ A3 ) )
=> ( ( ord_less_eq_set_nat @ A3 @ B7 )
=> ( member_nat @ ( the_inv_into_nat_nat @ A3 @ F @ X5 ) @ B7 ) ) ) ) ).
% the_inv_into_into
thf(fact_1221_subset__emptyI,axiom,
! [A3: set_nat] :
( ! [X: nat] :
~ ( member_nat @ X @ A3 )
=> ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1222_subset__Image1__Image1__iff,axiom,
! [R4: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( order_4861654808422542329on_nat @ ( field_nat @ R4 ) @ R4 )
=> ( ( member_nat @ A @ ( field_nat @ R4 ) )
=> ( ( member_nat @ B @ ( field_nat @ R4 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R4 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R4 @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R4 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1223_ImageI,axiom,
! [A: nat,B: nat,R4: set_Pr1261947904930325089at_nat,A3: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 )
=> ( ( member_nat @ A @ A3 )
=> ( member_nat @ B @ ( image_nat_nat @ R4 @ A3 ) ) ) ) ).
% ImageI
thf(fact_1224_ImageE,axiom,
! [B: nat,R4: set_Pr1261947904930325089at_nat,A3: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ R4 @ A3 ) )
=> ~ ! [X: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ B ) @ R4 )
=> ~ ( member_nat @ X @ A3 ) ) ) ).
% ImageE
thf(fact_1225_rev__ImageI,axiom,
! [A: nat,A3: set_nat,B: nat,R4: set_Pr1261947904930325089at_nat] :
( ( member_nat @ A @ A3 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 )
=> ( member_nat @ B @ ( image_nat_nat @ R4 @ A3 ) ) ) ) ).
% rev_ImageI
thf(fact_1226_Partial__order__eq__Image1__Image1__iff,axiom,
! [R4: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
( ( order_5251275573222108571on_nat @ ( field_nat @ R4 ) @ R4 )
=> ( ( member_nat @ A @ ( field_nat @ R4 ) )
=> ( ( member_nat @ B @ ( field_nat @ R4 ) )
=> ( ( ( image_nat_nat @ R4 @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
= ( image_nat_nat @ R4 @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) )
= ( A = B ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_1227_wf__fo__intp_Oelims_I2_J,axiom,
! [X5: fo_fmla_a_b,Xa: product_prod_b_nat > set_list_a] :
( ( wf_fo_intp_a_b @ X5 @ Xa )
=> ( ! [R: b,Ts: list_fo_term_a] :
( ( X5
= ( fo_Pred_b_a @ R @ Ts ) )
=> ~ ( finite_finite_list_a @ ( Xa @ ( product_Pair_b_nat @ R @ ( size_s8359284885213242922term_a @ Ts ) ) ) ) )
=> ( ! [B3: $o] :
( X5
!= ( fo_Bool_a_b @ B3 ) )
=> ( ! [T3: fo_term_a,T4: fo_term_a] :
( X5
!= ( fo_Eqa_a_b @ T3 @ T4 ) )
=> ( ! [Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Neg_a_b @ Phi2 ) )
=> ~ ( wf_fo_intp_a_b @ Phi2 @ Xa ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Conj_a_b @ Phi2 @ Psi2 ) )
=> ~ ( ( wf_fo_intp_a_b @ Phi2 @ Xa )
& ( wf_fo_intp_a_b @ Psi2 @ Xa ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Disj_a_b @ Phi2 @ Psi2 ) )
=> ~ ( ( wf_fo_intp_a_b @ Phi2 @ Xa )
& ( wf_fo_intp_a_b @ Psi2 @ Xa ) ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Exists_a_b @ N @ Phi2 ) )
=> ~ ( wf_fo_intp_a_b @ Phi2 @ Xa ) )
=> ~ ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Forall_a_b @ N @ Phi2 ) )
=> ~ ( wf_fo_intp_a_b @ Phi2 @ Xa ) ) ) ) ) ) ) ) ) ) ).
% wf_fo_intp.elims(2)
thf(fact_1228_wf__fo__intp_Oelims_I1_J,axiom,
! [X5: fo_fmla_a_b,Xa: product_prod_b_nat > set_list_a,Y: $o] :
( ( ( wf_fo_intp_a_b @ X5 @ Xa )
= Y )
=> ( ! [R: b,Ts: list_fo_term_a] :
( ( X5
= ( fo_Pred_b_a @ R @ Ts ) )
=> ( Y
= ( ~ ( finite_finite_list_a @ ( Xa @ ( product_Pair_b_nat @ R @ ( size_s8359284885213242922term_a @ Ts ) ) ) ) ) ) )
=> ( ( ? [B3: $o] :
( X5
= ( fo_Bool_a_b @ B3 ) )
=> ~ Y )
=> ( ( ? [T3: fo_term_a,T4: fo_term_a] :
( X5
= ( fo_Eqa_a_b @ T3 @ T4 ) )
=> ~ Y )
=> ( ! [Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Neg_a_b @ Phi2 ) )
=> ( Y
= ( ~ ( wf_fo_intp_a_b @ Phi2 @ Xa ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Conj_a_b @ Phi2 @ Psi2 ) )
=> ( Y
= ( ~ ( ( wf_fo_intp_a_b @ Phi2 @ Xa )
& ( wf_fo_intp_a_b @ Psi2 @ Xa ) ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Disj_a_b @ Phi2 @ Psi2 ) )
=> ( Y
= ( ~ ( ( wf_fo_intp_a_b @ Phi2 @ Xa )
& ( wf_fo_intp_a_b @ Psi2 @ Xa ) ) ) ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Exists_a_b @ N @ Phi2 ) )
=> ( Y
= ( ~ ( wf_fo_intp_a_b @ Phi2 @ Xa ) ) ) )
=> ~ ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Forall_a_b @ N @ Phi2 ) )
=> ( Y
= ( ~ ( wf_fo_intp_a_b @ Phi2 @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% wf_fo_intp.elims(1)
thf(fact_1229_wf__fo__intp_Osimps_I2_J,axiom,
! [B: $o,I: product_prod_b_nat > set_list_a] : ( wf_fo_intp_a_b @ ( fo_Bool_a_b @ B ) @ I ) ).
% wf_fo_intp.simps(2)
thf(fact_1230_wf__fo__intp_Osimps_I7_J,axiom,
! [N2: nat,Phi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a] :
( ( wf_fo_intp_a_b @ ( fo_Exists_a_b @ N2 @ Phi ) @ I )
= ( wf_fo_intp_a_b @ Phi @ I ) ) ).
% wf_fo_intp.simps(7)
thf(fact_1231_wf__fo__intp_Osimps_I4_J,axiom,
! [Phi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a] :
( ( wf_fo_intp_a_b @ ( fo_Neg_a_b @ Phi ) @ I )
= ( wf_fo_intp_a_b @ Phi @ I ) ) ).
% wf_fo_intp.simps(4)
thf(fact_1232_wf__fo__intp_Osimps_I6_J,axiom,
! [Phi: fo_fmla_a_b,Psi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a] :
( ( wf_fo_intp_a_b @ ( fo_Disj_a_b @ Phi @ Psi ) @ I )
= ( ( wf_fo_intp_a_b @ Phi @ I )
& ( wf_fo_intp_a_b @ Psi @ I ) ) ) ).
% wf_fo_intp.simps(6)
thf(fact_1233_wf__fo__intp_Osimps_I5_J,axiom,
! [Phi: fo_fmla_a_b,Psi: fo_fmla_a_b,I: product_prod_b_nat > set_list_a] :
( ( wf_fo_intp_a_b @ ( fo_Conj_a_b @ Phi @ Psi ) @ I )
= ( ( wf_fo_intp_a_b @ Phi @ I )
& ( wf_fo_intp_a_b @ Psi @ I ) ) ) ).
% wf_fo_intp.simps(5)
thf(fact_1234_infinite__finite__induct,axiom,
! [P: set_nat > $o,A3: set_nat] :
( ! [A10: set_nat] :
( ~ ( finite_finite_nat @ A10 )
=> ( P @ A10 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ X @ F3 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1235_finite__ne__induct,axiom,
! [F4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( F4 != bot_bot_set_nat )
=> ( ! [X: nat] : ( P @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
=> ( ! [X: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ X @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1236_finite__induct,axiom,
! [F4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ X @ F3 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_1237_finite__subset__induct_H,axiom,
! [F4: set_nat,A3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat @ F4 @ A3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A5 @ A3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A3 )
=> ( ~ ( member_nat @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1238_finite__subset__induct,axiom,
! [F4: set_nat,A3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat @ F4 @ A3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A5 @ A3 )
=> ( ~ ( member_nat @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1239_finite__empty__induct,axiom,
! [A3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A3 )
=> ( ( P @ A3 )
=> ( ! [A5: nat,A10: set_nat] :
( ( finite_finite_nat @ A10 )
=> ( ( member_nat @ A5 @ A10 )
=> ( ( P @ A10 )
=> ( P @ ( minus_minus_set_nat @ A10 @ ( insert_nat2 @ A5 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1240_remove__induct,axiom,
! [P: set_nat > $o,B7: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B7 )
=> ( P @ B7 ) )
=> ( ! [A10: set_nat] :
( ( finite_finite_nat @ A10 )
=> ( ( A10 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A10 @ B7 )
=> ( ! [X8: nat] :
( ( member_nat @ X8 @ A10 )
=> ( P @ ( minus_minus_set_nat @ A10 @ ( insert_nat2 @ X8 @ bot_bot_set_nat ) ) ) )
=> ( P @ A10 ) ) ) ) )
=> ( P @ B7 ) ) ) ) ).
% remove_induct
thf(fact_1241_finite__remove__induct,axiom,
! [B7: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B7 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A10: set_nat] :
( ( finite_finite_nat @ A10 )
=> ( ( A10 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A10 @ B7 )
=> ( ! [X8: nat] :
( ( member_nat @ X8 @ A10 )
=> ( P @ ( minus_minus_set_nat @ A10 @ ( insert_nat2 @ X8 @ bot_bot_set_nat ) ) ) )
=> ( P @ A10 ) ) ) ) )
=> ( P @ B7 ) ) ) ) ).
% finite_remove_induct
thf(fact_1242_wf__fo__intp_Oelims_I3_J,axiom,
! [X5: fo_fmla_a_b,Xa: product_prod_b_nat > set_list_a] :
( ~ ( wf_fo_intp_a_b @ X5 @ Xa )
=> ( ! [R: b,Ts: list_fo_term_a] :
( ( X5
= ( fo_Pred_b_a @ R @ Ts ) )
=> ( finite_finite_list_a @ ( Xa @ ( product_Pair_b_nat @ R @ ( size_s8359284885213242922term_a @ Ts ) ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Neg_a_b @ Phi2 ) )
=> ( wf_fo_intp_a_b @ Phi2 @ Xa ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Conj_a_b @ Phi2 @ Psi2 ) )
=> ( ( wf_fo_intp_a_b @ Phi2 @ Xa )
& ( wf_fo_intp_a_b @ Psi2 @ Xa ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Disj_a_b @ Phi2 @ Psi2 ) )
=> ( ( wf_fo_intp_a_b @ Phi2 @ Xa )
& ( wf_fo_intp_a_b @ Psi2 @ Xa ) ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Exists_a_b @ N @ Phi2 ) )
=> ( wf_fo_intp_a_b @ Phi2 @ Xa ) )
=> ~ ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Forall_a_b @ N @ Phi2 ) )
=> ( wf_fo_intp_a_b @ Phi2 @ Xa ) ) ) ) ) ) ) ) ).
% wf_fo_intp.elims(3)
thf(fact_1243_wf__fo__intp_Opelims_I1_J,axiom,
! [X5: fo_fmla_a_b,Xa: product_prod_b_nat > set_list_a,Y: $o] :
( ( ( wf_fo_intp_a_b @ X5 @ Xa )
= Y )
=> ( ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ X5 @ Xa ) )
=> ( ! [R: b,Ts: list_fo_term_a] :
( ( X5
= ( fo_Pred_b_a @ R @ Ts ) )
=> ( ( Y
= ( finite_finite_list_a @ ( Xa @ ( product_Pair_b_nat @ R @ ( size_s8359284885213242922term_a @ Ts ) ) ) ) )
=> ~ ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ ( fo_Pred_b_a @ R @ Ts ) @ Xa ) ) ) )
=> ( ! [B3: $o] :
( ( X5
= ( fo_Bool_a_b @ B3 ) )
=> ( Y
=> ~ ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ ( fo_Bool_a_b @ B3 ) @ Xa ) ) ) )
=> ( ! [T3: fo_term_a,T4: fo_term_a] :
( ( X5
= ( fo_Eqa_a_b @ T3 @ T4 ) )
=> ( Y
=> ~ ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ ( fo_Eqa_a_b @ T3 @ T4 ) @ Xa ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Neg_a_b @ Phi2 ) )
=> ( ( Y
= ( wf_fo_intp_a_b @ Phi2 @ Xa ) )
=> ~ ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ ( fo_Neg_a_b @ Phi2 ) @ Xa ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Conj_a_b @ Phi2 @ Psi2 ) )
=> ( ( Y
= ( ( wf_fo_intp_a_b @ Phi2 @ Xa )
& ( wf_fo_intp_a_b @ Psi2 @ Xa ) ) )
=> ~ ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ ( fo_Conj_a_b @ Phi2 @ Psi2 ) @ Xa ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Disj_a_b @ Phi2 @ Psi2 ) )
=> ( ( Y
= ( ( wf_fo_intp_a_b @ Phi2 @ Xa )
& ( wf_fo_intp_a_b @ Psi2 @ Xa ) ) )
=> ~ ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ ( fo_Disj_a_b @ Phi2 @ Psi2 ) @ Xa ) ) ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Exists_a_b @ N @ Phi2 ) )
=> ( ( Y
= ( wf_fo_intp_a_b @ Phi2 @ Xa ) )
=> ~ ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ ( fo_Exists_a_b @ N @ Phi2 ) @ Xa ) ) ) )
=> ~ ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Forall_a_b @ N @ Phi2 ) )
=> ( ( Y
= ( wf_fo_intp_a_b @ Phi2 @ Xa ) )
=> ~ ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ ( fo_Forall_a_b @ N @ Phi2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% wf_fo_intp.pelims(1)
thf(fact_1244_wf__fo__intp_Opelims_I2_J,axiom,
! [X5: fo_fmla_a_b,Xa: product_prod_b_nat > set_list_a] :
( ( wf_fo_intp_a_b @ X5 @ Xa )
=> ( ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ X5 @ Xa ) )
=> ( ! [R: b,Ts: list_fo_term_a] :
( ( X5
= ( fo_Pred_b_a @ R @ Ts ) )
=> ( ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ ( fo_Pred_b_a @ R @ Ts ) @ Xa ) )
=> ~ ( finite_finite_list_a @ ( Xa @ ( product_Pair_b_nat @ R @ ( size_s8359284885213242922term_a @ Ts ) ) ) ) ) )
=> ( ! [B3: $o] :
( ( X5
= ( fo_Bool_a_b @ B3 ) )
=> ~ ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ ( fo_Bool_a_b @ B3 ) @ Xa ) ) )
=> ( ! [T3: fo_term_a,T4: fo_term_a] :
( ( X5
= ( fo_Eqa_a_b @ T3 @ T4 ) )
=> ~ ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ ( fo_Eqa_a_b @ T3 @ T4 ) @ Xa ) ) )
=> ( ! [Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Neg_a_b @ Phi2 ) )
=> ( ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ ( fo_Neg_a_b @ Phi2 ) @ Xa ) )
=> ~ ( wf_fo_intp_a_b @ Phi2 @ Xa ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Conj_a_b @ Phi2 @ Psi2 ) )
=> ( ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ ( fo_Conj_a_b @ Phi2 @ Psi2 ) @ Xa ) )
=> ~ ( ( wf_fo_intp_a_b @ Phi2 @ Xa )
& ( wf_fo_intp_a_b @ Psi2 @ Xa ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Disj_a_b @ Phi2 @ Psi2 ) )
=> ( ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ ( fo_Disj_a_b @ Phi2 @ Psi2 ) @ Xa ) )
=> ~ ( ( wf_fo_intp_a_b @ Phi2 @ Xa )
& ( wf_fo_intp_a_b @ Psi2 @ Xa ) ) ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Exists_a_b @ N @ Phi2 ) )
=> ( ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ ( fo_Exists_a_b @ N @ Phi2 ) @ Xa ) )
=> ~ ( wf_fo_intp_a_b @ Phi2 @ Xa ) ) )
=> ~ ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Forall_a_b @ N @ Phi2 ) )
=> ( ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ ( fo_Forall_a_b @ N @ Phi2 ) @ Xa ) )
=> ~ ( wf_fo_intp_a_b @ Phi2 @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).
% wf_fo_intp.pelims(2)
thf(fact_1245_finite__fv__fo__fmla,axiom,
! [Phi: fo_fmla_a_b] : ( finite_finite_nat @ ( fv_fo_fmla_a_b @ Phi ) ) ).
% finite_fv_fo_fmla
thf(fact_1246_wf__fo__intp_Opelims_I3_J,axiom,
! [X5: fo_fmla_a_b,Xa: product_prod_b_nat > set_list_a] :
( ~ ( wf_fo_intp_a_b @ X5 @ Xa )
=> ( ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ X5 @ Xa ) )
=> ( ! [R: b,Ts: list_fo_term_a] :
( ( X5
= ( fo_Pred_b_a @ R @ Ts ) )
=> ( ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ ( fo_Pred_b_a @ R @ Ts ) @ Xa ) )
=> ( finite_finite_list_a @ ( Xa @ ( product_Pair_b_nat @ R @ ( size_s8359284885213242922term_a @ Ts ) ) ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Neg_a_b @ Phi2 ) )
=> ( ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ ( fo_Neg_a_b @ Phi2 ) @ Xa ) )
=> ( wf_fo_intp_a_b @ Phi2 @ Xa ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Conj_a_b @ Phi2 @ Psi2 ) )
=> ( ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ ( fo_Conj_a_b @ Phi2 @ Psi2 ) @ Xa ) )
=> ( ( wf_fo_intp_a_b @ Phi2 @ Xa )
& ( wf_fo_intp_a_b @ Psi2 @ Xa ) ) ) )
=> ( ! [Phi2: fo_fmla_a_b,Psi2: fo_fmla_a_b] :
( ( X5
= ( fo_Disj_a_b @ Phi2 @ Psi2 ) )
=> ( ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ ( fo_Disj_a_b @ Phi2 @ Psi2 ) @ Xa ) )
=> ( ( wf_fo_intp_a_b @ Phi2 @ Xa )
& ( wf_fo_intp_a_b @ Psi2 @ Xa ) ) ) )
=> ( ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Exists_a_b @ N @ Phi2 ) )
=> ( ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ ( fo_Exists_a_b @ N @ Phi2 ) @ Xa ) )
=> ( wf_fo_intp_a_b @ Phi2 @ Xa ) ) )
=> ~ ! [N: nat,Phi2: fo_fmla_a_b] :
( ( X5
= ( fo_Forall_a_b @ N @ Phi2 ) )
=> ( ( accp_P650863959989046617list_a @ wf_fo_intp_rel_a_b @ ( produc2741640913913151572list_a @ ( fo_Forall_a_b @ N @ Phi2 ) @ Xa ) )
=> ( wf_fo_intp_a_b @ Phi2 @ Xa ) ) ) ) ) ) ) ) ) ) ).
% wf_fo_intp.pelims(3)
thf(fact_1247_Inf__fin_Oremove,axiom,
! [A3: set_nat,X5: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ X5 @ A3 )
=> ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A3 )
= X5 ) )
& ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A3 )
= ( inf_inf_nat @ X5 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1248_Inf__fin_OboundedI,axiom,
! [A3: set_nat,X5: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A3 )
=> ( ord_less_eq_nat @ X5 @ A5 ) )
=> ( ord_less_eq_nat @ X5 @ ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1249_Inf__fin_OboundedE,axiom,
! [A3: set_nat,X5: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X5 @ ( lattic5238388535129920115in_nat @ A3 ) )
=> ! [A13: nat] :
( ( member_nat @ A13 @ A3 )
=> ( ord_less_eq_nat @ X5 @ A13 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1250_Inf__fin_Oclosed,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ! [X: nat,Y5: nat] : ( member_nat @ ( inf_inf_nat @ X @ Y5 ) @ ( insert_nat2 @ X @ ( insert_nat2 @ Y5 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic5238388535129920115in_nat @ A3 ) @ A3 ) ) ) ) ).
% Inf_fin.closed
thf(fact_1251_Inf__fin_Oinsert__not__elem,axiom,
! [A3: set_nat,X5: nat] :
( ( finite_finite_nat @ A3 )
=> ( ~ ( member_nat @ X5 @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X5 @ A3 ) )
= ( inf_inf_nat @ X5 @ ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1252_Sup__fin_Oremove,axiom,
! [A3: set_nat,X5: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ X5 @ A3 )
=> ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A3 )
= X5 ) )
& ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A3 )
= ( sup_sup_nat @ X5 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1253_Sup__fin_OboundedE,axiom,
! [A3: set_nat,X5: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ X5 )
=> ! [A13: nat] :
( ( member_nat @ A13 @ A3 )
=> ( ord_less_eq_nat @ A13 @ X5 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1254_Sup__fin_OboundedI,axiom,
! [A3: set_nat,X5: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A3 )
=> ( ord_less_eq_nat @ A5 @ X5 ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ X5 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1255_Sup__fin_Oinsert__not__elem,axiom,
! [A3: set_nat,X5: nat] :
( ( finite_finite_nat @ A3 )
=> ( ~ ( member_nat @ X5 @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X5 @ A3 ) )
= ( sup_sup_nat @ X5 @ ( lattic1093996805478795353in_nat @ A3 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1256_Sup__fin_Oclosed,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ! [X: nat,Y5: nat] : ( member_nat @ ( sup_sup_nat @ X @ Y5 ) @ ( insert_nat2 @ X @ ( insert_nat2 @ Y5 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ A3 ) ) ) ) ).
% Sup_fin.closed
thf(fact_1257_finite__update__induct,axiom,
! [F: nat > sum_sum_a_nat,C: sum_sum_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [A2: nat] :
( ( F @ A2 )
!= C ) ) )
=> ( ( P
@ ^ [A2: nat] : C )
=> ( ! [A5: nat,B3: sum_sum_a_nat,F5: nat > sum_sum_a_nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [C6: nat] :
( ( F5 @ C6 )
!= C ) ) )
=> ( ( ( F5 @ A5 )
= C )
=> ( ( B3 != C )
=> ( ( P @ F5 )
=> ( P @ ( fun_up180537416982607344_a_nat @ F5 @ A5 @ B3 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_update_induct
thf(fact_1258_imageE,axiom,
! [B: nat,F: nat > nat,A3: set_nat] :
( ( member_nat @ B @ ( image_nat_nat2 @ F @ A3 ) )
=> ~ ! [X: nat] :
( ( B
= ( F @ X ) )
=> ~ ( member_nat @ X @ A3 ) ) ) ).
% imageE
thf(fact_1259_Compr__image__eq,axiom,
! [F: nat > nat,A3: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat2 @ F @ A3 ) )
& ( P @ X3 ) ) )
= ( image_nat_nat2 @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_1260_insert__compr,axiom,
( insert_nat2
= ( ^ [A2: nat,B8: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( X3 = A2 )
| ( member_nat @ X3 @ B8 ) ) ) ) ) ).
% insert_compr
thf(fact_1261_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A8: set_nat,B8: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A8 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B8 ) ) ) ) ) ).
% sup_set_def
thf(fact_1262_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A8: set_nat,B8: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A8 )
| ( member_nat @ X3 @ B8 ) ) ) ) ) ).
% Un_def
thf(fact_1263_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A8: set_nat,B8: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A8 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B8 ) ) ) ) ) ).
% minus_set_def
thf(fact_1264_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A8: set_nat,B8: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A8 )
& ~ ( member_nat @ X3 @ B8 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1265_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A8: set_nat,B8: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A8 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B8 ) ) ) ) ) ).
% inf_set_def
thf(fact_1266_Int__Collect,axiom,
! [X5: nat,A3: set_nat,P: nat > $o] :
( ( member_nat @ X5 @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X5 @ A3 )
& ( P @ X5 ) ) ) ).
% Int_Collect
thf(fact_1267_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A8: set_nat,B8: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A8 )
& ( member_nat @ X3 @ B8 ) ) ) ) ) ).
% Int_def
thf(fact_1268_Collect__subset,axiom,
! [A3: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( P @ X3 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_1269_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A8: set_nat,B8: set_nat] :
( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A8 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B8 ) ) ) ) ).
% less_eq_set_def
thf(fact_1270_Id__onI,axiom,
! [A: nat,A3: set_nat] :
( ( member_nat @ A @ A3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( id_on_nat @ A3 ) ) ) ).
% Id_onI
thf(fact_1271_Id__onE,axiom,
! [C: product_prod_nat_nat,A3: set_nat] :
( ( member8440522571783428010at_nat @ C @ ( id_on_nat @ A3 ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( C
!= ( product_Pair_nat_nat @ X @ X ) ) ) ) ).
% Id_onE
thf(fact_1272_Id__on__eqI,axiom,
! [A: nat,B: nat,A3: set_nat] :
( ( A = B )
=> ( ( member_nat @ A @ A3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( id_on_nat @ A3 ) ) ) ) ).
% Id_on_eqI
thf(fact_1273_Id__on__iff,axiom,
! [X5: nat,Y: nat,A3: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y ) @ ( id_on_nat @ A3 ) )
= ( ( X5 = Y )
& ( member_nat @ X5 @ A3 ) ) ) ).
% Id_on_iff
thf(fact_1274_Set_Omember__filter,axiom,
! [X5: nat,P: nat > $o,A3: set_nat] :
( ( member_nat @ X5 @ ( filter_nat @ P @ A3 ) )
= ( ( member_nat @ X5 @ A3 )
& ( P @ X5 ) ) ) ).
% Set.member_filter
thf(fact_1275_Set_Ofilter__def,axiom,
( filter_nat
= ( ^ [P4: nat > $o,A8: set_nat] :
( collect_nat
@ ^ [A2: nat] :
( ( member_nat @ A2 @ A8 )
& ( P4 @ A2 ) ) ) ) ) ).
% Set.filter_def
thf(fact_1276_in__set__remove1,axiom,
! [A: nat,B: nat,Xs: list_nat] :
( ( A != B )
=> ( ( member_nat @ A @ ( set_nat2 @ ( remove1_nat @ B @ Xs ) ) )
= ( member_nat @ A @ ( set_nat2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_1277_remove1__idem,axiom,
! [X5: nat,Xs: list_nat] :
( ~ ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
=> ( ( remove1_nat @ X5 @ Xs )
= Xs ) ) ).
% remove1_idem
% Helper facts (3)
thf(help_If_3_1_If_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_T,axiom,
! [X5: sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( if_Sum_sum_a_nat @ $false @ X5 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_T,axiom,
! [X5: sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( if_Sum_sum_a_nat @ $true @ X5 @ Y )
= X5 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
esat_a_b @ phi @ i @ sigma @ x ).
%------------------------------------------------------------------------------