TPTP Problem File: ITP039^1.p
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%------------------------------------------------------------------------------
% File : ITP039^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Coincidence problem prob_472__7214396_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Coincidence/prob_472__7214396_1 [Des21]
% Status : Theorem
% Rating : 0.20 v8.2.0, 0.23 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 507 ( 165 unt; 152 typ; 0 def)
% Number of atoms : 782 ( 357 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3134 ( 150 ~; 0 |; 45 &;2594 @)
% ( 0 <=>; 345 =>; 0 <=; 0 <~>)
% Maximal formula depth : 33 ( 9 avg)
% Number of types : 27 ( 26 usr)
% Number of type conns : 754 ( 754 >; 0 *; 0 +; 0 <<)
% Number of symbols : 128 ( 126 usr; 5 con; 0-10 aty)
% Number of variables : 1586 ( 99 ^;1462 !; 25 ?;1586 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:39:39.768
%------------------------------------------------------------------------------
% Could-be-implicit typings (26)
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image_1121381717real_c: ( produc866628903_sc_sz > produc190496183real_c ) > set_Pr1774599175_sc_sz > set_Pr1389752855real_c ).
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image_1313166277_sc_sz: ( produc866628903_sc_sz > produc866628903_sc_sz ) > set_Pr1774599175_sc_sz > set_Pr1774599175_sc_sz ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_Mt__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J_Mt__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J_J,type,
image_768765061real_c: ( produc866628903_sc_sz > produc640052711real_c ) > set_Pr1774599175_sc_sz > set_Pr754230471real_c ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_Mt__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_Mt__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_J_Mt__Denotational____Semantics__Ointerp__Ointerp____ext_Itf__sf_Mtf__sc_Mtf__sz_Mt__Product____Type__Ounit_J_J,type,
image_1551908392t_unit: ( produc866628903_sc_sz > produc999471430t_unit ) > set_Pr1774599175_sc_sz > set_Pr2041213564t_unit ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J_Mt__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J_J_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J,type,
image_124598413real_c: ( produc640052711real_c > finite1398487019real_c ) > set_Pr754230471real_c > set_Fi1407883041real_c ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J_Mt__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J_Mt__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J_J,type,
image_2041365445real_c: ( produc640052711real_c > produc640052711real_c ) > set_Pr754230471real_c > set_Pr754230471real_c ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_Mt__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_J_Mt__Denotational____Semantics__Ointerp__Ointerp____ext_Itf__sf_Mtf__sc_Mtf__sz_Mt__Product____Type__Ounit_J_J_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J,type,
image_1269062456real_c: ( produc999471430t_unit > finite1398487019real_c ) > set_Pr2041213564t_unit > set_Fi1407883041real_c ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_Mt__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_J_Mt__Denotational____Semantics__Ointerp__Ointerp____ext_Itf__sf_Mtf__sc_Mtf__sz_Mt__Product____Type__Ounit_J_J_001t__Product____Type__Oprod_It__Denotational____Semantics__Ointerp__Ointerp____ext_Itf__sf_Mtf__sc_Mtf__sz_Mt__Product____Type__Ounit_J_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_Mt__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_J_J,type,
image_1426741113_sc_sz: ( produc999471430t_unit > produc1821101996_sc_sz ) > set_Pr2041213564t_unit > set_Pr1169339874_sc_sz ).
thf(sy_c_Static__Semantics_OFVF_001tf__a_001tf__b_001tf__c,type,
static_FVF_a_b_c: formula_a_b_c > set_Sum_sum_c_c ).
thf(sy_c_Static__Semantics_OSIGF_001tf__a_001tf__b_001tf__c,type,
static_SIGF_a_b_c: formula_a_b_c > set_Su1783761653um_b_c ).
thf(sy_c_Syntax_OEquiv_001tf__a_001tf__b_001tf__c,type,
equiv_a_b_c: formula_a_b_c > formula_a_b_c > formula_a_b_c ).
thf(sy_c_Syntax_OImplies_001tf__a_001tf__b_001tf__c,type,
implies_a_b_c: formula_a_b_c > formula_a_b_c > formula_a_b_c ).
thf(sy_c_Syntax_OPredicational_001tf__sc_001tf__sf_001tf__sz,type,
predic913887884_sf_sz: sc > formula_sf_sc_sz ).
thf(sy_c_Syntax_Oids_OP_001tf__sc_001tf__sf_001tf__sz,type,
p_sc_sf_sz: sc > formula_sf_sc_sz ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Denotational____Semantics__Ointerp__Ointerp____ext_Itf__sf_Mtf__sc_Mtf__sz_Mt__Product____Type__Ounit_J,type,
member754147591t_unit: denota610675952t_unit > set_De208295462t_unit > $o ).
thf(sy_c_member_001t__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J,type,
member1261661570real_c: finite1398487019real_c > set_Fi1407883041real_c > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Denotational____Semantics__Ointerp__Ointerp____ext_Itf__sf_Mtf__sc_Mtf__sz_Mt__Product____Type__Ounit_J_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_Mt__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_J_J,type,
member1057565379_sc_sz: produc1821101996_sc_sz > set_Pr1169339874_sc_sz > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J,type,
member1895684704real_c: produc190496183real_c > set_Pr1389752855real_c > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__sz_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__sz_J_J,type,
member630771088eal_sz: produc1149990247eal_sz > set_Pr60491591eal_sz > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_Mt__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_J,type,
member245028432_sc_sz: produc866628903_sc_sz > set_Pr1774599175_sc_sz > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J_Mt__Product____Type__Oprod_It__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_Mt__Finite____Cartesian____Product__Ovec_It__Real__Oreal_Mtf__c_J_J_J,type,
member346808080real_c: produc640052711real_c > set_Pr754230471real_c > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_Mt__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_J_Mt__Denotational____Semantics__Ointerp__Ointerp____ext_Itf__sf_Mtf__sc_Mtf__sz_Mt__Product____Type__Ounit_J_J,type,
member235934813t_unit: produc999471430t_unit > set_Pr2041213564t_unit > $o ).
thf(sy_v__092_060phi_062,type,
phi: formula_a_b_c ).
% Relevant facts (354)
thf(fact_0_coincide__fml__def,axiom,
( coinci501627771_a_b_c
= ( ^ [Phi: formula_a_b_c] :
! [Nu: produc190496183real_c,Nu2: produc190496183real_c,I: denota231621370t_unit,J: denota231621370t_unit] :
( ( denota69106024_a_b_c @ I @ J @ ( static_SIGF_a_b_c @ Phi ) )
=> ( ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVF_a_b_c @ Phi ) )
=> ( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ Phi ) )
= ( member1895684704real_c @ Nu2 @ ( denota968303861_a_b_c @ J @ Phi ) ) ) ) ) ) ) ).
% coincide_fml_def
thf(fact_1_ids_Ocoincide__fml__def,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc,Phi2: formula_a_b_c] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( coinci501627771_a_b_c @ Phi2 )
= ( ! [Nu: produc190496183real_c,Nu2: produc190496183real_c,I: denota231621370t_unit,J: denota231621370t_unit] :
( ( denota69106024_a_b_c @ I @ J @ ( static_SIGF_a_b_c @ Phi2 ) )
=> ( ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVF_a_b_c @ Phi2 ) )
=> ( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ Phi2 ) )
= ( member1895684704real_c @ Nu2 @ ( denota968303861_a_b_c @ J @ Phi2 ) ) ) ) ) ) ) ) ).
% ids.coincide_fml_def
thf(fact_2_Iagree__comm,axiom,
! [A: denota231621370t_unit,B: denota231621370t_unit,V: set_Su1783761653um_b_c] :
( ( denota69106024_a_b_c @ A @ B @ V )
=> ( denota69106024_a_b_c @ B @ A @ V ) ) ).
% Iagree_comm
thf(fact_3_Iagree__refl,axiom,
! [I2: denota231621370t_unit,A: set_Su1783761653um_b_c] : ( denota69106024_a_b_c @ I2 @ I2 @ A ) ).
% Iagree_refl
thf(fact_4_agree__comm,axiom,
! [A: produc190496183real_c,B: produc190496183real_c,V: set_Sum_sum_c_c] :
( ( denota1997846518gree_c @ A @ B @ V )
=> ( denota1997846518gree_c @ B @ A @ V ) ) ).
% agree_comm
thf(fact_5_agree__refl,axiom,
! [Nu3: produc190496183real_c,A: set_Sum_sum_c_c] : ( denota1997846518gree_c @ Nu3 @ Nu3 @ A ) ).
% agree_refl
thf(fact_6_iff__to__impl,axiom,
! [Nu3: produc190496183real_c,I2: denota231621370t_unit,A: formula_a_b_c,B: formula_a_b_c] :
( ( ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I2 @ A ) )
= ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I2 @ B ) ) )
= ( ( ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I2 @ A ) )
=> ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I2 @ B ) ) )
& ( ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I2 @ B ) )
=> ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I2 @ A ) ) ) ) ) ).
% iff_to_impl
thf(fact_7_iff__sem,axiom,
! [Nu3: produc190496183real_c,I2: denota231621370t_unit,A: formula_a_b_c,B: formula_a_b_c] :
( ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I2 @ ( equiv_a_b_c @ A @ B ) ) )
= ( ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I2 @ A ) )
= ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I2 @ B ) ) ) ) ).
% iff_sem
thf(fact_8_impl__sem,axiom,
! [Nu3: produc190496183real_c,I2: denota231621370t_unit,A: formula_a_b_c,B: formula_a_b_c] :
( ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I2 @ ( implies_a_b_c @ A @ B ) ) )
= ( ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I2 @ A ) )
=> ( member1895684704real_c @ Nu3 @ ( denota968303861_a_b_c @ I2 @ B ) ) ) ) ).
% impl_sem
thf(fact_9_P__def,axiom,
p_sc_sf_sz = predic913887884_sf_sz ).
% P_def
thf(fact_10_seq__sem_Ocases,axiom,
! [X: produc1821101996_sc_sz] :
~ ! [I3: denota610675952t_unit,S: produc866628903_sc_sz] :
( X
!= ( produc789536734_sc_sz @ I3 @ S ) ) ).
% seq_sem.cases
thf(fact_11_ids_Oseq__sem_Ocases,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc,X: produc1821101996_sc_sz] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ~ ! [I3: denota610675952t_unit,S: produc866628903_sc_sz] :
( X
!= ( produc789536734_sc_sz @ I3 @ S ) ) ) ).
% ids.seq_sem.cases
thf(fact_12_ids_Oseq__sem_Oinduct,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc,P: denota610675952t_unit > produc866628903_sc_sz > $o,A0: denota610675952t_unit,A1: produc866628903_sc_sz] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ! [I3: denota610675952t_unit,X_1: produc866628903_sc_sz] : ( P @ I3 @ X_1 )
=> ( P @ A0 @ A1 ) ) ) ).
% ids.seq_sem.induct
thf(fact_13_ids_OP__def,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc,P2: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( p_sc_sf_sz @ P2 )
= ( predic913887884_sf_sz @ P2 ) ) ) ).
% ids.P_def
thf(fact_14_prod_Oinject,axiom,
! [X1: list_f1238882004_sc_sz,X2: list_f1238882004_sc_sz,Y1: list_f1238882004_sc_sz,Y2: list_f1238882004_sc_sz] :
( ( ( produc1822718231_sc_sz @ X1 @ X2 )
= ( produc1822718231_sc_sz @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_15_prod_Oinject,axiom,
! [X1: finite1398487019real_c,X2: finite1398487019real_c,Y1: finite1398487019real_c,Y2: finite1398487019real_c] :
( ( ( produc394644079real_c @ X1 @ X2 )
= ( produc394644079real_c @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_16_prod_Oinject,axiom,
! [X1: produc866628903_sc_sz,X2: denota610675952t_unit,Y1: produc866628903_sc_sz,Y2: denota610675952t_unit] :
( ( ( produc502616768t_unit @ X1 @ X2 )
= ( produc502616768t_unit @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_17_prod_Oinject,axiom,
! [X1: produc190496183real_c,X2: produc190496183real_c,Y1: produc190496183real_c,Y2: produc190496183real_c] :
( ( ( produc1687028567real_c @ X1 @ X2 )
= ( produc1687028567real_c @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_18_prod_Oinject,axiom,
! [X1: denota610675952t_unit,X2: produc866628903_sc_sz,Y1: denota610675952t_unit,Y2: produc866628903_sc_sz] :
( ( ( produc789536734_sc_sz @ X1 @ X2 )
= ( produc789536734_sc_sz @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_19_old_Oprod_Oinject,axiom,
! [A2: list_f1238882004_sc_sz,B2: list_f1238882004_sc_sz,A3: list_f1238882004_sc_sz,B3: list_f1238882004_sc_sz] :
( ( ( produc1822718231_sc_sz @ A2 @ B2 )
= ( produc1822718231_sc_sz @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_20_old_Oprod_Oinject,axiom,
! [A2: finite1398487019real_c,B2: finite1398487019real_c,A3: finite1398487019real_c,B3: finite1398487019real_c] :
( ( ( produc394644079real_c @ A2 @ B2 )
= ( produc394644079real_c @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_21_old_Oprod_Oinject,axiom,
! [A2: produc866628903_sc_sz,B2: denota610675952t_unit,A3: produc866628903_sc_sz,B3: denota610675952t_unit] :
( ( ( produc502616768t_unit @ A2 @ B2 )
= ( produc502616768t_unit @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_22_old_Oprod_Oinject,axiom,
! [A2: produc190496183real_c,B2: produc190496183real_c,A3: produc190496183real_c,B3: produc190496183real_c] :
( ( ( produc1687028567real_c @ A2 @ B2 )
= ( produc1687028567real_c @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_23_old_Oprod_Oinject,axiom,
! [A2: denota610675952t_unit,B2: produc866628903_sc_sz,A3: denota610675952t_unit,B3: produc866628903_sc_sz] :
( ( ( produc789536734_sc_sz @ A2 @ B2 )
= ( produc789536734_sc_sz @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_24_prod__cases3,axiom,
! [Y: produc640052711real_c] :
~ ! [A4: produc190496183real_c,B4: finite1398487019real_c,C: finite1398487019real_c] :
( Y
!= ( produc1687028567real_c @ A4 @ ( produc394644079real_c @ B4 @ C ) ) ) ).
% prod_cases3
thf(fact_25_prod__cases3,axiom,
! [Y: produc1821101996_sc_sz] :
~ ! [A4: denota610675952t_unit,B4: list_f1238882004_sc_sz,C: list_f1238882004_sc_sz] :
( Y
!= ( produc789536734_sc_sz @ A4 @ ( produc1822718231_sc_sz @ B4 @ C ) ) ) ).
% prod_cases3
thf(fact_26_old_Oprod_Oinducts,axiom,
! [P: produc866628903_sc_sz > $o,Prod: produc866628903_sc_sz] :
( ! [A4: list_f1238882004_sc_sz,B4: list_f1238882004_sc_sz] : ( P @ ( produc1822718231_sc_sz @ A4 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_27_old_Oprod_Oinducts,axiom,
! [P: produc190496183real_c > $o,Prod: produc190496183real_c] :
( ! [A4: finite1398487019real_c,B4: finite1398487019real_c] : ( P @ ( produc394644079real_c @ A4 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_28_old_Oprod_Oinducts,axiom,
! [P: produc999471430t_unit > $o,Prod: produc999471430t_unit] :
( ! [A4: produc866628903_sc_sz,B4: denota610675952t_unit] : ( P @ ( produc502616768t_unit @ A4 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_29_old_Oprod_Oinducts,axiom,
! [P: produc640052711real_c > $o,Prod: produc640052711real_c] :
( ! [A4: produc190496183real_c,B4: produc190496183real_c] : ( P @ ( produc1687028567real_c @ A4 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_30_old_Oprod_Oinducts,axiom,
! [P: produc1821101996_sc_sz > $o,Prod: produc1821101996_sc_sz] :
( ! [A4: denota610675952t_unit,B4: produc866628903_sc_sz] : ( P @ ( produc789536734_sc_sz @ A4 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_31_old_Oprod_Oexhaust,axiom,
! [Y: produc866628903_sc_sz] :
~ ! [A4: list_f1238882004_sc_sz,B4: list_f1238882004_sc_sz] :
( Y
!= ( produc1822718231_sc_sz @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_32_old_Oprod_Oexhaust,axiom,
! [Y: produc190496183real_c] :
~ ! [A4: finite1398487019real_c,B4: finite1398487019real_c] :
( Y
!= ( produc394644079real_c @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_33_old_Oprod_Oexhaust,axiom,
! [Y: produc999471430t_unit] :
~ ! [A4: produc866628903_sc_sz,B4: denota610675952t_unit] :
( Y
!= ( produc502616768t_unit @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_34_old_Oprod_Oexhaust,axiom,
! [Y: produc640052711real_c] :
~ ! [A4: produc190496183real_c,B4: produc190496183real_c] :
( Y
!= ( produc1687028567real_c @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_35_old_Oprod_Oexhaust,axiom,
! [Y: produc1821101996_sc_sz] :
~ ! [A4: denota610675952t_unit,B4: produc866628903_sc_sz] :
( Y
!= ( produc789536734_sc_sz @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_36_Pair__inject,axiom,
! [A2: list_f1238882004_sc_sz,B2: list_f1238882004_sc_sz,A3: list_f1238882004_sc_sz,B3: list_f1238882004_sc_sz] :
( ( ( produc1822718231_sc_sz @ A2 @ B2 )
= ( produc1822718231_sc_sz @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_37_Pair__inject,axiom,
! [A2: finite1398487019real_c,B2: finite1398487019real_c,A3: finite1398487019real_c,B3: finite1398487019real_c] :
( ( ( produc394644079real_c @ A2 @ B2 )
= ( produc394644079real_c @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_38_Pair__inject,axiom,
! [A2: produc866628903_sc_sz,B2: denota610675952t_unit,A3: produc866628903_sc_sz,B3: denota610675952t_unit] :
( ( ( produc502616768t_unit @ A2 @ B2 )
= ( produc502616768t_unit @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_39_Pair__inject,axiom,
! [A2: produc190496183real_c,B2: produc190496183real_c,A3: produc190496183real_c,B3: produc190496183real_c] :
( ( ( produc1687028567real_c @ A2 @ B2 )
= ( produc1687028567real_c @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_40_Pair__inject,axiom,
! [A2: denota610675952t_unit,B2: produc866628903_sc_sz,A3: denota610675952t_unit,B3: produc866628903_sc_sz] :
( ( ( produc789536734_sc_sz @ A2 @ B2 )
= ( produc789536734_sc_sz @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_41_prod__cases,axiom,
! [P: produc866628903_sc_sz > $o,P2: produc866628903_sc_sz] :
( ! [A4: list_f1238882004_sc_sz,B4: list_f1238882004_sc_sz] : ( P @ ( produc1822718231_sc_sz @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_42_prod__cases,axiom,
! [P: produc190496183real_c > $o,P2: produc190496183real_c] :
( ! [A4: finite1398487019real_c,B4: finite1398487019real_c] : ( P @ ( produc394644079real_c @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_43_prod__cases,axiom,
! [P: produc999471430t_unit > $o,P2: produc999471430t_unit] :
( ! [A4: produc866628903_sc_sz,B4: denota610675952t_unit] : ( P @ ( produc502616768t_unit @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_44_prod__cases,axiom,
! [P: produc640052711real_c > $o,P2: produc640052711real_c] :
( ! [A4: produc190496183real_c,B4: produc190496183real_c] : ( P @ ( produc1687028567real_c @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_45_prod__cases,axiom,
! [P: produc1821101996_sc_sz > $o,P2: produc1821101996_sc_sz] :
( ! [A4: denota610675952t_unit,B4: produc866628903_sc_sz] : ( P @ ( produc789536734_sc_sz @ A4 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_46_surj__pair,axiom,
! [P2: produc866628903_sc_sz] :
? [X3: list_f1238882004_sc_sz,Y3: list_f1238882004_sc_sz] :
( P2
= ( produc1822718231_sc_sz @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_47_surj__pair,axiom,
! [P2: produc190496183real_c] :
? [X3: finite1398487019real_c,Y3: finite1398487019real_c] :
( P2
= ( produc394644079real_c @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_48_surj__pair,axiom,
! [P2: produc999471430t_unit] :
? [X3: produc866628903_sc_sz,Y3: denota610675952t_unit] :
( P2
= ( produc502616768t_unit @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_49_surj__pair,axiom,
! [P2: produc640052711real_c] :
? [X3: produc190496183real_c,Y3: produc190496183real_c] :
( P2
= ( produc1687028567real_c @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_50_surj__pair,axiom,
! [P2: produc1821101996_sc_sz] :
? [X3: denota610675952t_unit,Y3: produc866628903_sc_sz] :
( P2
= ( produc789536734_sc_sz @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_51_prod__induct3,axiom,
! [P: produc640052711real_c > $o,X: produc640052711real_c] :
( ! [A4: produc190496183real_c,B4: finite1398487019real_c,C: finite1398487019real_c] : ( P @ ( produc1687028567real_c @ A4 @ ( produc394644079real_c @ B4 @ C ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_52_prod__induct3,axiom,
! [P: produc1821101996_sc_sz > $o,X: produc1821101996_sc_sz] :
( ! [A4: denota610675952t_unit,B4: list_f1238882004_sc_sz,C: list_f1238882004_sc_sz] : ( P @ ( produc789536734_sc_sz @ A4 @ ( produc1822718231_sc_sz @ B4 @ C ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_53_mem__Collect__eq,axiom,
! [A2: finite1398487019real_c,P: finite1398487019real_c > $o] :
( ( member1261661570real_c @ A2 @ ( collec230941376real_c @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_54_mem__Collect__eq,axiom,
! [A2: produc1821101996_sc_sz,P: produc1821101996_sc_sz > $o] :
( ( member1057565379_sc_sz @ A2 @ ( collec1199592193_sc_sz @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_55_mem__Collect__eq,axiom,
! [A2: produc640052711real_c,P: produc640052711real_c > $o] :
( ( member346808080real_c @ A2 @ ( collec1126017874real_c @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_56_mem__Collect__eq,axiom,
! [A2: produc999471430t_unit,P: produc999471430t_unit > $o] :
( ( member235934813t_unit @ A2 @ ( collec377961627t_unit @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_57_mem__Collect__eq,axiom,
! [A2: produc866628903_sc_sz,P: produc866628903_sc_sz > $o] :
( ( member245028432_sc_sz @ A2 @ ( collec2036662930_sc_sz @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_58_mem__Collect__eq,axiom,
! [A2: produc190496183real_c,P: produc190496183real_c > $o] :
( ( member1895684704real_c @ A2 @ ( collec1643251106real_c @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A: set_Fi1407883041real_c] :
( ( collec230941376real_c
@ ^ [X4: finite1398487019real_c] : ( member1261661570real_c @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
! [A: set_Pr1169339874_sc_sz] :
( ( collec1199592193_sc_sz
@ ^ [X4: produc1821101996_sc_sz] : ( member1057565379_sc_sz @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_61_Collect__mem__eq,axiom,
! [A: set_Pr754230471real_c] :
( ( collec1126017874real_c
@ ^ [X4: produc640052711real_c] : ( member346808080real_c @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_62_Collect__mem__eq,axiom,
! [A: set_Pr2041213564t_unit] :
( ( collec377961627t_unit
@ ^ [X4: produc999471430t_unit] : ( member235934813t_unit @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_63_Collect__mem__eq,axiom,
! [A: set_Pr1774599175_sc_sz] :
( ( collec2036662930_sc_sz
@ ^ [X4: produc866628903_sc_sz] : ( member245028432_sc_sz @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_64_Collect__mem__eq,axiom,
! [A: set_Pr1389752855real_c] :
( ( collec1643251106real_c
@ ^ [X4: produc190496183real_c] : ( member1895684704real_c @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_65_Collect__cong,axiom,
! [P: produc190496183real_c > $o,Q: produc190496183real_c > $o] :
( ! [X3: produc190496183real_c] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec1643251106real_c @ P )
= ( collec1643251106real_c @ Q ) ) ) ).
% Collect_cong
thf(fact_66_ids_Ovne23,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid2 != Vid3 ) ) ).
% ids.vne23
thf(fact_67_ids_Ovne13,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid1 != Vid3 ) ) ).
% ids.vne13
thf(fact_68_ids_Ovne12,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid1 != Vid2 ) ) ).
% ids.vne12
thf(fact_69_ids_Opne34,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid3 != Pid4 ) ) ).
% ids.pne34
thf(fact_70_ids_Opne24,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid2 != Pid4 ) ) ).
% ids.pne24
thf(fact_71_ids_Opne23,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid2 != Pid3 ) ) ).
% ids.pne23
thf(fact_72_ids_Oid__simps_I24_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid4 != Pid3 ) ) ).
% ids.id_simps(24)
thf(fact_73_ids_Oid__simps_I23_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid4 != Pid2 ) ) ).
% ids.id_simps(23)
thf(fact_74_ids_Oid__simps_I22_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid4 != Pid1 ) ) ).
% ids.id_simps(22)
thf(fact_75_ids_Oid__simps_I21_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid3 != Pid1 ) ) ).
% ids.id_simps(21)
thf(fact_76_ids_Oid__simps_I20_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid3 != Pid2 ) ) ).
% ids.id_simps(20)
thf(fact_77_ids_Oid__simps_I19_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid2 != Pid1 ) ) ).
% ids.id_simps(19)
thf(fact_78_ids_Oid__simps_I18_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid3 != Fid1 ) ) ).
% ids.id_simps(18)
thf(fact_79_ids_Oid__simps_I17_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid3 != Fid2 ) ) ).
% ids.id_simps(17)
thf(fact_80_ids_Oid__simps_I16_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid2 != Fid1 ) ) ).
% ids.id_simps(16)
thf(fact_81_ids_Oid__simps_I15_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid3 != Vid1 ) ) ).
% ids.id_simps(15)
thf(fact_82_ids_Oid__simps_I14_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid3 != Vid2 ) ) ).
% ids.id_simps(14)
thf(fact_83_ids_Oid__simps_I13_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid2 != Vid1 ) ) ).
% ids.id_simps(13)
thf(fact_84_ids_Oid__simps_I12_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid3 != Pid4 ) ) ).
% ids.id_simps(12)
thf(fact_85_ids_Oid__simps_I11_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid2 != Pid4 ) ) ).
% ids.id_simps(11)
thf(fact_86_ids_Oid__simps_I10_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid1 != Pid4 ) ) ).
% ids.id_simps(10)
thf(fact_87_ids_Oid__simps_I9_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid1 != Pid3 ) ) ).
% ids.id_simps(9)
thf(fact_88_ids_Oid__simps_I8_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid2 != Pid3 ) ) ).
% ids.id_simps(8)
thf(fact_89_ids_Oid__simps_I7_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid1 != Pid2 ) ) ).
% ids.id_simps(7)
thf(fact_90_ids_Oid__simps_I6_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid1 != Fid3 ) ) ).
% ids.id_simps(6)
thf(fact_91_ids_Oid__simps_I5_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid2 != Fid3 ) ) ).
% ids.id_simps(5)
thf(fact_92_ids_Oid__simps_I4_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid1 != Fid2 ) ) ).
% ids.id_simps(4)
thf(fact_93_ids_Oid__simps_I3_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid1 != Vid3 ) ) ).
% ids.id_simps(3)
thf(fact_94_ids_Oid__simps_I2_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid2 != Vid3 ) ) ).
% ids.id_simps(2)
thf(fact_95_ids_Oid__simps_I1_J,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Vid1 != Vid2 ) ) ).
% ids.id_simps(1)
thf(fact_96_ids__def,axiom,
( ids_sz_sf_sc
= ( ^ [Vid12: sz,Vid22: sz,Vid32: sz,Fid12: sf,Fid22: sf,Fid32: sf,Pid12: sc,Pid22: sc,Pid32: sc,Pid42: sc] :
( ( Vid12 != Vid22 )
& ( Vid22 != Vid32 )
& ( Vid12 != Vid32 )
& ( Fid12 != Fid22 )
& ( Fid22 != Fid32 )
& ( Fid12 != Fid32 )
& ( Pid12 != Pid22 )
& ( Pid22 != Pid32 )
& ( Pid12 != Pid32 )
& ( Pid12 != Pid42 )
& ( Pid22 != Pid42 )
& ( Pid32 != Pid42 ) ) ) ) ).
% ids_def
thf(fact_97_ids_Ofne12,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid1 != Fid2 ) ) ).
% ids.fne12
thf(fact_98_ids_Ofne13,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid1 != Fid3 ) ) ).
% ids.fne13
thf(fact_99_ids_Ofne23,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Fid2 != Fid3 ) ) ).
% ids.fne23
thf(fact_100_ids_Ointro,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( Vid1 != Vid2 )
=> ( ( Vid2 != Vid3 )
=> ( ( Vid1 != Vid3 )
=> ( ( Fid1 != Fid2 )
=> ( ( Fid2 != Fid3 )
=> ( ( Fid1 != Fid3 )
=> ( ( Pid1 != Pid2 )
=> ( ( Pid2 != Pid3 )
=> ( ( Pid1 != Pid3 )
=> ( ( Pid1 != Pid4 )
=> ( ( Pid2 != Pid4 )
=> ( ( Pid3 != Pid4 )
=> ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% ids.intro
thf(fact_101_ids_Opne12,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid1 != Pid2 ) ) ).
% ids.pne12
thf(fact_102_ids_Opne13,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid1 != Pid3 ) ) ).
% ids.pne13
thf(fact_103_ids_Opne14,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( Pid1 != Pid4 ) ) ).
% ids.pne14
thf(fact_104_curry__conv,axiom,
( produc1086673900c_sz_o
= ( ^ [F: produc1821101996_sc_sz > $o,A5: denota610675952t_unit,B5: produc866628903_sc_sz] : ( F @ ( produc789536734_sc_sz @ A5 @ B5 ) ) ) ) ).
% curry_conv
thf(fact_105_ssubst__Pair__rhs,axiom,
! [R: list_f1238882004_sc_sz,S2: list_f1238882004_sc_sz,R2: set_Pr1774599175_sc_sz,S3: list_f1238882004_sc_sz] :
( ( member245028432_sc_sz @ ( produc1822718231_sc_sz @ R @ S2 ) @ R2 )
=> ( ( S3 = S2 )
=> ( member245028432_sc_sz @ ( produc1822718231_sc_sz @ R @ S3 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_106_ssubst__Pair__rhs,axiom,
! [R: produc866628903_sc_sz,S2: denota610675952t_unit,R2: set_Pr2041213564t_unit,S3: denota610675952t_unit] :
( ( member235934813t_unit @ ( produc502616768t_unit @ R @ S2 ) @ R2 )
=> ( ( S3 = S2 )
=> ( member235934813t_unit @ ( produc502616768t_unit @ R @ S3 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_107_ssubst__Pair__rhs,axiom,
! [R: produc190496183real_c,S2: produc190496183real_c,R2: set_Pr754230471real_c,S3: produc190496183real_c] :
( ( member346808080real_c @ ( produc1687028567real_c @ R @ S2 ) @ R2 )
=> ( ( S3 = S2 )
=> ( member346808080real_c @ ( produc1687028567real_c @ R @ S3 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_108_ssubst__Pair__rhs,axiom,
! [R: finite1398487019real_c,S2: finite1398487019real_c,R2: set_Pr1389752855real_c,S3: finite1398487019real_c] :
( ( member1895684704real_c @ ( produc394644079real_c @ R @ S2 ) @ R2 )
=> ( ( S3 = S2 )
=> ( member1895684704real_c @ ( produc394644079real_c @ R @ S3 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_109_ssubst__Pair__rhs,axiom,
! [R: denota610675952t_unit,S2: produc866628903_sc_sz,R2: set_Pr1169339874_sc_sz,S3: produc866628903_sc_sz] :
( ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ R @ S2 ) @ R2 )
=> ( ( S3 = S2 )
=> ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ R @ S3 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_110_curryI,axiom,
! [F2: produc866628903_sc_sz > $o,A2: list_f1238882004_sc_sz,B2: list_f1238882004_sc_sz] :
( ( F2 @ ( produc1822718231_sc_sz @ A2 @ B2 ) )
=> ( produc880541325c_sz_o @ F2 @ A2 @ B2 ) ) ).
% curryI
thf(fact_111_curryI,axiom,
! [F2: produc190496183real_c > $o,A2: finite1398487019real_c,B2: finite1398487019real_c] :
( ( F2 @ ( produc394644079real_c @ A2 @ B2 ) )
=> ( produc1716060341al_c_o @ F2 @ A2 @ B2 ) ) ).
% curryI
thf(fact_112_curryI,axiom,
! [F2: produc999471430t_unit > $o,A2: produc866628903_sc_sz,B2: denota610675952t_unit] :
( ( F2 @ ( produc502616768t_unit @ A2 @ B2 ) )
=> ( produc38128778unit_o @ F2 @ A2 @ B2 ) ) ).
% curryI
thf(fact_113_curryI,axiom,
! [F2: produc640052711real_c > $o,A2: produc190496183real_c,B2: produc190496183real_c] :
( ( F2 @ ( produc1687028567real_c @ A2 @ B2 ) )
=> ( produc440529997al_c_o @ F2 @ A2 @ B2 ) ) ).
% curryI
thf(fact_114_curryI,axiom,
! [F2: produc1821101996_sc_sz > $o,A2: denota610675952t_unit,B2: produc866628903_sc_sz] :
( ( F2 @ ( produc789536734_sc_sz @ A2 @ B2 ) )
=> ( produc1086673900c_sz_o @ F2 @ A2 @ B2 ) ) ).
% curryI
thf(fact_115_swap__simp,axiom,
! [X: list_f1238882004_sc_sz,Y: list_f1238882004_sc_sz] :
( ( produc1145837239_sc_sz @ ( produc1822718231_sc_sz @ X @ Y ) )
= ( produc1822718231_sc_sz @ Y @ X ) ) ).
% swap_simp
thf(fact_116_swap__simp,axiom,
! [X: produc190496183real_c,Y: produc190496183real_c] :
( ( produc350724343real_c @ ( produc1687028567real_c @ X @ Y ) )
= ( produc1687028567real_c @ Y @ X ) ) ).
% swap_simp
thf(fact_117_swap__simp,axiom,
! [X: finite1398487019real_c,Y: finite1398487019real_c] :
( ( produc704157711real_c @ ( produc394644079real_c @ X @ Y ) )
= ( produc394644079real_c @ Y @ X ) ) ).
% swap_simp
thf(fact_118_swap__simp,axiom,
! [X: produc866628903_sc_sz,Y: denota610675952t_unit] :
( ( produc245128480t_unit @ ( produc502616768t_unit @ X @ Y ) )
= ( produc789536734_sc_sz @ Y @ X ) ) ).
% swap_simp
thf(fact_119_swap__simp,axiom,
! [X: denota610675952t_unit,Y: produc866628903_sc_sz] :
( ( produc532048446_sc_sz @ ( produc789536734_sc_sz @ X @ Y ) )
= ( produc502616768t_unit @ Y @ X ) ) ).
% swap_simp
thf(fact_120_ids_Ovalid__def,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc,Phi2: formula_sf_sc_sz] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( denota1946510237_sc_sz @ Phi2 )
= ( ! [I: denota610675952t_unit,Nu: produc1149990247eal_sz] :
( ( denota1579475975_sc_sz @ I )
=> ( member630771088eal_sz @ Nu @ ( denota1711272799_sc_sz @ I @ Phi2 ) ) ) ) ) ) ).
% ids.valid_def
thf(fact_121_swap__swap,axiom,
! [P2: produc999471430t_unit] :
( ( produc532048446_sc_sz @ ( produc245128480t_unit @ P2 ) )
= P2 ) ).
% swap_swap
thf(fact_122_swap__swap,axiom,
! [P2: produc1821101996_sc_sz] :
( ( produc245128480t_unit @ ( produc532048446_sc_sz @ P2 ) )
= P2 ) ).
% swap_swap
thf(fact_123_swap__swap,axiom,
! [P2: produc190496183real_c] :
( ( produc704157711real_c @ ( produc704157711real_c @ P2 ) )
= P2 ) ).
% swap_swap
thf(fact_124_curryE,axiom,
! [F2: produc866628903_sc_sz > $o,A2: list_f1238882004_sc_sz,B2: list_f1238882004_sc_sz] :
( ( produc880541325c_sz_o @ F2 @ A2 @ B2 )
=> ( F2 @ ( produc1822718231_sc_sz @ A2 @ B2 ) ) ) ).
% curryE
thf(fact_125_curryE,axiom,
! [F2: produc190496183real_c > $o,A2: finite1398487019real_c,B2: finite1398487019real_c] :
( ( produc1716060341al_c_o @ F2 @ A2 @ B2 )
=> ( F2 @ ( produc394644079real_c @ A2 @ B2 ) ) ) ).
% curryE
thf(fact_126_curryE,axiom,
! [F2: produc999471430t_unit > $o,A2: produc866628903_sc_sz,B2: denota610675952t_unit] :
( ( produc38128778unit_o @ F2 @ A2 @ B2 )
=> ( F2 @ ( produc502616768t_unit @ A2 @ B2 ) ) ) ).
% curryE
thf(fact_127_curryE,axiom,
! [F2: produc640052711real_c > $o,A2: produc190496183real_c,B2: produc190496183real_c] :
( ( produc440529997al_c_o @ F2 @ A2 @ B2 )
=> ( F2 @ ( produc1687028567real_c @ A2 @ B2 ) ) ) ).
% curryE
thf(fact_128_curryE,axiom,
! [F2: produc1821101996_sc_sz > $o,A2: denota610675952t_unit,B2: produc866628903_sc_sz] :
( ( produc1086673900c_sz_o @ F2 @ A2 @ B2 )
=> ( F2 @ ( produc789536734_sc_sz @ A2 @ B2 ) ) ) ).
% curryE
thf(fact_129_curryD,axiom,
! [F2: produc866628903_sc_sz > $o,A2: list_f1238882004_sc_sz,B2: list_f1238882004_sc_sz] :
( ( produc880541325c_sz_o @ F2 @ A2 @ B2 )
=> ( F2 @ ( produc1822718231_sc_sz @ A2 @ B2 ) ) ) ).
% curryD
thf(fact_130_curryD,axiom,
! [F2: produc190496183real_c > $o,A2: finite1398487019real_c,B2: finite1398487019real_c] :
( ( produc1716060341al_c_o @ F2 @ A2 @ B2 )
=> ( F2 @ ( produc394644079real_c @ A2 @ B2 ) ) ) ).
% curryD
thf(fact_131_curryD,axiom,
! [F2: produc999471430t_unit > $o,A2: produc866628903_sc_sz,B2: denota610675952t_unit] :
( ( produc38128778unit_o @ F2 @ A2 @ B2 )
=> ( F2 @ ( produc502616768t_unit @ A2 @ B2 ) ) ) ).
% curryD
thf(fact_132_curryD,axiom,
! [F2: produc640052711real_c > $o,A2: produc190496183real_c,B2: produc190496183real_c] :
( ( produc440529997al_c_o @ F2 @ A2 @ B2 )
=> ( F2 @ ( produc1687028567real_c @ A2 @ B2 ) ) ) ).
% curryD
thf(fact_133_curryD,axiom,
! [F2: produc1821101996_sc_sz > $o,A2: denota610675952t_unit,B2: produc866628903_sc_sz] :
( ( produc1086673900c_sz_o @ F2 @ A2 @ B2 )
=> ( F2 @ ( produc789536734_sc_sz @ A2 @ B2 ) ) ) ).
% curryD
thf(fact_134_pair__in__swap__image,axiom,
! [Y: list_f1238882004_sc_sz,X: list_f1238882004_sc_sz,A: set_Pr1774599175_sc_sz] :
( ( member245028432_sc_sz @ ( produc1822718231_sc_sz @ Y @ X ) @ ( image_1313166277_sc_sz @ produc1145837239_sc_sz @ A ) )
= ( member245028432_sc_sz @ ( produc1822718231_sc_sz @ X @ Y ) @ A ) ) ).
% pair_in_swap_image
thf(fact_135_pair__in__swap__image,axiom,
! [Y: produc190496183real_c,X: produc190496183real_c,A: set_Pr754230471real_c] :
( ( member346808080real_c @ ( produc1687028567real_c @ Y @ X ) @ ( image_2041365445real_c @ produc350724343real_c @ A ) )
= ( member346808080real_c @ ( produc1687028567real_c @ X @ Y ) @ A ) ) ).
% pair_in_swap_image
thf(fact_136_pair__in__swap__image,axiom,
! [Y: finite1398487019real_c,X: finite1398487019real_c,A: set_Pr1389752855real_c] :
( ( member1895684704real_c @ ( produc394644079real_c @ Y @ X ) @ ( image_738229701real_c @ produc704157711real_c @ A ) )
= ( member1895684704real_c @ ( produc394644079real_c @ X @ Y ) @ A ) ) ).
% pair_in_swap_image
thf(fact_137_pair__in__swap__image,axiom,
! [Y: produc866628903_sc_sz,X: denota610675952t_unit,A: set_Pr1169339874_sc_sz] :
( ( member235934813t_unit @ ( produc502616768t_unit @ Y @ X ) @ ( image_904562809t_unit @ produc532048446_sc_sz @ A ) )
= ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ X @ Y ) @ A ) ) ).
% pair_in_swap_image
thf(fact_138_pair__in__swap__image,axiom,
! [Y: denota610675952t_unit,X: produc866628903_sc_sz,A: set_Pr2041213564t_unit] :
( ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ Y @ X ) @ ( image_1426741113_sc_sz @ produc245128480t_unit @ A ) )
= ( member235934813t_unit @ ( produc502616768t_unit @ X @ Y ) @ A ) ) ).
% pair_in_swap_image
thf(fact_139_internal__case__prod__def,axiom,
produc383750338real_c = produc1575513068real_c ).
% internal_case_prod_def
thf(fact_140_internal__case__prod__def,axiom,
produc198687651c_sz_o = produc619560141c_sz_o ).
% internal_case_prod_def
thf(fact_141_internal__case__prod__def,axiom,
produc1894848802real_c = produc237521996real_c ).
% internal_case_prod_def
thf(fact_142_internal__case__prod__def,axiom,
produc1442549981real_c = produc107375667real_c ).
% internal_case_prod_def
thf(fact_143_internal__case__prod__def,axiom,
produc2099580355_sc_sz = produc435149849_sc_sz ).
% internal_case_prod_def
thf(fact_144_apsnd__conv,axiom,
! [F2: list_f1238882004_sc_sz > list_f1238882004_sc_sz,X: list_f1238882004_sc_sz,Y: list_f1238882004_sc_sz] :
( ( produc233219286_sc_sz @ F2 @ ( produc1822718231_sc_sz @ X @ Y ) )
= ( produc1822718231_sc_sz @ X @ ( F2 @ Y ) ) ) ).
% apsnd_conv
thf(fact_145_apsnd__conv,axiom,
! [F2: finite1398487019real_c > finite1398487019real_c,X: finite1398487019real_c,Y: finite1398487019real_c] :
( ( produc981295297real_c @ F2 @ ( produc394644079real_c @ X @ Y ) )
= ( produc394644079real_c @ X @ ( F2 @ Y ) ) ) ).
% apsnd_conv
thf(fact_146_apsnd__conv,axiom,
! [F2: denota610675952t_unit > denota610675952t_unit,X: produc866628903_sc_sz,Y: denota610675952t_unit] :
( ( produc793407943_sc_sz @ F2 @ ( produc502616768t_unit @ X @ Y ) )
= ( produc502616768t_unit @ X @ ( F2 @ Y ) ) ) ).
% apsnd_conv
thf(fact_147_apsnd__conv,axiom,
! [F2: produc190496183real_c > produc190496183real_c,X: produc190496183real_c,Y: produc190496183real_c] :
( ( produc2089853305real_c @ F2 @ ( produc1687028567real_c @ X @ Y ) )
= ( produc1687028567real_c @ X @ ( F2 @ Y ) ) ) ).
% apsnd_conv
thf(fact_148_apsnd__conv,axiom,
! [F2: produc866628903_sc_sz > produc866628903_sc_sz,X: denota610675952t_unit,Y: produc866628903_sc_sz] :
( ( produc1959054510t_unit @ F2 @ ( produc789536734_sc_sz @ X @ Y ) )
= ( produc789536734_sc_sz @ X @ ( F2 @ Y ) ) ) ).
% apsnd_conv
thf(fact_149_mem__case__prodI2,axiom,
! [P2: produc1821101996_sc_sz,Z: produc190496183real_c,C2: denota610675952t_unit > produc866628903_sc_sz > set_Pr1389752855real_c] :
( ! [A4: denota610675952t_unit,B4: produc866628903_sc_sz] :
( ( P2
= ( produc789536734_sc_sz @ A4 @ B4 ) )
=> ( member1895684704real_c @ Z @ ( C2 @ A4 @ B4 ) ) )
=> ( member1895684704real_c @ Z @ ( produc237521996real_c @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_150_mem__case__prodI2,axiom,
! [P2: produc190496183real_c,Z: finite1398487019real_c,C2: finite1398487019real_c > finite1398487019real_c > set_Fi1407883041real_c] :
( ! [A4: finite1398487019real_c,B4: finite1398487019real_c] :
( ( P2
= ( produc394644079real_c @ A4 @ B4 ) )
=> ( member1261661570real_c @ Z @ ( C2 @ A4 @ B4 ) ) )
=> ( member1261661570real_c @ Z @ ( produc29022117real_c @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_151_mem__case__prodI2,axiom,
! [P2: produc866628903_sc_sz,Z: finite1398487019real_c,C2: list_f1238882004_sc_sz > list_f1238882004_sc_sz > set_Fi1407883041real_c] :
( ! [A4: list_f1238882004_sc_sz,B4: list_f1238882004_sc_sz] :
( ( P2
= ( produc1822718231_sc_sz @ A4 @ B4 ) )
=> ( member1261661570real_c @ Z @ ( C2 @ A4 @ B4 ) ) )
=> ( member1261661570real_c @ Z @ ( produc1449736269real_c @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_152_mem__case__prodI2,axiom,
! [P2: produc190496183real_c,Z: produc190496183real_c,C2: finite1398487019real_c > finite1398487019real_c > set_Pr1389752855real_c] :
( ! [A4: finite1398487019real_c,B4: finite1398487019real_c] :
( ( P2
= ( produc394644079real_c @ A4 @ B4 ) )
=> ( member1895684704real_c @ Z @ ( C2 @ A4 @ B4 ) ) )
=> ( member1895684704real_c @ Z @ ( produc923381907real_c @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_153_mem__case__prodI2,axiom,
! [P2: produc866628903_sc_sz,Z: produc190496183real_c,C2: list_f1238882004_sc_sz > list_f1238882004_sc_sz > set_Pr1389752855real_c] :
( ! [A4: list_f1238882004_sc_sz,B4: list_f1238882004_sc_sz] :
( ( P2
= ( produc1822718231_sc_sz @ A4 @ B4 ) )
=> ( member1895684704real_c @ Z @ ( C2 @ A4 @ B4 ) ) )
=> ( member1895684704real_c @ Z @ ( produc6841835real_c @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_154_mem__case__prodI2,axiom,
! [P2: produc190496183real_c,Z: produc866628903_sc_sz,C2: finite1398487019real_c > finite1398487019real_c > set_Pr1774599175_sc_sz] :
( ! [A4: finite1398487019real_c,B4: finite1398487019real_c] :
( ( P2
= ( produc394644079real_c @ A4 @ B4 ) )
=> ( member245028432_sc_sz @ Z @ ( C2 @ A4 @ B4 ) ) )
=> ( member245028432_sc_sz @ Z @ ( produc642283651_sc_sz @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_155_mem__case__prodI2,axiom,
! [P2: produc640052711real_c,Z: finite1398487019real_c,C2: produc190496183real_c > produc190496183real_c > set_Fi1407883041real_c] :
( ! [A4: produc190496183real_c,B4: produc190496183real_c] :
( ( P2
= ( produc1687028567real_c @ A4 @ B4 ) )
=> ( member1261661570real_c @ Z @ ( C2 @ A4 @ B4 ) ) )
=> ( member1261661570real_c @ Z @ ( produc1066779277real_c @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_156_mem__case__prodI2,axiom,
! [P2: produc1821101996_sc_sz,Z: finite1398487019real_c,C2: denota610675952t_unit > produc866628903_sc_sz > set_Fi1407883041real_c] :
( ! [A4: denota610675952t_unit,B4: produc866628903_sc_sz] :
( ( P2
= ( produc789536734_sc_sz @ A4 @ B4 ) )
=> ( member1261661570real_c @ Z @ ( C2 @ A4 @ B4 ) ) )
=> ( member1261661570real_c @ Z @ ( produc2068697644real_c @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_157_mem__case__prodI2,axiom,
! [P2: produc999471430t_unit,Z: finite1398487019real_c,C2: produc866628903_sc_sz > denota610675952t_unit > set_Fi1407883041real_c] :
( ! [A4: produc866628903_sc_sz,B4: denota610675952t_unit] :
( ( P2
= ( produc502616768t_unit @ A4 @ B4 ) )
=> ( member1261661570real_c @ Z @ ( C2 @ A4 @ B4 ) ) )
=> ( member1261661570real_c @ Z @ ( produc1368849166real_c @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_158_mem__case__prodI2,axiom,
! [P2: produc866628903_sc_sz,Z: produc866628903_sc_sz,C2: list_f1238882004_sc_sz > list_f1238882004_sc_sz > set_Pr1774599175_sc_sz] :
( ! [A4: list_f1238882004_sc_sz,B4: list_f1238882004_sc_sz] :
( ( P2
= ( produc1822718231_sc_sz @ A4 @ B4 ) )
=> ( member245028432_sc_sz @ Z @ ( C2 @ A4 @ B4 ) ) )
=> ( member245028432_sc_sz @ Z @ ( produc1563580891_sc_sz @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_159_mem__case__prodI,axiom,
! [Z: produc190496183real_c,C2: denota610675952t_unit > produc866628903_sc_sz > set_Pr1389752855real_c,A2: denota610675952t_unit,B2: produc866628903_sc_sz] :
( ( member1895684704real_c @ Z @ ( C2 @ A2 @ B2 ) )
=> ( member1895684704real_c @ Z @ ( produc237521996real_c @ C2 @ ( produc789536734_sc_sz @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_160_mem__case__prodI,axiom,
! [Z: finite1398487019real_c,C2: finite1398487019real_c > finite1398487019real_c > set_Fi1407883041real_c,A2: finite1398487019real_c,B2: finite1398487019real_c] :
( ( member1261661570real_c @ Z @ ( C2 @ A2 @ B2 ) )
=> ( member1261661570real_c @ Z @ ( produc29022117real_c @ C2 @ ( produc394644079real_c @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_161_mem__case__prodI,axiom,
! [Z: finite1398487019real_c,C2: list_f1238882004_sc_sz > list_f1238882004_sc_sz > set_Fi1407883041real_c,A2: list_f1238882004_sc_sz,B2: list_f1238882004_sc_sz] :
( ( member1261661570real_c @ Z @ ( C2 @ A2 @ B2 ) )
=> ( member1261661570real_c @ Z @ ( produc1449736269real_c @ C2 @ ( produc1822718231_sc_sz @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_162_mem__case__prodI,axiom,
! [Z: produc190496183real_c,C2: finite1398487019real_c > finite1398487019real_c > set_Pr1389752855real_c,A2: finite1398487019real_c,B2: finite1398487019real_c] :
( ( member1895684704real_c @ Z @ ( C2 @ A2 @ B2 ) )
=> ( member1895684704real_c @ Z @ ( produc923381907real_c @ C2 @ ( produc394644079real_c @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_163_mem__case__prodI,axiom,
! [Z: produc190496183real_c,C2: list_f1238882004_sc_sz > list_f1238882004_sc_sz > set_Pr1389752855real_c,A2: list_f1238882004_sc_sz,B2: list_f1238882004_sc_sz] :
( ( member1895684704real_c @ Z @ ( C2 @ A2 @ B2 ) )
=> ( member1895684704real_c @ Z @ ( produc6841835real_c @ C2 @ ( produc1822718231_sc_sz @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_164_mem__case__prodI,axiom,
! [Z: produc866628903_sc_sz,C2: finite1398487019real_c > finite1398487019real_c > set_Pr1774599175_sc_sz,A2: finite1398487019real_c,B2: finite1398487019real_c] :
( ( member245028432_sc_sz @ Z @ ( C2 @ A2 @ B2 ) )
=> ( member245028432_sc_sz @ Z @ ( produc642283651_sc_sz @ C2 @ ( produc394644079real_c @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_165_mem__case__prodI,axiom,
! [Z: finite1398487019real_c,C2: produc190496183real_c > produc190496183real_c > set_Fi1407883041real_c,A2: produc190496183real_c,B2: produc190496183real_c] :
( ( member1261661570real_c @ Z @ ( C2 @ A2 @ B2 ) )
=> ( member1261661570real_c @ Z @ ( produc1066779277real_c @ C2 @ ( produc1687028567real_c @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_166_mem__case__prodI,axiom,
! [Z: finite1398487019real_c,C2: denota610675952t_unit > produc866628903_sc_sz > set_Fi1407883041real_c,A2: denota610675952t_unit,B2: produc866628903_sc_sz] :
( ( member1261661570real_c @ Z @ ( C2 @ A2 @ B2 ) )
=> ( member1261661570real_c @ Z @ ( produc2068697644real_c @ C2 @ ( produc789536734_sc_sz @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_167_mem__case__prodI,axiom,
! [Z: finite1398487019real_c,C2: produc866628903_sc_sz > denota610675952t_unit > set_Fi1407883041real_c,A2: produc866628903_sc_sz,B2: denota610675952t_unit] :
( ( member1261661570real_c @ Z @ ( C2 @ A2 @ B2 ) )
=> ( member1261661570real_c @ Z @ ( produc1368849166real_c @ C2 @ ( produc502616768t_unit @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_168_mem__case__prodI,axiom,
! [Z: produc866628903_sc_sz,C2: list_f1238882004_sc_sz > list_f1238882004_sc_sz > set_Pr1774599175_sc_sz,A2: list_f1238882004_sc_sz,B2: list_f1238882004_sc_sz] :
( ( member245028432_sc_sz @ Z @ ( C2 @ A2 @ B2 ) )
=> ( member245028432_sc_sz @ Z @ ( produc1563580891_sc_sz @ C2 @ ( produc1822718231_sc_sz @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_169_case__prodI2,axiom,
! [P2: produc866628903_sc_sz,C2: list_f1238882004_sc_sz > list_f1238882004_sc_sz > $o] :
( ! [A4: list_f1238882004_sc_sz,B4: list_f1238882004_sc_sz] :
( ( P2
= ( produc1822718231_sc_sz @ A4 @ B4 ) )
=> ( C2 @ A4 @ B4 ) )
=> ( produc2039065452c_sz_o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_170_case__prodI2,axiom,
! [P2: produc190496183real_c,C2: finite1398487019real_c > finite1398487019real_c > $o] :
( ! [A4: finite1398487019real_c,B4: finite1398487019real_c] :
( ( P2
= ( produc394644079real_c @ A4 @ B4 ) )
=> ( C2 @ A4 @ B4 ) )
=> ( produc353381908al_c_o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_171_case__prodI2,axiom,
! [P2: produc999471430t_unit,C2: produc866628903_sc_sz > denota610675952t_unit > $o] :
( ! [A4: produc866628903_sc_sz,B4: denota610675952t_unit] :
( ( P2
= ( produc502616768t_unit @ A4 @ B4 ) )
=> ( C2 @ A4 @ B4 ) )
=> ( produc1718498667unit_o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_172_case__prodI2,axiom,
! [P2: produc640052711real_c,C2: produc190496183real_c > produc190496183real_c > $o] :
( ! [A4: produc190496183real_c,B4: produc190496183real_c] :
( ( P2
= ( produc1687028567real_c @ A4 @ B4 ) )
=> ( C2 @ A4 @ B4 ) )
=> ( produc1726730540al_c_o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_173_case__prodI2,axiom,
! [P2: produc1821101996_sc_sz,C2: denota610675952t_unit > produc866628903_sc_sz > $o] :
( ! [A4: denota610675952t_unit,B4: produc866628903_sc_sz] :
( ( P2
= ( produc789536734_sc_sz @ A4 @ B4 ) )
=> ( C2 @ A4 @ B4 ) )
=> ( produc619560141c_sz_o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_174_case__prodI,axiom,
! [F2: list_f1238882004_sc_sz > list_f1238882004_sc_sz > $o,A2: list_f1238882004_sc_sz,B2: list_f1238882004_sc_sz] :
( ( F2 @ A2 @ B2 )
=> ( produc2039065452c_sz_o @ F2 @ ( produc1822718231_sc_sz @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_175_case__prodI,axiom,
! [F2: finite1398487019real_c > finite1398487019real_c > $o,A2: finite1398487019real_c,B2: finite1398487019real_c] :
( ( F2 @ A2 @ B2 )
=> ( produc353381908al_c_o @ F2 @ ( produc394644079real_c @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_176_case__prodI,axiom,
! [F2: produc866628903_sc_sz > denota610675952t_unit > $o,A2: produc866628903_sc_sz,B2: denota610675952t_unit] :
( ( F2 @ A2 @ B2 )
=> ( produc1718498667unit_o @ F2 @ ( produc502616768t_unit @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_177_case__prodI,axiom,
! [F2: produc190496183real_c > produc190496183real_c > $o,A2: produc190496183real_c,B2: produc190496183real_c] :
( ( F2 @ A2 @ B2 )
=> ( produc1726730540al_c_o @ F2 @ ( produc1687028567real_c @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_178_case__prodI,axiom,
! [F2: denota610675952t_unit > produc866628903_sc_sz > $o,A2: denota610675952t_unit,B2: produc866628903_sc_sz] :
( ( F2 @ A2 @ B2 )
=> ( produc619560141c_sz_o @ F2 @ ( produc789536734_sc_sz @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_179_case__prod__conv,axiom,
! [F2: denota610675952t_unit > produc866628903_sc_sz > produc190496183real_c,A2: denota610675952t_unit,B2: produc866628903_sc_sz] :
( ( produc1575513068real_c @ F2 @ ( produc789536734_sc_sz @ A2 @ B2 ) )
= ( F2 @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_180_case__prod__conv,axiom,
! [F2: denota610675952t_unit > produc866628903_sc_sz > $o,A2: denota610675952t_unit,B2: produc866628903_sc_sz] :
( ( produc619560141c_sz_o @ F2 @ ( produc789536734_sc_sz @ A2 @ B2 ) )
= ( F2 @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_181_case__prod__conv,axiom,
! [F2: denota610675952t_unit > produc866628903_sc_sz > set_Pr1389752855real_c,A2: denota610675952t_unit,B2: produc866628903_sc_sz] :
( ( produc237521996real_c @ F2 @ ( produc789536734_sc_sz @ A2 @ B2 ) )
= ( F2 @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_182_case__prod__conv,axiom,
! [F2: finite1398487019real_c > finite1398487019real_c > produc190496183real_c,A2: finite1398487019real_c,B2: finite1398487019real_c] :
( ( produc107375667real_c @ F2 @ ( produc394644079real_c @ A2 @ B2 ) )
= ( F2 @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_183_case__prod__conv,axiom,
! [F2: produc866628903_sc_sz > denota610675952t_unit > produc1821101996_sc_sz,A2: produc866628903_sc_sz,B2: denota610675952t_unit] :
( ( produc435149849_sc_sz @ F2 @ ( produc502616768t_unit @ A2 @ B2 ) )
= ( F2 @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_184_curry__case__prod,axiom,
! [F2: denota610675952t_unit > produc866628903_sc_sz > produc190496183real_c] :
( ( produc753314315real_c @ ( produc1575513068real_c @ F2 ) )
= F2 ) ).
% curry_case_prod
thf(fact_185_curry__case__prod,axiom,
! [F2: denota610675952t_unit > produc866628903_sc_sz > $o] :
( ( produc1086673900c_sz_o @ ( produc619560141c_sz_o @ F2 ) )
= F2 ) ).
% curry_case_prod
thf(fact_186_curry__case__prod,axiom,
! [F2: denota610675952t_unit > produc866628903_sc_sz > set_Pr1389752855real_c] :
( ( produc1827049067real_c @ ( produc237521996real_c @ F2 ) )
= F2 ) ).
% curry_case_prod
thf(fact_187_curry__case__prod,axiom,
! [F2: finite1398487019real_c > finite1398487019real_c > produc190496183real_c] :
( ( produc1119115220real_c @ ( produc107375667real_c @ F2 ) )
= F2 ) ).
% curry_case_prod
thf(fact_188_curry__case__prod,axiom,
! [F2: produc866628903_sc_sz > denota610675952t_unit > produc1821101996_sc_sz] :
( ( produc295339066_sc_sz @ ( produc435149849_sc_sz @ F2 ) )
= F2 ) ).
% curry_case_prod
thf(fact_189_case__prod__curry,axiom,
! [F2: produc1821101996_sc_sz > produc190496183real_c] :
( ( produc1575513068real_c @ ( produc753314315real_c @ F2 ) )
= F2 ) ).
% case_prod_curry
thf(fact_190_case__prod__curry,axiom,
! [F2: produc1821101996_sc_sz > $o] :
( ( produc619560141c_sz_o @ ( produc1086673900c_sz_o @ F2 ) )
= F2 ) ).
% case_prod_curry
thf(fact_191_case__prod__curry,axiom,
! [F2: produc1821101996_sc_sz > set_Pr1389752855real_c] :
( ( produc237521996real_c @ ( produc1827049067real_c @ F2 ) )
= F2 ) ).
% case_prod_curry
thf(fact_192_case__prod__curry,axiom,
! [F2: produc190496183real_c > produc190496183real_c] :
( ( produc107375667real_c @ ( produc1119115220real_c @ F2 ) )
= F2 ) ).
% case_prod_curry
thf(fact_193_case__prod__curry,axiom,
! [F2: produc999471430t_unit > produc1821101996_sc_sz] :
( ( produc435149849_sc_sz @ ( produc295339066_sc_sz @ F2 ) )
= F2 ) ).
% case_prod_curry
thf(fact_194_case__swap,axiom,
! [F2: denota610675952t_unit > produc866628903_sc_sz > produc190496183real_c,P2: produc1821101996_sc_sz] :
( ( produc645763722real_c
@ ^ [Y4: produc866628903_sc_sz,X4: denota610675952t_unit] : ( F2 @ X4 @ Y4 )
@ ( produc532048446_sc_sz @ P2 ) )
= ( produc1575513068real_c @ F2 @ P2 ) ) ).
% case_swap
thf(fact_195_case__swap,axiom,
! [F2: denota610675952t_unit > produc866628903_sc_sz > $o,P2: produc1821101996_sc_sz] :
( ( produc1718498667unit_o
@ ^ [Y4: produc866628903_sc_sz,X4: denota610675952t_unit] : ( F2 @ X4 @ Y4 )
@ ( produc532048446_sc_sz @ P2 ) )
= ( produc619560141c_sz_o @ F2 @ P2 ) ) ).
% case_swap
thf(fact_196_case__swap,axiom,
! [F2: denota610675952t_unit > produc866628903_sc_sz > set_Pr1389752855real_c,P2: produc1821101996_sc_sz] :
( ( produc593754858real_c
@ ^ [Y4: produc866628903_sc_sz,X4: denota610675952t_unit] : ( F2 @ X4 @ Y4 )
@ ( produc532048446_sc_sz @ P2 ) )
= ( produc237521996real_c @ F2 @ P2 ) ) ).
% case_swap
thf(fact_197_case__swap,axiom,
! [F2: produc866628903_sc_sz > denota610675952t_unit > produc1821101996_sc_sz,P2: produc999471430t_unit] :
( ( produc448466487_sc_sz
@ ^ [Y4: denota610675952t_unit,X4: produc866628903_sc_sz] : ( F2 @ X4 @ Y4 )
@ ( produc245128480t_unit @ P2 ) )
= ( produc435149849_sc_sz @ F2 @ P2 ) ) ).
% case_swap
thf(fact_198_case__swap,axiom,
! [F2: produc866628903_sc_sz > denota610675952t_unit > produc190496183real_c,P2: produc999471430t_unit] :
( ( produc1575513068real_c
@ ^ [Y4: denota610675952t_unit,X4: produc866628903_sc_sz] : ( F2 @ X4 @ Y4 )
@ ( produc245128480t_unit @ P2 ) )
= ( produc645763722real_c @ F2 @ P2 ) ) ).
% case_swap
thf(fact_199_case__swap,axiom,
! [F2: produc866628903_sc_sz > denota610675952t_unit > $o,P2: produc999471430t_unit] :
( ( produc619560141c_sz_o
@ ^ [Y4: denota610675952t_unit,X4: produc866628903_sc_sz] : ( F2 @ X4 @ Y4 )
@ ( produc245128480t_unit @ P2 ) )
= ( produc1718498667unit_o @ F2 @ P2 ) ) ).
% case_swap
thf(fact_200_case__swap,axiom,
! [F2: produc866628903_sc_sz > denota610675952t_unit > set_Pr1389752855real_c,P2: produc999471430t_unit] :
( ( produc237521996real_c
@ ^ [Y4: denota610675952t_unit,X4: produc866628903_sc_sz] : ( F2 @ X4 @ Y4 )
@ ( produc245128480t_unit @ P2 ) )
= ( produc593754858real_c @ F2 @ P2 ) ) ).
% case_swap
thf(fact_201_case__swap,axiom,
! [F2: finite1398487019real_c > finite1398487019real_c > produc190496183real_c,P2: produc190496183real_c] :
( ( produc107375667real_c
@ ^ [Y4: finite1398487019real_c,X4: finite1398487019real_c] : ( F2 @ X4 @ Y4 )
@ ( produc704157711real_c @ P2 ) )
= ( produc107375667real_c @ F2 @ P2 ) ) ).
% case_swap
thf(fact_202_case__swap,axiom,
! [F2: denota610675952t_unit > produc866628903_sc_sz > produc1821101996_sc_sz,P2: produc1821101996_sc_sz] :
( ( produc435149849_sc_sz
@ ^ [Y4: produc866628903_sc_sz,X4: denota610675952t_unit] : ( F2 @ X4 @ Y4 )
@ ( produc532048446_sc_sz @ P2 ) )
= ( produc448466487_sc_sz @ F2 @ P2 ) ) ).
% case_swap
thf(fact_203_pair__imageI,axiom,
! [A2: finite1398487019real_c,B2: finite1398487019real_c,A: set_Pr1389752855real_c,F2: finite1398487019real_c > finite1398487019real_c > produc190496183real_c] :
( ( member1895684704real_c @ ( produc394644079real_c @ A2 @ B2 ) @ A )
=> ( member1895684704real_c @ ( F2 @ A2 @ B2 ) @ ( image_738229701real_c @ ( produc107375667real_c @ F2 ) @ A ) ) ) ).
% pair_imageI
thf(fact_204_pair__imageI,axiom,
! [A2: denota610675952t_unit,B2: produc866628903_sc_sz,A: set_Pr1169339874_sc_sz,F2: denota610675952t_unit > produc866628903_sc_sz > produc190496183real_c] :
( ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ A2 @ B2 ) @ A )
=> ( member1895684704real_c @ ( F2 @ A2 @ B2 ) @ ( image_386905412real_c @ ( produc1575513068real_c @ F2 ) @ A ) ) ) ).
% pair_imageI
thf(fact_205_pair__imageI,axiom,
! [A2: finite1398487019real_c,B2: finite1398487019real_c,A: set_Pr1389752855real_c,F2: finite1398487019real_c > finite1398487019real_c > finite1398487019real_c] :
( ( member1895684704real_c @ ( produc394644079real_c @ A2 @ B2 ) @ A )
=> ( member1261661570real_c @ ( F2 @ A2 @ B2 ) @ ( image_14429021real_c @ ( produc567249007real_c @ F2 ) @ A ) ) ) ).
% pair_imageI
thf(fact_206_pair__imageI,axiom,
! [A2: list_f1238882004_sc_sz,B2: list_f1238882004_sc_sz,A: set_Pr1774599175_sc_sz,F2: list_f1238882004_sc_sz > list_f1238882004_sc_sz > finite1398487019real_c] :
( ( member245028432_sc_sz @ ( produc1822718231_sc_sz @ A2 @ B2 ) @ A )
=> ( member1261661570real_c @ ( F2 @ A2 @ B2 ) @ ( image_1332486093real_c @ ( produc236226839real_c @ F2 ) @ A ) ) ) ).
% pair_imageI
thf(fact_207_pair__imageI,axiom,
! [A2: list_f1238882004_sc_sz,B2: list_f1238882004_sc_sz,A: set_Pr1774599175_sc_sz,F2: list_f1238882004_sc_sz > list_f1238882004_sc_sz > produc190496183real_c] :
( ( member245028432_sc_sz @ ( produc1822718231_sc_sz @ A2 @ B2 ) @ A )
=> ( member1895684704real_c @ ( F2 @ A2 @ B2 ) @ ( image_1121381717real_c @ ( produc126255499real_c @ F2 ) @ A ) ) ) ).
% pair_imageI
thf(fact_208_pair__imageI,axiom,
! [A2: finite1398487019real_c,B2: finite1398487019real_c,A: set_Pr1389752855real_c,F2: finite1398487019real_c > finite1398487019real_c > produc866628903_sc_sz] :
( ( member1895684704real_c @ ( produc394644079real_c @ A2 @ B2 ) @ A )
=> ( member245028432_sc_sz @ ( F2 @ A2 @ B2 ) @ ( image_388766773_sc_sz @ ( produc1341776291_sc_sz @ F2 ) @ A ) ) ) ).
% pair_imageI
thf(fact_209_pair__imageI,axiom,
! [A2: produc190496183real_c,B2: produc190496183real_c,A: set_Pr754230471real_c,F2: produc190496183real_c > produc190496183real_c > finite1398487019real_c] :
( ( member346808080real_c @ ( produc1687028567real_c @ A2 @ B2 ) @ A )
=> ( member1261661570real_c @ ( F2 @ A2 @ B2 ) @ ( image_124598413real_c @ ( produc344593239real_c @ F2 ) @ A ) ) ) ).
% pair_imageI
thf(fact_210_pair__imageI,axiom,
! [A2: denota610675952t_unit,B2: produc866628903_sc_sz,A: set_Pr1169339874_sc_sz,F2: denota610675952t_unit > produc866628903_sc_sz > $o] :
( ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ A2 @ B2 ) @ A )
=> ( member_o @ ( F2 @ A2 @ B2 ) @ ( image_1641704997c_sz_o @ ( produc619560141c_sz_o @ F2 ) @ A ) ) ) ).
% pair_imageI
thf(fact_211_pair__imageI,axiom,
! [A2: denota610675952t_unit,B2: produc866628903_sc_sz,A: set_Pr1169339874_sc_sz,F2: denota610675952t_unit > produc866628903_sc_sz > finite1398487019real_c] :
( ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ A2 @ B2 ) @ A )
=> ( member1261661570real_c @ ( F2 @ A2 @ B2 ) @ ( image_1268968350real_c @ ( produc1920474870real_c @ F2 ) @ A ) ) ) ).
% pair_imageI
thf(fact_212_pair__imageI,axiom,
! [A2: produc866628903_sc_sz,B2: denota610675952t_unit,A: set_Pr2041213564t_unit,F2: produc866628903_sc_sz > denota610675952t_unit > finite1398487019real_c] :
( ( member235934813t_unit @ ( produc502616768t_unit @ A2 @ B2 ) @ A )
=> ( member1261661570real_c @ ( F2 @ A2 @ B2 ) @ ( image_1269062456real_c @ ( produc1753599960real_c @ F2 ) @ A ) ) ) ).
% pair_imageI
thf(fact_213_curry__def,axiom,
( produc1086673900c_sz_o
= ( ^ [C3: produc1821101996_sc_sz > $o,X4: denota610675952t_unit,Y4: produc866628903_sc_sz] : ( C3 @ ( produc789536734_sc_sz @ X4 @ Y4 ) ) ) ) ).
% curry_def
thf(fact_214_curry__K,axiom,
! [C2: $o] :
( ( produc1086673900c_sz_o
@ ^ [X4: produc1821101996_sc_sz] : C2 )
= ( ^ [X4: denota610675952t_unit,Y4: produc866628903_sc_sz] : C2 ) ) ).
% curry_K
thf(fact_215_old_Oprod_Ocase,axiom,
! [F2: denota610675952t_unit > produc866628903_sc_sz > produc190496183real_c,X1: denota610675952t_unit,X2: produc866628903_sc_sz] :
( ( produc1575513068real_c @ F2 @ ( produc789536734_sc_sz @ X1 @ X2 ) )
= ( F2 @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_216_old_Oprod_Ocase,axiom,
! [F2: denota610675952t_unit > produc866628903_sc_sz > $o,X1: denota610675952t_unit,X2: produc866628903_sc_sz] :
( ( produc619560141c_sz_o @ F2 @ ( produc789536734_sc_sz @ X1 @ X2 ) )
= ( F2 @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_217_old_Oprod_Ocase,axiom,
! [F2: denota610675952t_unit > produc866628903_sc_sz > set_Pr1389752855real_c,X1: denota610675952t_unit,X2: produc866628903_sc_sz] :
( ( produc237521996real_c @ F2 @ ( produc789536734_sc_sz @ X1 @ X2 ) )
= ( F2 @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_218_old_Oprod_Ocase,axiom,
! [F2: finite1398487019real_c > finite1398487019real_c > produc190496183real_c,X1: finite1398487019real_c,X2: finite1398487019real_c] :
( ( produc107375667real_c @ F2 @ ( produc394644079real_c @ X1 @ X2 ) )
= ( F2 @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_219_old_Oprod_Ocase,axiom,
! [F2: produc866628903_sc_sz > denota610675952t_unit > produc1821101996_sc_sz,X1: produc866628903_sc_sz,X2: denota610675952t_unit] :
( ( produc435149849_sc_sz @ F2 @ ( produc502616768t_unit @ X1 @ X2 ) )
= ( F2 @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_220_mem__case__prodE,axiom,
! [Z: produc190496183real_c,C2: denota610675952t_unit > produc866628903_sc_sz > set_Pr1389752855real_c,P2: produc1821101996_sc_sz] :
( ( member1895684704real_c @ Z @ ( produc237521996real_c @ C2 @ P2 ) )
=> ~ ! [X3: denota610675952t_unit,Y3: produc866628903_sc_sz] :
( ( P2
= ( produc789536734_sc_sz @ X3 @ Y3 ) )
=> ~ ( member1895684704real_c @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_221_mem__case__prodE,axiom,
! [Z: finite1398487019real_c,C2: finite1398487019real_c > finite1398487019real_c > set_Fi1407883041real_c,P2: produc190496183real_c] :
( ( member1261661570real_c @ Z @ ( produc29022117real_c @ C2 @ P2 ) )
=> ~ ! [X3: finite1398487019real_c,Y3: finite1398487019real_c] :
( ( P2
= ( produc394644079real_c @ X3 @ Y3 ) )
=> ~ ( member1261661570real_c @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_222_mem__case__prodE,axiom,
! [Z: finite1398487019real_c,C2: list_f1238882004_sc_sz > list_f1238882004_sc_sz > set_Fi1407883041real_c,P2: produc866628903_sc_sz] :
( ( member1261661570real_c @ Z @ ( produc1449736269real_c @ C2 @ P2 ) )
=> ~ ! [X3: list_f1238882004_sc_sz,Y3: list_f1238882004_sc_sz] :
( ( P2
= ( produc1822718231_sc_sz @ X3 @ Y3 ) )
=> ~ ( member1261661570real_c @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_223_mem__case__prodE,axiom,
! [Z: produc190496183real_c,C2: finite1398487019real_c > finite1398487019real_c > set_Pr1389752855real_c,P2: produc190496183real_c] :
( ( member1895684704real_c @ Z @ ( produc923381907real_c @ C2 @ P2 ) )
=> ~ ! [X3: finite1398487019real_c,Y3: finite1398487019real_c] :
( ( P2
= ( produc394644079real_c @ X3 @ Y3 ) )
=> ~ ( member1895684704real_c @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_224_mem__case__prodE,axiom,
! [Z: produc190496183real_c,C2: list_f1238882004_sc_sz > list_f1238882004_sc_sz > set_Pr1389752855real_c,P2: produc866628903_sc_sz] :
( ( member1895684704real_c @ Z @ ( produc6841835real_c @ C2 @ P2 ) )
=> ~ ! [X3: list_f1238882004_sc_sz,Y3: list_f1238882004_sc_sz] :
( ( P2
= ( produc1822718231_sc_sz @ X3 @ Y3 ) )
=> ~ ( member1895684704real_c @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_225_mem__case__prodE,axiom,
! [Z: produc866628903_sc_sz,C2: finite1398487019real_c > finite1398487019real_c > set_Pr1774599175_sc_sz,P2: produc190496183real_c] :
( ( member245028432_sc_sz @ Z @ ( produc642283651_sc_sz @ C2 @ P2 ) )
=> ~ ! [X3: finite1398487019real_c,Y3: finite1398487019real_c] :
( ( P2
= ( produc394644079real_c @ X3 @ Y3 ) )
=> ~ ( member245028432_sc_sz @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_226_mem__case__prodE,axiom,
! [Z: finite1398487019real_c,C2: produc190496183real_c > produc190496183real_c > set_Fi1407883041real_c,P2: produc640052711real_c] :
( ( member1261661570real_c @ Z @ ( produc1066779277real_c @ C2 @ P2 ) )
=> ~ ! [X3: produc190496183real_c,Y3: produc190496183real_c] :
( ( P2
= ( produc1687028567real_c @ X3 @ Y3 ) )
=> ~ ( member1261661570real_c @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_227_mem__case__prodE,axiom,
! [Z: finite1398487019real_c,C2: denota610675952t_unit > produc866628903_sc_sz > set_Fi1407883041real_c,P2: produc1821101996_sc_sz] :
( ( member1261661570real_c @ Z @ ( produc2068697644real_c @ C2 @ P2 ) )
=> ~ ! [X3: denota610675952t_unit,Y3: produc866628903_sc_sz] :
( ( P2
= ( produc789536734_sc_sz @ X3 @ Y3 ) )
=> ~ ( member1261661570real_c @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_228_mem__case__prodE,axiom,
! [Z: finite1398487019real_c,C2: produc866628903_sc_sz > denota610675952t_unit > set_Fi1407883041real_c,P2: produc999471430t_unit] :
( ( member1261661570real_c @ Z @ ( produc1368849166real_c @ C2 @ P2 ) )
=> ~ ! [X3: produc866628903_sc_sz,Y3: denota610675952t_unit] :
( ( P2
= ( produc502616768t_unit @ X3 @ Y3 ) )
=> ~ ( member1261661570real_c @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_229_mem__case__prodE,axiom,
! [Z: produc866628903_sc_sz,C2: list_f1238882004_sc_sz > list_f1238882004_sc_sz > set_Pr1774599175_sc_sz,P2: produc866628903_sc_sz] :
( ( member245028432_sc_sz @ Z @ ( produc1563580891_sc_sz @ C2 @ P2 ) )
=> ~ ! [X3: list_f1238882004_sc_sz,Y3: list_f1238882004_sc_sz] :
( ( P2
= ( produc1822718231_sc_sz @ X3 @ Y3 ) )
=> ~ ( member245028432_sc_sz @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_230_prod_Ocase__distrib,axiom,
! [H: $o > $o,F2: denota610675952t_unit > produc866628903_sc_sz > $o,Prod: produc1821101996_sc_sz] :
( ( H @ ( produc619560141c_sz_o @ F2 @ Prod ) )
= ( produc619560141c_sz_o
@ ^ [X12: denota610675952t_unit,X22: produc866628903_sc_sz] : ( H @ ( F2 @ X12 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_231_prod_Ocase__distrib,axiom,
! [H: produc190496183real_c > produc190496183real_c,F2: finite1398487019real_c > finite1398487019real_c > produc190496183real_c,Prod: produc190496183real_c] :
( ( H @ ( produc107375667real_c @ F2 @ Prod ) )
= ( produc107375667real_c
@ ^ [X12: finite1398487019real_c,X22: finite1398487019real_c] : ( H @ ( F2 @ X12 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_232_prod_Ocase__distrib,axiom,
! [H: produc190496183real_c > $o,F2: denota610675952t_unit > produc866628903_sc_sz > produc190496183real_c,Prod: produc1821101996_sc_sz] :
( ( H @ ( produc1575513068real_c @ F2 @ Prod ) )
= ( produc619560141c_sz_o
@ ^ [X12: denota610675952t_unit,X22: produc866628903_sc_sz] : ( H @ ( F2 @ X12 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_233_prod_Ocase__distrib,axiom,
! [H: $o > produc190496183real_c,F2: denota610675952t_unit > produc866628903_sc_sz > $o,Prod: produc1821101996_sc_sz] :
( ( H @ ( produc619560141c_sz_o @ F2 @ Prod ) )
= ( produc1575513068real_c
@ ^ [X12: denota610675952t_unit,X22: produc866628903_sc_sz] : ( H @ ( F2 @ X12 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_234_prod_Ocase__distrib,axiom,
! [H: $o > set_Pr1389752855real_c,F2: denota610675952t_unit > produc866628903_sc_sz > $o,Prod: produc1821101996_sc_sz] :
( ( H @ ( produc619560141c_sz_o @ F2 @ Prod ) )
= ( produc237521996real_c
@ ^ [X12: denota610675952t_unit,X22: produc866628903_sc_sz] : ( H @ ( F2 @ X12 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_235_prod_Ocase__distrib,axiom,
! [H: set_Pr1389752855real_c > $o,F2: denota610675952t_unit > produc866628903_sc_sz > set_Pr1389752855real_c,Prod: produc1821101996_sc_sz] :
( ( H @ ( produc237521996real_c @ F2 @ Prod ) )
= ( produc619560141c_sz_o
@ ^ [X12: denota610675952t_unit,X22: produc866628903_sc_sz] : ( H @ ( F2 @ X12 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_236_prod_Ocase__distrib,axiom,
! [H: produc190496183real_c > produc190496183real_c,F2: denota610675952t_unit > produc866628903_sc_sz > produc190496183real_c,Prod: produc1821101996_sc_sz] :
( ( H @ ( produc1575513068real_c @ F2 @ Prod ) )
= ( produc1575513068real_c
@ ^ [X12: denota610675952t_unit,X22: produc866628903_sc_sz] : ( H @ ( F2 @ X12 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_237_prod_Ocase__distrib,axiom,
! [H: produc190496183real_c > set_Pr1389752855real_c,F2: denota610675952t_unit > produc866628903_sc_sz > produc190496183real_c,Prod: produc1821101996_sc_sz] :
( ( H @ ( produc1575513068real_c @ F2 @ Prod ) )
= ( produc237521996real_c
@ ^ [X12: denota610675952t_unit,X22: produc866628903_sc_sz] : ( H @ ( F2 @ X12 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_238_prod_Ocase__distrib,axiom,
! [H: set_Pr1389752855real_c > produc190496183real_c,F2: denota610675952t_unit > produc866628903_sc_sz > set_Pr1389752855real_c,Prod: produc1821101996_sc_sz] :
( ( H @ ( produc237521996real_c @ F2 @ Prod ) )
= ( produc1575513068real_c
@ ^ [X12: denota610675952t_unit,X22: produc866628903_sc_sz] : ( H @ ( F2 @ X12 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_239_prod_Ocase__distrib,axiom,
! [H: set_Pr1389752855real_c > set_Pr1389752855real_c,F2: denota610675952t_unit > produc866628903_sc_sz > set_Pr1389752855real_c,Prod: produc1821101996_sc_sz] :
( ( H @ ( produc237521996real_c @ F2 @ Prod ) )
= ( produc237521996real_c
@ ^ [X12: denota610675952t_unit,X22: produc866628903_sc_sz] : ( H @ ( F2 @ X12 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_240_cond__case__prod__eta,axiom,
! [F2: denota610675952t_unit > produc866628903_sc_sz > produc190496183real_c,G: produc1821101996_sc_sz > produc190496183real_c] :
( ! [X3: denota610675952t_unit,Y3: produc866628903_sc_sz] :
( ( F2 @ X3 @ Y3 )
= ( G @ ( produc789536734_sc_sz @ X3 @ Y3 ) ) )
=> ( ( produc1575513068real_c @ F2 )
= G ) ) ).
% cond_case_prod_eta
thf(fact_241_cond__case__prod__eta,axiom,
! [F2: denota610675952t_unit > produc866628903_sc_sz > $o,G: produc1821101996_sc_sz > $o] :
( ! [X3: denota610675952t_unit,Y3: produc866628903_sc_sz] :
( ( F2 @ X3 @ Y3 )
= ( G @ ( produc789536734_sc_sz @ X3 @ Y3 ) ) )
=> ( ( produc619560141c_sz_o @ F2 )
= G ) ) ).
% cond_case_prod_eta
thf(fact_242_cond__case__prod__eta,axiom,
! [F2: denota610675952t_unit > produc866628903_sc_sz > set_Pr1389752855real_c,G: produc1821101996_sc_sz > set_Pr1389752855real_c] :
( ! [X3: denota610675952t_unit,Y3: produc866628903_sc_sz] :
( ( F2 @ X3 @ Y3 )
= ( G @ ( produc789536734_sc_sz @ X3 @ Y3 ) ) )
=> ( ( produc237521996real_c @ F2 )
= G ) ) ).
% cond_case_prod_eta
thf(fact_243_cond__case__prod__eta,axiom,
! [F2: finite1398487019real_c > finite1398487019real_c > produc190496183real_c,G: produc190496183real_c > produc190496183real_c] :
( ! [X3: finite1398487019real_c,Y3: finite1398487019real_c] :
( ( F2 @ X3 @ Y3 )
= ( G @ ( produc394644079real_c @ X3 @ Y3 ) ) )
=> ( ( produc107375667real_c @ F2 )
= G ) ) ).
% cond_case_prod_eta
thf(fact_244_cond__case__prod__eta,axiom,
! [F2: produc866628903_sc_sz > denota610675952t_unit > produc1821101996_sc_sz,G: produc999471430t_unit > produc1821101996_sc_sz] :
( ! [X3: produc866628903_sc_sz,Y3: denota610675952t_unit] :
( ( F2 @ X3 @ Y3 )
= ( G @ ( produc502616768t_unit @ X3 @ Y3 ) ) )
=> ( ( produc435149849_sc_sz @ F2 )
= G ) ) ).
% cond_case_prod_eta
thf(fact_245_case__prod__eta,axiom,
! [F2: produc1821101996_sc_sz > produc190496183real_c] :
( ( produc1575513068real_c
@ ^ [X4: denota610675952t_unit,Y4: produc866628903_sc_sz] : ( F2 @ ( produc789536734_sc_sz @ X4 @ Y4 ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_246_case__prod__eta,axiom,
! [F2: produc1821101996_sc_sz > $o] :
( ( produc619560141c_sz_o
@ ^ [X4: denota610675952t_unit,Y4: produc866628903_sc_sz] : ( F2 @ ( produc789536734_sc_sz @ X4 @ Y4 ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_247_case__prod__eta,axiom,
! [F2: produc1821101996_sc_sz > set_Pr1389752855real_c] :
( ( produc237521996real_c
@ ^ [X4: denota610675952t_unit,Y4: produc866628903_sc_sz] : ( F2 @ ( produc789536734_sc_sz @ X4 @ Y4 ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_248_case__prod__eta,axiom,
! [F2: produc190496183real_c > produc190496183real_c] :
( ( produc107375667real_c
@ ^ [X4: finite1398487019real_c,Y4: finite1398487019real_c] : ( F2 @ ( produc394644079real_c @ X4 @ Y4 ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_249_case__prod__eta,axiom,
! [F2: produc999471430t_unit > produc1821101996_sc_sz] :
( ( produc435149849_sc_sz
@ ^ [X4: produc866628903_sc_sz,Y4: denota610675952t_unit] : ( F2 @ ( produc502616768t_unit @ X4 @ Y4 ) ) )
= F2 ) ).
% case_prod_eta
thf(fact_250_case__prodE2,axiom,
! [Q: produc190496183real_c > $o,P: denota610675952t_unit > produc866628903_sc_sz > produc190496183real_c,Z: produc1821101996_sc_sz] :
( ( Q @ ( produc1575513068real_c @ P @ Z ) )
=> ~ ! [X3: denota610675952t_unit,Y3: produc866628903_sc_sz] :
( ( Z
= ( produc789536734_sc_sz @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_251_case__prodE2,axiom,
! [Q: $o > $o,P: denota610675952t_unit > produc866628903_sc_sz > $o,Z: produc1821101996_sc_sz] :
( ( Q @ ( produc619560141c_sz_o @ P @ Z ) )
=> ~ ! [X3: denota610675952t_unit,Y3: produc866628903_sc_sz] :
( ( Z
= ( produc789536734_sc_sz @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_252_case__prodE2,axiom,
! [Q: set_Pr1389752855real_c > $o,P: denota610675952t_unit > produc866628903_sc_sz > set_Pr1389752855real_c,Z: produc1821101996_sc_sz] :
( ( Q @ ( produc237521996real_c @ P @ Z ) )
=> ~ ! [X3: denota610675952t_unit,Y3: produc866628903_sc_sz] :
( ( Z
= ( produc789536734_sc_sz @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_253_case__prodE2,axiom,
! [Q: produc190496183real_c > $o,P: finite1398487019real_c > finite1398487019real_c > produc190496183real_c,Z: produc190496183real_c] :
( ( Q @ ( produc107375667real_c @ P @ Z ) )
=> ~ ! [X3: finite1398487019real_c,Y3: finite1398487019real_c] :
( ( Z
= ( produc394644079real_c @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_254_case__prodE2,axiom,
! [Q: produc1821101996_sc_sz > $o,P: produc866628903_sc_sz > denota610675952t_unit > produc1821101996_sc_sz,Z: produc999471430t_unit] :
( ( Q @ ( produc435149849_sc_sz @ P @ Z ) )
=> ~ ! [X3: produc866628903_sc_sz,Y3: denota610675952t_unit] :
( ( Z
= ( produc502616768t_unit @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_255_case__prodE,axiom,
! [C2: list_f1238882004_sc_sz > list_f1238882004_sc_sz > $o,P2: produc866628903_sc_sz] :
( ( produc2039065452c_sz_o @ C2 @ P2 )
=> ~ ! [X3: list_f1238882004_sc_sz,Y3: list_f1238882004_sc_sz] :
( ( P2
= ( produc1822718231_sc_sz @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_256_case__prodE,axiom,
! [C2: finite1398487019real_c > finite1398487019real_c > $o,P2: produc190496183real_c] :
( ( produc353381908al_c_o @ C2 @ P2 )
=> ~ ! [X3: finite1398487019real_c,Y3: finite1398487019real_c] :
( ( P2
= ( produc394644079real_c @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_257_case__prodE,axiom,
! [C2: produc866628903_sc_sz > denota610675952t_unit > $o,P2: produc999471430t_unit] :
( ( produc1718498667unit_o @ C2 @ P2 )
=> ~ ! [X3: produc866628903_sc_sz,Y3: denota610675952t_unit] :
( ( P2
= ( produc502616768t_unit @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_258_case__prodE,axiom,
! [C2: produc190496183real_c > produc190496183real_c > $o,P2: produc640052711real_c] :
( ( produc1726730540al_c_o @ C2 @ P2 )
=> ~ ! [X3: produc190496183real_c,Y3: produc190496183real_c] :
( ( P2
= ( produc1687028567real_c @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_259_case__prodE,axiom,
! [C2: denota610675952t_unit > produc866628903_sc_sz > $o,P2: produc1821101996_sc_sz] :
( ( produc619560141c_sz_o @ C2 @ P2 )
=> ~ ! [X3: denota610675952t_unit,Y3: produc866628903_sc_sz] :
( ( P2
= ( produc789536734_sc_sz @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_260_case__prodD,axiom,
! [F2: list_f1238882004_sc_sz > list_f1238882004_sc_sz > $o,A2: list_f1238882004_sc_sz,B2: list_f1238882004_sc_sz] :
( ( produc2039065452c_sz_o @ F2 @ ( produc1822718231_sc_sz @ A2 @ B2 ) )
=> ( F2 @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_261_case__prodD,axiom,
! [F2: finite1398487019real_c > finite1398487019real_c > $o,A2: finite1398487019real_c,B2: finite1398487019real_c] :
( ( produc353381908al_c_o @ F2 @ ( produc394644079real_c @ A2 @ B2 ) )
=> ( F2 @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_262_case__prodD,axiom,
! [F2: produc866628903_sc_sz > denota610675952t_unit > $o,A2: produc866628903_sc_sz,B2: denota610675952t_unit] :
( ( produc1718498667unit_o @ F2 @ ( produc502616768t_unit @ A2 @ B2 ) )
=> ( F2 @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_263_case__prodD,axiom,
! [F2: produc190496183real_c > produc190496183real_c > $o,A2: produc190496183real_c,B2: produc190496183real_c] :
( ( produc1726730540al_c_o @ F2 @ ( produc1687028567real_c @ A2 @ B2 ) )
=> ( F2 @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_264_case__prodD,axiom,
! [F2: denota610675952t_unit > produc866628903_sc_sz > $o,A2: denota610675952t_unit,B2: produc866628903_sc_sz] :
( ( produc619560141c_sz_o @ F2 @ ( produc789536734_sc_sz @ A2 @ B2 ) )
=> ( F2 @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_265_image__ident,axiom,
! [Y5: set_Pr1389752855real_c] :
( ( image_738229701real_c
@ ^ [X4: produc190496183real_c] : X4
@ Y5 )
= Y5 ) ).
% image_ident
thf(fact_266_image__eqI,axiom,
! [B2: produc190496183real_c,F2: produc190496183real_c > produc190496183real_c,X: produc190496183real_c,A: set_Pr1389752855real_c] :
( ( B2
= ( F2 @ X ) )
=> ( ( member1895684704real_c @ X @ A )
=> ( member1895684704real_c @ B2 @ ( image_738229701real_c @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_267_image__eqI,axiom,
! [B2: finite1398487019real_c,F2: finite1398487019real_c > finite1398487019real_c,X: finite1398487019real_c,A: set_Fi1407883041real_c] :
( ( B2
= ( F2 @ X ) )
=> ( ( member1261661570real_c @ X @ A )
=> ( member1261661570real_c @ B2 @ ( image_1514048989real_c @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_268_image__eqI,axiom,
! [B2: finite1398487019real_c,F2: produc190496183real_c > finite1398487019real_c,X: produc190496183real_c,A: set_Pr1389752855real_c] :
( ( B2
= ( F2 @ X ) )
=> ( ( member1895684704real_c @ X @ A )
=> ( member1261661570real_c @ B2 @ ( image_14429021real_c @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_269_image__eqI,axiom,
! [B2: produc190496183real_c,F2: finite1398487019real_c > produc190496183real_c,X: finite1398487019real_c,A: set_Fi1407883041real_c] :
( ( B2
= ( F2 @ X ) )
=> ( ( member1261661570real_c @ X @ A )
=> ( member1895684704real_c @ B2 @ ( image_1461287749real_c @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_270_image__eqI,axiom,
! [B2: produc866628903_sc_sz,F2: finite1398487019real_c > produc866628903_sc_sz,X: finite1398487019real_c,A: set_Fi1407883041real_c] :
( ( B2
= ( F2 @ X ) )
=> ( ( member1261661570real_c @ X @ A )
=> ( member245028432_sc_sz @ B2 @ ( image_290183605_sc_sz @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_271_image__eqI,axiom,
! [B2: finite1398487019real_c,F2: produc866628903_sc_sz > finite1398487019real_c,X: produc866628903_sc_sz,A: set_Pr1774599175_sc_sz] :
( ( B2
= ( F2 @ X ) )
=> ( ( member245028432_sc_sz @ X @ A )
=> ( member1261661570real_c @ B2 @ ( image_1332486093real_c @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_272_image__eqI,axiom,
! [B2: produc866628903_sc_sz,F2: produc190496183real_c > produc866628903_sc_sz,X: produc190496183real_c,A: set_Pr1389752855real_c] :
( ( B2
= ( F2 @ X ) )
=> ( ( member1895684704real_c @ X @ A )
=> ( member245028432_sc_sz @ B2 @ ( image_388766773_sc_sz @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_273_image__eqI,axiom,
! [B2: produc640052711real_c,F2: finite1398487019real_c > produc640052711real_c,X: finite1398487019real_c,A: set_Fi1407883041real_c] :
( ( B2
= ( F2 @ X ) )
=> ( ( member1261661570real_c @ X @ A )
=> ( member346808080real_c @ B2 @ ( image_1746662005real_c @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_274_image__eqI,axiom,
! [B2: finite1398487019real_c,F2: produc640052711real_c > finite1398487019real_c,X: produc640052711real_c,A: set_Pr754230471real_c] :
( ( B2
= ( F2 @ X ) )
=> ( ( member346808080real_c @ X @ A )
=> ( member1261661570real_c @ B2 @ ( image_124598413real_c @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_275_image__eqI,axiom,
! [B2: produc190496183real_c,F2: produc866628903_sc_sz > produc190496183real_c,X: produc866628903_sc_sz,A: set_Pr1774599175_sc_sz] :
( ( B2
= ( F2 @ X ) )
=> ( ( member245028432_sc_sz @ X @ A )
=> ( member1895684704real_c @ B2 @ ( image_1121381717real_c @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_276_split__part,axiom,
! [P: $o,Q: denota610675952t_unit > produc866628903_sc_sz > $o] :
( ( produc619560141c_sz_o
@ ^ [A5: denota610675952t_unit,B5: produc866628903_sc_sz] :
( P
& ( Q @ A5 @ B5 ) ) )
= ( ^ [Ab: produc1821101996_sc_sz] :
( P
& ( produc619560141c_sz_o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_277_prod_Odisc__eq__case,axiom,
! [Prod: produc1821101996_sc_sz] :
( produc619560141c_sz_o
@ ^ [Uu: denota610675952t_unit,Uv: produc866628903_sc_sz] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_278_rev__image__eqI,axiom,
! [X: produc190496183real_c,A: set_Pr1389752855real_c,B2: produc190496183real_c,F2: produc190496183real_c > produc190496183real_c] :
( ( member1895684704real_c @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member1895684704real_c @ B2 @ ( image_738229701real_c @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_279_rev__image__eqI,axiom,
! [X: finite1398487019real_c,A: set_Fi1407883041real_c,B2: finite1398487019real_c,F2: finite1398487019real_c > finite1398487019real_c] :
( ( member1261661570real_c @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member1261661570real_c @ B2 @ ( image_1514048989real_c @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_280_rev__image__eqI,axiom,
! [X: produc190496183real_c,A: set_Pr1389752855real_c,B2: finite1398487019real_c,F2: produc190496183real_c > finite1398487019real_c] :
( ( member1895684704real_c @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member1261661570real_c @ B2 @ ( image_14429021real_c @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_281_rev__image__eqI,axiom,
! [X: finite1398487019real_c,A: set_Fi1407883041real_c,B2: produc190496183real_c,F2: finite1398487019real_c > produc190496183real_c] :
( ( member1261661570real_c @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member1895684704real_c @ B2 @ ( image_1461287749real_c @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_282_rev__image__eqI,axiom,
! [X: finite1398487019real_c,A: set_Fi1407883041real_c,B2: produc866628903_sc_sz,F2: finite1398487019real_c > produc866628903_sc_sz] :
( ( member1261661570real_c @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member245028432_sc_sz @ B2 @ ( image_290183605_sc_sz @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_283_rev__image__eqI,axiom,
! [X: produc866628903_sc_sz,A: set_Pr1774599175_sc_sz,B2: finite1398487019real_c,F2: produc866628903_sc_sz > finite1398487019real_c] :
( ( member245028432_sc_sz @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member1261661570real_c @ B2 @ ( image_1332486093real_c @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_284_rev__image__eqI,axiom,
! [X: produc190496183real_c,A: set_Pr1389752855real_c,B2: produc866628903_sc_sz,F2: produc190496183real_c > produc866628903_sc_sz] :
( ( member1895684704real_c @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member245028432_sc_sz @ B2 @ ( image_388766773_sc_sz @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_285_rev__image__eqI,axiom,
! [X: finite1398487019real_c,A: set_Fi1407883041real_c,B2: produc640052711real_c,F2: finite1398487019real_c > produc640052711real_c] :
( ( member1261661570real_c @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member346808080real_c @ B2 @ ( image_1746662005real_c @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_286_rev__image__eqI,axiom,
! [X: produc640052711real_c,A: set_Pr754230471real_c,B2: finite1398487019real_c,F2: produc640052711real_c > finite1398487019real_c] :
( ( member346808080real_c @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member1261661570real_c @ B2 @ ( image_124598413real_c @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_287_rev__image__eqI,axiom,
! [X: produc866628903_sc_sz,A: set_Pr1774599175_sc_sz,B2: produc190496183real_c,F2: produc866628903_sc_sz > produc190496183real_c] :
( ( member245028432_sc_sz @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member1895684704real_c @ B2 @ ( image_1121381717real_c @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_288_ball__imageD,axiom,
! [F2: produc1821101996_sc_sz > produc190496183real_c,A: set_Pr1169339874_sc_sz,P: produc190496183real_c > $o] :
( ! [X3: produc190496183real_c] :
( ( member1895684704real_c @ X3 @ ( image_386905412real_c @ F2 @ A ) )
=> ( P @ X3 ) )
=> ! [X5: produc1821101996_sc_sz] :
( ( member1057565379_sc_sz @ X5 @ A )
=> ( P @ ( F2 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_289_ball__imageD,axiom,
! [F2: produc1821101996_sc_sz > produc999471430t_unit,A: set_Pr1169339874_sc_sz,P: produc999471430t_unit > $o] :
( ! [X3: produc999471430t_unit] :
( ( member235934813t_unit @ X3 @ ( image_904562809t_unit @ F2 @ A ) )
=> ( P @ X3 ) )
=> ! [X5: produc1821101996_sc_sz] :
( ( member1057565379_sc_sz @ X5 @ A )
=> ( P @ ( F2 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_290_ball__imageD,axiom,
! [F2: produc190496183real_c > produc1821101996_sc_sz,A: set_Pr1389752855real_c,P: produc1821101996_sc_sz > $o] :
( ! [X3: produc1821101996_sc_sz] :
( ( member1057565379_sc_sz @ X3 @ ( image_1259833374_sc_sz @ F2 @ A ) )
=> ( P @ X3 ) )
=> ! [X5: produc190496183real_c] :
( ( member1895684704real_c @ X5 @ A )
=> ( P @ ( F2 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_291_ball__imageD,axiom,
! [F2: produc190496183real_c > produc190496183real_c,A: set_Pr1389752855real_c,P: produc190496183real_c > $o] :
( ! [X3: produc190496183real_c] :
( ( member1895684704real_c @ X3 @ ( image_738229701real_c @ F2 @ A ) )
=> ( P @ X3 ) )
=> ! [X5: produc190496183real_c] :
( ( member1895684704real_c @ X5 @ A )
=> ( P @ ( F2 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_292_ball__imageD,axiom,
! [F2: produc999471430t_unit > produc1821101996_sc_sz,A: set_Pr2041213564t_unit,P: produc1821101996_sc_sz > $o] :
( ! [X3: produc1821101996_sc_sz] :
( ( member1057565379_sc_sz @ X3 @ ( image_1426741113_sc_sz @ F2 @ A ) )
=> ( P @ X3 ) )
=> ! [X5: produc999471430t_unit] :
( ( member235934813t_unit @ X5 @ A )
=> ( P @ ( F2 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_293_image__cong,axiom,
! [M: set_Pr1389752855real_c,N: set_Pr1389752855real_c,F2: produc190496183real_c > produc1821101996_sc_sz,G: produc190496183real_c > produc1821101996_sc_sz] :
( ( M = N )
=> ( ! [X3: produc190496183real_c] :
( ( member1895684704real_c @ X3 @ N )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_1259833374_sc_sz @ F2 @ M )
= ( image_1259833374_sc_sz @ G @ N ) ) ) ) ).
% image_cong
thf(fact_294_image__cong,axiom,
! [M: set_Pr1389752855real_c,N: set_Pr1389752855real_c,F2: produc190496183real_c > produc190496183real_c,G: produc190496183real_c > produc190496183real_c] :
( ( M = N )
=> ( ! [X3: produc190496183real_c] :
( ( member1895684704real_c @ X3 @ N )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_738229701real_c @ F2 @ M )
= ( image_738229701real_c @ G @ N ) ) ) ) ).
% image_cong
thf(fact_295_image__cong,axiom,
! [M: set_Pr1169339874_sc_sz,N: set_Pr1169339874_sc_sz,F2: produc1821101996_sc_sz > produc190496183real_c,G: produc1821101996_sc_sz > produc190496183real_c] :
( ( M = N )
=> ( ! [X3: produc1821101996_sc_sz] :
( ( member1057565379_sc_sz @ X3 @ N )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_386905412real_c @ F2 @ M )
= ( image_386905412real_c @ G @ N ) ) ) ) ).
% image_cong
thf(fact_296_image__cong,axiom,
! [M: set_Pr1169339874_sc_sz,N: set_Pr1169339874_sc_sz,F2: produc1821101996_sc_sz > produc999471430t_unit,G: produc1821101996_sc_sz > produc999471430t_unit] :
( ( M = N )
=> ( ! [X3: produc1821101996_sc_sz] :
( ( member1057565379_sc_sz @ X3 @ N )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_904562809t_unit @ F2 @ M )
= ( image_904562809t_unit @ G @ N ) ) ) ) ).
% image_cong
thf(fact_297_image__cong,axiom,
! [M: set_Pr2041213564t_unit,N: set_Pr2041213564t_unit,F2: produc999471430t_unit > produc1821101996_sc_sz,G: produc999471430t_unit > produc1821101996_sc_sz] :
( ( M = N )
=> ( ! [X3: produc999471430t_unit] :
( ( member235934813t_unit @ X3 @ N )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_1426741113_sc_sz @ F2 @ M )
= ( image_1426741113_sc_sz @ G @ N ) ) ) ) ).
% image_cong
thf(fact_298_bex__imageD,axiom,
! [F2: produc1821101996_sc_sz > produc190496183real_c,A: set_Pr1169339874_sc_sz,P: produc190496183real_c > $o] :
( ? [X5: produc190496183real_c] :
( ( member1895684704real_c @ X5 @ ( image_386905412real_c @ F2 @ A ) )
& ( P @ X5 ) )
=> ? [X3: produc1821101996_sc_sz] :
( ( member1057565379_sc_sz @ X3 @ A )
& ( P @ ( F2 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_299_bex__imageD,axiom,
! [F2: produc1821101996_sc_sz > produc999471430t_unit,A: set_Pr1169339874_sc_sz,P: produc999471430t_unit > $o] :
( ? [X5: produc999471430t_unit] :
( ( member235934813t_unit @ X5 @ ( image_904562809t_unit @ F2 @ A ) )
& ( P @ X5 ) )
=> ? [X3: produc1821101996_sc_sz] :
( ( member1057565379_sc_sz @ X3 @ A )
& ( P @ ( F2 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_300_bex__imageD,axiom,
! [F2: produc190496183real_c > produc1821101996_sc_sz,A: set_Pr1389752855real_c,P: produc1821101996_sc_sz > $o] :
( ? [X5: produc1821101996_sc_sz] :
( ( member1057565379_sc_sz @ X5 @ ( image_1259833374_sc_sz @ F2 @ A ) )
& ( P @ X5 ) )
=> ? [X3: produc190496183real_c] :
( ( member1895684704real_c @ X3 @ A )
& ( P @ ( F2 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_301_bex__imageD,axiom,
! [F2: produc190496183real_c > produc190496183real_c,A: set_Pr1389752855real_c,P: produc190496183real_c > $o] :
( ? [X5: produc190496183real_c] :
( ( member1895684704real_c @ X5 @ ( image_738229701real_c @ F2 @ A ) )
& ( P @ X5 ) )
=> ? [X3: produc190496183real_c] :
( ( member1895684704real_c @ X3 @ A )
& ( P @ ( F2 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_302_bex__imageD,axiom,
! [F2: produc999471430t_unit > produc1821101996_sc_sz,A: set_Pr2041213564t_unit,P: produc1821101996_sc_sz > $o] :
( ? [X5: produc1821101996_sc_sz] :
( ( member1057565379_sc_sz @ X5 @ ( image_1426741113_sc_sz @ F2 @ A ) )
& ( P @ X5 ) )
=> ? [X3: produc999471430t_unit] :
( ( member235934813t_unit @ X3 @ A )
& ( P @ ( F2 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_303_image__iff,axiom,
! [Z: produc190496183real_c,F2: produc1821101996_sc_sz > produc190496183real_c,A: set_Pr1169339874_sc_sz] :
( ( member1895684704real_c @ Z @ ( image_386905412real_c @ F2 @ A ) )
= ( ? [X4: produc1821101996_sc_sz] :
( ( member1057565379_sc_sz @ X4 @ A )
& ( Z
= ( F2 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_304_image__iff,axiom,
! [Z: produc190496183real_c,F2: produc190496183real_c > produc190496183real_c,A: set_Pr1389752855real_c] :
( ( member1895684704real_c @ Z @ ( image_738229701real_c @ F2 @ A ) )
= ( ? [X4: produc190496183real_c] :
( ( member1895684704real_c @ X4 @ A )
& ( Z
= ( F2 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_305_image__iff,axiom,
! [Z: produc1821101996_sc_sz,F2: produc190496183real_c > produc1821101996_sc_sz,A: set_Pr1389752855real_c] :
( ( member1057565379_sc_sz @ Z @ ( image_1259833374_sc_sz @ F2 @ A ) )
= ( ? [X4: produc190496183real_c] :
( ( member1895684704real_c @ X4 @ A )
& ( Z
= ( F2 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_306_image__iff,axiom,
! [Z: produc1821101996_sc_sz,F2: produc999471430t_unit > produc1821101996_sc_sz,A: set_Pr2041213564t_unit] :
( ( member1057565379_sc_sz @ Z @ ( image_1426741113_sc_sz @ F2 @ A ) )
= ( ? [X4: produc999471430t_unit] :
( ( member235934813t_unit @ X4 @ A )
& ( Z
= ( F2 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_307_image__iff,axiom,
! [Z: produc999471430t_unit,F2: produc1821101996_sc_sz > produc999471430t_unit,A: set_Pr1169339874_sc_sz] :
( ( member235934813t_unit @ Z @ ( image_904562809t_unit @ F2 @ A ) )
= ( ? [X4: produc1821101996_sc_sz] :
( ( member1057565379_sc_sz @ X4 @ A )
& ( Z
= ( F2 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_308_imageI,axiom,
! [X: produc866628903_sc_sz,A: set_Pr1774599175_sc_sz,F2: produc866628903_sc_sz > produc190496183real_c] :
( ( member245028432_sc_sz @ X @ A )
=> ( member1895684704real_c @ ( F2 @ X ) @ ( image_1121381717real_c @ F2 @ A ) ) ) ).
% imageI
thf(fact_309_imageI,axiom,
! [X: produc866628903_sc_sz,A: set_Pr1774599175_sc_sz,F2: produc866628903_sc_sz > finite1398487019real_c] :
( ( member245028432_sc_sz @ X @ A )
=> ( member1261661570real_c @ ( F2 @ X ) @ ( image_1332486093real_c @ F2 @ A ) ) ) ).
% imageI
thf(fact_310_imageI,axiom,
! [X: produc866628903_sc_sz,A: set_Pr1774599175_sc_sz,F2: produc866628903_sc_sz > produc1821101996_sc_sz] :
( ( member245028432_sc_sz @ X @ A )
=> ( member1057565379_sc_sz @ ( F2 @ X ) @ ( image_226055310_sc_sz @ F2 @ A ) ) ) ).
% imageI
thf(fact_311_imageI,axiom,
! [X: produc866628903_sc_sz,A: set_Pr1774599175_sc_sz,F2: produc866628903_sc_sz > produc640052711real_c] :
( ( member245028432_sc_sz @ X @ A )
=> ( member346808080real_c @ ( F2 @ X ) @ ( image_768765061real_c @ F2 @ A ) ) ) ).
% imageI
thf(fact_312_imageI,axiom,
! [X: produc866628903_sc_sz,A: set_Pr1774599175_sc_sz,F2: produc866628903_sc_sz > produc999471430t_unit] :
( ( member245028432_sc_sz @ X @ A )
=> ( member235934813t_unit @ ( F2 @ X ) @ ( image_1551908392t_unit @ F2 @ A ) ) ) ).
% imageI
thf(fact_313_imageI,axiom,
! [X: produc866628903_sc_sz,A: set_Pr1774599175_sc_sz,F2: produc866628903_sc_sz > produc866628903_sc_sz] :
( ( member245028432_sc_sz @ X @ A )
=> ( member245028432_sc_sz @ ( F2 @ X ) @ ( image_1313166277_sc_sz @ F2 @ A ) ) ) ).
% imageI
thf(fact_314_imageI,axiom,
! [X: produc190496183real_c,A: set_Pr1389752855real_c,F2: produc190496183real_c > produc190496183real_c] :
( ( member1895684704real_c @ X @ A )
=> ( member1895684704real_c @ ( F2 @ X ) @ ( image_738229701real_c @ F2 @ A ) ) ) ).
% imageI
thf(fact_315_Compr__image__eq,axiom,
! [F2: produc190496183real_c > produc190496183real_c,A: set_Pr1389752855real_c,P: produc190496183real_c > $o] :
( ( collec1643251106real_c
@ ^ [X4: produc190496183real_c] :
( ( member1895684704real_c @ X4 @ ( image_738229701real_c @ F2 @ A ) )
& ( P @ X4 ) ) )
= ( image_738229701real_c @ F2
@ ( collec1643251106real_c
@ ^ [X4: produc190496183real_c] :
( ( member1895684704real_c @ X4 @ A )
& ( P @ ( F2 @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_316_imageE,axiom,
! [B2: produc190496183real_c,F2: produc190496183real_c > produc190496183real_c,A: set_Pr1389752855real_c] :
( ( member1895684704real_c @ B2 @ ( image_738229701real_c @ F2 @ A ) )
=> ~ ! [X3: produc190496183real_c] :
( ( B2
= ( F2 @ X3 ) )
=> ~ ( member1895684704real_c @ X3 @ A ) ) ) ).
% imageE
thf(fact_317_case__prod__Pair__iden,axiom,
! [P2: produc1821101996_sc_sz] :
( ( produc448466487_sc_sz @ produc789536734_sc_sz @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_318_pred__equals__eq2,axiom,
! [R2: set_Pr1389752855real_c,S4: set_Pr1389752855real_c] :
( ( ( ^ [X4: finite1398487019real_c,Y4: finite1398487019real_c] : ( member1895684704real_c @ ( produc394644079real_c @ X4 @ Y4 ) @ R2 ) )
= ( ^ [X4: finite1398487019real_c,Y4: finite1398487019real_c] : ( member1895684704real_c @ ( produc394644079real_c @ X4 @ Y4 ) @ S4 ) ) )
= ( R2 = S4 ) ) ).
% pred_equals_eq2
thf(fact_319_pred__equals__eq2,axiom,
! [R2: set_Pr1169339874_sc_sz,S4: set_Pr1169339874_sc_sz] :
( ( ( ^ [X4: denota610675952t_unit,Y4: produc866628903_sc_sz] : ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ X4 @ Y4 ) @ R2 ) )
= ( ^ [X4: denota610675952t_unit,Y4: produc866628903_sc_sz] : ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ X4 @ Y4 ) @ S4 ) ) )
= ( R2 = S4 ) ) ).
% pred_equals_eq2
thf(fact_320_mem__Sigma__iff,axiom,
! [A2: produc190496183real_c,B2: produc190496183real_c,A: set_Pr1389752855real_c,B: produc190496183real_c > set_Pr1389752855real_c] :
( ( member346808080real_c @ ( produc1687028567real_c @ A2 @ B2 ) @ ( produc1017530376real_c @ A @ B ) )
= ( ( member1895684704real_c @ A2 @ A )
& ( member1895684704real_c @ B2 @ ( B @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_321_mem__Sigma__iff,axiom,
! [A2: finite1398487019real_c,B2: finite1398487019real_c,A: set_Fi1407883041real_c,B: finite1398487019real_c > set_Fi1407883041real_c] :
( ( member1895684704real_c @ ( produc394644079real_c @ A2 @ B2 ) @ ( produc563345056real_c @ A @ B ) )
= ( ( member1261661570real_c @ A2 @ A )
& ( member1261661570real_c @ B2 @ ( B @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_322_mem__Sigma__iff,axiom,
! [A2: denota610675952t_unit,B2: produc866628903_sc_sz,A: set_De208295462t_unit,B: denota610675952t_unit > set_Pr1774599175_sc_sz] :
( ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ A2 @ B2 ) @ ( produc159603053_sc_sz @ A @ B ) )
= ( ( member754147591t_unit @ A2 @ A )
& ( member245028432_sc_sz @ B2 @ ( B @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_323_SigmaI,axiom,
! [A2: finite1398487019real_c,A: set_Fi1407883041real_c,B2: finite1398487019real_c,B: finite1398487019real_c > set_Fi1407883041real_c] :
( ( member1261661570real_c @ A2 @ A )
=> ( ( member1261661570real_c @ B2 @ ( B @ A2 ) )
=> ( member1895684704real_c @ ( produc394644079real_c @ A2 @ B2 ) @ ( produc563345056real_c @ A @ B ) ) ) ) ).
% SigmaI
thf(fact_324_SigmaI,axiom,
! [A2: produc190496183real_c,A: set_Pr1389752855real_c,B2: produc190496183real_c,B: produc190496183real_c > set_Pr1389752855real_c] :
( ( member1895684704real_c @ A2 @ A )
=> ( ( member1895684704real_c @ B2 @ ( B @ A2 ) )
=> ( member346808080real_c @ ( produc1687028567real_c @ A2 @ B2 ) @ ( produc1017530376real_c @ A @ B ) ) ) ) ).
% SigmaI
thf(fact_325_SigmaI,axiom,
! [A2: denota610675952t_unit,A: set_De208295462t_unit,B2: produc866628903_sc_sz,B: denota610675952t_unit > set_Pr1774599175_sc_sz] :
( ( member754147591t_unit @ A2 @ A )
=> ( ( member245028432_sc_sz @ B2 @ ( B @ A2 ) )
=> ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ A2 @ B2 ) @ ( produc159603053_sc_sz @ A @ B ) ) ) ) ).
% SigmaI
thf(fact_326_SigmaE2,axiom,
! [A2: produc190496183real_c,B2: produc190496183real_c,A: set_Pr1389752855real_c,B: produc190496183real_c > set_Pr1389752855real_c] :
( ( member346808080real_c @ ( produc1687028567real_c @ A2 @ B2 ) @ ( produc1017530376real_c @ A @ B ) )
=> ~ ( ( member1895684704real_c @ A2 @ A )
=> ~ ( member1895684704real_c @ B2 @ ( B @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_327_SigmaE2,axiom,
! [A2: finite1398487019real_c,B2: finite1398487019real_c,A: set_Fi1407883041real_c,B: finite1398487019real_c > set_Fi1407883041real_c] :
( ( member1895684704real_c @ ( produc394644079real_c @ A2 @ B2 ) @ ( produc563345056real_c @ A @ B ) )
=> ~ ( ( member1261661570real_c @ A2 @ A )
=> ~ ( member1261661570real_c @ B2 @ ( B @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_328_SigmaE2,axiom,
! [A2: denota610675952t_unit,B2: produc866628903_sc_sz,A: set_De208295462t_unit,B: denota610675952t_unit > set_Pr1774599175_sc_sz] :
( ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ A2 @ B2 ) @ ( produc159603053_sc_sz @ A @ B ) )
=> ~ ( ( member754147591t_unit @ A2 @ A )
=> ~ ( member245028432_sc_sz @ B2 @ ( B @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_329_SigmaD2,axiom,
! [A2: finite1398487019real_c,B2: finite1398487019real_c,A: set_Fi1407883041real_c,B: finite1398487019real_c > set_Fi1407883041real_c] :
( ( member1895684704real_c @ ( produc394644079real_c @ A2 @ B2 ) @ ( produc563345056real_c @ A @ B ) )
=> ( member1261661570real_c @ B2 @ ( B @ A2 ) ) ) ).
% SigmaD2
thf(fact_330_SigmaD2,axiom,
! [A2: denota610675952t_unit,B2: produc866628903_sc_sz,A: set_De208295462t_unit,B: denota610675952t_unit > set_Pr1774599175_sc_sz] :
( ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ A2 @ B2 ) @ ( produc159603053_sc_sz @ A @ B ) )
=> ( member245028432_sc_sz @ B2 @ ( B @ A2 ) ) ) ).
% SigmaD2
thf(fact_331_SigmaD1,axiom,
! [A2: finite1398487019real_c,B2: finite1398487019real_c,A: set_Fi1407883041real_c,B: finite1398487019real_c > set_Fi1407883041real_c] :
( ( member1895684704real_c @ ( produc394644079real_c @ A2 @ B2 ) @ ( produc563345056real_c @ A @ B ) )
=> ( member1261661570real_c @ A2 @ A ) ) ).
% SigmaD1
thf(fact_332_SigmaD1,axiom,
! [A2: denota610675952t_unit,B2: produc866628903_sc_sz,A: set_De208295462t_unit,B: denota610675952t_unit > set_Pr1774599175_sc_sz] :
( ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ A2 @ B2 ) @ ( produc159603053_sc_sz @ A @ B ) )
=> ( member754147591t_unit @ A2 @ A ) ) ).
% SigmaD1
thf(fact_333_SigmaE,axiom,
! [C2: produc640052711real_c,A: set_Pr1389752855real_c,B: produc190496183real_c > set_Pr1389752855real_c] :
( ( member346808080real_c @ C2 @ ( produc1017530376real_c @ A @ B ) )
=> ~ ! [X3: produc190496183real_c] :
( ( member1895684704real_c @ X3 @ A )
=> ! [Y3: produc190496183real_c] :
( ( member1895684704real_c @ Y3 @ ( B @ X3 ) )
=> ( C2
!= ( produc1687028567real_c @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_334_SigmaE,axiom,
! [C2: produc190496183real_c,A: set_Fi1407883041real_c,B: finite1398487019real_c > set_Fi1407883041real_c] :
( ( member1895684704real_c @ C2 @ ( produc563345056real_c @ A @ B ) )
=> ~ ! [X3: finite1398487019real_c] :
( ( member1261661570real_c @ X3 @ A )
=> ! [Y3: finite1398487019real_c] :
( ( member1261661570real_c @ Y3 @ ( B @ X3 ) )
=> ( C2
!= ( produc394644079real_c @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_335_SigmaE,axiom,
! [C2: produc1821101996_sc_sz,A: set_De208295462t_unit,B: denota610675952t_unit > set_Pr1774599175_sc_sz] :
( ( member1057565379_sc_sz @ C2 @ ( produc159603053_sc_sz @ A @ B ) )
=> ~ ! [X3: denota610675952t_unit] :
( ( member754147591t_unit @ X3 @ A )
=> ! [Y3: produc866628903_sc_sz] :
( ( member245028432_sc_sz @ Y3 @ ( B @ X3 ) )
=> ( C2
!= ( produc789536734_sc_sz @ X3 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_336_swap__product,axiom,
! [A: set_Pr1774599175_sc_sz,B: set_De208295462t_unit] :
( ( image_1426741113_sc_sz
@ ( produc435149849_sc_sz
@ ^ [I4: produc866628903_sc_sz,J2: denota610675952t_unit] : ( produc789536734_sc_sz @ J2 @ I4 ) )
@ ( produc2020166735t_unit @ A
@ ^ [Uu: produc866628903_sc_sz] : B ) )
= ( produc159603053_sc_sz @ B
@ ^ [Uu: denota610675952t_unit] : A ) ) ).
% swap_product
thf(fact_337_member__product,axiom,
! [X: produc190496183real_c,A: set_Fi1407883041real_c,B: set_Fi1407883041real_c] :
( ( member1895684704real_c @ X @ ( produc3751594real_c @ A @ B ) )
= ( member1895684704real_c @ X
@ ( produc563345056real_c @ A
@ ^ [Uu: finite1398487019real_c] : B ) ) ) ).
% member_product
thf(fact_338_map__prod__simp,axiom,
! [F2: denota610675952t_unit > denota610675952t_unit,G: produc866628903_sc_sz > produc866628903_sc_sz,A2: denota610675952t_unit,B2: produc866628903_sc_sz] :
( ( produc403791768_sc_sz @ F2 @ G @ ( produc789536734_sc_sz @ A2 @ B2 ) )
= ( produc789536734_sc_sz @ ( F2 @ A2 ) @ ( G @ B2 ) ) ) ).
% map_prod_simp
thf(fact_339_in__inv__image,axiom,
! [X: finite1398487019real_c,Y: finite1398487019real_c,R: set_Pr1389752855real_c,F2: finite1398487019real_c > finite1398487019real_c] :
( ( member1895684704real_c @ ( produc394644079real_c @ X @ Y ) @ ( inv_im1827249322real_c @ R @ F2 ) )
= ( member1895684704real_c @ ( produc394644079real_c @ ( F2 @ X ) @ ( F2 @ Y ) ) @ R ) ) ).
% in_inv_image
thf(fact_340_map__prod__imageI,axiom,
! [A2: finite1398487019real_c,B2: finite1398487019real_c,R2: set_Pr1389752855real_c,F2: finite1398487019real_c > finite1398487019real_c,G: finite1398487019real_c > finite1398487019real_c] :
( ( member1895684704real_c @ ( produc394644079real_c @ A2 @ B2 ) @ R2 )
=> ( member1895684704real_c @ ( produc394644079real_c @ ( F2 @ A2 ) @ ( G @ B2 ) ) @ ( image_738229701real_c @ ( produc368345510real_c @ F2 @ G ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_341_map__prod__imageI,axiom,
! [A2: finite1398487019real_c,B2: finite1398487019real_c,R2: set_Pr1389752855real_c,F2: finite1398487019real_c > denota610675952t_unit,G: finite1398487019real_c > produc866628903_sc_sz] :
( ( member1895684704real_c @ ( produc394644079real_c @ A2 @ B2 ) @ R2 )
=> ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ ( F2 @ A2 ) @ ( G @ B2 ) ) @ ( image_1259833374_sc_sz @ ( produc1784590311_sc_sz @ F2 @ G ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_342_map__prod__imageI,axiom,
! [A2: denota610675952t_unit,B2: produc866628903_sc_sz,R2: set_Pr1169339874_sc_sz,F2: denota610675952t_unit > finite1398487019real_c,G: produc866628903_sc_sz > finite1398487019real_c] :
( ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ A2 @ B2 ) @ R2 )
=> ( member1895684704real_c @ ( produc394644079real_c @ ( F2 @ A2 ) @ ( G @ B2 ) ) @ ( image_386905412real_c @ ( produc1612557567real_c @ F2 @ G ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_343_map__prod__imageI,axiom,
! [A2: denota610675952t_unit,B2: produc866628903_sc_sz,R2: set_Pr1169339874_sc_sz,F2: denota610675952t_unit > denota610675952t_unit,G: produc866628903_sc_sz > produc866628903_sc_sz] :
( ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ A2 @ B2 ) @ R2 )
=> ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ ( F2 @ A2 ) @ ( G @ B2 ) ) @ ( image_1726193375_sc_sz @ ( produc403791768_sc_sz @ F2 @ G ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_344_prod__fun__imageE,axiom,
! [C2: produc190496183real_c,F2: finite1398487019real_c > finite1398487019real_c,G: finite1398487019real_c > finite1398487019real_c,R2: set_Pr1389752855real_c] :
( ( member1895684704real_c @ C2 @ ( image_738229701real_c @ ( produc368345510real_c @ F2 @ G ) @ R2 ) )
=> ~ ! [X3: finite1398487019real_c,Y3: finite1398487019real_c] :
( ( C2
= ( produc394644079real_c @ ( F2 @ X3 ) @ ( G @ Y3 ) ) )
=> ~ ( member1895684704real_c @ ( produc394644079real_c @ X3 @ Y3 ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_345_prod__fun__imageE,axiom,
! [C2: produc190496183real_c,F2: denota610675952t_unit > finite1398487019real_c,G: produc866628903_sc_sz > finite1398487019real_c,R2: set_Pr1169339874_sc_sz] :
( ( member1895684704real_c @ C2 @ ( image_386905412real_c @ ( produc1612557567real_c @ F2 @ G ) @ R2 ) )
=> ~ ! [X3: denota610675952t_unit,Y3: produc866628903_sc_sz] :
( ( C2
= ( produc394644079real_c @ ( F2 @ X3 ) @ ( G @ Y3 ) ) )
=> ~ ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ X3 @ Y3 ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_346_prod__fun__imageE,axiom,
! [C2: produc1821101996_sc_sz,F2: finite1398487019real_c > denota610675952t_unit,G: finite1398487019real_c > produc866628903_sc_sz,R2: set_Pr1389752855real_c] :
( ( member1057565379_sc_sz @ C2 @ ( image_1259833374_sc_sz @ ( produc1784590311_sc_sz @ F2 @ G ) @ R2 ) )
=> ~ ! [X3: finite1398487019real_c,Y3: finite1398487019real_c] :
( ( C2
= ( produc789536734_sc_sz @ ( F2 @ X3 ) @ ( G @ Y3 ) ) )
=> ~ ( member1895684704real_c @ ( produc394644079real_c @ X3 @ Y3 ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_347_prod__fun__imageE,axiom,
! [C2: produc1821101996_sc_sz,F2: denota610675952t_unit > denota610675952t_unit,G: produc866628903_sc_sz > produc866628903_sc_sz,R2: set_Pr1169339874_sc_sz] :
( ( member1057565379_sc_sz @ C2 @ ( image_1726193375_sc_sz @ ( produc403791768_sc_sz @ F2 @ G ) @ R2 ) )
=> ~ ! [X3: denota610675952t_unit,Y3: produc866628903_sc_sz] :
( ( C2
= ( produc789536734_sc_sz @ ( F2 @ X3 ) @ ( G @ Y3 ) ) )
=> ~ ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ X3 @ Y3 ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_348_Vagree__univ,axiom,
! [A2: finite1398487019real_c,B2: finite1398487019real_c,C2: finite1398487019real_c,D: finite1398487019real_c] :
( ( denota1997846518gree_c @ ( produc394644079real_c @ A2 @ B2 ) @ ( produc394644079real_c @ C2 @ D ) @ top_to1660572043um_c_c )
=> ( ( A2 = C2 )
& ( B2 = D ) ) ) ).
% Vagree_univ
thf(fact_349_agree__UNIV__eq,axiom,
! [Nu3: produc190496183real_c,Omega: produc190496183real_c] :
( ( denota1997846518gree_c @ Nu3 @ Omega @ top_to1660572043um_c_c )
=> ( Nu3 = Omega ) ) ).
% agree_UNIV_eq
thf(fact_350_top__empty__eq2,axiom,
( top_to964419200al_c_o
= ( ^ [X4: finite1398487019real_c,Y4: finite1398487019real_c] : ( member1895684704real_c @ ( produc394644079real_c @ X4 @ Y4 ) @ top_to226056039real_c ) ) ) ).
% top_empty_eq2
thf(fact_351_top__empty__eq2,axiom,
( top_to1399053141c_sz_o
= ( ^ [X4: denota610675952t_unit,Y4: produc866628903_sc_sz] : ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ X4 @ Y4 ) @ top_to1499569298_sc_sz ) ) ) ).
% top_empty_eq2
thf(fact_352_fst__conv,axiom,
! [X1: denota610675952t_unit,X2: produc866628903_sc_sz] :
( ( produc1503602930_sc_sz @ ( produc789536734_sc_sz @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_353_fst__eqD,axiom,
! [X: denota610675952t_unit,Y: produc866628903_sc_sz,A2: denota610675952t_unit] :
( ( ( produc1503602930_sc_sz @ ( produc789536734_sc_sz @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
% Conjectures (1)
thf(conj_0,conjecture,
( ( ! [Nu: produc190496183real_c,Nu2: produc190496183real_c,I: denota231621370t_unit,J: denota231621370t_unit] :
( ( denota69106024_a_b_c @ I @ J @ ( static_SIGF_a_b_c @ phi ) )
=> ( ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVF_a_b_c @ phi ) )
=> ( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ phi ) )
= ( member1895684704real_c @ Nu2 @ ( denota968303861_a_b_c @ J @ phi ) ) ) ) ) )
= ( ! [Nu: produc190496183real_c,Nu2: produc190496183real_c,I: denota231621370t_unit,J: denota231621370t_unit] :
( ( denota69106024_a_b_c @ I @ J @ ( static_SIGF_a_b_c @ phi ) )
=> ( ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVF_a_b_c @ phi ) )
=> ( ( member1895684704real_c @ Nu @ ( denota968303861_a_b_c @ I @ phi ) )
= ( member1895684704real_c @ Nu2 @ ( denota968303861_a_b_c @ J @ phi ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------