TPTP Problem File: SLH0609^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Safe_Range_RC/0017_Preliminaries/prob_00455_019885__16458780_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1777 ( 782 unt; 494 typ; 0 def)
% Number of atoms : 3527 (1541 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 10447 ( 466 ~; 48 |; 293 &;8545 @)
% ( 0 <=>;1095 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 6 avg)
% Number of types : 71 ( 70 usr)
% Number of type conns : 1833 (1833 >; 0 *; 0 +; 0 <<)
% Number of symbols : 427 ( 424 usr; 52 con; 0-4 aty)
% Number of variables : 3781 ( 606 ^;3061 !; 114 ?;3781 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:24:54.864
%------------------------------------------------------------------------------
% Could-be-implicit typings (70)
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thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
insert8933621553020740791od_a_a: produc3498347346309940967od_a_a > set_Pr8600417178894128327od_a_a > set_Pr8600417178894128327od_a_a ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
insert4534936382041156343od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__empty_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
is_emp2937470224744679417od_a_a: set_Product_prod_a_a > $o ).
thf(sy_c_Set_Ois__empty_001tf__a,type,
is_empty_a: set_a > $o ).
thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
is_sin3171834905898671131od_a_a: set_Product_prod_a_a > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
the_el8589169208993665564od_a_a: set_Product_prod_a_a > product_prod_a_a ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_Set_Ovimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
vimage889120284960592251od_a_a: ( product_prod_a_a > product_prod_a_a ) > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Set_Ovimage_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_001tf__a,type,
vimage1181244665347930962_a_a_a: ( product_prod_a_a > a ) > set_a > set_Product_prod_a_a ).
thf(sy_c_Set_Ovimage_001tf__a_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J,type,
vimage434944399782712955_a_a_a: ( a > produc3802892049952890430_a_a_a ) > set_Pr8876520727511657886_a_a_a > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
vimage5143925195038468708od_a_a: ( a > product_prod_a_a ) > set_Product_prod_a_a > set_a ).
thf(sy_c_Set_Ovimage_001tf__a_001tf__a,type,
vimage_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Transitive__Closure_Osymclp_001tf__a,type,
transitive_symclp_a: ( a > a > $o ) > a > a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
accp_l288472794138438844od_a_a: ( list_l2076173926988081843od_a_a > list_l2076173926988081843od_a_a > $o ) > list_l2076173926988081843od_a_a > $o ).
thf(sy_c_Wellfounded_Olex__prod_001tf__a_001tf__a,type,
lex_prod_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr8600417178894128327od_a_a ).
thf(sy_c_Wellfounded_Omax__ext_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
max_ex7229726517752475106od_a_a: set_Pr8600417178894128327od_a_a > set_Pr5931134355633282247od_a_a ).
thf(sy_c_Wellfounded_Omax__ext_001tf__a,type,
max_ext_a: set_Product_prod_a_a > set_Pr5845495582615845127_set_a ).
thf(sy_c_Wellfounded_Omax__extp_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
max_ex6057718940173624378od_a_a: ( product_prod_a_a > product_prod_a_a > $o ) > set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_Wellfounded_Omax__extp_001tf__a,type,
max_extp_a: ( a > a > $o ) > set_a > set_a > $o ).
thf(sy_c_Wfrec_Osame__fst_001tf__a_001tf__a,type,
same_fst_a_a: ( a > $o ) > ( a > set_Product_prod_a_a ) > set_Pr8600417178894128327od_a_a ).
thf(sy_c_member_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J,type,
member1716570166360300819_a_a_a: ( product_prod_a_a > a ) > set_Pr952751117562918450_a_a_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member1957775702407316389od_a_a: ( a > product_prod_a_a ) > set_a_6829686330177631172od_a_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member6824001069763096534od_a_a: list_P1396940483166286381od_a_a > set_li8827807065578854541od_a_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member8183384484874023062od_a_a: option5210160422955383789od_a_a > set_op7160277562814721357od_a_a > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Set__Oset_Itf__a_J_J,type,
member_option_set_a: option_set_a > set_option_set_a > $o ).
thf(sy_c_member_001t__Option__Ooption_Itf__a_J,type,
member_option_a: option_a > set_option_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_Eo_J_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member1420838311466244943od_a_a: produc5831972324637314982od_a_a > set_Pr1368515532014665606od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
member7466972457876170832od_o_o: product_prod_o_o > set_Product_prod_o_o > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member1370157277785437840od_a_a: produc5715173734304154471od_a_a > set_Pr8563485649468363847od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member8191768239178080336list_a: produc9164743771328383783list_a > set_Pr4048851178543822343list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member6330455413206600464od_a_a: produc3498347346309940967od_a_a > set_Pr8600417178894128327od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J,type,
member2829916517802586983_a_a_a: produc3802892049952890430_a_a_a > set_Pr8876520727511657886_a_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_J,type,
member4804927706910867312od_a_a: produc6759108136775012935od_a_a > set_Pr5629751886588192039od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_Mt__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member4903802553882211088od_a_a: produc7983190928304305895od_a_a > set_Pr5931134355633282247od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
member7983343339038529360_set_a: produc1703568184450464039_set_a > set_Pr5845495582615845127_set_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member3071122053849602553od_a_a: produc4044097585999906000od_a_a > set_Pr5530083903271594800od_a_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member153658015593001526od_a_a: set_li8827807065578854541od_a_a > set_se4694998945225767661od_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
member_set_list_a: set_list_a > set_set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member1816616512716248880od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J,type,
member2792204094846457616od_a_a: set_se5735800977113168103od_a_a > set_se5775600271191333063od_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A,type,
a2: set_a ).
thf(sy_v_X____,type,
x: set_a ).
thf(sy_v_a____,type,
a3: a ).
thf(sy_v_b____,type,
b: a ).
thf(sy_v_x____,type,
x2: a ).
thf(sy_v_xys,type,
xys: list_P1396940483166286381od_a_a ).
thf(sy_v_y____,type,
y: a ).
% Relevant facts (1273)
thf(fact_0__092_060open_062a_A_092_060in_062_AX_092_060close_062,axiom,
member_a @ a3 @ x ).
% \<open>a \<in> X\<close>
thf(fact_1__092_060open_062a_A_092_060in_062_AA_092_060close_062,axiom,
member_a @ a3 @ a2 ).
% \<open>a \<in> A\<close>
thf(fact_2__092_060open_062_Ia_M_Ab_J_A_092_060in_062_Asymcl_A_Iset_Axys_J_092_060close_062,axiom,
member1426531477525435216od_a_a @ ( product_Pair_a_a @ a3 @ b ) @ ( symcl_a @ ( set_Product_prod_a_a2 @ xys ) ) ).
% \<open>(a, b) \<in> symcl (set xys)\<close>
thf(fact_3_assms_I2_J,axiom,
xys != nil_Product_prod_a_a ).
% assms(2)
thf(fact_4__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_Ay_O_A_Ix_M_Ay_J_A_092_060in_062_Aset_Axys_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X: a,Y: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( set_Product_prod_a_a2 @ xys ) ) ).
% \<open>\<And>thesis. (\<And>x y. (x, y) \<in> set xys \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_5__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062a_O_A_092_060lbrakk_062a_A_092_060in_062_AA_059_Aa_A_092_060in_062_AX_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [A: a] :
( ( member_a @ A @ a2 )
=> ~ ( member_a @ A @ x ) ) ).
% \<open>\<And>thesis. (\<And>a. \<lbrakk>a \<in> A; a \<in> X\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_6__092_060open_062_Ix_M_Ay_J_A_092_060in_062_Aset_Axys_092_060close_062,axiom,
member1426531477525435216od_a_a @ ( product_Pair_a_a @ x2 @ y ) @ ( set_Product_prod_a_a2 @ xys ) ).
% \<open>(x, y) \<in> set xys\<close>
thf(fact_7_split__part,axiom,
! [P: $o,Q: a > a > $o] :
( ( produc6436628058953941356_a_a_o
@ ^ [A2: a,B: a] :
( P
& ( Q @ A2 @ B ) ) )
= ( ^ [Ab: product_prod_a_a] :
( P
& ( produc6436628058953941356_a_a_o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_8__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062b_O_A_Ia_M_Ab_J_A_092_060in_062_Asymcl_A_Iset_Axys_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [B2: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ a3 @ B2 ) @ ( symcl_a @ ( set_Product_prod_a_a2 @ xys ) ) ) ).
% \<open>\<And>thesis. (\<And>b. (a, b) \<in> symcl (set xys) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_9_prod_Odisc__eq__case,axiom,
! [Prod: product_prod_a_a] :
( produc6436628058953941356_a_a_o
@ ^ [Uu: a,Uv: a] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_10_prod_Ocase__distrib,axiom,
! [H: $o > $o,F: a > a > $o,Prod: product_prod_a_a] :
( ( H @ ( produc6436628058953941356_a_a_o @ F @ Prod ) )
= ( produc6436628058953941356_a_a_o
@ ^ [X1: a,X2: a] : ( H @ ( F @ X1 @ X2 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_11_prod_Ocase__distrib,axiom,
! [H: $o > a,F: a > a > $o,Prod: product_prod_a_a] :
( ( H @ ( produc6436628058953941356_a_a_o @ F @ Prod ) )
= ( produc8815886927560695506_a_a_a
@ ^ [X1: a,X2: a] : ( H @ ( F @ X1 @ X2 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_12_prod_Ocase__distrib,axiom,
! [H: a > $o,F: a > a > a,Prod: product_prod_a_a] :
( ( H @ ( produc8815886927560695506_a_a_a @ F @ Prod ) )
= ( produc6436628058953941356_a_a_o
@ ^ [X1: a,X2: a] : ( H @ ( F @ X1 @ X2 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_13_prod_Ocase__distrib,axiom,
! [H: a > a,F: a > a > a,Prod: product_prod_a_a] :
( ( H @ ( produc8815886927560695506_a_a_a @ F @ Prod ) )
= ( produc8815886927560695506_a_a_a
@ ^ [X1: a,X2: a] : ( H @ ( F @ X1 @ X2 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_14_prod_Ocase__distrib,axiom,
! [H: $o > set_a,F: a > a > $o,Prod: product_prod_a_a] :
( ( H @ ( produc6436628058953941356_a_a_o @ F @ Prod ) )
= ( produc9217457822752978994_set_a
@ ^ [X1: a,X2: a] : ( H @ ( F @ X1 @ X2 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_15_prod_Ocase__distrib,axiom,
! [H: set_a > $o,F: a > a > set_a,Prod: product_prod_a_a] :
( ( H @ ( produc9217457822752978994_set_a @ F @ Prod ) )
= ( produc6436628058953941356_a_a_o
@ ^ [X1: a,X2: a] : ( H @ ( F @ X1 @ X2 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_16_prod_Ocase__distrib,axiom,
! [H: set_a > a,F: a > a > set_a,Prod: product_prod_a_a] :
( ( H @ ( produc9217457822752978994_set_a @ F @ Prod ) )
= ( produc8815886927560695506_a_a_a
@ ^ [X1: a,X2: a] : ( H @ ( F @ X1 @ X2 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_17_prod_Ocase__distrib,axiom,
! [H: a > set_a,F: a > a > a,Prod: product_prod_a_a] :
( ( H @ ( produc8815886927560695506_a_a_a @ F @ Prod ) )
= ( produc9217457822752978994_set_a
@ ^ [X1: a,X2: a] : ( H @ ( F @ X1 @ X2 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_18_prod_Ocase__distrib,axiom,
! [H: set_a > set_a,F: a > a > set_a,Prod: product_prod_a_a] :
( ( H @ ( produc9217457822752978994_set_a @ F @ Prod ) )
= ( produc9217457822752978994_set_a
@ ^ [X1: a,X2: a] : ( H @ ( F @ X1 @ X2 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_19_prod_Ocase__distrib,axiom,
! [H: $o > set_Product_prod_a_a,F: a > a > $o,Prod: product_prod_a_a] :
( ( H @ ( produc6436628058953941356_a_a_o @ F @ Prod ) )
= ( produc5766521956407364827od_a_a
@ ^ [X1: a,X2: a] : ( H @ ( F @ X1 @ X2 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_20_case__prod__app,axiom,
( produc5089340801457792962_a_a_o
= ( ^ [F2: a > a > product_prod_a_a > $o,X3: product_prod_a_a,Y2: product_prod_a_a] :
( produc6436628058953941356_a_a_o
@ ^ [L: a,R: a] : ( F2 @ L @ R @ Y2 )
@ X3 ) ) ) ).
% case_prod_app
thf(fact_21_case__prod__app,axiom,
( produc2520556867391656337_set_a
= ( ^ [F2: a > a > set_a > set_a,X3: product_prod_a_a,Y2: set_a] :
( produc9217457822752978994_set_a
@ ^ [L: a,R: a] : ( F2 @ L @ R @ Y2 )
@ X3 ) ) ) ).
% case_prod_app
thf(fact_22_symcl__def,axiom,
( symcl_286300493019922782od_a_a
= ( ^ [R: set_Pr5931134355633282247od_a_a] :
( collec3859266087184377170od_a_a
@ ( produc4361984678274043180_a_a_o
@ ^ [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a] :
( ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ X3 @ Y2 ) @ R )
| ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ Y2 @ X3 ) @ R ) ) ) ) ) ) ).
% symcl_def
thf(fact_23_symcl__def,axiom,
( symcl_set_a
= ( ^ [R: set_Pr5845495582615845127_set_a] :
( collec8259436133773553042_set_a
@ ( produc2941226113218366828et_a_o
@ ^ [X3: set_a,Y2: set_a] :
( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X3 @ Y2 ) @ R )
| ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ Y2 @ X3 ) @ R ) ) ) ) ) ) ).
% symcl_def
thf(fact_24_symcl__def,axiom,
( symcl_a
= ( ^ [R: set_Product_prod_a_a] :
( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X3: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R )
| ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y2 @ X3 ) @ R ) ) ) ) ) ) ).
% symcl_def
thf(fact_25_find__None__iff,axiom,
! [P: list_a > $o,Xs: list_list_a] :
( ( ( find_list_a @ P @ Xs )
= none_list_a )
= ( ~ ? [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% find_None_iff
thf(fact_26_find__None__iff,axiom,
! [P: $o > $o,Xs: list_o] :
( ( ( find_o @ P @ Xs )
= none_o )
= ( ~ ? [X3: $o] :
( ( member_o @ X3 @ ( set_o2 @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% find_None_iff
thf(fact_27_find__None__iff,axiom,
! [P: produc3498347346309940967od_a_a > $o,Xs: list_P6569952146052023789od_a_a] :
( ( ( find_P7890441392728764675od_a_a @ P @ Xs )
= none_P5538296941832000840od_a_a )
= ( ~ ? [X3: produc3498347346309940967od_a_a] :
( ( member6330455413206600464od_a_a @ X3 @ ( set_Pr6493056779096584572od_a_a @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% find_None_iff
thf(fact_28_find__None__iff,axiom,
! [P: list_P1396940483166286381od_a_a > $o,Xs: list_l2076173926988081843od_a_a] :
( ( ( find_l5372068402991683145od_a_a @ P @ Xs )
= none_l1738640210340310542od_a_a )
= ( ~ ? [X3: list_P1396940483166286381od_a_a] :
( ( member6824001069763096534od_a_a @ X3 @ ( set_li4974249543031949378od_a_a @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% find_None_iff
thf(fact_29_find__None__iff,axiom,
! [P: set_a > $o,Xs: list_set_a] :
( ( ( find_set_a @ P @ Xs )
= none_set_a )
= ( ~ ? [X3: set_a] :
( ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% find_None_iff
thf(fact_30_find__None__iff,axiom,
! [P: set_Product_prod_a_a > $o,Xs: list_s9060204159073123853od_a_a] :
( ( ( find_s2827193664816301859od_a_a @ P @ Xs )
= none_s5714488769912451432od_a_a )
= ( ~ ? [X3: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X3 @ ( set_se8408754101646271900od_a_a @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% find_None_iff
thf(fact_31_find__None__iff,axiom,
! [P: a > $o,Xs: list_a] :
( ( ( find_a @ P @ Xs )
= none_a )
= ( ~ ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% find_None_iff
thf(fact_32_find__None__iff,axiom,
! [P: product_prod_a_a > $o,Xs: list_P1396940483166286381od_a_a] :
( ( ( find_P2248954792982286147od_a_a @ P @ Xs )
= none_P1914705783780380040od_a_a )
= ( ~ ? [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% find_None_iff
thf(fact_33_find__None__iff2,axiom,
! [P: list_a > $o,Xs: list_list_a] :
( ( none_list_a
= ( find_list_a @ P @ Xs ) )
= ( ~ ? [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% find_None_iff2
thf(fact_34_find__None__iff2,axiom,
! [P: $o > $o,Xs: list_o] :
( ( none_o
= ( find_o @ P @ Xs ) )
= ( ~ ? [X3: $o] :
( ( member_o @ X3 @ ( set_o2 @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% find_None_iff2
thf(fact_35_find__None__iff2,axiom,
! [P: produc3498347346309940967od_a_a > $o,Xs: list_P6569952146052023789od_a_a] :
( ( none_P5538296941832000840od_a_a
= ( find_P7890441392728764675od_a_a @ P @ Xs ) )
= ( ~ ? [X3: produc3498347346309940967od_a_a] :
( ( member6330455413206600464od_a_a @ X3 @ ( set_Pr6493056779096584572od_a_a @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% find_None_iff2
thf(fact_36_find__None__iff2,axiom,
! [P: list_P1396940483166286381od_a_a > $o,Xs: list_l2076173926988081843od_a_a] :
( ( none_l1738640210340310542od_a_a
= ( find_l5372068402991683145od_a_a @ P @ Xs ) )
= ( ~ ? [X3: list_P1396940483166286381od_a_a] :
( ( member6824001069763096534od_a_a @ X3 @ ( set_li4974249543031949378od_a_a @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% find_None_iff2
thf(fact_37_find__None__iff2,axiom,
! [P: set_a > $o,Xs: list_set_a] :
( ( none_set_a
= ( find_set_a @ P @ Xs ) )
= ( ~ ? [X3: set_a] :
( ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% find_None_iff2
thf(fact_38_find__None__iff2,axiom,
! [P: set_Product_prod_a_a > $o,Xs: list_s9060204159073123853od_a_a] :
( ( none_s5714488769912451432od_a_a
= ( find_s2827193664816301859od_a_a @ P @ Xs ) )
= ( ~ ? [X3: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X3 @ ( set_se8408754101646271900od_a_a @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% find_None_iff2
thf(fact_39_find__None__iff2,axiom,
! [P: a > $o,Xs: list_a] :
( ( none_a
= ( find_a @ P @ Xs ) )
= ( ~ ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% find_None_iff2
thf(fact_40_find__None__iff2,axiom,
! [P: product_prod_a_a > $o,Xs: list_P1396940483166286381od_a_a] :
( ( none_P1914705783780380040od_a_a
= ( find_P2248954792982286147od_a_a @ P @ Xs ) )
= ( ~ ? [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% find_None_iff2
thf(fact_41_case__prodI,axiom,
! [F: set_Product_prod_a_a > produc5715173734304154471od_a_a > $o,A3: set_Product_prod_a_a,B3: produc5715173734304154471od_a_a] :
( ( F @ A3 @ B3 )
=> ( produc6861823381246321356_a_a_o @ F @ ( produc9141123965022120375od_a_a @ A3 @ B3 ) ) ) ).
% case_prodI
thf(fact_42_case__prodI,axiom,
! [F: set_Product_prod_a_a > set_Product_prod_a_a > $o,A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( F @ A3 @ B3 )
=> ( produc4361984678274043180_a_a_o @ F @ ( produc3372137660326521687od_a_a @ A3 @ B3 ) ) ) ).
% case_prodI
thf(fact_43_case__prodI,axiom,
! [F: set_a > set_a > $o,A3: set_a,B3: set_a] :
( ( F @ A3 @ B3 )
=> ( produc2941226113218366828et_a_o @ F @ ( produc9088192753505129239_set_a @ A3 @ B3 ) ) ) ).
% case_prodI
thf(fact_44_case__prodI,axiom,
! [F: ( product_prod_a_a > product_prod_a_a > $o ) > list_P1396940483166286381od_a_a > $o,A3: product_prod_a_a > product_prod_a_a > $o,B3: list_P1396940483166286381od_a_a] :
( ( F @ A3 @ B3 )
=> ( produc8373039017049881389_a_a_o @ F @ ( produc7609767723203257750od_a_a @ A3 @ B3 ) ) ) ).
% case_prodI
thf(fact_45_case__prodI,axiom,
! [F: a > product_prod_a_a > $o,A3: a,B3: product_prod_a_a] :
( ( F @ A3 @ B3 )
=> ( produc5856822985862792195_a_a_o @ F @ ( produc431845341423274048od_a_a @ A3 @ B3 ) ) ) ).
% case_prodI
thf(fact_46_case__prodI,axiom,
! [F: a > a > $o,A3: a,B3: a] :
( ( F @ A3 @ B3 )
=> ( produc6436628058953941356_a_a_o @ F @ ( product_Pair_a_a @ A3 @ B3 ) ) ) ).
% case_prodI
thf(fact_47_case__prodI2,axiom,
! [P2: produc6759108136775012935od_a_a,C: set_Product_prod_a_a > produc5715173734304154471od_a_a > $o] :
( ! [A: set_Product_prod_a_a,B2: produc5715173734304154471od_a_a] :
( ( P2
= ( produc9141123965022120375od_a_a @ A @ B2 ) )
=> ( C @ A @ B2 ) )
=> ( produc6861823381246321356_a_a_o @ C @ P2 ) ) ).
% case_prodI2
thf(fact_48_case__prodI2,axiom,
! [P2: produc7983190928304305895od_a_a,C: set_Product_prod_a_a > set_Product_prod_a_a > $o] :
( ! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( ( P2
= ( produc3372137660326521687od_a_a @ A @ B2 ) )
=> ( C @ A @ B2 ) )
=> ( produc4361984678274043180_a_a_o @ C @ P2 ) ) ).
% case_prodI2
thf(fact_49_case__prodI2,axiom,
! [P2: produc1703568184450464039_set_a,C: set_a > set_a > $o] :
( ! [A: set_a,B2: set_a] :
( ( P2
= ( produc9088192753505129239_set_a @ A @ B2 ) )
=> ( C @ A @ B2 ) )
=> ( produc2941226113218366828et_a_o @ C @ P2 ) ) ).
% case_prodI2
thf(fact_50_case__prodI2,axiom,
! [P2: produc5831972324637314982od_a_a,C: ( product_prod_a_a > product_prod_a_a > $o ) > list_P1396940483166286381od_a_a > $o] :
( ! [A: product_prod_a_a > product_prod_a_a > $o,B2: list_P1396940483166286381od_a_a] :
( ( P2
= ( produc7609767723203257750od_a_a @ A @ B2 ) )
=> ( C @ A @ B2 ) )
=> ( produc8373039017049881389_a_a_o @ C @ P2 ) ) ).
% case_prodI2
thf(fact_51_case__prodI2,axiom,
! [P2: produc4044097585999906000od_a_a,C: a > product_prod_a_a > $o] :
( ! [A: a,B2: product_prod_a_a] :
( ( P2
= ( produc431845341423274048od_a_a @ A @ B2 ) )
=> ( C @ A @ B2 ) )
=> ( produc5856822985862792195_a_a_o @ C @ P2 ) ) ).
% case_prodI2
thf(fact_52_case__prodI2,axiom,
! [P2: product_prod_a_a,C: a > a > $o] :
( ! [A: a,B2: a] :
( ( P2
= ( product_Pair_a_a @ A @ B2 ) )
=> ( C @ A @ B2 ) )
=> ( produc6436628058953941356_a_a_o @ C @ P2 ) ) ).
% case_prodI2
thf(fact_53_case__prod__conv,axiom,
! [F: a > a > set_Product_prod_a_a,A3: a,B3: a] :
( ( produc5766521956407364827od_a_a @ F @ ( product_Pair_a_a @ A3 @ B3 ) )
= ( F @ A3 @ B3 ) ) ).
% case_prod_conv
thf(fact_54_case__prod__conv,axiom,
! [F: a > a > set_a,A3: a,B3: a] :
( ( produc9217457822752978994_set_a @ F @ ( product_Pair_a_a @ A3 @ B3 ) )
= ( F @ A3 @ B3 ) ) ).
% case_prod_conv
thf(fact_55_case__prod__conv,axiom,
! [F: a > a > product_prod_a_a > $o,A3: a,B3: a] :
( ( produc5089340801457792962_a_a_o @ F @ ( product_Pair_a_a @ A3 @ B3 ) )
= ( F @ A3 @ B3 ) ) ).
% case_prod_conv
thf(fact_56_case__prod__conv,axiom,
! [F: a > a > set_a > set_a,A3: a,B3: a] :
( ( produc2520556867391656337_set_a @ F @ ( product_Pair_a_a @ A3 @ B3 ) )
= ( F @ A3 @ B3 ) ) ).
% case_prod_conv
thf(fact_57_case__prod__conv,axiom,
! [F: a > a > a,A3: a,B3: a] :
( ( produc8815886927560695506_a_a_a @ F @ ( product_Pair_a_a @ A3 @ B3 ) )
= ( F @ A3 @ B3 ) ) ).
% case_prod_conv
thf(fact_58_case__prod__conv,axiom,
! [F: a > a > $o,A3: a,B3: a] :
( ( produc6436628058953941356_a_a_o @ F @ ( product_Pair_a_a @ A3 @ B3 ) )
= ( F @ A3 @ B3 ) ) ).
% case_prod_conv
thf(fact_59_prod_Oinject,axiom,
! [X12: set_Product_prod_a_a,X22: produc5715173734304154471od_a_a,Y1: set_Product_prod_a_a,Y22: produc5715173734304154471od_a_a] :
( ( ( produc9141123965022120375od_a_a @ X12 @ X22 )
= ( produc9141123965022120375od_a_a @ Y1 @ Y22 ) )
= ( ( X12 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_60_prod_Oinject,axiom,
! [X12: set_Product_prod_a_a,X22: set_Product_prod_a_a,Y1: set_Product_prod_a_a,Y22: set_Product_prod_a_a] :
( ( ( produc3372137660326521687od_a_a @ X12 @ X22 )
= ( produc3372137660326521687od_a_a @ Y1 @ Y22 ) )
= ( ( X12 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_61_prod_Oinject,axiom,
! [X12: set_a,X22: set_a,Y1: set_a,Y22: set_a] :
( ( ( produc9088192753505129239_set_a @ X12 @ X22 )
= ( produc9088192753505129239_set_a @ Y1 @ Y22 ) )
= ( ( X12 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_62_prod_Oinject,axiom,
! [X12: product_prod_a_a > product_prod_a_a > $o,X22: list_P1396940483166286381od_a_a,Y1: product_prod_a_a > product_prod_a_a > $o,Y22: list_P1396940483166286381od_a_a] :
( ( ( produc7609767723203257750od_a_a @ X12 @ X22 )
= ( produc7609767723203257750od_a_a @ Y1 @ Y22 ) )
= ( ( X12 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_63_prod_Oinject,axiom,
! [X12: a,X22: product_prod_a_a,Y1: a,Y22: product_prod_a_a] :
( ( ( produc431845341423274048od_a_a @ X12 @ X22 )
= ( produc431845341423274048od_a_a @ Y1 @ Y22 ) )
= ( ( X12 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_64_prod_Oinject,axiom,
! [X12: a,X22: a,Y1: a,Y22: a] :
( ( ( product_Pair_a_a @ X12 @ X22 )
= ( product_Pair_a_a @ Y1 @ Y22 ) )
= ( ( X12 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_65_old_Oprod_Oinject,axiom,
! [A3: set_Product_prod_a_a,B3: produc5715173734304154471od_a_a,A4: set_Product_prod_a_a,B4: produc5715173734304154471od_a_a] :
( ( ( produc9141123965022120375od_a_a @ A3 @ B3 )
= ( produc9141123965022120375od_a_a @ A4 @ B4 ) )
= ( ( A3 = A4 )
& ( B3 = B4 ) ) ) ).
% old.prod.inject
thf(fact_66_old_Oprod_Oinject,axiom,
! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a,A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ( produc3372137660326521687od_a_a @ A3 @ B3 )
= ( produc3372137660326521687od_a_a @ A4 @ B4 ) )
= ( ( A3 = A4 )
& ( B3 = B4 ) ) ) ).
% old.prod.inject
thf(fact_67_old_Oprod_Oinject,axiom,
! [A3: set_a,B3: set_a,A4: set_a,B4: set_a] :
( ( ( produc9088192753505129239_set_a @ A3 @ B3 )
= ( produc9088192753505129239_set_a @ A4 @ B4 ) )
= ( ( A3 = A4 )
& ( B3 = B4 ) ) ) ).
% old.prod.inject
thf(fact_68_old_Oprod_Oinject,axiom,
! [A3: product_prod_a_a > product_prod_a_a > $o,B3: list_P1396940483166286381od_a_a,A4: product_prod_a_a > product_prod_a_a > $o,B4: list_P1396940483166286381od_a_a] :
( ( ( produc7609767723203257750od_a_a @ A3 @ B3 )
= ( produc7609767723203257750od_a_a @ A4 @ B4 ) )
= ( ( A3 = A4 )
& ( B3 = B4 ) ) ) ).
% old.prod.inject
thf(fact_69_old_Oprod_Oinject,axiom,
! [A3: a,B3: product_prod_a_a,A4: a,B4: product_prod_a_a] :
( ( ( produc431845341423274048od_a_a @ A3 @ B3 )
= ( produc431845341423274048od_a_a @ A4 @ B4 ) )
= ( ( A3 = A4 )
& ( B3 = B4 ) ) ) ).
% old.prod.inject
thf(fact_70_old_Oprod_Oinject,axiom,
! [A3: a,B3: a,A4: a,B4: a] :
( ( ( product_Pair_a_a @ A3 @ B3 )
= ( product_Pair_a_a @ A4 @ B4 ) )
= ( ( A3 = A4 )
& ( B3 = B4 ) ) ) ).
% old.prod.inject
thf(fact_71_case__prodI2_H,axiom,
! [P2: product_prod_a_a,C: a > a > product_prod_a_a > $o,X4: product_prod_a_a] :
( ! [A: a,B2: a] :
( ( ( product_Pair_a_a @ A @ B2 )
= P2 )
=> ( C @ A @ B2 @ X4 ) )
=> ( produc5089340801457792962_a_a_o @ C @ P2 @ X4 ) ) ).
% case_prodI2'
thf(fact_72_mem__case__prodI,axiom,
! [Z: a,C: a > a > set_a,A3: a,B3: a] :
( ( member_a @ Z @ ( C @ A3 @ B3 ) )
=> ( member_a @ Z @ ( produc9217457822752978994_set_a @ C @ ( product_Pair_a_a @ A3 @ B3 ) ) ) ) ).
% mem_case_prodI
thf(fact_73_mem__case__prodI,axiom,
! [Z: product_prod_a_a,C: a > a > set_Product_prod_a_a,A3: a,B3: a] :
( ( member1426531477525435216od_a_a @ Z @ ( C @ A3 @ B3 ) )
=> ( member1426531477525435216od_a_a @ Z @ ( produc5766521956407364827od_a_a @ C @ ( product_Pair_a_a @ A3 @ B3 ) ) ) ) ).
% mem_case_prodI
thf(fact_74_mem__case__prodI,axiom,
! [Z: $o,C: a > a > set_o,A3: a,B3: a] :
( ( member_o @ Z @ ( C @ A3 @ B3 ) )
=> ( member_o @ Z @ ( produc850716608257783884_set_o @ C @ ( product_Pair_a_a @ A3 @ B3 ) ) ) ) ).
% mem_case_prodI
thf(fact_75_mem__case__prodI,axiom,
! [Z: list_a,C: a > a > set_list_a,A3: a,B3: a] :
( ( member_list_a @ Z @ ( C @ A3 @ B3 ) )
=> ( member_list_a @ Z @ ( produc8229661187788642616list_a @ C @ ( product_Pair_a_a @ A3 @ B3 ) ) ) ) ).
% mem_case_prodI
thf(fact_76_mem__case__prodI,axiom,
! [Z: set_a,C: a > a > set_set_a,A3: a,B3: a] :
( ( member_set_a @ Z @ ( C @ A3 @ B3 ) )
=> ( member_set_a @ Z @ ( produc2818859882669968786_set_a @ C @ ( product_Pair_a_a @ A3 @ B3 ) ) ) ) ).
% mem_case_prodI
thf(fact_77_mem__case__prodI,axiom,
! [Z: a,C: set_a > set_a > set_a,A3: set_a,B3: set_a] :
( ( member_a @ Z @ ( C @ A3 @ B3 ) )
=> ( member_a @ Z @ ( produc4474087116711199794_set_a @ C @ ( produc9088192753505129239_set_a @ A3 @ B3 ) ) ) ) ).
% mem_case_prodI
thf(fact_78_mem__case__prodI,axiom,
! [Z: $o,C: set_a > set_a > set_o,A3: set_a,B3: set_a] :
( ( member_o @ Z @ ( C @ A3 @ B3 ) )
=> ( member_o @ Z @ ( produc4110613151828132940_set_o @ C @ ( produc9088192753505129239_set_a @ A3 @ B3 ) ) ) ) ).
% mem_case_prodI
thf(fact_79_mem__case__prodI,axiom,
! [Z: a,C: a > product_prod_a_a > set_a,A3: a,B3: product_prod_a_a] :
( ( member_a @ Z @ ( C @ A3 @ B3 ) )
=> ( member_a @ Z @ ( produc1173728711619330889_set_a @ C @ ( produc431845341423274048od_a_a @ A3 @ B3 ) ) ) ) ).
% mem_case_prodI
thf(fact_80_mem__case__prodI,axiom,
! [Z: $o,C: a > product_prod_a_a > set_o,A3: a,B3: product_prod_a_a] :
( ( member_o @ Z @ ( C @ A3 @ B3 ) )
=> ( member_o @ Z @ ( produc1957367346529820387_set_o @ C @ ( produc431845341423274048od_a_a @ A3 @ B3 ) ) ) ) ).
% mem_case_prodI
thf(fact_81_mem__case__prodI,axiom,
! [Z: list_P1396940483166286381od_a_a,C: a > a > set_li8827807065578854541od_a_a,A3: a,B3: a] :
( ( member6824001069763096534od_a_a @ Z @ ( C @ A3 @ B3 ) )
=> ( member6824001069763096534od_a_a @ Z @ ( produc8624483484282574433od_a_a @ C @ ( product_Pair_a_a @ A3 @ B3 ) ) ) ) ).
% mem_case_prodI
thf(fact_82_mem__case__prodI2,axiom,
! [P2: product_prod_a_a,Z: a,C: a > a > set_a] :
( ! [A: a,B2: a] :
( ( P2
= ( product_Pair_a_a @ A @ B2 ) )
=> ( member_a @ Z @ ( C @ A @ B2 ) ) )
=> ( member_a @ Z @ ( produc9217457822752978994_set_a @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_83_mem__case__prodI2,axiom,
! [P2: product_prod_a_a,Z: product_prod_a_a,C: a > a > set_Product_prod_a_a] :
( ! [A: a,B2: a] :
( ( P2
= ( product_Pair_a_a @ A @ B2 ) )
=> ( member1426531477525435216od_a_a @ Z @ ( C @ A @ B2 ) ) )
=> ( member1426531477525435216od_a_a @ Z @ ( produc5766521956407364827od_a_a @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_84_mem__case__prodI2,axiom,
! [P2: product_prod_a_a,Z: $o,C: a > a > set_o] :
( ! [A: a,B2: a] :
( ( P2
= ( product_Pair_a_a @ A @ B2 ) )
=> ( member_o @ Z @ ( C @ A @ B2 ) ) )
=> ( member_o @ Z @ ( produc850716608257783884_set_o @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_85_mem__case__prodI2,axiom,
! [P2: product_prod_a_a,Z: list_a,C: a > a > set_list_a] :
( ! [A: a,B2: a] :
( ( P2
= ( product_Pair_a_a @ A @ B2 ) )
=> ( member_list_a @ Z @ ( C @ A @ B2 ) ) )
=> ( member_list_a @ Z @ ( produc8229661187788642616list_a @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_86_mem__case__prodI2,axiom,
! [P2: product_prod_a_a,Z: set_a,C: a > a > set_set_a] :
( ! [A: a,B2: a] :
( ( P2
= ( product_Pair_a_a @ A @ B2 ) )
=> ( member_set_a @ Z @ ( C @ A @ B2 ) ) )
=> ( member_set_a @ Z @ ( produc2818859882669968786_set_a @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_87_mem__case__prodI2,axiom,
! [P2: produc1703568184450464039_set_a,Z: a,C: set_a > set_a > set_a] :
( ! [A: set_a,B2: set_a] :
( ( P2
= ( produc9088192753505129239_set_a @ A @ B2 ) )
=> ( member_a @ Z @ ( C @ A @ B2 ) ) )
=> ( member_a @ Z @ ( produc4474087116711199794_set_a @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_88_mem__case__prodI2,axiom,
! [P2: produc1703568184450464039_set_a,Z: $o,C: set_a > set_a > set_o] :
( ! [A: set_a,B2: set_a] :
( ( P2
= ( produc9088192753505129239_set_a @ A @ B2 ) )
=> ( member_o @ Z @ ( C @ A @ B2 ) ) )
=> ( member_o @ Z @ ( produc4110613151828132940_set_o @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_89_mem__case__prodI2,axiom,
! [P2: produc4044097585999906000od_a_a,Z: a,C: a > product_prod_a_a > set_a] :
( ! [A: a,B2: product_prod_a_a] :
( ( P2
= ( produc431845341423274048od_a_a @ A @ B2 ) )
=> ( member_a @ Z @ ( C @ A @ B2 ) ) )
=> ( member_a @ Z @ ( produc1173728711619330889_set_a @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_90_mem__case__prodI2,axiom,
! [P2: produc4044097585999906000od_a_a,Z: $o,C: a > product_prod_a_a > set_o] :
( ! [A: a,B2: product_prod_a_a] :
( ( P2
= ( produc431845341423274048od_a_a @ A @ B2 ) )
=> ( member_o @ Z @ ( C @ A @ B2 ) ) )
=> ( member_o @ Z @ ( produc1957367346529820387_set_o @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_91_mem__case__prodI2,axiom,
! [P2: product_prod_a_a,Z: list_P1396940483166286381od_a_a,C: a > a > set_li8827807065578854541od_a_a] :
( ! [A: a,B2: a] :
( ( P2
= ( product_Pair_a_a @ A @ B2 ) )
=> ( member6824001069763096534od_a_a @ Z @ ( C @ A @ B2 ) ) )
=> ( member6824001069763096534od_a_a @ Z @ ( produc8624483484282574433od_a_a @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_92_old_Oprod_Oexhaust,axiom,
! [Y3: produc6759108136775012935od_a_a] :
~ ! [A: set_Product_prod_a_a,B2: produc5715173734304154471od_a_a] :
( Y3
!= ( produc9141123965022120375od_a_a @ A @ B2 ) ) ).
% old.prod.exhaust
thf(fact_93_old_Oprod_Oexhaust,axiom,
! [Y3: produc7983190928304305895od_a_a] :
~ ! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] :
( Y3
!= ( produc3372137660326521687od_a_a @ A @ B2 ) ) ).
% old.prod.exhaust
thf(fact_94_old_Oprod_Oexhaust,axiom,
! [Y3: produc1703568184450464039_set_a] :
~ ! [A: set_a,B2: set_a] :
( Y3
!= ( produc9088192753505129239_set_a @ A @ B2 ) ) ).
% old.prod.exhaust
thf(fact_95_old_Oprod_Oexhaust,axiom,
! [Y3: produc5831972324637314982od_a_a] :
~ ! [A: product_prod_a_a > product_prod_a_a > $o,B2: list_P1396940483166286381od_a_a] :
( Y3
!= ( produc7609767723203257750od_a_a @ A @ B2 ) ) ).
% old.prod.exhaust
thf(fact_96_old_Oprod_Oexhaust,axiom,
! [Y3: produc4044097585999906000od_a_a] :
~ ! [A: a,B2: product_prod_a_a] :
( Y3
!= ( produc431845341423274048od_a_a @ A @ B2 ) ) ).
% old.prod.exhaust
thf(fact_97_old_Oprod_Oexhaust,axiom,
! [Y3: product_prod_a_a] :
~ ! [A: a,B2: a] :
( Y3
!= ( product_Pair_a_a @ A @ B2 ) ) ).
% old.prod.exhaust
thf(fact_98_surj__pair,axiom,
! [P2: produc6759108136775012935od_a_a] :
? [X: set_Product_prod_a_a,Y: produc5715173734304154471od_a_a] :
( P2
= ( produc9141123965022120375od_a_a @ X @ Y ) ) ).
% surj_pair
thf(fact_99_surj__pair,axiom,
! [P2: produc7983190928304305895od_a_a] :
? [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( P2
= ( produc3372137660326521687od_a_a @ X @ Y ) ) ).
% surj_pair
thf(fact_100_surj__pair,axiom,
! [P2: produc1703568184450464039_set_a] :
? [X: set_a,Y: set_a] :
( P2
= ( produc9088192753505129239_set_a @ X @ Y ) ) ).
% surj_pair
thf(fact_101_surj__pair,axiom,
! [P2: produc5831972324637314982od_a_a] :
? [X: product_prod_a_a > product_prod_a_a > $o,Y: list_P1396940483166286381od_a_a] :
( P2
= ( produc7609767723203257750od_a_a @ X @ Y ) ) ).
% surj_pair
thf(fact_102_surj__pair,axiom,
! [P2: produc4044097585999906000od_a_a] :
? [X: a,Y: product_prod_a_a] :
( P2
= ( produc431845341423274048od_a_a @ X @ Y ) ) ).
% surj_pair
thf(fact_103_surj__pair,axiom,
! [P2: product_prod_a_a] :
? [X: a,Y: a] :
( P2
= ( product_Pair_a_a @ X @ Y ) ) ).
% surj_pair
thf(fact_104_prod__cases,axiom,
! [P: produc6759108136775012935od_a_a > $o,P2: produc6759108136775012935od_a_a] :
( ! [A: set_Product_prod_a_a,B2: produc5715173734304154471od_a_a] : ( P @ ( produc9141123965022120375od_a_a @ A @ B2 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_105_prod__cases,axiom,
! [P: produc7983190928304305895od_a_a > $o,P2: produc7983190928304305895od_a_a] :
( ! [A: set_Product_prod_a_a,B2: set_Product_prod_a_a] : ( P @ ( produc3372137660326521687od_a_a @ A @ B2 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_106_prod__cases,axiom,
! [P: produc1703568184450464039_set_a > $o,P2: produc1703568184450464039_set_a] :
( ! [A: set_a,B2: set_a] : ( P @ ( produc9088192753505129239_set_a @ A @ B2 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_107_prod__cases,axiom,
! [P: produc5831972324637314982od_a_a > $o,P2: produc5831972324637314982od_a_a] :
( ! [A: product_prod_a_a > product_prod_a_a > $o,B2: list_P1396940483166286381od_a_a] : ( P @ ( produc7609767723203257750od_a_a @ A @ B2 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_108_prod__cases,axiom,
! [P: produc4044097585999906000od_a_a > $o,P2: produc4044097585999906000od_a_a] :
( ! [A: a,B2: product_prod_a_a] : ( P @ ( produc431845341423274048od_a_a @ A @ B2 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_109_prod__cases,axiom,
! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
( ! [A: a,B2: a] : ( P @ ( product_Pair_a_a @ A @ B2 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_110_Pair__inject,axiom,
! [A3: set_Product_prod_a_a,B3: produc5715173734304154471od_a_a,A4: set_Product_prod_a_a,B4: produc5715173734304154471od_a_a] :
( ( ( produc9141123965022120375od_a_a @ A3 @ B3 )
= ( produc9141123965022120375od_a_a @ A4 @ B4 ) )
=> ~ ( ( A3 = A4 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_111_Pair__inject,axiom,
! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a,A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ( produc3372137660326521687od_a_a @ A3 @ B3 )
= ( produc3372137660326521687od_a_a @ A4 @ B4 ) )
=> ~ ( ( A3 = A4 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_112_Pair__inject,axiom,
! [A3: set_a,B3: set_a,A4: set_a,B4: set_a] :
( ( ( produc9088192753505129239_set_a @ A3 @ B3 )
= ( produc9088192753505129239_set_a @ A4 @ B4 ) )
=> ~ ( ( A3 = A4 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_113_Pair__inject,axiom,
! [A3: product_prod_a_a > product_prod_a_a > $o,B3: list_P1396940483166286381od_a_a,A4: product_prod_a_a > product_prod_a_a > $o,B4: list_P1396940483166286381od_a_a] :
( ( ( produc7609767723203257750od_a_a @ A3 @ B3 )
= ( produc7609767723203257750od_a_a @ A4 @ B4 ) )
=> ~ ( ( A3 = A4 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_114_Pair__inject,axiom,
! [A3: a,B3: product_prod_a_a,A4: a,B4: product_prod_a_a] :
( ( ( produc431845341423274048od_a_a @ A3 @ B3 )
= ( produc431845341423274048od_a_a @ A4 @ B4 ) )
=> ~ ( ( A3 = A4 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_115_Pair__inject,axiom,
! [A3: a,B3: a,A4: a,B4: a] :
( ( ( product_Pair_a_a @ A3 @ B3 )
= ( product_Pair_a_a @ A4 @ B4 ) )
=> ~ ( ( A3 = A4 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_116_case__prodD_H,axiom,
! [R2: a > a > product_prod_a_a > $o,A3: a,B3: a,C: product_prod_a_a] :
( ( produc5089340801457792962_a_a_o @ R2 @ ( product_Pair_a_a @ A3 @ B3 ) @ C )
=> ( R2 @ A3 @ B3 @ C ) ) ).
% case_prodD'
thf(fact_117_case__prodE_H,axiom,
! [C: a > a > product_prod_a_a > $o,P2: product_prod_a_a,Z: product_prod_a_a] :
( ( produc5089340801457792962_a_a_o @ C @ P2 @ Z )
=> ~ ! [X: a,Y: a] :
( ( P2
= ( product_Pair_a_a @ X @ Y ) )
=> ~ ( C @ X @ Y @ Z ) ) ) ).
% case_prodE'
thf(fact_118_prod__cases3,axiom,
! [Y3: produc6759108136775012935od_a_a] :
~ ! [A: set_Product_prod_a_a,B2: list_P1396940483166286381od_a_a,C2: list_P1396940483166286381od_a_a] :
( Y3
!= ( produc9141123965022120375od_a_a @ A @ ( produc1014507885701610967od_a_a @ B2 @ C2 ) ) ) ).
% prod_cases3
thf(fact_119_prod__cases3,axiom,
! [Y3: produc4044097585999906000od_a_a] :
~ ! [A: a,B2: a,C2: a] :
( Y3
!= ( produc431845341423274048od_a_a @ A @ ( product_Pair_a_a @ B2 @ C2 ) ) ) ).
% prod_cases3
thf(fact_120_prod__induct3,axiom,
! [P: produc6759108136775012935od_a_a > $o,X4: produc6759108136775012935od_a_a] :
( ! [A: set_Product_prod_a_a,B2: list_P1396940483166286381od_a_a,C2: list_P1396940483166286381od_a_a] : ( P @ ( produc9141123965022120375od_a_a @ A @ ( produc1014507885701610967od_a_a @ B2 @ C2 ) ) )
=> ( P @ X4 ) ) ).
% prod_induct3
thf(fact_121_prod__induct3,axiom,
! [P: produc4044097585999906000od_a_a > $o,X4: produc4044097585999906000od_a_a] :
( ! [A: a,B2: a,C2: a] : ( P @ ( produc431845341423274048od_a_a @ A @ ( product_Pair_a_a @ B2 @ C2 ) ) )
=> ( P @ X4 ) ) ).
% prod_induct3
thf(fact_122_mem__case__prodE,axiom,
! [Z: a,C: a > a > set_a,P2: product_prod_a_a] :
( ( member_a @ Z @ ( produc9217457822752978994_set_a @ C @ P2 ) )
=> ~ ! [X: a,Y: a] :
( ( P2
= ( product_Pair_a_a @ X @ Y ) )
=> ~ ( member_a @ Z @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_123_mem__case__prodE,axiom,
! [Z: product_prod_a_a,C: a > a > set_Product_prod_a_a,P2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z @ ( produc5766521956407364827od_a_a @ C @ P2 ) )
=> ~ ! [X: a,Y: a] :
( ( P2
= ( product_Pair_a_a @ X @ Y ) )
=> ~ ( member1426531477525435216od_a_a @ Z @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_124_mem__case__prodE,axiom,
! [Z: $o,C: a > a > set_o,P2: product_prod_a_a] :
( ( member_o @ Z @ ( produc850716608257783884_set_o @ C @ P2 ) )
=> ~ ! [X: a,Y: a] :
( ( P2
= ( product_Pair_a_a @ X @ Y ) )
=> ~ ( member_o @ Z @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_125_mem__case__prodE,axiom,
! [Z: list_a,C: a > a > set_list_a,P2: product_prod_a_a] :
( ( member_list_a @ Z @ ( produc8229661187788642616list_a @ C @ P2 ) )
=> ~ ! [X: a,Y: a] :
( ( P2
= ( product_Pair_a_a @ X @ Y ) )
=> ~ ( member_list_a @ Z @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_126_mem__case__prodE,axiom,
! [Z: set_a,C: a > a > set_set_a,P2: product_prod_a_a] :
( ( member_set_a @ Z @ ( produc2818859882669968786_set_a @ C @ P2 ) )
=> ~ ! [X: a,Y: a] :
( ( P2
= ( product_Pair_a_a @ X @ Y ) )
=> ~ ( member_set_a @ Z @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_127_mem__case__prodE,axiom,
! [Z: a,C: set_a > set_a > set_a,P2: produc1703568184450464039_set_a] :
( ( member_a @ Z @ ( produc4474087116711199794_set_a @ C @ P2 ) )
=> ~ ! [X: set_a,Y: set_a] :
( ( P2
= ( produc9088192753505129239_set_a @ X @ Y ) )
=> ~ ( member_a @ Z @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_128_mem__case__prodE,axiom,
! [Z: $o,C: set_a > set_a > set_o,P2: produc1703568184450464039_set_a] :
( ( member_o @ Z @ ( produc4110613151828132940_set_o @ C @ P2 ) )
=> ~ ! [X: set_a,Y: set_a] :
( ( P2
= ( produc9088192753505129239_set_a @ X @ Y ) )
=> ~ ( member_o @ Z @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_129_mem__case__prodE,axiom,
! [Z: a,C: a > product_prod_a_a > set_a,P2: produc4044097585999906000od_a_a] :
( ( member_a @ Z @ ( produc1173728711619330889_set_a @ C @ P2 ) )
=> ~ ! [X: a,Y: product_prod_a_a] :
( ( P2
= ( produc431845341423274048od_a_a @ X @ Y ) )
=> ~ ( member_a @ Z @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_130_mem__case__prodE,axiom,
! [Z: $o,C: a > product_prod_a_a > set_o,P2: produc4044097585999906000od_a_a] :
( ( member_o @ Z @ ( produc1957367346529820387_set_o @ C @ P2 ) )
=> ~ ! [X: a,Y: product_prod_a_a] :
( ( P2
= ( produc431845341423274048od_a_a @ X @ Y ) )
=> ~ ( member_o @ Z @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_131_mem__case__prodE,axiom,
! [Z: list_P1396940483166286381od_a_a,C: a > a > set_li8827807065578854541od_a_a,P2: product_prod_a_a] :
( ( member6824001069763096534od_a_a @ Z @ ( produc8624483484282574433od_a_a @ C @ P2 ) )
=> ~ ! [X: a,Y: a] :
( ( P2
= ( product_Pair_a_a @ X @ Y ) )
=> ~ ( member6824001069763096534od_a_a @ Z @ ( C @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_132_case__prod__Pair__iden,axiom,
! [P2: produc6759108136775012935od_a_a] :
( ( produc5447416582098204539od_a_a @ produc9141123965022120375od_a_a @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_133_case__prod__Pair__iden,axiom,
! [P2: produc7983190928304305895od_a_a] :
( ( produc1806814694415540859od_a_a @ produc3372137660326521687od_a_a @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_134_case__prod__Pair__iden,axiom,
! [P2: produc1703568184450464039_set_a] :
( ( produc373481016724630779_set_a @ produc9088192753505129239_set_a @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_135_case__prod__Pair__iden,axiom,
! [P2: produc5831972324637314982od_a_a] :
( ( produc8597769208330846011od_a_a @ produc7609767723203257750od_a_a @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_136_case__prod__Pair__iden,axiom,
! [P2: produc4044097585999906000od_a_a] :
( ( produc5605962277697872315od_a_a @ produc431845341423274048od_a_a @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_137_case__prod__Pair__iden,axiom,
! [P2: product_prod_a_a] :
( ( produc408267641121961211od_a_a @ product_Pair_a_a @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_138_in__symclI_I1_J,axiom,
! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a,R3: set_Pr5931134355633282247od_a_a] :
( ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ A3 @ B3 ) @ R3 )
=> ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ A3 @ B3 ) @ ( symcl_286300493019922782od_a_a @ R3 ) ) ) ).
% in_symclI(1)
thf(fact_139_in__symclI_I1_J,axiom,
! [A3: set_a,B3: set_a,R3: set_Pr5845495582615845127_set_a] :
( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A3 @ B3 ) @ R3 )
=> ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A3 @ B3 ) @ ( symcl_set_a @ R3 ) ) ) ).
% in_symclI(1)
thf(fact_140_in__symclI_I1_J,axiom,
! [A3: a,B3: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ R3 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ ( symcl_a @ R3 ) ) ) ).
% in_symclI(1)
thf(fact_141_in__symclI_I2_J,axiom,
! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a,R3: set_Pr5931134355633282247od_a_a] :
( ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ A3 @ B3 ) @ R3 )
=> ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ B3 @ A3 ) @ ( symcl_286300493019922782od_a_a @ R3 ) ) ) ).
% in_symclI(2)
thf(fact_142_in__symclI_I2_J,axiom,
! [A3: set_a,B3: set_a,R3: set_Pr5845495582615845127_set_a] :
( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A3 @ B3 ) @ R3 )
=> ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ B3 @ A3 ) @ ( symcl_set_a @ R3 ) ) ) ).
% in_symclI(2)
thf(fact_143_in__symclI_I2_J,axiom,
! [A3: a,B3: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ R3 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ A3 ) @ ( symcl_a @ R3 ) ) ) ).
% in_symclI(2)
thf(fact_144_mem__Collect__eq,axiom,
! [A3: a,P: a > $o] :
( ( member_a @ A3 @ ( collect_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_145_mem__Collect__eq,axiom,
! [A3: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A3 @ ( collec3336397797384452498od_a_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_146_mem__Collect__eq,axiom,
! [A3: $o,P: $o > $o] :
( ( member_o @ A3 @ ( collect_o @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_147_mem__Collect__eq,axiom,
! [A3: list_a,P: list_a > $o] :
( ( member_list_a @ A3 @ ( collect_list_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_148_mem__Collect__eq,axiom,
! [A3: set_a,P: set_a > $o] :
( ( member_set_a @ A3 @ ( collect_set_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_149_mem__Collect__eq,axiom,
! [A3: a > a,P: ( a > a ) > $o] :
( ( member_a_a @ A3 @ ( collect_a_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_150_mem__Collect__eq,axiom,
! [A3: list_P1396940483166286381od_a_a,P: list_P1396940483166286381od_a_a > $o] :
( ( member6824001069763096534od_a_a @ A3 @ ( collec2774578310585404696od_a_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_151_mem__Collect__eq,axiom,
! [A3: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( member1816616512716248880od_a_a @ A3 @ ( collec1673347964119250290od_a_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_152_mem__Collect__eq,axiom,
! [A3: product_prod_a_a > a,P: ( product_prod_a_a > a ) > $o] :
( ( member1716570166360300819_a_a_a @ A3 @ ( collec2886379341470944849_a_a_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_153_mem__Collect__eq,axiom,
! [A3: a > product_prod_a_a,P: ( a > product_prod_a_a ) > $o] :
( ( member1957775702407316389od_a_a @ A3 @ ( collec3127584877517960419od_a_a @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_154_Collect__mem__eq,axiom,
! [A5: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_155_Collect__mem__eq,axiom,
! [A5: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_156_Collect__mem__eq,axiom,
! [A5: set_o] :
( ( collect_o
@ ^ [X3: $o] : ( member_o @ X3 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_157_Collect__mem__eq,axiom,
! [A5: set_list_a] :
( ( collect_list_a
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_158_Collect__mem__eq,axiom,
! [A5: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_159_Collect__mem__eq,axiom,
! [A5: set_a_a] :
( ( collect_a_a
@ ^ [X3: a > a] : ( member_a_a @ X3 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_160_Collect__mem__eq,axiom,
! [A5: set_li8827807065578854541od_a_a] :
( ( collec2774578310585404696od_a_a
@ ^ [X3: list_P1396940483166286381od_a_a] : ( member6824001069763096534od_a_a @ X3 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_161_Collect__mem__eq,axiom,
! [A5: set_se5735800977113168103od_a_a] :
( ( collec1673347964119250290od_a_a
@ ^ [X3: set_Product_prod_a_a] : ( member1816616512716248880od_a_a @ X3 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_162_Collect__mem__eq,axiom,
! [A5: set_Pr952751117562918450_a_a_a] :
( ( collec2886379341470944849_a_a_a
@ ^ [X3: product_prod_a_a > a] : ( member1716570166360300819_a_a_a @ X3 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_163_Collect__mem__eq,axiom,
! [A5: set_a_6829686330177631172od_a_a] :
( ( collec3127584877517960419od_a_a
@ ^ [X3: a > product_prod_a_a] : ( member1957775702407316389od_a_a @ X3 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_164_Collect__cong,axiom,
! [P: ( product_prod_a_a > product_prod_a_a ) > $o,Q: ( product_prod_a_a > product_prod_a_a ) > $o] :
( ! [X: product_prod_a_a > product_prod_a_a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collec8125451137695935482od_a_a @ P )
= ( collec8125451137695935482od_a_a @ Q ) ) ) ).
% Collect_cong
thf(fact_165_Collect__cong,axiom,
! [P: ( product_prod_a_a > a ) > $o,Q: ( product_prod_a_a > a ) > $o] :
( ! [X: product_prod_a_a > a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collec2886379341470944849_a_a_a @ P )
= ( collec2886379341470944849_a_a_a @ Q ) ) ) ).
% Collect_cong
thf(fact_166_Collect__cong,axiom,
! [P: ( a > product_prod_a_a ) > $o,Q: ( a > product_prod_a_a ) > $o] :
( ! [X: a > product_prod_a_a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collec3127584877517960419od_a_a @ P )
= ( collec3127584877517960419od_a_a @ Q ) ) ) ).
% Collect_cong
thf(fact_167_Collect__cong,axiom,
! [P: ( a > a ) > $o,Q: ( a > a ) > $o] :
( ! [X: a > a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_a_a @ P )
= ( collect_a_a @ Q ) ) ) ).
% Collect_cong
thf(fact_168_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_169_find_Osimps_I1_J,axiom,
! [Uu2: list_P1396940483166286381od_a_a > $o] :
( ( find_l5372068402991683145od_a_a @ Uu2 @ nil_li8564743906319921459od_a_a )
= none_l1738640210340310542od_a_a ) ).
% find.simps(1)
thf(fact_170_find_Osimps_I1_J,axiom,
! [Uu2: nat > $o] :
( ( find_nat @ Uu2 @ nil_nat )
= none_nat ) ).
% find.simps(1)
thf(fact_171_find_Osimps_I1_J,axiom,
! [Uu2: a > $o] :
( ( find_a @ Uu2 @ nil_a )
= none_a ) ).
% find.simps(1)
thf(fact_172_find_Osimps_I1_J,axiom,
! [Uu2: set_a > $o] :
( ( find_set_a @ Uu2 @ nil_set_a )
= none_set_a ) ).
% find.simps(1)
thf(fact_173_find_Osimps_I1_J,axiom,
! [Uu2: set_Product_prod_a_a > $o] :
( ( find_s2827193664816301859od_a_a @ Uu2 @ nil_se7151937861063948045od_a_a )
= none_s5714488769912451432od_a_a ) ).
% find.simps(1)
thf(fact_174_find_Osimps_I1_J,axiom,
! [Uu2: product_prod_a_a > $o] :
( ( find_P2248954792982286147od_a_a @ Uu2 @ nil_Product_prod_a_a )
= none_P1914705783780380040od_a_a ) ).
% find.simps(1)
thf(fact_175_old_Oprod_Ocase,axiom,
! [F: a > a > set_Product_prod_a_a,X12: a,X22: a] :
( ( produc5766521956407364827od_a_a @ F @ ( product_Pair_a_a @ X12 @ X22 ) )
= ( F @ X12 @ X22 ) ) ).
% old.prod.case
thf(fact_176_old_Oprod_Ocase,axiom,
! [F: a > a > set_a,X12: a,X22: a] :
( ( produc9217457822752978994_set_a @ F @ ( product_Pair_a_a @ X12 @ X22 ) )
= ( F @ X12 @ X22 ) ) ).
% old.prod.case
thf(fact_177_old_Oprod_Ocase,axiom,
! [F: a > a > product_prod_a_a > $o,X12: a,X22: a] :
( ( produc5089340801457792962_a_a_o @ F @ ( product_Pair_a_a @ X12 @ X22 ) )
= ( F @ X12 @ X22 ) ) ).
% old.prod.case
thf(fact_178_old_Oprod_Ocase,axiom,
! [F: a > a > set_a > set_a,X12: a,X22: a] :
( ( produc2520556867391656337_set_a @ F @ ( product_Pair_a_a @ X12 @ X22 ) )
= ( F @ X12 @ X22 ) ) ).
% old.prod.case
thf(fact_179_old_Oprod_Ocase,axiom,
! [F: a > a > a,X12: a,X22: a] :
( ( produc8815886927560695506_a_a_a @ F @ ( product_Pair_a_a @ X12 @ X22 ) )
= ( F @ X12 @ X22 ) ) ).
% old.prod.case
thf(fact_180_old_Oprod_Ocase,axiom,
! [F: a > a > $o,X12: a,X22: a] :
( ( produc6436628058953941356_a_a_o @ F @ ( product_Pair_a_a @ X12 @ X22 ) )
= ( F @ X12 @ X22 ) ) ).
% old.prod.case
thf(fact_181_find__cong,axiom,
! [Xs: list_list_a,Ys: list_list_a,P: list_a > $o,Q: list_a > $o] :
( ( Xs = Ys )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Ys ) )
=> ( ( P @ X )
= ( Q @ X ) ) )
=> ( ( find_list_a @ P @ Xs )
= ( find_list_a @ Q @ Ys ) ) ) ) ).
% find_cong
thf(fact_182_find__cong,axiom,
! [Xs: list_o,Ys: list_o,P: $o > $o,Q: $o > $o] :
( ( Xs = Ys )
=> ( ! [X: $o] :
( ( member_o @ X @ ( set_o2 @ Ys ) )
=> ( ( P @ X )
= ( Q @ X ) ) )
=> ( ( find_o @ P @ Xs )
= ( find_o @ Q @ Ys ) ) ) ) ).
% find_cong
thf(fact_183_find__cong,axiom,
! [Xs: list_s9060204159073123853od_a_a,Ys: list_s9060204159073123853od_a_a,P: set_Product_prod_a_a > $o,Q: set_Product_prod_a_a > $o] :
( ( Xs = Ys )
=> ( ! [X: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X @ ( set_se8408754101646271900od_a_a @ Ys ) )
=> ( ( P @ X )
= ( Q @ X ) ) )
=> ( ( find_s2827193664816301859od_a_a @ P @ Xs )
= ( find_s2827193664816301859od_a_a @ Q @ Ys ) ) ) ) ).
% find_cong
thf(fact_184_find__cong,axiom,
! [Xs: list_set_a,Ys: list_set_a,P: set_a > $o,Q: set_a > $o] :
( ( Xs = Ys )
=> ( ! [X: set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Ys ) )
=> ( ( P @ X )
= ( Q @ X ) ) )
=> ( ( find_set_a @ P @ Xs )
= ( find_set_a @ Q @ Ys ) ) ) ) ).
% find_cong
thf(fact_185_find__cong,axiom,
! [Xs: list_P6569952146052023789od_a_a,Ys: list_P6569952146052023789od_a_a,P: produc3498347346309940967od_a_a > $o,Q: produc3498347346309940967od_a_a > $o] :
( ( Xs = Ys )
=> ( ! [X: produc3498347346309940967od_a_a] :
( ( member6330455413206600464od_a_a @ X @ ( set_Pr6493056779096584572od_a_a @ Ys ) )
=> ( ( P @ X )
= ( Q @ X ) ) )
=> ( ( find_P7890441392728764675od_a_a @ P @ Xs )
= ( find_P7890441392728764675od_a_a @ Q @ Ys ) ) ) ) ).
% find_cong
thf(fact_186_find__cong,axiom,
! [Xs: list_l2076173926988081843od_a_a,Ys: list_l2076173926988081843od_a_a,P: list_P1396940483166286381od_a_a > $o,Q: list_P1396940483166286381od_a_a > $o] :
( ( Xs = Ys )
=> ( ! [X: list_P1396940483166286381od_a_a] :
( ( member6824001069763096534od_a_a @ X @ ( set_li4974249543031949378od_a_a @ Ys ) )
=> ( ( P @ X )
= ( Q @ X ) ) )
=> ( ( find_l5372068402991683145od_a_a @ P @ Xs )
= ( find_l5372068402991683145od_a_a @ Q @ Ys ) ) ) ) ).
% find_cong
thf(fact_187_find__cong,axiom,
! [Xs: list_a,Ys: list_a,P: a > $o,Q: a > $o] :
( ( Xs = Ys )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Ys ) )
=> ( ( P @ X )
= ( Q @ X ) ) )
=> ( ( find_a @ P @ Xs )
= ( find_a @ Q @ Ys ) ) ) ) ).
% find_cong
thf(fact_188_find__cong,axiom,
! [Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a,P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( Xs = Ys )
=> ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Ys ) )
=> ( ( P @ X )
= ( Q @ X ) ) )
=> ( ( find_P2248954792982286147od_a_a @ P @ Xs )
= ( find_P2248954792982286147od_a_a @ Q @ Ys ) ) ) ) ).
% find_cong
thf(fact_189_cond__case__prod__eta,axiom,
! [F: a > a > set_Product_prod_a_a,G: product_prod_a_a > set_Product_prod_a_a] :
( ! [X: a,Y: a] :
( ( F @ X @ Y )
= ( G @ ( product_Pair_a_a @ X @ Y ) ) )
=> ( ( produc5766521956407364827od_a_a @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_190_cond__case__prod__eta,axiom,
! [F: a > a > set_a,G: product_prod_a_a > set_a] :
( ! [X: a,Y: a] :
( ( F @ X @ Y )
= ( G @ ( product_Pair_a_a @ X @ Y ) ) )
=> ( ( produc9217457822752978994_set_a @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_191_cond__case__prod__eta,axiom,
! [F: a > a > product_prod_a_a > $o,G: product_prod_a_a > product_prod_a_a > $o] :
( ! [X: a,Y: a] :
( ( F @ X @ Y )
= ( G @ ( product_Pair_a_a @ X @ Y ) ) )
=> ( ( produc5089340801457792962_a_a_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_192_cond__case__prod__eta,axiom,
! [F: a > a > set_a > set_a,G: product_prod_a_a > set_a > set_a] :
( ! [X: a,Y: a] :
( ( F @ X @ Y )
= ( G @ ( product_Pair_a_a @ X @ Y ) ) )
=> ( ( produc2520556867391656337_set_a @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_193_cond__case__prod__eta,axiom,
! [F: a > a > a,G: product_prod_a_a > a] :
( ! [X: a,Y: a] :
( ( F @ X @ Y )
= ( G @ ( product_Pair_a_a @ X @ Y ) ) )
=> ( ( produc8815886927560695506_a_a_a @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_194_cond__case__prod__eta,axiom,
! [F: a > a > $o,G: product_prod_a_a > $o] :
( ! [X: a,Y: a] :
( ( F @ X @ Y )
= ( G @ ( product_Pair_a_a @ X @ Y ) ) )
=> ( ( produc6436628058953941356_a_a_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_195_case__prod__eta,axiom,
! [F: product_prod_a_a > set_Product_prod_a_a] :
( ( produc5766521956407364827od_a_a
@ ^ [X3: a,Y2: a] : ( F @ ( product_Pair_a_a @ X3 @ Y2 ) ) )
= F ) ).
% case_prod_eta
thf(fact_196_case__prod__eta,axiom,
! [F: product_prod_a_a > set_a] :
( ( produc9217457822752978994_set_a
@ ^ [X3: a,Y2: a] : ( F @ ( product_Pair_a_a @ X3 @ Y2 ) ) )
= F ) ).
% case_prod_eta
thf(fact_197_case__prod__eta,axiom,
! [F: product_prod_a_a > product_prod_a_a > $o] :
( ( produc5089340801457792962_a_a_o
@ ^ [X3: a,Y2: a] : ( F @ ( product_Pair_a_a @ X3 @ Y2 ) ) )
= F ) ).
% case_prod_eta
thf(fact_198_case__prod__eta,axiom,
! [F: product_prod_a_a > set_a > set_a] :
( ( produc2520556867391656337_set_a
@ ^ [X3: a,Y2: a] : ( F @ ( product_Pair_a_a @ X3 @ Y2 ) ) )
= F ) ).
% case_prod_eta
thf(fact_199_case__prod__eta,axiom,
! [F: product_prod_a_a > a] :
( ( produc8815886927560695506_a_a_a
@ ^ [X3: a,Y2: a] : ( F @ ( product_Pair_a_a @ X3 @ Y2 ) ) )
= F ) ).
% case_prod_eta
thf(fact_200_case__prod__eta,axiom,
! [F: product_prod_a_a > $o] :
( ( produc6436628058953941356_a_a_o
@ ^ [X3: a,Y2: a] : ( F @ ( product_Pair_a_a @ X3 @ Y2 ) ) )
= F ) ).
% case_prod_eta
thf(fact_201_case__prodE2,axiom,
! [Q: set_Product_prod_a_a > $o,P: a > a > set_Product_prod_a_a,Z: product_prod_a_a] :
( ( Q @ ( produc5766521956407364827od_a_a @ P @ Z ) )
=> ~ ! [X: a,Y: a] :
( ( Z
= ( product_Pair_a_a @ X @ Y ) )
=> ~ ( Q @ ( P @ X @ Y ) ) ) ) ).
% case_prodE2
thf(fact_202_case__prodE2,axiom,
! [Q: set_a > $o,P: a > a > set_a,Z: product_prod_a_a] :
( ( Q @ ( produc9217457822752978994_set_a @ P @ Z ) )
=> ~ ! [X: a,Y: a] :
( ( Z
= ( product_Pair_a_a @ X @ Y ) )
=> ~ ( Q @ ( P @ X @ Y ) ) ) ) ).
% case_prodE2
thf(fact_203_case__prodE2,axiom,
! [Q: ( product_prod_a_a > $o ) > $o,P: a > a > product_prod_a_a > $o,Z: product_prod_a_a] :
( ( Q @ ( produc5089340801457792962_a_a_o @ P @ Z ) )
=> ~ ! [X: a,Y: a] :
( ( Z
= ( product_Pair_a_a @ X @ Y ) )
=> ~ ( Q @ ( P @ X @ Y ) ) ) ) ).
% case_prodE2
thf(fact_204_case__prodE2,axiom,
! [Q: ( set_a > set_a ) > $o,P: a > a > set_a > set_a,Z: product_prod_a_a] :
( ( Q @ ( produc2520556867391656337_set_a @ P @ Z ) )
=> ~ ! [X: a,Y: a] :
( ( Z
= ( product_Pair_a_a @ X @ Y ) )
=> ~ ( Q @ ( P @ X @ Y ) ) ) ) ).
% case_prodE2
thf(fact_205_case__prodE2,axiom,
! [Q: a > $o,P: a > a > a,Z: product_prod_a_a] :
( ( Q @ ( produc8815886927560695506_a_a_a @ P @ Z ) )
=> ~ ! [X: a,Y: a] :
( ( Z
= ( product_Pair_a_a @ X @ Y ) )
=> ~ ( Q @ ( P @ X @ Y ) ) ) ) ).
% case_prodE2
thf(fact_206_case__prodE2,axiom,
! [Q: $o > $o,P: a > a > $o,Z: product_prod_a_a] :
( ( Q @ ( produc6436628058953941356_a_a_o @ P @ Z ) )
=> ~ ! [X: a,Y: a] :
( ( Z
= ( product_Pair_a_a @ X @ Y ) )
=> ~ ( Q @ ( P @ X @ Y ) ) ) ) ).
% case_prodE2
thf(fact_207_case__prodE,axiom,
! [C: set_Product_prod_a_a > produc5715173734304154471od_a_a > $o,P2: produc6759108136775012935od_a_a] :
( ( produc6861823381246321356_a_a_o @ C @ P2 )
=> ~ ! [X: set_Product_prod_a_a,Y: produc5715173734304154471od_a_a] :
( ( P2
= ( produc9141123965022120375od_a_a @ X @ Y ) )
=> ~ ( C @ X @ Y ) ) ) ).
% case_prodE
thf(fact_208_case__prodE,axiom,
! [C: set_Product_prod_a_a > set_Product_prod_a_a > $o,P2: produc7983190928304305895od_a_a] :
( ( produc4361984678274043180_a_a_o @ C @ P2 )
=> ~ ! [X: set_Product_prod_a_a,Y: set_Product_prod_a_a] :
( ( P2
= ( produc3372137660326521687od_a_a @ X @ Y ) )
=> ~ ( C @ X @ Y ) ) ) ).
% case_prodE
thf(fact_209_case__prodE,axiom,
! [C: set_a > set_a > $o,P2: produc1703568184450464039_set_a] :
( ( produc2941226113218366828et_a_o @ C @ P2 )
=> ~ ! [X: set_a,Y: set_a] :
( ( P2
= ( produc9088192753505129239_set_a @ X @ Y ) )
=> ~ ( C @ X @ Y ) ) ) ).
% case_prodE
thf(fact_210_case__prodE,axiom,
! [C: ( product_prod_a_a > product_prod_a_a > $o ) > list_P1396940483166286381od_a_a > $o,P2: produc5831972324637314982od_a_a] :
( ( produc8373039017049881389_a_a_o @ C @ P2 )
=> ~ ! [X: product_prod_a_a > product_prod_a_a > $o,Y: list_P1396940483166286381od_a_a] :
( ( P2
= ( produc7609767723203257750od_a_a @ X @ Y ) )
=> ~ ( C @ X @ Y ) ) ) ).
% case_prodE
thf(fact_211_case__prodE,axiom,
! [C: a > product_prod_a_a > $o,P2: produc4044097585999906000od_a_a] :
( ( produc5856822985862792195_a_a_o @ C @ P2 )
=> ~ ! [X: a,Y: product_prod_a_a] :
( ( P2
= ( produc431845341423274048od_a_a @ X @ Y ) )
=> ~ ( C @ X @ Y ) ) ) ).
% case_prodE
thf(fact_212_case__prodE,axiom,
! [C: a > a > $o,P2: product_prod_a_a] :
( ( produc6436628058953941356_a_a_o @ C @ P2 )
=> ~ ! [X: a,Y: a] :
( ( P2
= ( product_Pair_a_a @ X @ Y ) )
=> ~ ( C @ X @ Y ) ) ) ).
% case_prodE
thf(fact_213_case__prodD,axiom,
! [F: set_Product_prod_a_a > produc5715173734304154471od_a_a > $o,A3: set_Product_prod_a_a,B3: produc5715173734304154471od_a_a] :
( ( produc6861823381246321356_a_a_o @ F @ ( produc9141123965022120375od_a_a @ A3 @ B3 ) )
=> ( F @ A3 @ B3 ) ) ).
% case_prodD
thf(fact_214_case__prodD,axiom,
! [F: set_Product_prod_a_a > set_Product_prod_a_a > $o,A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( produc4361984678274043180_a_a_o @ F @ ( produc3372137660326521687od_a_a @ A3 @ B3 ) )
=> ( F @ A3 @ B3 ) ) ).
% case_prodD
thf(fact_215_case__prodD,axiom,
! [F: set_a > set_a > $o,A3: set_a,B3: set_a] :
( ( produc2941226113218366828et_a_o @ F @ ( produc9088192753505129239_set_a @ A3 @ B3 ) )
=> ( F @ A3 @ B3 ) ) ).
% case_prodD
thf(fact_216_case__prodD,axiom,
! [F: ( product_prod_a_a > product_prod_a_a > $o ) > list_P1396940483166286381od_a_a > $o,A3: product_prod_a_a > product_prod_a_a > $o,B3: list_P1396940483166286381od_a_a] :
( ( produc8373039017049881389_a_a_o @ F @ ( produc7609767723203257750od_a_a @ A3 @ B3 ) )
=> ( F @ A3 @ B3 ) ) ).
% case_prodD
thf(fact_217_case__prodD,axiom,
! [F: a > product_prod_a_a > $o,A3: a,B3: product_prod_a_a] :
( ( produc5856822985862792195_a_a_o @ F @ ( produc431845341423274048od_a_a @ A3 @ B3 ) )
=> ( F @ A3 @ B3 ) ) ).
% case_prodD
thf(fact_218_case__prodD,axiom,
! [F: a > a > $o,A3: a,B3: a] :
( ( produc6436628058953941356_a_a_o @ F @ ( product_Pair_a_a @ A3 @ B3 ) )
=> ( F @ A3 @ B3 ) ) ).
% case_prodD
thf(fact_219__092_060open_062_Ia_M_Ax_J_A_092_060in_062_Atransymcl_A_Iset_Axys_J_092_060close_062,axiom,
member1426531477525435216od_a_a @ ( product_Pair_a_a @ a3 @ x2 ) @ ( transymcl_a @ ( set_Product_prod_a_a2 @ xys ) ) ).
% \<open>(a, x) \<in> transymcl (set xys)\<close>
thf(fact_220_x_I1_J,axiom,
( ( class_a @ x2 @ ( set_Product_prod_a_a2 @ xys ) )
= ( some_set_a @ x ) ) ).
% x(1)
thf(fact_221_split__cong,axiom,
! [Q2: product_prod_a_a,F: a > a > set_Product_prod_a_a,G: a > a > set_Product_prod_a_a,P2: product_prod_a_a] :
( ! [X: a,Y: a] :
( ( ( product_Pair_a_a @ X @ Y )
= Q2 )
=> ( ( F @ X @ Y )
= ( G @ X @ Y ) ) )
=> ( ( P2 = Q2 )
=> ( ( produc5766521956407364827od_a_a @ F @ P2 )
= ( produc5766521956407364827od_a_a @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_222_split__cong,axiom,
! [Q2: product_prod_a_a,F: a > a > set_a,G: a > a > set_a,P2: product_prod_a_a] :
( ! [X: a,Y: a] :
( ( ( product_Pair_a_a @ X @ Y )
= Q2 )
=> ( ( F @ X @ Y )
= ( G @ X @ Y ) ) )
=> ( ( P2 = Q2 )
=> ( ( produc9217457822752978994_set_a @ F @ P2 )
= ( produc9217457822752978994_set_a @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_223_split__cong,axiom,
! [Q2: product_prod_a_a,F: a > a > product_prod_a_a > $o,G: a > a > product_prod_a_a > $o,P2: product_prod_a_a] :
( ! [X: a,Y: a] :
( ( ( product_Pair_a_a @ X @ Y )
= Q2 )
=> ( ( F @ X @ Y )
= ( G @ X @ Y ) ) )
=> ( ( P2 = Q2 )
=> ( ( produc5089340801457792962_a_a_o @ F @ P2 )
= ( produc5089340801457792962_a_a_o @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_224_split__cong,axiom,
! [Q2: product_prod_a_a,F: a > a > set_a > set_a,G: a > a > set_a > set_a,P2: product_prod_a_a] :
( ! [X: a,Y: a] :
( ( ( product_Pair_a_a @ X @ Y )
= Q2 )
=> ( ( F @ X @ Y )
= ( G @ X @ Y ) ) )
=> ( ( P2 = Q2 )
=> ( ( produc2520556867391656337_set_a @ F @ P2 )
= ( produc2520556867391656337_set_a @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_225_split__cong,axiom,
! [Q2: product_prod_a_a,F: a > a > a,G: a > a > a,P2: product_prod_a_a] :
( ! [X: a,Y: a] :
( ( ( product_Pair_a_a @ X @ Y )
= Q2 )
=> ( ( F @ X @ Y )
= ( G @ X @ Y ) ) )
=> ( ( P2 = Q2 )
=> ( ( produc8815886927560695506_a_a_a @ F @ P2 )
= ( produc8815886927560695506_a_a_a @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_226_split__cong,axiom,
! [Q2: product_prod_a_a,F: a > a > $o,G: a > a > $o,P2: product_prod_a_a] :
( ! [X: a,Y: a] :
( ( ( product_Pair_a_a @ X @ Y )
= Q2 )
=> ( ( F @ X @ Y )
= ( G @ X @ Y ) ) )
=> ( ( P2 = Q2 )
=> ( ( produc6436628058953941356_a_a_o @ F @ P2 )
= ( produc6436628058953941356_a_a_o @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_227_x_I2_J,axiom,
( ( inf_inf_set_a @ x @ a2 )
!= bot_bot_set_a ) ).
% x(2)
thf(fact_228_internal__case__prod__def,axiom,
produc2194622835577206661od_a_a = produc5766521956407364827od_a_a ).
% internal_case_prod_def
thf(fact_229_internal__case__prod__def,axiom,
produc4792916124136496348_set_a = produc9217457822752978994_set_a ).
% internal_case_prod_def
thf(fact_230_internal__case__prod__def,axiom,
produc8939780266206559640_a_a_o = produc5089340801457792962_a_a_o ).
% internal_case_prod_def
thf(fact_231_internal__case__prod__def,axiom,
produc9070753104417976935_set_a = produc2520556867391656337_set_a ).
% internal_case_prod_def
thf(fact_232_internal__case__prod__def,axiom,
produc3106313966426205052_a_a_a = produc8815886927560695506_a_a_a ).
% internal_case_prod_def
thf(fact_233_internal__case__prod__def,axiom,
produc7516648808376468246_a_a_o = produc6436628058953941356_a_a_o ).
% internal_case_prod_def
thf(fact_234_pred__equals__eq2,axiom,
! [R2: set_Pr5629751886588192039od_a_a,S: set_Pr5629751886588192039od_a_a] :
( ( ( ^ [X3: set_Product_prod_a_a,Y2: produc5715173734304154471od_a_a] : ( member4804927706910867312od_a_a @ ( produc9141123965022120375od_a_a @ X3 @ Y2 ) @ R2 ) )
= ( ^ [X3: set_Product_prod_a_a,Y2: produc5715173734304154471od_a_a] : ( member4804927706910867312od_a_a @ ( produc9141123965022120375od_a_a @ X3 @ Y2 ) @ S ) ) )
= ( R2 = S ) ) ).
% pred_equals_eq2
thf(fact_235_pred__equals__eq2,axiom,
! [R2: set_Pr5931134355633282247od_a_a,S: set_Pr5931134355633282247od_a_a] :
( ( ( ^ [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a] : ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ X3 @ Y2 ) @ R2 ) )
= ( ^ [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a] : ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ X3 @ Y2 ) @ S ) ) )
= ( R2 = S ) ) ).
% pred_equals_eq2
thf(fact_236_pred__equals__eq2,axiom,
! [R2: set_Pr5845495582615845127_set_a,S: set_Pr5845495582615845127_set_a] :
( ( ( ^ [X3: set_a,Y2: set_a] : ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X3 @ Y2 ) @ R2 ) )
= ( ^ [X3: set_a,Y2: set_a] : ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X3 @ Y2 ) @ S ) ) )
= ( R2 = S ) ) ).
% pred_equals_eq2
thf(fact_237_pred__equals__eq2,axiom,
! [R2: set_Pr1368515532014665606od_a_a,S: set_Pr1368515532014665606od_a_a] :
( ( ( ^ [X3: product_prod_a_a > product_prod_a_a > $o,Y2: list_P1396940483166286381od_a_a] : ( member1420838311466244943od_a_a @ ( produc7609767723203257750od_a_a @ X3 @ Y2 ) @ R2 ) )
= ( ^ [X3: product_prod_a_a > product_prod_a_a > $o,Y2: list_P1396940483166286381od_a_a] : ( member1420838311466244943od_a_a @ ( produc7609767723203257750od_a_a @ X3 @ Y2 ) @ S ) ) )
= ( R2 = S ) ) ).
% pred_equals_eq2
thf(fact_238_pred__equals__eq2,axiom,
! [R2: set_Pr5530083903271594800od_a_a,S: set_Pr5530083903271594800od_a_a] :
( ( ( ^ [X3: a,Y2: product_prod_a_a] : ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ X3 @ Y2 ) @ R2 ) )
= ( ^ [X3: a,Y2: product_prod_a_a] : ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ X3 @ Y2 ) @ S ) ) )
= ( R2 = S ) ) ).
% pred_equals_eq2
thf(fact_239_pred__equals__eq2,axiom,
! [R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ( ^ [X3: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R2 ) )
= ( ^ [X3: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ S ) ) )
= ( R2 = S ) ) ).
% pred_equals_eq2
thf(fact_240_internal__case__prod__conv,axiom,
! [C: a > a > $o,A3: a,B3: a] :
( ( produc7516648808376468246_a_a_o @ C @ ( product_Pair_a_a @ A3 @ B3 ) )
= ( C @ A3 @ B3 ) ) ).
% internal_case_prod_conv
thf(fact_241_list__ex1__simps_I1_J,axiom,
! [P: list_P1396940483166286381od_a_a > $o] :
~ ( list_e3550399131106442742od_a_a @ P @ nil_li8564743906319921459od_a_a ) ).
% list_ex1_simps(1)
thf(fact_242_list__ex1__simps_I1_J,axiom,
! [P: set_Product_prod_a_a > $o] :
~ ( list_e4689149996104398160od_a_a @ P @ nil_se7151937861063948045od_a_a ) ).
% list_ex1_simps(1)
thf(fact_243_list__ex1__simps_I1_J,axiom,
! [P: nat > $o] :
~ ( list_ex1_nat @ P @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_244_list__ex1__simps_I1_J,axiom,
! [P: a > $o] :
~ ( list_ex1_a @ P @ nil_a ) ).
% list_ex1_simps(1)
thf(fact_245_list__ex1__simps_I1_J,axiom,
! [P: product_prod_a_a > $o] :
~ ( list_e7533892409833140592od_a_a @ P @ nil_Product_prod_a_a ) ).
% list_ex1_simps(1)
thf(fact_246_product_Osimps_I1_J,axiom,
! [Uu2: list_P1396940483166286381od_a_a] :
( ( produc9150955665926335749od_a_a @ nil_Product_prod_a_a @ Uu2 )
= nil_Pr2171190902156203245od_a_a ) ).
% product.simps(1)
thf(fact_247_product_Osimps_I1_J,axiom,
! [Uu2: list_a] :
( ( product_a_a @ nil_a @ Uu2 )
= nil_Product_prod_a_a ) ).
% product.simps(1)
thf(fact_248_assms_I1_J,axiom,
! [X5: set_a] :
( ( member_set_a @ X5 @ ( classes_a @ ( set_Product_prod_a_a2 @ xys ) ) )
=> ( ( inf_inf_set_a @ X5 @ a2 )
!= bot_bot_set_a ) ) ).
% assms(1)
thf(fact_249_set__empty2,axiom,
! [Xs: list_s9060204159073123853od_a_a] :
( ( bot_bo777872063958040403od_a_a
= ( set_se8408754101646271900od_a_a @ Xs ) )
= ( Xs = nil_se7151937861063948045od_a_a ) ) ).
% set_empty2
thf(fact_250_set__empty2,axiom,
! [Xs: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% set_empty2
thf(fact_251_set__empty2,axiom,
! [Xs: list_P6569952146052023789od_a_a] :
( ( bot_bo510284599550014259od_a_a
= ( set_Pr6493056779096584572od_a_a @ Xs ) )
= ( Xs = nil_Pr2171190902156203245od_a_a ) ) ).
% set_empty2
thf(fact_252_set__empty2,axiom,
! [Xs: list_l2076173926988081843od_a_a] :
( ( bot_bo6558180029120801017od_a_a
= ( set_li4974249543031949378od_a_a @ Xs ) )
= ( Xs = nil_li8564743906319921459od_a_a ) ) ).
% set_empty2
thf(fact_253_set__empty2,axiom,
! [Xs: list_list_a] :
( ( bot_bot_set_list_a
= ( set_list_a2 @ Xs ) )
= ( Xs = nil_list_a ) ) ).
% set_empty2
thf(fact_254_set__empty2,axiom,
! [Xs: list_set_a] :
( ( bot_bot_set_set_a
= ( set_set_a2 @ Xs ) )
= ( Xs = nil_set_a ) ) ).
% set_empty2
thf(fact_255_set__empty2,axiom,
! [Xs: list_P1396940483166286381od_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( set_Product_prod_a_a2 @ Xs ) )
= ( Xs = nil_Product_prod_a_a ) ) ).
% set_empty2
thf(fact_256_set__empty2,axiom,
! [Xs: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% set_empty2
thf(fact_257_set__empty,axiom,
! [Xs: list_s9060204159073123853od_a_a] :
( ( ( set_se8408754101646271900od_a_a @ Xs )
= bot_bo777872063958040403od_a_a )
= ( Xs = nil_se7151937861063948045od_a_a ) ) ).
% set_empty
thf(fact_258_set__empty,axiom,
! [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= bot_bot_set_nat )
= ( Xs = nil_nat ) ) ).
% set_empty
thf(fact_259_set__empty,axiom,
! [Xs: list_P6569952146052023789od_a_a] :
( ( ( set_Pr6493056779096584572od_a_a @ Xs )
= bot_bo510284599550014259od_a_a )
= ( Xs = nil_Pr2171190902156203245od_a_a ) ) ).
% set_empty
thf(fact_260_set__empty,axiom,
! [Xs: list_l2076173926988081843od_a_a] :
( ( ( set_li4974249543031949378od_a_a @ Xs )
= bot_bo6558180029120801017od_a_a )
= ( Xs = nil_li8564743906319921459od_a_a ) ) ).
% set_empty
thf(fact_261_set__empty,axiom,
! [Xs: list_list_a] :
( ( ( set_list_a2 @ Xs )
= bot_bot_set_list_a )
= ( Xs = nil_list_a ) ) ).
% set_empty
thf(fact_262_set__empty,axiom,
! [Xs: list_set_a] :
( ( ( set_set_a2 @ Xs )
= bot_bot_set_set_a )
= ( Xs = nil_set_a ) ) ).
% set_empty
thf(fact_263_set__empty,axiom,
! [Xs: list_P1396940483166286381od_a_a] :
( ( ( set_Product_prod_a_a2 @ Xs )
= bot_bo3357376287454694259od_a_a )
= ( Xs = nil_Product_prod_a_a ) ) ).
% set_empty
thf(fact_264_set__empty,axiom,
! [Xs: list_a] :
( ( ( set_a2 @ Xs )
= bot_bot_set_a )
= ( Xs = nil_a ) ) ).
% set_empty
thf(fact_265__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062X_O_A_092_060lbrakk_062class_Ax_A_Iset_Axys_J_A_061_ASome_AX_059_AX_A_092_060inter_062_AA_A_092_060noteq_062_A_123_125_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X6: set_a] :
( ( ( class_a @ x2 @ ( set_Product_prod_a_a2 @ xys ) )
= ( some_set_a @ X6 ) )
=> ( ( inf_inf_set_a @ X6 @ a2 )
= bot_bot_set_a ) ) ).
% \<open>\<And>thesis. (\<And>X. \<lbrakk>class x (set xys) = Some X; X \<inter> A \<noteq> {}\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_266_classes__nonempty,axiom,
! [Qeq: set_Pr8600417178894128327od_a_a] :
~ ( member1816616512716248880od_a_a @ bot_bo3357376287454694259od_a_a @ ( classe2514596149140685704od_a_a @ Qeq ) ) ).
% classes_nonempty
thf(fact_267_classes__nonempty,axiom,
! [Qeq: set_Pr4048851178543822343list_a] :
~ ( member_set_list_a @ bot_bot_set_list_a @ ( classes_list_a @ Qeq ) ) ).
% classes_nonempty
thf(fact_268_classes__nonempty,axiom,
! [Qeq: set_Pr5845495582615845127_set_a] :
~ ( member_set_set_a @ bot_bot_set_set_a @ ( classes_set_a @ Qeq ) ) ).
% classes_nonempty
thf(fact_269_classes__nonempty,axiom,
! [Qeq: set_Product_prod_a_a] :
~ ( member_set_a @ bot_bot_set_a @ ( classes_a @ Qeq ) ) ).
% classes_nonempty
thf(fact_270_classes__disjoint,axiom,
! [X7: set_Product_prod_a_a,Qeq: set_Pr8600417178894128327od_a_a,Y4: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X7 @ ( classe2514596149140685704od_a_a @ Qeq ) )
=> ( ( member1816616512716248880od_a_a @ Y4 @ ( classe2514596149140685704od_a_a @ Qeq ) )
=> ( ( X7 = Y4 )
| ( ( inf_in8905007599844390133od_a_a @ X7 @ Y4 )
= bot_bo3357376287454694259od_a_a ) ) ) ) ).
% classes_disjoint
thf(fact_271_classes__disjoint,axiom,
! [X7: set_list_a,Qeq: set_Pr4048851178543822343list_a,Y4: set_list_a] :
( ( member_set_list_a @ X7 @ ( classes_list_a @ Qeq ) )
=> ( ( member_set_list_a @ Y4 @ ( classes_list_a @ Qeq ) )
=> ( ( X7 = Y4 )
| ( ( inf_inf_set_list_a @ X7 @ Y4 )
= bot_bot_set_list_a ) ) ) ) ).
% classes_disjoint
thf(fact_272_classes__disjoint,axiom,
! [X7: set_set_a,Qeq: set_Pr5845495582615845127_set_a,Y4: set_set_a] :
( ( member_set_set_a @ X7 @ ( classes_set_a @ Qeq ) )
=> ( ( member_set_set_a @ Y4 @ ( classes_set_a @ Qeq ) )
=> ( ( X7 = Y4 )
| ( ( inf_inf_set_set_a @ X7 @ Y4 )
= bot_bot_set_set_a ) ) ) ) ).
% classes_disjoint
thf(fact_273_classes__disjoint,axiom,
! [X7: set_a,Qeq: set_Product_prod_a_a,Y4: set_a] :
( ( member_set_a @ X7 @ ( classes_a @ Qeq ) )
=> ( ( member_set_a @ Y4 @ ( classes_a @ Qeq ) )
=> ( ( X7 = Y4 )
| ( ( inf_inf_set_a @ X7 @ Y4 )
= bot_bot_set_a ) ) ) ) ).
% classes_disjoint
thf(fact_274_class__Some__eq,axiom,
! [X4: list_P1396940483166286381od_a_a,Qeq: set_Pr8563485649468363847od_a_a,X7: set_li8827807065578854541od_a_a] :
( ( ( class_2041412566464829660od_a_a @ X4 @ Qeq )
= ( some_s6382682511002504690od_a_a @ X7 ) )
= ( ( member153658015593001526od_a_a @ X7 @ ( classe2953274524819481614od_a_a @ Qeq ) )
& ( member6824001069763096534od_a_a @ X4 @ X7 ) ) ) ).
% class_Some_eq
thf(fact_275_class__Some__eq,axiom,
! [X4: list_a,Qeq: set_Pr4048851178543822343list_a,X7: set_list_a] :
( ( ( class_list_a @ X4 @ Qeq )
= ( some_set_list_a @ X7 ) )
= ( ( member_set_list_a @ X7 @ ( classes_list_a @ Qeq ) )
& ( member_list_a @ X4 @ X7 ) ) ) ).
% class_Some_eq
thf(fact_276_class__Some__eq,axiom,
! [X4: $o,Qeq: set_Product_prod_o_o,X7: set_o] :
( ( ( class_o @ X4 @ Qeq )
= ( some_set_o @ X7 ) )
= ( ( member_set_o @ X7 @ ( classes_o @ Qeq ) )
& ( member_o @ X4 @ X7 ) ) ) ).
% class_Some_eq
thf(fact_277_class__Some__eq,axiom,
! [X4: set_Product_prod_a_a,Qeq: set_Pr5931134355633282247od_a_a,X7: set_se5735800977113168103od_a_a] :
( ( ( class_4263231457671734070od_a_a @ X4 @ Qeq )
= ( some_s8397874356805898444od_a_a @ X7 ) )
= ( ( member2792204094846457616od_a_a @ X7 @ ( classe739942612566943592od_a_a @ Qeq ) )
& ( member1816616512716248880od_a_a @ X4 @ X7 ) ) ) ).
% class_Some_eq
thf(fact_278_class__Some__eq,axiom,
! [X4: set_a,Qeq: set_Pr5845495582615845127_set_a,X7: set_set_a] :
( ( ( class_set_a @ X4 @ Qeq )
= ( some_set_set_a @ X7 ) )
= ( ( member_set_set_a @ X7 @ ( classes_set_a @ Qeq ) )
& ( member_set_a @ X4 @ X7 ) ) ) ).
% class_Some_eq
thf(fact_279_class__Some__eq,axiom,
! [X4: product_prod_a_a,Qeq: set_Pr8600417178894128327od_a_a,X7: set_Product_prod_a_a] :
( ( ( class_6315440051758036822od_a_a @ X4 @ Qeq )
= ( some_s3837607710808129772od_a_a @ X7 ) )
= ( ( member1816616512716248880od_a_a @ X7 @ ( classe2514596149140685704od_a_a @ Qeq ) )
& ( member1426531477525435216od_a_a @ X4 @ X7 ) ) ) ).
% class_Some_eq
thf(fact_280_class__Some__eq,axiom,
! [X4: a,Qeq: set_Product_prod_a_a,X7: set_a] :
( ( ( class_a @ X4 @ Qeq )
= ( some_set_a @ X7 ) )
= ( ( member_set_a @ X7 @ ( classes_a @ Qeq ) )
& ( member_a @ X4 @ X7 ) ) ) ).
% class_Some_eq
thf(fact_281_transymcl__trans,axiom,
! [X4: set_Product_prod_a_a,Y3: set_Product_prod_a_a,R3: set_Pr5931134355633282247od_a_a,Z: set_Product_prod_a_a] :
( ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ X4 @ Y3 ) @ ( transy7732602298517359145od_a_a @ R3 ) )
=> ( ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ Y3 @ Z ) @ ( transy7732602298517359145od_a_a @ R3 ) )
=> ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ X4 @ Z ) @ ( transy7732602298517359145od_a_a @ R3 ) ) ) ) ).
% transymcl_trans
thf(fact_282_transymcl__trans,axiom,
! [X4: set_a,Y3: set_a,R3: set_Pr5845495582615845127_set_a,Z: set_a] :
( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X4 @ Y3 ) @ ( transymcl_set_a @ R3 ) )
=> ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ Y3 @ Z ) @ ( transymcl_set_a @ R3 ) )
=> ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X4 @ Z ) @ ( transymcl_set_a @ R3 ) ) ) ) ).
% transymcl_trans
thf(fact_283_transymcl__trans,axiom,
! [X4: product_prod_a_a,Y3: product_prod_a_a,R3: set_Pr8600417178894128327od_a_a,Z: product_prod_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X4 @ Y3 ) @ ( transy2380204104860363337od_a_a @ R3 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ Z ) @ ( transy2380204104860363337od_a_a @ R3 ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X4 @ Z ) @ ( transy2380204104860363337od_a_a @ R3 ) ) ) ) ).
% transymcl_trans
thf(fact_284_transymcl__trans,axiom,
! [X4: a,Y3: a,R3: set_Product_prod_a_a,Z: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ ( transymcl_a @ R3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Z ) @ ( transymcl_a @ R3 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Z ) @ ( transymcl_a @ R3 ) ) ) ) ).
% transymcl_trans
thf(fact_285_transymcl__sym,axiom,
! [X4: set_Product_prod_a_a,Y3: set_Product_prod_a_a,R3: set_Pr5931134355633282247od_a_a] :
( ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ X4 @ Y3 ) @ ( transy7732602298517359145od_a_a @ R3 ) )
=> ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ Y3 @ X4 ) @ ( transy7732602298517359145od_a_a @ R3 ) ) ) ).
% transymcl_sym
thf(fact_286_transymcl__sym,axiom,
! [X4: set_a,Y3: set_a,R3: set_Pr5845495582615845127_set_a] :
( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X4 @ Y3 ) @ ( transymcl_set_a @ R3 ) )
=> ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ Y3 @ X4 ) @ ( transymcl_set_a @ R3 ) ) ) ).
% transymcl_sym
thf(fact_287_transymcl__sym,axiom,
! [X4: product_prod_a_a,Y3: product_prod_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X4 @ Y3 ) @ ( transy2380204104860363337od_a_a @ R3 ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ X4 ) @ ( transy2380204104860363337od_a_a @ R3 ) ) ) ).
% transymcl_sym
thf(fact_288_transymcl__sym,axiom,
! [X4: a,Y3: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ ( transymcl_a @ R3 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X4 ) @ ( transymcl_a @ R3 ) ) ) ).
% transymcl_sym
thf(fact_289_transymcl__self_I1_J,axiom,
! [X4: set_Product_prod_a_a,Y3: set_Product_prod_a_a,R3: set_Pr5931134355633282247od_a_a] :
( ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ X4 @ Y3 ) @ R3 )
=> ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ X4 @ X4 ) @ ( transy7732602298517359145od_a_a @ R3 ) ) ) ).
% transymcl_self(1)
thf(fact_290_transymcl__self_I1_J,axiom,
! [X4: set_a,Y3: set_a,R3: set_Pr5845495582615845127_set_a] :
( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X4 @ Y3 ) @ R3 )
=> ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X4 @ X4 ) @ ( transymcl_set_a @ R3 ) ) ) ).
% transymcl_self(1)
thf(fact_291_transymcl__self_I1_J,axiom,
! [X4: product_prod_a_a,Y3: product_prod_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X4 @ Y3 ) @ R3 )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X4 @ X4 ) @ ( transy2380204104860363337od_a_a @ R3 ) ) ) ).
% transymcl_self(1)
thf(fact_292_transymcl__self_I1_J,axiom,
! [X4: a,Y3: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R3 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ X4 ) @ ( transymcl_a @ R3 ) ) ) ).
% transymcl_self(1)
thf(fact_293_transymcl__self_I2_J,axiom,
! [X4: set_Product_prod_a_a,Y3: set_Product_prod_a_a,R3: set_Pr5931134355633282247od_a_a] :
( ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ X4 @ Y3 ) @ R3 )
=> ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ Y3 @ Y3 ) @ ( transy7732602298517359145od_a_a @ R3 ) ) ) ).
% transymcl_self(2)
thf(fact_294_transymcl__self_I2_J,axiom,
! [X4: set_a,Y3: set_a,R3: set_Pr5845495582615845127_set_a] :
( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X4 @ Y3 ) @ R3 )
=> ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ Y3 @ Y3 ) @ ( transymcl_set_a @ R3 ) ) ) ).
% transymcl_self(2)
thf(fact_295_transymcl__self_I2_J,axiom,
! [X4: product_prod_a_a,Y3: product_prod_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X4 @ Y3 ) @ R3 )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ Y3 ) @ ( transy2380204104860363337od_a_a @ R3 ) ) ) ).
% transymcl_self(2)
thf(fact_296_transymcl__self_I2_J,axiom,
! [X4: a,Y3: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R3 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Y3 ) @ ( transymcl_a @ R3 ) ) ) ).
% transymcl_self(2)
thf(fact_297_transymcl__into_I1_J,axiom,
! [X4: set_Product_prod_a_a,Y3: set_Product_prod_a_a,R3: set_Pr5931134355633282247od_a_a] :
( ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ X4 @ Y3 ) @ R3 )
=> ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ X4 @ Y3 ) @ ( transy7732602298517359145od_a_a @ R3 ) ) ) ).
% transymcl_into(1)
thf(fact_298_transymcl__into_I1_J,axiom,
! [X4: set_a,Y3: set_a,R3: set_Pr5845495582615845127_set_a] :
( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X4 @ Y3 ) @ R3 )
=> ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X4 @ Y3 ) @ ( transymcl_set_a @ R3 ) ) ) ).
% transymcl_into(1)
thf(fact_299_transymcl__into_I1_J,axiom,
! [X4: product_prod_a_a,Y3: product_prod_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X4 @ Y3 ) @ R3 )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X4 @ Y3 ) @ ( transy2380204104860363337od_a_a @ R3 ) ) ) ).
% transymcl_into(1)
thf(fact_300_transymcl__into_I1_J,axiom,
! [X4: a,Y3: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R3 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ ( transymcl_a @ R3 ) ) ) ).
% transymcl_into(1)
thf(fact_301_transymcl__into_I2_J,axiom,
! [X4: set_Product_prod_a_a,Y3: set_Product_prod_a_a,R3: set_Pr5931134355633282247od_a_a] :
( ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ X4 @ Y3 ) @ R3 )
=> ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ Y3 @ X4 ) @ ( transy7732602298517359145od_a_a @ R3 ) ) ) ).
% transymcl_into(2)
thf(fact_302_transymcl__into_I2_J,axiom,
! [X4: set_a,Y3: set_a,R3: set_Pr5845495582615845127_set_a] :
( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X4 @ Y3 ) @ R3 )
=> ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ Y3 @ X4 ) @ ( transymcl_set_a @ R3 ) ) ) ).
% transymcl_into(2)
thf(fact_303_transymcl__into_I2_J,axiom,
! [X4: product_prod_a_a,Y3: product_prod_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X4 @ Y3 ) @ R3 )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ X4 ) @ ( transy2380204104860363337od_a_a @ R3 ) ) ) ).
% transymcl_into(2)
thf(fact_304_transymcl__into_I2_J,axiom,
! [X4: a,Y3: a,R3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R3 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X4 ) @ ( transymcl_a @ R3 ) ) ) ).
% transymcl_into(2)
thf(fact_305_edge__same__class,axiom,
! [X7: set_li8827807065578854541od_a_a,Qeq: set_Pr8563485649468363847od_a_a,A3: list_P1396940483166286381od_a_a,B3: list_P1396940483166286381od_a_a] :
( ( member153658015593001526od_a_a @ X7 @ ( classe2953274524819481614od_a_a @ Qeq ) )
=> ( ( member1370157277785437840od_a_a @ ( produc1014507885701610967od_a_a @ A3 @ B3 ) @ Qeq )
=> ( ( member6824001069763096534od_a_a @ A3 @ X7 )
= ( member6824001069763096534od_a_a @ B3 @ X7 ) ) ) ) ).
% edge_same_class
thf(fact_306_edge__same__class,axiom,
! [X7: set_list_a,Qeq: set_Pr4048851178543822343list_a,A3: list_a,B3: list_a] :
( ( member_set_list_a @ X7 @ ( classes_list_a @ Qeq ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A3 @ B3 ) @ Qeq )
=> ( ( member_list_a @ A3 @ X7 )
= ( member_list_a @ B3 @ X7 ) ) ) ) ).
% edge_same_class
thf(fact_307_edge__same__class,axiom,
! [X7: set_o,Qeq: set_Product_prod_o_o,A3: $o,B3: $o] :
( ( member_set_o @ X7 @ ( classes_o @ Qeq ) )
=> ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A3 @ B3 ) @ Qeq )
=> ( ( member_o @ A3 @ X7 )
= ( member_o @ B3 @ X7 ) ) ) ) ).
% edge_same_class
thf(fact_308_edge__same__class,axiom,
! [X7: set_se5735800977113168103od_a_a,Qeq: set_Pr5931134355633282247od_a_a,A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( member2792204094846457616od_a_a @ X7 @ ( classe739942612566943592od_a_a @ Qeq ) )
=> ( ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ A3 @ B3 ) @ Qeq )
=> ( ( member1816616512716248880od_a_a @ A3 @ X7 )
= ( member1816616512716248880od_a_a @ B3 @ X7 ) ) ) ) ).
% edge_same_class
thf(fact_309_edge__same__class,axiom,
! [X7: set_set_a,Qeq: set_Pr5845495582615845127_set_a,A3: set_a,B3: set_a] :
( ( member_set_set_a @ X7 @ ( classes_set_a @ Qeq ) )
=> ( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A3 @ B3 ) @ Qeq )
=> ( ( member_set_a @ A3 @ X7 )
= ( member_set_a @ B3 @ X7 ) ) ) ) ).
% edge_same_class
thf(fact_310_edge__same__class,axiom,
! [X7: set_Product_prod_a_a,Qeq: set_Pr8600417178894128327od_a_a,A3: product_prod_a_a,B3: product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X7 @ ( classe2514596149140685704od_a_a @ Qeq ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A3 @ B3 ) @ Qeq )
=> ( ( member1426531477525435216od_a_a @ A3 @ X7 )
= ( member1426531477525435216od_a_a @ B3 @ X7 ) ) ) ) ).
% edge_same_class
thf(fact_311_edge__same__class,axiom,
! [X7: set_a,Qeq: set_Product_prod_a_a,A3: a,B3: a] :
( ( member_set_a @ X7 @ ( classes_a @ Qeq ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ Qeq )
=> ( ( member_a @ A3 @ X7 )
= ( member_a @ B3 @ X7 ) ) ) ) ).
% edge_same_class
thf(fact_312_empty__set,axiom,
( bot_bo777872063958040403od_a_a
= ( set_se8408754101646271900od_a_a @ nil_se7151937861063948045od_a_a ) ) ).
% empty_set
thf(fact_313_empty__set,axiom,
( bot_bot_set_nat
= ( set_nat2 @ nil_nat ) ) ).
% empty_set
thf(fact_314_empty__set,axiom,
( bot_bo510284599550014259od_a_a
= ( set_Pr6493056779096584572od_a_a @ nil_Pr2171190902156203245od_a_a ) ) ).
% empty_set
thf(fact_315_empty__set,axiom,
( bot_bo6558180029120801017od_a_a
= ( set_li4974249543031949378od_a_a @ nil_li8564743906319921459od_a_a ) ) ).
% empty_set
thf(fact_316_empty__set,axiom,
( bot_bot_set_list_a
= ( set_list_a2 @ nil_list_a ) ) ).
% empty_set
thf(fact_317_empty__set,axiom,
( bot_bot_set_set_a
= ( set_set_a2 @ nil_set_a ) ) ).
% empty_set
thf(fact_318_empty__set,axiom,
( bot_bo3357376287454694259od_a_a
= ( set_Product_prod_a_a2 @ nil_Product_prod_a_a ) ) ).
% empty_set
thf(fact_319_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_320_list__ex1__iff,axiom,
( list_ex1_list_a
= ( ^ [P3: list_a > $o,Xs2: list_list_a] :
? [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs2 ) )
& ( P3 @ X3 )
& ! [Y2: list_a] :
( ( ( member_list_a @ Y2 @ ( set_list_a2 @ Xs2 ) )
& ( P3 @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_321_list__ex1__iff,axiom,
( list_ex1_o
= ( ^ [P3: $o > $o,Xs2: list_o] :
? [X3: $o] :
( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
& ( P3 @ X3 )
& ! [Y2: $o] :
( ( ( member_o @ Y2 @ ( set_o2 @ Xs2 ) )
& ( P3 @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_322_list__ex1__iff,axiom,
( list_e4689149996104398160od_a_a
= ( ^ [P3: set_Product_prod_a_a > $o,Xs2: list_s9060204159073123853od_a_a] :
? [X3: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X3 @ ( set_se8408754101646271900od_a_a @ Xs2 ) )
& ( P3 @ X3 )
& ! [Y2: set_Product_prod_a_a] :
( ( ( member1816616512716248880od_a_a @ Y2 @ ( set_se8408754101646271900od_a_a @ Xs2 ) )
& ( P3 @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_323_list__ex1__iff,axiom,
( list_ex1_set_a
= ( ^ [P3: set_a > $o,Xs2: list_set_a] :
? [X3: set_a] :
( ( member_set_a @ X3 @ ( set_set_a2 @ Xs2 ) )
& ( P3 @ X3 )
& ! [Y2: set_a] :
( ( ( member_set_a @ Y2 @ ( set_set_a2 @ Xs2 ) )
& ( P3 @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_324_list__ex1__iff,axiom,
( list_e8816154552257947952od_a_a
= ( ^ [P3: produc3498347346309940967od_a_a > $o,Xs2: list_P6569952146052023789od_a_a] :
? [X3: produc3498347346309940967od_a_a] :
( ( member6330455413206600464od_a_a @ X3 @ ( set_Pr6493056779096584572od_a_a @ Xs2 ) )
& ( P3 @ X3 )
& ! [Y2: produc3498347346309940967od_a_a] :
( ( ( member6330455413206600464od_a_a @ Y2 @ ( set_Pr6493056779096584572od_a_a @ Xs2 ) )
& ( P3 @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_325_list__ex1__iff,axiom,
( list_e3550399131106442742od_a_a
= ( ^ [P3: list_P1396940483166286381od_a_a > $o,Xs2: list_l2076173926988081843od_a_a] :
? [X3: list_P1396940483166286381od_a_a] :
( ( member6824001069763096534od_a_a @ X3 @ ( set_li4974249543031949378od_a_a @ Xs2 ) )
& ( P3 @ X3 )
& ! [Y2: list_P1396940483166286381od_a_a] :
( ( ( member6824001069763096534od_a_a @ Y2 @ ( set_li4974249543031949378od_a_a @ Xs2 ) )
& ( P3 @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_326_list__ex1__iff,axiom,
( list_ex1_a
= ( ^ [P3: a > $o,Xs2: list_a] :
? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
& ( P3 @ X3 )
& ! [Y2: a] :
( ( ( member_a @ Y2 @ ( set_a2 @ Xs2 ) )
& ( P3 @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_327_list__ex1__iff,axiom,
( list_e7533892409833140592od_a_a
= ( ^ [P3: product_prod_a_a > $o,Xs2: list_P1396940483166286381od_a_a] :
? [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs2 ) )
& ( P3 @ X3 )
& ! [Y2: product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ Y2 @ ( set_Product_prod_a_a2 @ Xs2 ) )
& ( P3 @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_328_not__None__eq,axiom,
! [X4: option3362143566709901261od_a_a] :
( ( X4 != none_s5714488769912451432od_a_a )
= ( ? [Y2: set_Product_prod_a_a] :
( X4
= ( some_s3837607710808129772od_a_a @ Y2 ) ) ) ) ).
% not_None_eq
thf(fact_329_not__None__eq,axiom,
! [X4: option_a] :
( ( X4 != none_a )
= ( ? [Y2: a] :
( X4
= ( some_a @ Y2 ) ) ) ) ).
% not_None_eq
thf(fact_330_not__None__eq,axiom,
! [X4: option5210160422955383789od_a_a] :
( ( X4 != none_P1914705783780380040od_a_a )
= ( ? [Y2: product_prod_a_a] :
( X4
= ( some_P3592067295195376908od_a_a @ Y2 ) ) ) ) ).
% not_None_eq
thf(fact_331_not__None__eq,axiom,
! [X4: option_set_a] :
( ( X4 != none_set_a )
= ( ? [Y2: set_a] :
( X4
= ( some_set_a @ Y2 ) ) ) ) ).
% not_None_eq
thf(fact_332_not__Some__eq,axiom,
! [X4: option3362143566709901261od_a_a] :
( ( ! [Y2: set_Product_prod_a_a] :
( X4
!= ( some_s3837607710808129772od_a_a @ Y2 ) ) )
= ( X4 = none_s5714488769912451432od_a_a ) ) ).
% not_Some_eq
thf(fact_333_not__Some__eq,axiom,
! [X4: option_a] :
( ( ! [Y2: a] :
( X4
!= ( some_a @ Y2 ) ) )
= ( X4 = none_a ) ) ).
% not_Some_eq
thf(fact_334_not__Some__eq,axiom,
! [X4: option5210160422955383789od_a_a] :
( ( ! [Y2: product_prod_a_a] :
( X4
!= ( some_P3592067295195376908od_a_a @ Y2 ) ) )
= ( X4 = none_P1914705783780380040od_a_a ) ) ).
% not_Some_eq
thf(fact_335_not__Some__eq,axiom,
! [X4: option_set_a] :
( ( ! [Y2: set_a] :
( X4
!= ( some_set_a @ Y2 ) ) )
= ( X4 = none_set_a ) ) ).
% not_Some_eq
thf(fact_336_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X4 @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_337_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_list_a] :
( ( inf_inf_set_list_a @ X4 @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_338_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_set_a] :
( ( inf_inf_set_set_a @ X4 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_339_boolean__algebra_Oconj__zero__right,axiom,
! [X4: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ X4 @ bot_bo4160289986317612842_a_a_o )
= bot_bo4160289986317612842_a_a_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_340_boolean__algebra_Oconj__zero__right,axiom,
! [X4: a > $o] :
( ( inf_inf_a_o @ X4 @ bot_bot_a_o )
= bot_bot_a_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_341_boolean__algebra_Oconj__zero__right,axiom,
! [X4: a > a > $o] :
( ( inf_inf_a_a_o2 @ X4 @ bot_bot_a_a_o2 )
= bot_bot_a_a_o2 ) ).
% boolean_algebra.conj_zero_right
thf(fact_342_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_343_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ X4 )
= bot_bo3357376287454694259od_a_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_344_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ X4 )
= bot_bot_set_list_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_345_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X4 )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_346_boolean__algebra_Oconj__zero__left,axiom,
! [X4: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ bot_bo4160289986317612842_a_a_o @ X4 )
= bot_bo4160289986317612842_a_a_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_347_boolean__algebra_Oconj__zero__left,axiom,
! [X4: a > $o] :
( ( inf_inf_a_o @ bot_bot_a_o @ X4 )
= bot_bot_a_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_348_boolean__algebra_Oconj__zero__left,axiom,
! [X4: a > a > $o] :
( ( inf_inf_a_a_o2 @ bot_bot_a_a_o2 @ X4 )
= bot_bot_a_a_o2 ) ).
% boolean_algebra.conj_zero_left
thf(fact_349_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X4 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_350_inf__bot__right,axiom,
! [X4: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X4 @ bot_bo3357376287454694259od_a_a )
= bot_bo3357376287454694259od_a_a ) ).
% inf_bot_right
thf(fact_351_inf__bot__right,axiom,
! [X4: set_list_a] :
( ( inf_inf_set_list_a @ X4 @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% inf_bot_right
thf(fact_352_inf__bot__right,axiom,
! [X4: set_set_a] :
( ( inf_inf_set_set_a @ X4 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% inf_bot_right
thf(fact_353_inf__bot__right,axiom,
! [X4: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ X4 @ bot_bo4160289986317612842_a_a_o )
= bot_bo4160289986317612842_a_a_o ) ).
% inf_bot_right
thf(fact_354_inf__bot__right,axiom,
! [X4: a > $o] :
( ( inf_inf_a_o @ X4 @ bot_bot_a_o )
= bot_bot_a_o ) ).
% inf_bot_right
thf(fact_355_inf__bot__right,axiom,
! [X4: a > a > $o] :
( ( inf_inf_a_a_o2 @ X4 @ bot_bot_a_a_o2 )
= bot_bot_a_a_o2 ) ).
% inf_bot_right
thf(fact_356_inf__bot__right,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_357_inf__bot__left,axiom,
! [X4: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ bot_bo3357376287454694259od_a_a @ X4 )
= bot_bo3357376287454694259od_a_a ) ).
% inf_bot_left
thf(fact_358_inf__bot__left,axiom,
! [X4: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ X4 )
= bot_bot_set_list_a ) ).
% inf_bot_left
thf(fact_359_inf__bot__left,axiom,
! [X4: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X4 )
= bot_bot_set_set_a ) ).
% inf_bot_left
thf(fact_360_inf__bot__left,axiom,
! [X4: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ bot_bo4160289986317612842_a_a_o @ X4 )
= bot_bo4160289986317612842_a_a_o ) ).
% inf_bot_left
thf(fact_361_inf__bot__left,axiom,
! [X4: a > $o] :
( ( inf_inf_a_o @ bot_bot_a_o @ X4 )
= bot_bot_a_o ) ).
% inf_bot_left
thf(fact_362_inf__bot__left,axiom,
! [X4: a > a > $o] :
( ( inf_inf_a_a_o2 @ bot_bot_a_a_o2 @ X4 )
= bot_bot_a_a_o2 ) ).
% inf_bot_left
thf(fact_363_inf__bot__left,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X4 )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_364_can__select__set__list__ex1,axiom,
! [P: produc3498347346309940967od_a_a > $o,A5: list_P6569952146052023789od_a_a] :
( ( can_se7382453719581608548od_a_a @ P @ ( set_Pr6493056779096584572od_a_a @ A5 ) )
= ( list_e8816154552257947952od_a_a @ P @ A5 ) ) ).
% can_select_set_list_ex1
thf(fact_365_can__select__set__list__ex1,axiom,
! [P: list_P1396940483166286381od_a_a > $o,A5: list_l2076173926988081843od_a_a] :
( ( can_se3375833721033465130od_a_a @ P @ ( set_li4974249543031949378od_a_a @ A5 ) )
= ( list_e3550399131106442742od_a_a @ P @ A5 ) ) ).
% can_select_set_list_ex1
thf(fact_366_can__select__set__list__ex1,axiom,
! [P: a > $o,A5: list_a] :
( ( can_select_a @ P @ ( set_a2 @ A5 ) )
= ( list_ex1_a @ P @ A5 ) ) ).
% can_select_set_list_ex1
thf(fact_367_can__select__set__list__ex1,axiom,
! [P: product_prod_a_a > $o,A5: list_P1396940483166286381od_a_a] :
( ( can_se2521037559381195428od_a_a @ P @ ( set_Product_prod_a_a2 @ A5 ) )
= ( list_e7533892409833140592od_a_a @ P @ A5 ) ) ).
% can_select_set_list_ex1
thf(fact_368_option_Oinject,axiom,
! [X22: set_Product_prod_a_a,Y22: set_Product_prod_a_a] :
( ( ( some_s3837607710808129772od_a_a @ X22 )
= ( some_s3837607710808129772od_a_a @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_369_option_Oinject,axiom,
! [X22: product_prod_a_a,Y22: product_prod_a_a] :
( ( ( some_P3592067295195376908od_a_a @ X22 )
= ( some_P3592067295195376908od_a_a @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_370_option_Oinject,axiom,
! [X22: a,Y22: a] :
( ( ( some_a @ X22 )
= ( some_a @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_371_option_Oinject,axiom,
! [X22: set_a,Y22: set_a] :
( ( ( some_set_a @ X22 )
= ( some_set_a @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_372_Int__iff,axiom,
! [C: list_P1396940483166286381od_a_a,A5: set_li8827807065578854541od_a_a,B5: set_li8827807065578854541od_a_a] :
( ( member6824001069763096534od_a_a @ C @ ( inf_in7369770237913016699od_a_a @ A5 @ B5 ) )
= ( ( member6824001069763096534od_a_a @ C @ A5 )
& ( member6824001069763096534od_a_a @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_373_Int__iff,axiom,
! [C: $o,A5: set_o,B5: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A5 @ B5 ) )
= ( ( member_o @ C @ A5 )
& ( member_o @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_374_Int__iff,axiom,
! [C: set_Product_prod_a_a,A5: set_se5735800977113168103od_a_a,B5: set_se5735800977113168103od_a_a] :
( ( member1816616512716248880od_a_a @ C @ ( inf_in3339382566020358357od_a_a @ A5 @ B5 ) )
= ( ( member1816616512716248880od_a_a @ C @ A5 )
& ( member1816616512716248880od_a_a @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_375_Int__iff,axiom,
! [C: set_a,A5: set_set_a,B5: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A5 @ B5 ) )
= ( ( member_set_a @ C @ A5 )
& ( member_set_a @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_376_Int__iff,axiom,
! [C: list_a,A5: set_list_a,B5: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A5 @ B5 ) )
= ( ( member_list_a @ C @ A5 )
& ( member_list_a @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_377_Int__iff,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A5 @ B5 ) )
= ( ( member1426531477525435216od_a_a @ C @ A5 )
& ( member1426531477525435216od_a_a @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_378_Int__iff,axiom,
! [C: a,A5: set_a,B5: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A5 @ B5 ) )
= ( ( member_a @ C @ A5 )
& ( member_a @ C @ B5 ) ) ) ).
% Int_iff
thf(fact_379_IntI,axiom,
! [C: list_P1396940483166286381od_a_a,A5: set_li8827807065578854541od_a_a,B5: set_li8827807065578854541od_a_a] :
( ( member6824001069763096534od_a_a @ C @ A5 )
=> ( ( member6824001069763096534od_a_a @ C @ B5 )
=> ( member6824001069763096534od_a_a @ C @ ( inf_in7369770237913016699od_a_a @ A5 @ B5 ) ) ) ) ).
% IntI
thf(fact_380_IntI,axiom,
! [C: $o,A5: set_o,B5: set_o] :
( ( member_o @ C @ A5 )
=> ( ( member_o @ C @ B5 )
=> ( member_o @ C @ ( inf_inf_set_o @ A5 @ B5 ) ) ) ) ).
% IntI
thf(fact_381_IntI,axiom,
! [C: set_Product_prod_a_a,A5: set_se5735800977113168103od_a_a,B5: set_se5735800977113168103od_a_a] :
( ( member1816616512716248880od_a_a @ C @ A5 )
=> ( ( member1816616512716248880od_a_a @ C @ B5 )
=> ( member1816616512716248880od_a_a @ C @ ( inf_in3339382566020358357od_a_a @ A5 @ B5 ) ) ) ) ).
% IntI
thf(fact_382_IntI,axiom,
! [C: set_a,A5: set_set_a,B5: set_set_a] :
( ( member_set_a @ C @ A5 )
=> ( ( member_set_a @ C @ B5 )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A5 @ B5 ) ) ) ) ).
% IntI
thf(fact_383_IntI,axiom,
! [C: list_a,A5: set_list_a,B5: set_list_a] :
( ( member_list_a @ C @ A5 )
=> ( ( member_list_a @ C @ B5 )
=> ( member_list_a @ C @ ( inf_inf_set_list_a @ A5 @ B5 ) ) ) ) ).
% IntI
thf(fact_384_IntI,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A5 )
=> ( ( member1426531477525435216od_a_a @ C @ B5 )
=> ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A5 @ B5 ) ) ) ) ).
% IntI
thf(fact_385_IntI,axiom,
! [C: a,A5: set_a,B5: set_a] :
( ( member_a @ C @ A5 )
=> ( ( member_a @ C @ B5 )
=> ( member_a @ C @ ( inf_inf_set_a @ A5 @ B5 ) ) ) ) ).
% IntI
thf(fact_386_empty__iff,axiom,
! [C: list_P1396940483166286381od_a_a] :
~ ( member6824001069763096534od_a_a @ C @ bot_bo6558180029120801017od_a_a ) ).
% empty_iff
thf(fact_387_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_388_empty__iff,axiom,
! [C: set_Product_prod_a_a] :
~ ( member1816616512716248880od_a_a @ C @ bot_bo777872063958040403od_a_a ) ).
% empty_iff
thf(fact_389_empty__iff,axiom,
! [C: list_a] :
~ ( member_list_a @ C @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_390_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_391_empty__iff,axiom,
! [C: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ C @ bot_bo3357376287454694259od_a_a ) ).
% empty_iff
thf(fact_392_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_393_all__not__in__conv,axiom,
! [A5: set_li8827807065578854541od_a_a] :
( ( ! [X3: list_P1396940483166286381od_a_a] :
~ ( member6824001069763096534od_a_a @ X3 @ A5 ) )
= ( A5 = bot_bo6558180029120801017od_a_a ) ) ).
% all_not_in_conv
thf(fact_394_all__not__in__conv,axiom,
! [A5: set_o] :
( ( ! [X3: $o] :
~ ( member_o @ X3 @ A5 ) )
= ( A5 = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_395_all__not__in__conv,axiom,
! [A5: set_se5735800977113168103od_a_a] :
( ( ! [X3: set_Product_prod_a_a] :
~ ( member1816616512716248880od_a_a @ X3 @ A5 ) )
= ( A5 = bot_bo777872063958040403od_a_a ) ) ).
% all_not_in_conv
thf(fact_396_all__not__in__conv,axiom,
! [A5: set_list_a] :
( ( ! [X3: list_a] :
~ ( member_list_a @ X3 @ A5 ) )
= ( A5 = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_397_all__not__in__conv,axiom,
! [A5: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A5 ) )
= ( A5 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_398_all__not__in__conv,axiom,
! [A5: set_Product_prod_a_a] :
( ( ! [X3: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X3 @ A5 ) )
= ( A5 = bot_bo3357376287454694259od_a_a ) ) ).
% all_not_in_conv
thf(fact_399_all__not__in__conv,axiom,
! [A5: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A5 ) )
= ( A5 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_400_Collect__empty__eq,axiom,
! [P: ( product_prod_a_a > product_prod_a_a ) > $o] :
( ( ( collec8125451137695935482od_a_a @ P )
= bot_bo2841618473486996463od_a_a )
= ( ! [X3: product_prod_a_a > product_prod_a_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_401_Collect__empty__eq,axiom,
! [P: ( product_prod_a_a > a ) > $o] :
( ( ( collec2886379341470944849_a_a_a @ P )
= bot_bo728555429280175302_a_a_a )
= ( ! [X3: product_prod_a_a > a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_402_Collect__empty__eq,axiom,
! [P: ( a > product_prod_a_a ) > $o] :
( ( ( collec3127584877517960419od_a_a @ P )
= bot_bo6605490641894888024od_a_a )
= ( ! [X3: a > product_prod_a_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_403_Collect__empty__eq,axiom,
! [P: ( a > a ) > $o] :
( ( ( collect_a_a @ P )
= bot_bot_set_a_a )
= ( ! [X3: a > a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_404_Collect__empty__eq,axiom,
! [P: list_a > $o] :
( ( ( collect_list_a @ P )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_405_Collect__empty__eq,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_406_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_407_empty__Collect__eq,axiom,
! [P: ( product_prod_a_a > product_prod_a_a ) > $o] :
( ( bot_bo2841618473486996463od_a_a
= ( collec8125451137695935482od_a_a @ P ) )
= ( ! [X3: product_prod_a_a > product_prod_a_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_408_empty__Collect__eq,axiom,
! [P: ( product_prod_a_a > a ) > $o] :
( ( bot_bo728555429280175302_a_a_a
= ( collec2886379341470944849_a_a_a @ P ) )
= ( ! [X3: product_prod_a_a > a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_409_empty__Collect__eq,axiom,
! [P: ( a > product_prod_a_a ) > $o] :
( ( bot_bo6605490641894888024od_a_a
= ( collec3127584877517960419od_a_a @ P ) )
= ( ! [X3: a > product_prod_a_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_410_empty__Collect__eq,axiom,
! [P: ( a > a ) > $o] :
( ( bot_bot_set_a_a
= ( collect_a_a @ P ) )
= ( ! [X3: a > a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_411_empty__Collect__eq,axiom,
! [P: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P ) )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_412_empty__Collect__eq,axiom,
! [P: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P ) )
= ( ! [X3: set_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_413_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_414_inf_Oidem,axiom,
! [A3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A3 @ A3 )
= A3 ) ).
% inf.idem
thf(fact_415_inf_Oidem,axiom,
! [A3: a > a > $o] :
( ( inf_inf_a_a_o2 @ A3 @ A3 )
= A3 ) ).
% inf.idem
thf(fact_416_inf_Oidem,axiom,
! [A3: a > $o] :
( ( inf_inf_a_o @ A3 @ A3 )
= A3 ) ).
% inf.idem
thf(fact_417_inf_Oidem,axiom,
! [A3: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ A3 @ A3 )
= A3 ) ).
% inf.idem
thf(fact_418_inf_Oidem,axiom,
! [A3: set_list_a] :
( ( inf_inf_set_list_a @ A3 @ A3 )
= A3 ) ).
% inf.idem
thf(fact_419_inf_Oidem,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ A3 @ A3 )
= A3 ) ).
% inf.idem
thf(fact_420_inf__idem,axiom,
! [X4: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_421_inf__idem,axiom,
! [X4: a > a > $o] :
( ( inf_inf_a_a_o2 @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_422_inf__idem,axiom,
! [X4: a > $o] :
( ( inf_inf_a_o @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_423_inf__idem,axiom,
! [X4: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_424_inf__idem,axiom,
! [X4: set_list_a] :
( ( inf_inf_set_list_a @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_425_inf__idem,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_426_inf_Oleft__idem,axiom,
! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ A3 @ ( inf_in8905007599844390133od_a_a @ A3 @ B3 ) )
= ( inf_in8905007599844390133od_a_a @ A3 @ B3 ) ) ).
% inf.left_idem
thf(fact_427_inf_Oleft__idem,axiom,
! [A3: a > a > $o,B3: a > a > $o] :
( ( inf_inf_a_a_o2 @ A3 @ ( inf_inf_a_a_o2 @ A3 @ B3 ) )
= ( inf_inf_a_a_o2 @ A3 @ B3 ) ) ).
% inf.left_idem
thf(fact_428_inf_Oleft__idem,axiom,
! [A3: a > $o,B3: a > $o] :
( ( inf_inf_a_o @ A3 @ ( inf_inf_a_o @ A3 @ B3 ) )
= ( inf_inf_a_o @ A3 @ B3 ) ) ).
% inf.left_idem
thf(fact_429_inf_Oleft__idem,axiom,
! [A3: product_prod_a_a > $o,B3: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ A3 @ ( inf_in2559554923042384936_a_a_o @ A3 @ B3 ) )
= ( inf_in2559554923042384936_a_a_o @ A3 @ B3 ) ) ).
% inf.left_idem
thf(fact_430_inf_Oleft__idem,axiom,
! [A3: set_list_a,B3: set_list_a] :
( ( inf_inf_set_list_a @ A3 @ ( inf_inf_set_list_a @ A3 @ B3 ) )
= ( inf_inf_set_list_a @ A3 @ B3 ) ) ).
% inf.left_idem
thf(fact_431_inf_Oleft__idem,axiom,
! [A3: set_a,B3: set_a] :
( ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ A3 @ B3 ) )
= ( inf_inf_set_a @ A3 @ B3 ) ) ).
% inf.left_idem
thf(fact_432_inf__left__idem,axiom,
! [X4: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X4 @ ( inf_in8905007599844390133od_a_a @ X4 @ Y3 ) )
= ( inf_in8905007599844390133od_a_a @ X4 @ Y3 ) ) ).
% inf_left_idem
thf(fact_433_inf__left__idem,axiom,
! [X4: a > a > $o,Y3: a > a > $o] :
( ( inf_inf_a_a_o2 @ X4 @ ( inf_inf_a_a_o2 @ X4 @ Y3 ) )
= ( inf_inf_a_a_o2 @ X4 @ Y3 ) ) ).
% inf_left_idem
thf(fact_434_inf__left__idem,axiom,
! [X4: a > $o,Y3: a > $o] :
( ( inf_inf_a_o @ X4 @ ( inf_inf_a_o @ X4 @ Y3 ) )
= ( inf_inf_a_o @ X4 @ Y3 ) ) ).
% inf_left_idem
thf(fact_435_inf__left__idem,axiom,
! [X4: product_prod_a_a > $o,Y3: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ X4 @ ( inf_in2559554923042384936_a_a_o @ X4 @ Y3 ) )
= ( inf_in2559554923042384936_a_a_o @ X4 @ Y3 ) ) ).
% inf_left_idem
thf(fact_436_inf__left__idem,axiom,
! [X4: set_list_a,Y3: set_list_a] :
( ( inf_inf_set_list_a @ X4 @ ( inf_inf_set_list_a @ X4 @ Y3 ) )
= ( inf_inf_set_list_a @ X4 @ Y3 ) ) ).
% inf_left_idem
thf(fact_437_inf__left__idem,axiom,
! [X4: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ X4 @ Y3 ) )
= ( inf_inf_set_a @ X4 @ Y3 ) ) ).
% inf_left_idem
thf(fact_438_inf_Oright__idem,axiom,
! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( inf_in8905007599844390133od_a_a @ A3 @ B3 ) @ B3 )
= ( inf_in8905007599844390133od_a_a @ A3 @ B3 ) ) ).
% inf.right_idem
thf(fact_439_inf_Oright__idem,axiom,
! [A3: a > a > $o,B3: a > a > $o] :
( ( inf_inf_a_a_o2 @ ( inf_inf_a_a_o2 @ A3 @ B3 ) @ B3 )
= ( inf_inf_a_a_o2 @ A3 @ B3 ) ) ).
% inf.right_idem
thf(fact_440_inf_Oright__idem,axiom,
! [A3: a > $o,B3: a > $o] :
( ( inf_inf_a_o @ ( inf_inf_a_o @ A3 @ B3 ) @ B3 )
= ( inf_inf_a_o @ A3 @ B3 ) ) ).
% inf.right_idem
thf(fact_441_inf_Oright__idem,axiom,
! [A3: product_prod_a_a > $o,B3: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ ( inf_in2559554923042384936_a_a_o @ A3 @ B3 ) @ B3 )
= ( inf_in2559554923042384936_a_a_o @ A3 @ B3 ) ) ).
% inf.right_idem
thf(fact_442_inf_Oright__idem,axiom,
! [A3: set_list_a,B3: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A3 @ B3 ) @ B3 )
= ( inf_inf_set_list_a @ A3 @ B3 ) ) ).
% inf.right_idem
thf(fact_443_inf_Oright__idem,axiom,
! [A3: set_a,B3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ B3 )
= ( inf_inf_set_a @ A3 @ B3 ) ) ).
% inf.right_idem
thf(fact_444_inf__right__idem,axiom,
! [X4: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( inf_in8905007599844390133od_a_a @ X4 @ Y3 ) @ Y3 )
= ( inf_in8905007599844390133od_a_a @ X4 @ Y3 ) ) ).
% inf_right_idem
thf(fact_445_inf__right__idem,axiom,
! [X4: a > a > $o,Y3: a > a > $o] :
( ( inf_inf_a_a_o2 @ ( inf_inf_a_a_o2 @ X4 @ Y3 ) @ Y3 )
= ( inf_inf_a_a_o2 @ X4 @ Y3 ) ) ).
% inf_right_idem
thf(fact_446_inf__right__idem,axiom,
! [X4: a > $o,Y3: a > $o] :
( ( inf_inf_a_o @ ( inf_inf_a_o @ X4 @ Y3 ) @ Y3 )
= ( inf_inf_a_o @ X4 @ Y3 ) ) ).
% inf_right_idem
thf(fact_447_inf__right__idem,axiom,
! [X4: product_prod_a_a > $o,Y3: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ ( inf_in2559554923042384936_a_a_o @ X4 @ Y3 ) @ Y3 )
= ( inf_in2559554923042384936_a_a_o @ X4 @ Y3 ) ) ).
% inf_right_idem
thf(fact_448_inf__right__idem,axiom,
! [X4: set_list_a,Y3: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X4 @ Y3 ) @ Y3 )
= ( inf_inf_set_list_a @ X4 @ Y3 ) ) ).
% inf_right_idem
thf(fact_449_inf__right__idem,axiom,
! [X4: set_a,Y3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ Y3 )
= ( inf_inf_set_a @ X4 @ Y3 ) ) ).
% inf_right_idem
thf(fact_450_inf__apply,axiom,
( inf_inf_a_a_o2
= ( ^ [F2: a > a > $o,G2: a > a > $o,X3: a] : ( inf_inf_a_o @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% inf_apply
thf(fact_451_inf__apply,axiom,
( inf_inf_a_o
= ( ^ [F2: a > $o,G2: a > $o,X3: a] : ( inf_inf_o @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% inf_apply
thf(fact_452_inf__apply,axiom,
( inf_in2559554923042384936_a_a_o
= ( ^ [F2: product_prod_a_a > $o,G2: product_prod_a_a > $o,X3: product_prod_a_a] : ( inf_inf_o @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% inf_apply
thf(fact_453_bot__set__def,axiom,
( bot_bo2841618473486996463od_a_a
= ( collec8125451137695935482od_a_a @ bot_bo6742726475195524950_a_a_o ) ) ).
% bot_set_def
thf(fact_454_bot__set__def,axiom,
( bot_bo728555429280175302_a_a_a
= ( collec2886379341470944849_a_a_a @ bot_bo5643334865281271615_a_a_o ) ) ).
% bot_set_def
thf(fact_455_bot__set__def,axiom,
( bot_bo6605490641894888024od_a_a
= ( collec3127584877517960419od_a_a @ bot_bo3862914247785971757_a_a_o ) ) ).
% bot_set_def
thf(fact_456_bot__set__def,axiom,
( bot_bot_set_a_a
= ( collect_a_a @ bot_bot_a_a_o ) ) ).
% bot_set_def
thf(fact_457_bot__set__def,axiom,
( bot_bo3357376287454694259od_a_a
= ( collec3336397797384452498od_a_a @ bot_bo4160289986317612842_a_a_o ) ) ).
% bot_set_def
thf(fact_458_bot__set__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a @ bot_bot_list_a_o ) ) ).
% bot_set_def
thf(fact_459_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_460_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_461_inf__Int__eq2,axiom,
! [R2: set_Pr5629751886588192039od_a_a,S: set_Pr5629751886588192039od_a_a] :
( ( inf_in8944399846650179004_a_a_o
@ ^ [X3: set_Product_prod_a_a,Y2: produc5715173734304154471od_a_a] : ( member4804927706910867312od_a_a @ ( produc9141123965022120375od_a_a @ X3 @ Y2 ) @ R2 )
@ ^ [X3: set_Product_prod_a_a,Y2: produc5715173734304154471od_a_a] : ( member4804927706910867312od_a_a @ ( produc9141123965022120375od_a_a @ X3 @ Y2 ) @ S ) )
= ( ^ [X3: set_Product_prod_a_a,Y2: produc5715173734304154471od_a_a] : ( member4804927706910867312od_a_a @ ( produc9141123965022120375od_a_a @ X3 @ Y2 ) @ ( inf_in3952362152483845909od_a_a @ R2 @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_462_inf__Int__eq2,axiom,
! [R2: set_Pr5931134355633282247od_a_a,S: set_Pr5931134355633282247od_a_a] :
( ( inf_in6097490241675510300_a_a_o
@ ^ [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a] : ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ X3 @ Y2 ) @ R2 )
@ ^ [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a] : ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ X3 @ Y2 ) @ S ) )
= ( ^ [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a] : ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ X3 @ Y2 ) @ ( inf_in7177451468758028469od_a_a @ R2 @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_463_inf__Int__eq2,axiom,
! [R2: set_Pr5845495582615845127_set_a,S: set_Pr5845495582615845127_set_a] :
( ( inf_in1388326141828443228et_a_o
@ ^ [X3: set_a,Y2: set_a] : ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X3 @ Y2 ) @ R2 )
@ ^ [X3: set_a,Y2: set_a] : ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X3 @ Y2 ) @ S ) )
= ( ^ [X3: set_a,Y2: set_a] : ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X3 @ Y2 ) @ ( inf_in1230022433524902133_set_a @ R2 @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_464_inf__Int__eq2,axiom,
! [R2: set_Pr1368515532014665606od_a_a,S: set_Pr1368515532014665606od_a_a] :
( ( inf_in1174067067476203933_a_a_o
@ ^ [X3: product_prod_a_a > product_prod_a_a > $o,Y2: list_P1396940483166286381od_a_a] : ( member1420838311466244943od_a_a @ ( produc7609767723203257750od_a_a @ X3 @ Y2 ) @ R2 )
@ ^ [X3: product_prod_a_a > product_prod_a_a > $o,Y2: list_P1396940483166286381od_a_a] : ( member1420838311466244943od_a_a @ ( produc7609767723203257750od_a_a @ X3 @ Y2 ) @ S ) )
= ( ^ [X3: product_prod_a_a > product_prod_a_a > $o,Y2: list_P1396940483166286381od_a_a] : ( member1420838311466244943od_a_a @ ( produc7609767723203257750od_a_a @ X3 @ Y2 ) @ ( inf_in7357090814806848628od_a_a @ R2 @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_465_inf__Int__eq2,axiom,
! [R2: set_Pr5530083903271594800od_a_a,S: set_Pr5530083903271594800od_a_a] :
( ( inf_in2376972441324840307_a_a_o
@ ^ [X3: a,Y2: product_prod_a_a] : ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ X3 @ Y2 ) @ R2 )
@ ^ [X3: a,Y2: product_prod_a_a] : ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ X3 @ Y2 ) @ S ) )
= ( ^ [X3: a,Y2: product_prod_a_a] : ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ X3 @ Y2 ) @ ( inf_in6567511182131981598od_a_a @ R2 @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_466_inf__Int__eq2,axiom,
! [R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( inf_inf_a_a_o2
@ ^ [X3: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R2 )
@ ^ [X3: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ S ) )
= ( ^ [X3: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ ( inf_in8905007599844390133od_a_a @ R2 @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_467_bot__empty__eq2,axiom,
( bot_bo6261260381589903930_a_a_o
= ( ^ [X3: set_Product_prod_a_a,Y2: produc5715173734304154471od_a_a] : ( member4804927706910867312od_a_a @ ( produc9141123965022120375od_a_a @ X3 @ Y2 ) @ bot_bo902829537889496979od_a_a ) ) ) ).
% bot_empty_eq2
thf(fact_468_bot__empty__eq2,axiom,
( bot_bo3998387317251997850_a_a_o
= ( ^ [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a] : ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ X3 @ Y2 ) @ bot_bo8313862440737889075od_a_a ) ) ) ).
% bot_empty_eq2
thf(fact_469_bot__empty__eq2,axiom,
( bot_bo4296365073212884698et_a_o
= ( ^ [X3: set_a,Y2: set_a] : ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X3 @ Y2 ) @ bot_bo5799363139946352499_set_a ) ) ) ).
% bot_empty_eq2
thf(fact_470_bot__empty__eq2,axiom,
( bot_bo1008171968724062491_a_a_o
= ( ^ [X3: product_prod_a_a > product_prod_a_a > $o,Y2: list_P1396940483166286381od_a_a] : ( member1420838311466244943od_a_a @ ( produc7609767723203257750od_a_a @ X3 @ Y2 ) @ bot_bo1850226986289526770od_a_a ) ) ) ).
% bot_empty_eq2
thf(fact_471_bot__empty__eq2,axiom,
( bot_bo1636446413509968369_a_a_o
= ( ^ [X3: a,Y2: product_prod_a_a] : ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ X3 @ Y2 ) @ bot_bo4436838304982128028od_a_a ) ) ) ).
% bot_empty_eq2
thf(fact_472_bot__empty__eq2,axiom,
( bot_bot_a_a_o2
= ( ^ [X3: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ bot_bo3357376287454694259od_a_a ) ) ) ).
% bot_empty_eq2
thf(fact_473_classes__empty,axiom,
( ( classe2514596149140685704od_a_a @ bot_bo510284599550014259od_a_a )
= bot_bo777872063958040403od_a_a ) ).
% classes_empty
thf(fact_474_classes__empty,axiom,
( ( classes_a @ bot_bo3357376287454694259od_a_a )
= bot_bot_set_set_a ) ).
% classes_empty
thf(fact_475_inf__set__def,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ( inf_inf_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A6 )
@ ^ [X3: a] : ( member_a @ X3 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_476_inf__set__def,axiom,
( inf_in8905007599844390133od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( collec3336397797384452498od_a_a
@ ( inf_in2559554923042384936_a_a_o
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A6 )
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_477_inf__set__def,axiom,
( inf_inf_set_o
= ( ^ [A6: set_o,B6: set_o] :
( collect_o
@ ( inf_inf_o_o
@ ^ [X3: $o] : ( member_o @ X3 @ A6 )
@ ^ [X3: $o] : ( member_o @ X3 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_478_inf__set__def,axiom,
( inf_inf_set_set_a
= ( ^ [A6: set_set_a,B6: set_set_a] :
( collect_set_a
@ ( inf_inf_set_a_o
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A6 )
@ ^ [X3: set_a] : ( member_set_a @ X3 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_479_inf__set__def,axiom,
( inf_inf_set_list_a
= ( ^ [A6: set_list_a,B6: set_list_a] :
( collect_list_a
@ ( inf_inf_list_a_o
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A6 )
@ ^ [X3: list_a] : ( member_list_a @ X3 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_480_inf__set__def,axiom,
( inf_inf_set_a_a
= ( ^ [A6: set_a_a,B6: set_a_a] :
( collect_a_a
@ ( inf_inf_a_a_o
@ ^ [X3: a > a] : ( member_a_a @ X3 @ A6 )
@ ^ [X3: a > a] : ( member_a_a @ X3 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_481_inf__set__def,axiom,
( inf_in7369770237913016699od_a_a
= ( ^ [A6: set_li8827807065578854541od_a_a,B6: set_li8827807065578854541od_a_a] :
( collec2774578310585404696od_a_a
@ ( inf_in8223129354698103074_a_a_o
@ ^ [X3: list_P1396940483166286381od_a_a] : ( member6824001069763096534od_a_a @ X3 @ A6 )
@ ^ [X3: list_P1396940483166286381od_a_a] : ( member6824001069763096534od_a_a @ X3 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_482_inf__set__def,axiom,
( inf_in3339382566020358357od_a_a
= ( ^ [A6: set_se5735800977113168103od_a_a,B6: set_se5735800977113168103od_a_a] :
( collec1673347964119250290od_a_a
@ ( inf_in1700971893745756232_a_a_o
@ ^ [X3: set_Product_prod_a_a] : ( member1816616512716248880od_a_a @ X3 @ A6 )
@ ^ [X3: set_Product_prod_a_a] : ( member1816616512716248880od_a_a @ X3 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_483_inf__set__def,axiom,
( inf_in7780504178787840708_a_a_a
= ( ^ [A6: set_Pr952751117562918450_a_a_a,B6: set_Pr952751117562918450_a_a_a] :
( collec2886379341470944849_a_a_a
@ ( inf_in6383860893096143553_a_a_o
@ ^ [X3: product_prod_a_a > a] : ( member1716570166360300819_a_a_a @ X3 @ A6 )
@ ^ [X3: product_prod_a_a > a] : ( member1716570166360300819_a_a_a @ X3 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_484_inf__set__def,axiom,
( inf_in4434067354547777622od_a_a
= ( ^ [A6: set_a_6829686330177631172od_a_a,B6: set_a_6829686330177631172od_a_a] :
( collec3127584877517960419od_a_a
@ ( inf_in4603440275600843695_a_a_o
@ ^ [X3: a > product_prod_a_a] : ( member1957775702407316389od_a_a @ X3 @ A6 )
@ ^ [X3: a > product_prod_a_a] : ( member1957775702407316389od_a_a @ X3 @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_485_bot__empty__eq,axiom,
( bot_bo1171180605190437668_a_a_o
= ( ^ [X3: list_P1396940483166286381od_a_a] : ( member6824001069763096534od_a_a @ X3 @ bot_bo6558180029120801017od_a_a ) ) ) ).
% bot_empty_eq
thf(fact_486_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X3: $o] : ( member_o @ X3 @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_487_bot__empty__eq,axiom,
( bot_bo3434591380065676106_a_a_o
= ( ^ [X3: set_Product_prod_a_a] : ( member1816616512716248880od_a_a @ X3 @ bot_bo777872063958040403od_a_a ) ) ) ).
% bot_empty_eq
thf(fact_488_bot__empty__eq,axiom,
( bot_bot_list_a_o
= ( ^ [X3: list_a] : ( member_list_a @ X3 @ bot_bot_set_list_a ) ) ) ).
% bot_empty_eq
thf(fact_489_bot__empty__eq,axiom,
( bot_bot_set_a_o
= ( ^ [X3: set_a] : ( member_set_a @ X3 @ bot_bot_set_set_a ) ) ) ).
% bot_empty_eq
thf(fact_490_bot__empty__eq,axiom,
( bot_bo4160289986317612842_a_a_o
= ( ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ bot_bo3357376287454694259od_a_a ) ) ) ).
% bot_empty_eq
thf(fact_491_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_492_can__select__def,axiom,
( can_se3375833721033465130od_a_a
= ( ^ [P3: list_P1396940483166286381od_a_a > $o,A6: set_li8827807065578854541od_a_a] :
? [X3: list_P1396940483166286381od_a_a] :
( ( member6824001069763096534od_a_a @ X3 @ A6 )
& ( P3 @ X3 )
& ! [Y2: list_P1396940483166286381od_a_a] :
( ( ( member6824001069763096534od_a_a @ Y2 @ A6 )
& ( P3 @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_493_can__select__def,axiom,
( can_select_list_a
= ( ^ [P3: list_a > $o,A6: set_list_a] :
? [X3: list_a] :
( ( member_list_a @ X3 @ A6 )
& ( P3 @ X3 )
& ! [Y2: list_a] :
( ( ( member_list_a @ Y2 @ A6 )
& ( P3 @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_494_can__select__def,axiom,
( can_select_o
= ( ^ [P3: $o > $o,A6: set_o] :
? [X3: $o] :
( ( member_o @ X3 @ A6 )
& ( P3 @ X3 )
& ! [Y2: $o] :
( ( ( member_o @ Y2 @ A6 )
& ( P3 @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_495_can__select__def,axiom,
( can_se2091955547861200516od_a_a
= ( ^ [P3: set_Product_prod_a_a > $o,A6: set_se5735800977113168103od_a_a] :
? [X3: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X3 @ A6 )
& ( P3 @ X3 )
& ! [Y2: set_Product_prod_a_a] :
( ( ( member1816616512716248880od_a_a @ Y2 @ A6 )
& ( P3 @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_496_can__select__def,axiom,
( can_select_set_a
= ( ^ [P3: set_a > $o,A6: set_set_a] :
? [X3: set_a] :
( ( member_set_a @ X3 @ A6 )
& ( P3 @ X3 )
& ! [Y2: set_a] :
( ( ( member_set_a @ Y2 @ A6 )
& ( P3 @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_497_can__select__def,axiom,
( can_select_a
= ( ^ [P3: a > $o,A6: set_a] :
? [X3: a] :
( ( member_a @ X3 @ A6 )
& ( P3 @ X3 )
& ! [Y2: a] :
( ( ( member_a @ Y2 @ A6 )
& ( P3 @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_498_can__select__def,axiom,
( can_se2521037559381195428od_a_a
= ( ^ [P3: product_prod_a_a > $o,A6: set_Product_prod_a_a] :
? [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A6 )
& ( P3 @ X3 )
& ! [Y2: product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ Y2 @ A6 )
& ( P3 @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_499_inf__Int__eq,axiom,
! [R2: set_li8827807065578854541od_a_a,S: set_li8827807065578854541od_a_a] :
( ( inf_in8223129354698103074_a_a_o
@ ^ [X3: list_P1396940483166286381od_a_a] : ( member6824001069763096534od_a_a @ X3 @ R2 )
@ ^ [X3: list_P1396940483166286381od_a_a] : ( member6824001069763096534od_a_a @ X3 @ S ) )
= ( ^ [X3: list_P1396940483166286381od_a_a] : ( member6824001069763096534od_a_a @ X3 @ ( inf_in7369770237913016699od_a_a @ R2 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_500_inf__Int__eq,axiom,
! [R2: set_o,S: set_o] :
( ( inf_inf_o_o
@ ^ [X3: $o] : ( member_o @ X3 @ R2 )
@ ^ [X3: $o] : ( member_o @ X3 @ S ) )
= ( ^ [X3: $o] : ( member_o @ X3 @ ( inf_inf_set_o @ R2 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_501_inf__Int__eq,axiom,
! [R2: set_se5735800977113168103od_a_a,S: set_se5735800977113168103od_a_a] :
( ( inf_in1700971893745756232_a_a_o
@ ^ [X3: set_Product_prod_a_a] : ( member1816616512716248880od_a_a @ X3 @ R2 )
@ ^ [X3: set_Product_prod_a_a] : ( member1816616512716248880od_a_a @ X3 @ S ) )
= ( ^ [X3: set_Product_prod_a_a] : ( member1816616512716248880od_a_a @ X3 @ ( inf_in3339382566020358357od_a_a @ R2 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_502_inf__Int__eq,axiom,
! [R2: set_set_a,S: set_set_a] :
( ( inf_inf_set_a_o
@ ^ [X3: set_a] : ( member_set_a @ X3 @ R2 )
@ ^ [X3: set_a] : ( member_set_a @ X3 @ S ) )
= ( ^ [X3: set_a] : ( member_set_a @ X3 @ ( inf_inf_set_set_a @ R2 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_503_inf__Int__eq,axiom,
! [R2: set_list_a,S: set_list_a] :
( ( inf_inf_list_a_o
@ ^ [X3: list_a] : ( member_list_a @ X3 @ R2 )
@ ^ [X3: list_a] : ( member_list_a @ X3 @ S ) )
= ( ^ [X3: list_a] : ( member_list_a @ X3 @ ( inf_inf_set_list_a @ R2 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_504_inf__Int__eq,axiom,
! [R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( inf_in2559554923042384936_a_a_o
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ R2 )
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ S ) )
= ( ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ ( inf_in8905007599844390133od_a_a @ R2 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_505_inf__Int__eq,axiom,
! [R2: set_a,S: set_a] :
( ( inf_inf_a_o
@ ^ [X3: a] : ( member_a @ X3 @ R2 )
@ ^ [X3: a] : ( member_a @ X3 @ S ) )
= ( ^ [X3: a] : ( member_a @ X3 @ ( inf_inf_set_a @ R2 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_506_emptyE,axiom,
! [A3: list_P1396940483166286381od_a_a] :
~ ( member6824001069763096534od_a_a @ A3 @ bot_bo6558180029120801017od_a_a ) ).
% emptyE
thf(fact_507_emptyE,axiom,
! [A3: $o] :
~ ( member_o @ A3 @ bot_bot_set_o ) ).
% emptyE
thf(fact_508_emptyE,axiom,
! [A3: set_Product_prod_a_a] :
~ ( member1816616512716248880od_a_a @ A3 @ bot_bo777872063958040403od_a_a ) ).
% emptyE
thf(fact_509_emptyE,axiom,
! [A3: list_a] :
~ ( member_list_a @ A3 @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_510_emptyE,axiom,
! [A3: set_a] :
~ ( member_set_a @ A3 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_511_emptyE,axiom,
! [A3: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ A3 @ bot_bo3357376287454694259od_a_a ) ).
% emptyE
thf(fact_512_emptyE,axiom,
! [A3: a] :
~ ( member_a @ A3 @ bot_bot_set_a ) ).
% emptyE
thf(fact_513_equals0D,axiom,
! [A5: set_li8827807065578854541od_a_a,A3: list_P1396940483166286381od_a_a] :
( ( A5 = bot_bo6558180029120801017od_a_a )
=> ~ ( member6824001069763096534od_a_a @ A3 @ A5 ) ) ).
% equals0D
thf(fact_514_equals0D,axiom,
! [A5: set_o,A3: $o] :
( ( A5 = bot_bot_set_o )
=> ~ ( member_o @ A3 @ A5 ) ) ).
% equals0D
thf(fact_515_equals0D,axiom,
! [A5: set_se5735800977113168103od_a_a,A3: set_Product_prod_a_a] :
( ( A5 = bot_bo777872063958040403od_a_a )
=> ~ ( member1816616512716248880od_a_a @ A3 @ A5 ) ) ).
% equals0D
thf(fact_516_equals0D,axiom,
! [A5: set_list_a,A3: list_a] :
( ( A5 = bot_bot_set_list_a )
=> ~ ( member_list_a @ A3 @ A5 ) ) ).
% equals0D
thf(fact_517_equals0D,axiom,
! [A5: set_set_a,A3: set_a] :
( ( A5 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A3 @ A5 ) ) ).
% equals0D
thf(fact_518_equals0D,axiom,
! [A5: set_Product_prod_a_a,A3: product_prod_a_a] :
( ( A5 = bot_bo3357376287454694259od_a_a )
=> ~ ( member1426531477525435216od_a_a @ A3 @ A5 ) ) ).
% equals0D
thf(fact_519_equals0D,axiom,
! [A5: set_a,A3: a] :
( ( A5 = bot_bot_set_a )
=> ~ ( member_a @ A3 @ A5 ) ) ).
% equals0D
thf(fact_520_equals0I,axiom,
! [A5: set_li8827807065578854541od_a_a] :
( ! [Y: list_P1396940483166286381od_a_a] :
~ ( member6824001069763096534od_a_a @ Y @ A5 )
=> ( A5 = bot_bo6558180029120801017od_a_a ) ) ).
% equals0I
thf(fact_521_equals0I,axiom,
! [A5: set_o] :
( ! [Y: $o] :
~ ( member_o @ Y @ A5 )
=> ( A5 = bot_bot_set_o ) ) ).
% equals0I
thf(fact_522_equals0I,axiom,
! [A5: set_se5735800977113168103od_a_a] :
( ! [Y: set_Product_prod_a_a] :
~ ( member1816616512716248880od_a_a @ Y @ A5 )
=> ( A5 = bot_bo777872063958040403od_a_a ) ) ).
% equals0I
thf(fact_523_equals0I,axiom,
! [A5: set_list_a] :
( ! [Y: list_a] :
~ ( member_list_a @ Y @ A5 )
=> ( A5 = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_524_equals0I,axiom,
! [A5: set_set_a] :
( ! [Y: set_a] :
~ ( member_set_a @ Y @ A5 )
=> ( A5 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_525_equals0I,axiom,
! [A5: set_Product_prod_a_a] :
( ! [Y: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ Y @ A5 )
=> ( A5 = bot_bo3357376287454694259od_a_a ) ) ).
% equals0I
thf(fact_526_equals0I,axiom,
! [A5: set_a] :
( ! [Y: a] :
~ ( member_a @ Y @ A5 )
=> ( A5 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_527_ex__in__conv,axiom,
! [A5: set_li8827807065578854541od_a_a] :
( ( ? [X3: list_P1396940483166286381od_a_a] : ( member6824001069763096534od_a_a @ X3 @ A5 ) )
= ( A5 != bot_bo6558180029120801017od_a_a ) ) ).
% ex_in_conv
thf(fact_528_ex__in__conv,axiom,
! [A5: set_o] :
( ( ? [X3: $o] : ( member_o @ X3 @ A5 ) )
= ( A5 != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_529_ex__in__conv,axiom,
! [A5: set_se5735800977113168103od_a_a] :
( ( ? [X3: set_Product_prod_a_a] : ( member1816616512716248880od_a_a @ X3 @ A5 ) )
= ( A5 != bot_bo777872063958040403od_a_a ) ) ).
% ex_in_conv
thf(fact_530_ex__in__conv,axiom,
! [A5: set_list_a] :
( ( ? [X3: list_a] : ( member_list_a @ X3 @ A5 ) )
= ( A5 != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_531_ex__in__conv,axiom,
! [A5: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A5 ) )
= ( A5 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_532_ex__in__conv,axiom,
! [A5: set_Product_prod_a_a] :
( ( ? [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A5 ) )
= ( A5 != bot_bo3357376287454694259od_a_a ) ) ).
% ex_in_conv
thf(fact_533_ex__in__conv,axiom,
! [A5: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A5 ) )
= ( A5 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_534_inf__sup__aci_I4_J,axiom,
! [X4: set_Product_prod_a_a,Y3: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X4 @ ( inf_in8905007599844390133od_a_a @ X4 @ Y3 ) )
= ( inf_in8905007599844390133od_a_a @ X4 @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_535_inf__sup__aci_I4_J,axiom,
! [X4: a > a > $o,Y3: a > a > $o] :
( ( inf_inf_a_a_o2 @ X4 @ ( inf_inf_a_a_o2 @ X4 @ Y3 ) )
= ( inf_inf_a_a_o2 @ X4 @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_536_inf__sup__aci_I4_J,axiom,
! [X4: a > $o,Y3: a > $o] :
( ( inf_inf_a_o @ X4 @ ( inf_inf_a_o @ X4 @ Y3 ) )
= ( inf_inf_a_o @ X4 @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_537_inf__sup__aci_I4_J,axiom,
! [X4: product_prod_a_a > $o,Y3: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ X4 @ ( inf_in2559554923042384936_a_a_o @ X4 @ Y3 ) )
= ( inf_in2559554923042384936_a_a_o @ X4 @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_538_inf__sup__aci_I4_J,axiom,
! [X4: set_list_a,Y3: set_list_a] :
( ( inf_inf_set_list_a @ X4 @ ( inf_inf_set_list_a @ X4 @ Y3 ) )
= ( inf_inf_set_list_a @ X4 @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_539_inf__sup__aci_I4_J,axiom,
! [X4: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ X4 @ Y3 ) )
= ( inf_inf_set_a @ X4 @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_540_inf__sup__aci_I3_J,axiom,
! [X4: set_Product_prod_a_a,Y3: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ X4 @ ( inf_in8905007599844390133od_a_a @ Y3 @ Z ) )
= ( inf_in8905007599844390133od_a_a @ Y3 @ ( inf_in8905007599844390133od_a_a @ X4 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_541_inf__sup__aci_I3_J,axiom,
! [X4: a > a > $o,Y3: a > a > $o,Z: a > a > $o] :
( ( inf_inf_a_a_o2 @ X4 @ ( inf_inf_a_a_o2 @ Y3 @ Z ) )
= ( inf_inf_a_a_o2 @ Y3 @ ( inf_inf_a_a_o2 @ X4 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_542_inf__sup__aci_I3_J,axiom,
! [X4: a > $o,Y3: a > $o,Z: a > $o] :
( ( inf_inf_a_o @ X4 @ ( inf_inf_a_o @ Y3 @ Z ) )
= ( inf_inf_a_o @ Y3 @ ( inf_inf_a_o @ X4 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_543_inf__sup__aci_I3_J,axiom,
! [X4: product_prod_a_a > $o,Y3: product_prod_a_a > $o,Z: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ X4 @ ( inf_in2559554923042384936_a_a_o @ Y3 @ Z ) )
= ( inf_in2559554923042384936_a_a_o @ Y3 @ ( inf_in2559554923042384936_a_a_o @ X4 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_544_inf__sup__aci_I3_J,axiom,
! [X4: set_list_a,Y3: set_list_a,Z: set_list_a] :
( ( inf_inf_set_list_a @ X4 @ ( inf_inf_set_list_a @ Y3 @ Z ) )
= ( inf_inf_set_list_a @ Y3 @ ( inf_inf_set_list_a @ X4 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_545_inf__sup__aci_I3_J,axiom,
! [X4: set_a,Y3: set_a,Z: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z ) )
= ( inf_inf_set_a @ Y3 @ ( inf_inf_set_a @ X4 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_546_inf__sup__aci_I2_J,axiom,
! [X4: set_Product_prod_a_a,Y3: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( inf_in8905007599844390133od_a_a @ X4 @ Y3 ) @ Z )
= ( inf_in8905007599844390133od_a_a @ X4 @ ( inf_in8905007599844390133od_a_a @ Y3 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_547_inf__sup__aci_I2_J,axiom,
! [X4: a > a > $o,Y3: a > a > $o,Z: a > a > $o] :
( ( inf_inf_a_a_o2 @ ( inf_inf_a_a_o2 @ X4 @ Y3 ) @ Z )
= ( inf_inf_a_a_o2 @ X4 @ ( inf_inf_a_a_o2 @ Y3 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_548_inf__sup__aci_I2_J,axiom,
! [X4: a > $o,Y3: a > $o,Z: a > $o] :
( ( inf_inf_a_o @ ( inf_inf_a_o @ X4 @ Y3 ) @ Z )
= ( inf_inf_a_o @ X4 @ ( inf_inf_a_o @ Y3 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_549_inf__sup__aci_I2_J,axiom,
! [X4: product_prod_a_a > $o,Y3: product_prod_a_a > $o,Z: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ ( inf_in2559554923042384936_a_a_o @ X4 @ Y3 ) @ Z )
= ( inf_in2559554923042384936_a_a_o @ X4 @ ( inf_in2559554923042384936_a_a_o @ Y3 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_550_inf__sup__aci_I2_J,axiom,
! [X4: set_list_a,Y3: set_list_a,Z: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X4 @ Y3 ) @ Z )
= ( inf_inf_set_list_a @ X4 @ ( inf_inf_set_list_a @ Y3 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_551_inf__sup__aci_I2_J,axiom,
! [X4: set_a,Y3: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ Z )
= ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_552_inf__sup__aci_I1_J,axiom,
( inf_in8905007599844390133od_a_a
= ( ^ [X3: set_Product_prod_a_a,Y2: set_Product_prod_a_a] : ( inf_in8905007599844390133od_a_a @ Y2 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_553_inf__sup__aci_I1_J,axiom,
( inf_inf_a_a_o2
= ( ^ [X3: a > a > $o,Y2: a > a > $o] : ( inf_inf_a_a_o2 @ Y2 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_554_inf__sup__aci_I1_J,axiom,
( inf_inf_a_o
= ( ^ [X3: a > $o,Y2: a > $o] : ( inf_inf_a_o @ Y2 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_555_inf__sup__aci_I1_J,axiom,
( inf_in2559554923042384936_a_a_o
= ( ^ [X3: product_prod_a_a > $o,Y2: product_prod_a_a > $o] : ( inf_in2559554923042384936_a_a_o @ Y2 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_556_inf__sup__aci_I1_J,axiom,
( inf_inf_set_list_a
= ( ^ [X3: set_list_a,Y2: set_list_a] : ( inf_inf_set_list_a @ Y2 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_557_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X3: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_558_inf_Oassoc,axiom,
! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a,C: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( inf_in8905007599844390133od_a_a @ A3 @ B3 ) @ C )
= ( inf_in8905007599844390133od_a_a @ A3 @ ( inf_in8905007599844390133od_a_a @ B3 @ C ) ) ) ).
% inf.assoc
thf(fact_559_inf_Oassoc,axiom,
! [A3: a > a > $o,B3: a > a > $o,C: a > a > $o] :
( ( inf_inf_a_a_o2 @ ( inf_inf_a_a_o2 @ A3 @ B3 ) @ C )
= ( inf_inf_a_a_o2 @ A3 @ ( inf_inf_a_a_o2 @ B3 @ C ) ) ) ).
% inf.assoc
thf(fact_560_inf_Oassoc,axiom,
! [A3: a > $o,B3: a > $o,C: a > $o] :
( ( inf_inf_a_o @ ( inf_inf_a_o @ A3 @ B3 ) @ C )
= ( inf_inf_a_o @ A3 @ ( inf_inf_a_o @ B3 @ C ) ) ) ).
% inf.assoc
thf(fact_561_inf_Oassoc,axiom,
! [A3: product_prod_a_a > $o,B3: product_prod_a_a > $o,C: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ ( inf_in2559554923042384936_a_a_o @ A3 @ B3 ) @ C )
= ( inf_in2559554923042384936_a_a_o @ A3 @ ( inf_in2559554923042384936_a_a_o @ B3 @ C ) ) ) ).
% inf.assoc
thf(fact_562_inf_Oassoc,axiom,
! [A3: set_list_a,B3: set_list_a,C: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A3 @ B3 ) @ C )
= ( inf_inf_set_list_a @ A3 @ ( inf_inf_set_list_a @ B3 @ C ) ) ) ).
% inf.assoc
thf(fact_563_inf_Oassoc,axiom,
! [A3: set_a,B3: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ C )
= ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C ) ) ) ).
% inf.assoc
thf(fact_564_inf__assoc,axiom,
! [X4: set_Product_prod_a_a,Y3: set_Product_prod_a_a,Z: set_Product_prod_a_a] :
( ( inf_in8905007599844390133od_a_a @ ( inf_in8905007599844390133od_a_a @ X4 @ Y3 ) @ Z )
= ( inf_in8905007599844390133od_a_a @ X4 @ ( inf_in8905007599844390133od_a_a @ Y3 @ Z ) ) ) ).
% inf_assoc
thf(fact_565_inf__assoc,axiom,
! [X4: a > a > $o,Y3: a > a > $o,Z: a > a > $o] :
( ( inf_inf_a_a_o2 @ ( inf_inf_a_a_o2 @ X4 @ Y3 ) @ Z )
= ( inf_inf_a_a_o2 @ X4 @ ( inf_inf_a_a_o2 @ Y3 @ Z ) ) ) ).
% inf_assoc
thf(fact_566_inf__assoc,axiom,
! [X4: a > $o,Y3: a > $o,Z: a > $o] :
( ( inf_inf_a_o @ ( inf_inf_a_o @ X4 @ Y3 ) @ Z )
= ( inf_inf_a_o @ X4 @ ( inf_inf_a_o @ Y3 @ Z ) ) ) ).
% inf_assoc
thf(fact_567_inf__assoc,axiom,
! [X4: product_prod_a_a > $o,Y3: product_prod_a_a > $o,Z: product_prod_a_a > $o] :
( ( inf_in2559554923042384936_a_a_o @ ( inf_in2559554923042384936_a_a_o @ X4 @ Y3 ) @ Z )
= ( inf_in2559554923042384936_a_a_o @ X4 @ ( inf_in2559554923042384936_a_a_o @ Y3 @ Z ) ) ) ).
% inf_assoc
thf(fact_568_inf__assoc,axiom,
! [X4: set_list_a,Y3: set_list_a,Z: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X4 @ Y3 ) @ Z )
= ( inf_inf_set_list_a @ X4 @ ( inf_inf_set_list_a @ Y3 @ Z ) ) ) ).
% inf_assoc
thf(fact_569_inf__assoc,axiom,
! [X4: set_a,Y3: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ Z )
= ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z ) ) ) ).
% inf_assoc
thf(fact_570_inf_Ocommute,axiom,
( inf_in2559554923042384936_a_a_o
= ( ^ [A2: product_prod_a_a > $o,B: product_prod_a_a > $o] : ( inf_in2559554923042384936_a_a_o @ B @ A2 ) ) ) ).
% inf.commute
thf(fact_571_inf_Ocommute,axiom,
( inf_inf_set_list_a
= ( ^ [A2: set_list_a,B: set_list_a] : ( inf_inf_set_list_a @ B @ A2 ) ) ) ).
% inf.commute
thf(fact_572_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A2: set_a,B: set_a] : ( inf_inf_set_a @ B @ A2 ) ) ) ).
% inf.commute
thf(fact_573_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X3: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X3 ) ) ) ).
% inf_commute
thf(fact_574_boolean__algebra__cancel_Oinf1,axiom,
! [A5: set_a,K: set_a,A3: set_a,B3: set_a] :
( ( A5
= ( inf_inf_set_a @ K @ A3 ) )
=> ( ( inf_inf_set_a @ A5 @ B3 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_575_boolean__algebra__cancel_Oinf2,axiom,
! [B5: set_a,K: set_a,B3: set_a,A3: set_a] :
( ( B5
= ( inf_inf_set_a @ K @ B3 ) )
=> ( ( inf_inf_set_a @ A3 @ B5 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_576_inf_Oleft__commute,axiom,
! [B3: set_a,A3: set_a,C: set_a] :
( ( inf_inf_set_a @ B3 @ ( inf_inf_set_a @ A3 @ C ) )
= ( inf_inf_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C ) ) ) ).
% inf.left_commute
thf(fact_577_inf__left__commute,axiom,
! [X4: set_a,Y3: set_a,Z: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z ) )
= ( inf_inf_set_a @ Y3 @ ( inf_inf_set_a @ X4 @ Z ) ) ) ).
% inf_left_commute
thf(fact_578_IntE,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A5 @ B5 ) )
=> ~ ( ( member1426531477525435216od_a_a @ C @ A5 )
=> ~ ( member1426531477525435216od_a_a @ C @ B5 ) ) ) ).
% IntE
thf(fact_579_IntE,axiom,
! [C: a,A5: set_a,B5: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A5 @ B5 ) )
=> ~ ( ( member_a @ C @ A5 )
=> ~ ( member_a @ C @ B5 ) ) ) ).
% IntE
thf(fact_580_IntD1,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A5 @ B5 ) )
=> ( member1426531477525435216od_a_a @ C @ A5 ) ) ).
% IntD1
thf(fact_581_IntD1,axiom,
! [C: a,A5: set_a,B5: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A5 @ B5 ) )
=> ( member_a @ C @ A5 ) ) ).
% IntD1
thf(fact_582_IntD2,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A5 @ B5 ) )
=> ( member1426531477525435216od_a_a @ C @ B5 ) ) ).
% IntD2
thf(fact_583_IntD2,axiom,
! [C: a,A5: set_a,B5: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A5 @ B5 ) )
=> ( member_a @ C @ B5 ) ) ).
% IntD2
thf(fact_584_Int__assoc,axiom,
! [A5: set_a,B5: set_a,C3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ C3 )
= ( inf_inf_set_a @ A5 @ ( inf_inf_set_a @ B5 @ C3 ) ) ) ).
% Int_assoc
thf(fact_585_Int__absorb,axiom,
! [A5: set_a] :
( ( inf_inf_set_a @ A5 @ A5 )
= A5 ) ).
% Int_absorb
thf(fact_586_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B6: set_a] : ( inf_inf_set_a @ B6 @ A6 ) ) ) ).
% Int_commute
thf(fact_587_Int__left__absorb,axiom,
! [A5: set_a,B5: set_a] :
( ( inf_inf_set_a @ A5 @ ( inf_inf_set_a @ A5 @ B5 ) )
= ( inf_inf_set_a @ A5 @ B5 ) ) ).
% Int_left_absorb
thf(fact_588_Int__left__commute,axiom,
! [A5: set_a,B5: set_a,C3: set_a] :
( ( inf_inf_set_a @ A5 @ ( inf_inf_set_a @ B5 @ C3 ) )
= ( inf_inf_set_a @ B5 @ ( inf_inf_set_a @ A5 @ C3 ) ) ) ).
% Int_left_commute
thf(fact_589_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X3: a] : $false ) ) ).
% empty_def
thf(fact_590_Int__def,axiom,
( inf_in8905007599844390133od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A6 )
& ( member1426531477525435216od_a_a @ X3 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_591_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A6 )
& ( member_a @ X3 @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_592_Int__Collect,axiom,
! [X4: product_prod_a_a,A5: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ X4 @ ( inf_in8905007599844390133od_a_a @ A5 @ ( collec3336397797384452498od_a_a @ P ) ) )
= ( ( member1426531477525435216od_a_a @ X4 @ A5 )
& ( P @ X4 ) ) ) ).
% Int_Collect
thf(fact_593_Int__Collect,axiom,
! [X4: a,A5: set_a,P: a > $o] :
( ( member_a @ X4 @ ( inf_inf_set_a @ A5 @ ( collect_a @ P ) ) )
= ( ( member_a @ X4 @ A5 )
& ( P @ X4 ) ) ) ).
% Int_Collect
thf(fact_594_Collect__conj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
& ( Q @ X3 ) ) )
= ( inf_inf_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_595_Int__emptyI,axiom,
! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A5 )
=> ~ ( member1426531477525435216od_a_a @ X @ B5 ) )
=> ( ( inf_in8905007599844390133od_a_a @ A5 @ B5 )
= bot_bo3357376287454694259od_a_a ) ) ).
% Int_emptyI
thf(fact_596_Int__emptyI,axiom,
! [A5: set_a,B5: set_a] :
( ! [X: a] :
( ( member_a @ X @ A5 )
=> ~ ( member_a @ X @ B5 ) )
=> ( ( inf_inf_set_a @ A5 @ B5 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_597_disjoint__iff,axiom,
! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A5 @ B5 )
= bot_bo3357376287454694259od_a_a )
= ( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A5 )
=> ~ ( member1426531477525435216od_a_a @ X3 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_598_disjoint__iff,axiom,
! [A5: set_a,B5: set_a] :
( ( ( inf_inf_set_a @ A5 @ B5 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ~ ( member_a @ X3 @ B5 ) ) ) ) ).
% disjoint_iff
thf(fact_599_Int__empty__left,axiom,
! [B5: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B5 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_600_Int__empty__right,axiom,
! [A5: set_a] :
( ( inf_inf_set_a @ A5 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_601_disjoint__iff__not__equal,axiom,
! [A5: set_a,B5: set_a] :
( ( ( inf_inf_set_a @ A5 @ B5 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ B5 )
=> ( X3 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_602_combine__options__cases,axiom,
! [X4: option5210160422955383789od_a_a,P: option5210160422955383789od_a_a > option5210160422955383789od_a_a > $o,Y3: option5210160422955383789od_a_a] :
( ( ( X4 = none_P1914705783780380040od_a_a )
=> ( P @ X4 @ Y3 ) )
=> ( ( ( Y3 = none_P1914705783780380040od_a_a )
=> ( P @ X4 @ Y3 ) )
=> ( ! [A: product_prod_a_a,B2: product_prod_a_a] :
( ( X4
= ( some_P3592067295195376908od_a_a @ A ) )
=> ( ( Y3
= ( some_P3592067295195376908od_a_a @ B2 ) )
=> ( P @ X4 @ Y3 ) ) )
=> ( P @ X4 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_603_combine__options__cases,axiom,
! [X4: option5210160422955383789od_a_a,P: option5210160422955383789od_a_a > option_set_a > $o,Y3: option_set_a] :
( ( ( X4 = none_P1914705783780380040od_a_a )
=> ( P @ X4 @ Y3 ) )
=> ( ( ( Y3 = none_set_a )
=> ( P @ X4 @ Y3 ) )
=> ( ! [A: product_prod_a_a,B2: set_a] :
( ( X4
= ( some_P3592067295195376908od_a_a @ A ) )
=> ( ( Y3
= ( some_set_a @ B2 ) )
=> ( P @ X4 @ Y3 ) ) )
=> ( P @ X4 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_604_combine__options__cases,axiom,
! [X4: option_set_a,P: option_set_a > option5210160422955383789od_a_a > $o,Y3: option5210160422955383789od_a_a] :
( ( ( X4 = none_set_a )
=> ( P @ X4 @ Y3 ) )
=> ( ( ( Y3 = none_P1914705783780380040od_a_a )
=> ( P @ X4 @ Y3 ) )
=> ( ! [A: set_a,B2: product_prod_a_a] :
( ( X4
= ( some_set_a @ A ) )
=> ( ( Y3
= ( some_P3592067295195376908od_a_a @ B2 ) )
=> ( P @ X4 @ Y3 ) ) )
=> ( P @ X4 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_605_combine__options__cases,axiom,
! [X4: option_set_a,P: option_set_a > option_set_a > $o,Y3: option_set_a] :
( ( ( X4 = none_set_a )
=> ( P @ X4 @ Y3 ) )
=> ( ( ( Y3 = none_set_a )
=> ( P @ X4 @ Y3 ) )
=> ( ! [A: set_a,B2: set_a] :
( ( X4
= ( some_set_a @ A ) )
=> ( ( Y3
= ( some_set_a @ B2 ) )
=> ( P @ X4 @ Y3 ) ) )
=> ( P @ X4 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_606_split__option__all,axiom,
( ( ^ [P4: option5210160422955383789od_a_a > $o] :
! [X8: option5210160422955383789od_a_a] : ( P4 @ X8 ) )
= ( ^ [P3: option5210160422955383789od_a_a > $o] :
( ( P3 @ none_P1914705783780380040od_a_a )
& ! [X3: product_prod_a_a] : ( P3 @ ( some_P3592067295195376908od_a_a @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_607_split__option__all,axiom,
( ( ^ [P4: option_set_a > $o] :
! [X8: option_set_a] : ( P4 @ X8 ) )
= ( ^ [P3: option_set_a > $o] :
( ( P3 @ none_set_a )
& ! [X3: set_a] : ( P3 @ ( some_set_a @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_608_split__option__ex,axiom,
( ( ^ [P4: option5210160422955383789od_a_a > $o] :
? [X8: option5210160422955383789od_a_a] : ( P4 @ X8 ) )
= ( ^ [P3: option5210160422955383789od_a_a > $o] :
( ( P3 @ none_P1914705783780380040od_a_a )
| ? [X3: product_prod_a_a] : ( P3 @ ( some_P3592067295195376908od_a_a @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_609_split__option__ex,axiom,
( ( ^ [P4: option_set_a > $o] :
? [X8: option_set_a] : ( P4 @ X8 ) )
= ( ^ [P3: option_set_a > $o] :
( ( P3 @ none_set_a )
| ? [X3: set_a] : ( P3 @ ( some_set_a @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_610_option_Oexhaust,axiom,
! [Y3: option5210160422955383789od_a_a] :
( ( Y3 != none_P1914705783780380040od_a_a )
=> ~ ! [X23: product_prod_a_a] :
( Y3
!= ( some_P3592067295195376908od_a_a @ X23 ) ) ) ).
% option.exhaust
thf(fact_611_option_Oexhaust,axiom,
! [Y3: option_set_a] :
( ( Y3 != none_set_a )
=> ~ ! [X23: set_a] :
( Y3
!= ( some_set_a @ X23 ) ) ) ).
% option.exhaust
thf(fact_612_option_OdiscI,axiom,
! [Option: option5210160422955383789od_a_a,X22: product_prod_a_a] :
( ( Option
= ( some_P3592067295195376908od_a_a @ X22 ) )
=> ( Option != none_P1914705783780380040od_a_a ) ) ).
% option.discI
thf(fact_613_option_OdiscI,axiom,
! [Option: option_set_a,X22: set_a] :
( ( Option
= ( some_set_a @ X22 ) )
=> ( Option != none_set_a ) ) ).
% option.discI
thf(fact_614_option_Odistinct_I1_J,axiom,
! [X22: product_prod_a_a] :
( none_P1914705783780380040od_a_a
!= ( some_P3592067295195376908od_a_a @ X22 ) ) ).
% option.distinct(1)
thf(fact_615_option_Odistinct_I1_J,axiom,
! [X22: set_a] :
( none_set_a
!= ( some_set_a @ X22 ) ) ).
% option.distinct(1)
thf(fact_616_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A6: set_a] : ( A6 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_617_set__product,axiom,
! [Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a] :
( ( set_Pr6493056779096584572od_a_a @ ( produc9150955665926335749od_a_a @ Xs @ Ys ) )
= ( produc5899993699339346696od_a_a @ ( set_Product_prod_a_a2 @ Xs )
@ ^ [Uu: product_prod_a_a] : ( set_Product_prod_a_a2 @ Ys ) ) ) ).
% set_product
thf(fact_618_set__product,axiom,
! [Xs: list_a,Ys: list_a] :
( ( set_Product_prod_a_a2 @ ( product_a_a @ Xs @ Ys ) )
= ( product_Sigma_a_a @ ( set_a2 @ Xs )
@ ^ [Uu: a] : ( set_a2 @ Ys ) ) ) ).
% set_product
thf(fact_619_inv__image__def,axiom,
( inv_image_a_a
= ( ^ [R: set_Product_prod_a_a,F2: a > a] :
( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X3: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) @ R ) ) ) ) ) ).
% inv_image_def
thf(fact_620_symclp__symcl__eq,axiom,
! [R3: set_Product_prod_a_a] :
( ( transitive_symclp_a
@ ^ [X3: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R3 ) )
= ( ^ [X3: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ ( symcl_a @ R3 ) ) ) ) ).
% symclp_symcl_eq
thf(fact_621_bind__simps_I1_J,axiom,
! [F: product_prod_a_a > list_P1396940483166286381od_a_a] :
( ( bind_P8339977525905061191od_a_a @ nil_Product_prod_a_a @ F )
= nil_Product_prod_a_a ) ).
% bind_simps(1)
thf(fact_622_same__fst__def,axiom,
( same_fst_a_a
= ( ^ [P3: a > $o,R4: a > set_Product_prod_a_a] :
( collec10116633892588882od_a_a
@ ( produc4261069933152462124_a_a_o
@ ( produc5089340801457792962_a_a_o
@ ^ [X9: a,Y5: a] :
( produc6436628058953941356_a_a_o
@ ^ [X3: a,Y2: a] :
( ( X9 = X3 )
& ( P3 @ X3 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ Y2 ) @ ( R4 @ X3 ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_623_The__split__eq,axiom,
! [X4: a,Y3: a] :
( ( the_Product_prod_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X9: a,Y5: a] :
( ( X4 = X9 )
& ( Y3 = Y5 ) ) ) )
= ( product_Pair_a_a @ X4 @ Y3 ) ) ).
% The_split_eq
thf(fact_624_curry__case__prod,axiom,
! [F: a > a > $o] :
( ( product_curry_a_a_o @ ( produc6436628058953941356_a_a_o @ F ) )
= F ) ).
% curry_case_prod
thf(fact_625_curryI,axiom,
! [F: product_prod_a_a > $o,A3: a,B3: a] :
( ( F @ ( product_Pair_a_a @ A3 @ B3 ) )
=> ( product_curry_a_a_o @ F @ A3 @ B3 ) ) ).
% curryI
thf(fact_626_mem__Sigma__iff,axiom,
! [A3: a,B3: product_prod_a_a,A5: set_a,B5: a > set_Product_prod_a_a] :
( ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ A3 @ B3 ) @ ( produc6342321021181284593od_a_a @ A5 @ B5 ) )
= ( ( member_a @ A3 @ A5 )
& ( member1426531477525435216od_a_a @ B3 @ ( B5 @ A3 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_627_mem__Sigma__iff,axiom,
! [A3: product_prod_a_a,B3: a,A5: set_Product_prod_a_a,B5: product_prod_a_a > set_a] :
( ( member2829916517802586983_a_a_a @ ( produc5692536848587512110_a_a_a @ A3 @ B3 ) @ ( produc2379640491490746847_a_a_a @ A5 @ B5 ) )
= ( ( member1426531477525435216od_a_a @ A3 @ A5 )
& ( member_a @ B3 @ ( B5 @ A3 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_628_mem__Sigma__iff,axiom,
! [A3: product_prod_a_a,B3: product_prod_a_a,A5: set_Product_prod_a_a,B5: product_prod_a_a > set_Product_prod_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A3 @ B3 ) @ ( produc5899993699339346696od_a_a @ A5 @ B5 ) )
= ( ( member1426531477525435216od_a_a @ A3 @ A5 )
& ( member1426531477525435216od_a_a @ B3 @ ( B5 @ A3 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_629_mem__Sigma__iff,axiom,
! [A3: a,B3: a,A5: set_a,B5: a > set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ ( product_Sigma_a_a @ A5 @ B5 ) )
= ( ( member_a @ A3 @ A5 )
& ( member_a @ B3 @ ( B5 @ A3 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_630_SigmaI,axiom,
! [A3: a,A5: set_a,B3: product_prod_a_a,B5: a > set_Product_prod_a_a] :
( ( member_a @ A3 @ A5 )
=> ( ( member1426531477525435216od_a_a @ B3 @ ( B5 @ A3 ) )
=> ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ A3 @ B3 ) @ ( produc6342321021181284593od_a_a @ A5 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_631_SigmaI,axiom,
! [A3: product_prod_a_a,A5: set_Product_prod_a_a,B3: a,B5: product_prod_a_a > set_a] :
( ( member1426531477525435216od_a_a @ A3 @ A5 )
=> ( ( member_a @ B3 @ ( B5 @ A3 ) )
=> ( member2829916517802586983_a_a_a @ ( produc5692536848587512110_a_a_a @ A3 @ B3 ) @ ( produc2379640491490746847_a_a_a @ A5 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_632_SigmaI,axiom,
! [A3: product_prod_a_a,A5: set_Product_prod_a_a,B3: product_prod_a_a,B5: product_prod_a_a > set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A3 @ A5 )
=> ( ( member1426531477525435216od_a_a @ B3 @ ( B5 @ A3 ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A3 @ B3 ) @ ( produc5899993699339346696od_a_a @ A5 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_633_SigmaI,axiom,
! [A3: a,A5: set_a,B3: a,B5: a > set_a] :
( ( member_a @ A3 @ A5 )
=> ( ( member_a @ B3 @ ( B5 @ A3 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ ( product_Sigma_a_a @ A5 @ B5 ) ) ) ) ).
% SigmaI
thf(fact_634_in__inv__image,axiom,
! [X4: a,Y3: a,R3: set_Product_prod_a_a,F: a > a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ ( inv_image_a_a @ R3 @ F ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F @ X4 ) @ ( F @ Y3 ) ) @ R3 ) ) ).
% in_inv_image
thf(fact_635_case__prod__curry,axiom,
! [F: product_prod_a_a > $o] :
( ( produc6436628058953941356_a_a_o @ ( product_curry_a_a_o @ F ) )
= F ) ).
% case_prod_curry
thf(fact_636_Collect__case__prod,axiom,
! [P: a > $o,Q: a > $o] :
( ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [A2: a,B: a] :
( ( P @ A2 )
& ( Q @ B ) ) ) )
= ( product_Sigma_a_a @ ( collect_a @ P )
@ ^ [Uu: a] : ( collect_a @ Q ) ) ) ).
% Collect_case_prod
thf(fact_637_same__fstI,axiom,
! [P: a > $o,X4: a,Y6: a,Y3: a,R2: a > set_Product_prod_a_a] :
( ( P @ X4 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y6 @ Y3 ) @ ( R2 @ X4 ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ ( product_Pair_a_a @ X4 @ Y6 ) @ ( product_Pair_a_a @ X4 @ Y3 ) ) @ ( same_fst_a_a @ P @ R2 ) ) ) ) ).
% same_fstI
thf(fact_638_Times__empty,axiom,
! [A5: set_a,B5: set_a] :
( ( ( product_Sigma_a_a @ A5
@ ^ [Uu: a] : B5 )
= bot_bo3357376287454694259od_a_a )
= ( ( A5 = bot_bot_set_a )
| ( B5 = bot_bot_set_a ) ) ) ).
% Times_empty
thf(fact_639_SigmaE2,axiom,
! [A3: a,B3: product_prod_a_a,A5: set_a,B5: a > set_Product_prod_a_a] :
( ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ A3 @ B3 ) @ ( produc6342321021181284593od_a_a @ A5 @ B5 ) )
=> ~ ( ( member_a @ A3 @ A5 )
=> ~ ( member1426531477525435216od_a_a @ B3 @ ( B5 @ A3 ) ) ) ) ).
% SigmaE2
thf(fact_640_SigmaE2,axiom,
! [A3: product_prod_a_a,B3: a,A5: set_Product_prod_a_a,B5: product_prod_a_a > set_a] :
( ( member2829916517802586983_a_a_a @ ( produc5692536848587512110_a_a_a @ A3 @ B3 ) @ ( produc2379640491490746847_a_a_a @ A5 @ B5 ) )
=> ~ ( ( member1426531477525435216od_a_a @ A3 @ A5 )
=> ~ ( member_a @ B3 @ ( B5 @ A3 ) ) ) ) ).
% SigmaE2
thf(fact_641_SigmaE2,axiom,
! [A3: product_prod_a_a,B3: product_prod_a_a,A5: set_Product_prod_a_a,B5: product_prod_a_a > set_Product_prod_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A3 @ B3 ) @ ( produc5899993699339346696od_a_a @ A5 @ B5 ) )
=> ~ ( ( member1426531477525435216od_a_a @ A3 @ A5 )
=> ~ ( member1426531477525435216od_a_a @ B3 @ ( B5 @ A3 ) ) ) ) ).
% SigmaE2
thf(fact_642_SigmaE2,axiom,
! [A3: a,B3: a,A5: set_a,B5: a > set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ ( product_Sigma_a_a @ A5 @ B5 ) )
=> ~ ( ( member_a @ A3 @ A5 )
=> ~ ( member_a @ B3 @ ( B5 @ A3 ) ) ) ) ).
% SigmaE2
thf(fact_643_SigmaD2,axiom,
! [A3: a,B3: a,A5: set_a,B5: a > set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ ( product_Sigma_a_a @ A5 @ B5 ) )
=> ( member_a @ B3 @ ( B5 @ A3 ) ) ) ).
% SigmaD2
thf(fact_644_SigmaD1,axiom,
! [A3: a,B3: a,A5: set_a,B5: a > set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ ( product_Sigma_a_a @ A5 @ B5 ) )
=> ( member_a @ A3 @ A5 ) ) ).
% SigmaD1
thf(fact_645_SigmaE,axiom,
! [C: produc4044097585999906000od_a_a,A5: set_a,B5: a > set_Product_prod_a_a] :
( ( member3071122053849602553od_a_a @ C @ ( produc6342321021181284593od_a_a @ A5 @ B5 ) )
=> ~ ! [X: a] :
( ( member_a @ X @ A5 )
=> ! [Y: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y @ ( B5 @ X ) )
=> ( C
!= ( produc431845341423274048od_a_a @ X @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_646_SigmaE,axiom,
! [C: produc3802892049952890430_a_a_a,A5: set_Product_prod_a_a,B5: product_prod_a_a > set_a] :
( ( member2829916517802586983_a_a_a @ C @ ( produc2379640491490746847_a_a_a @ A5 @ B5 ) )
=> ~ ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A5 )
=> ! [Y: a] :
( ( member_a @ Y @ ( B5 @ X ) )
=> ( C
!= ( produc5692536848587512110_a_a_a @ X @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_647_SigmaE,axiom,
! [C: produc3498347346309940967od_a_a,A5: set_Product_prod_a_a,B5: product_prod_a_a > set_Product_prod_a_a] :
( ( member6330455413206600464od_a_a @ C @ ( produc5899993699339346696od_a_a @ A5 @ B5 ) )
=> ~ ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A5 )
=> ! [Y: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y @ ( B5 @ X ) )
=> ( C
!= ( produc7886510207707329367od_a_a @ X @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_648_SigmaE,axiom,
! [C: product_prod_a_a,A5: set_a,B5: a > set_a] :
( ( member1426531477525435216od_a_a @ C @ ( product_Sigma_a_a @ A5 @ B5 ) )
=> ~ ! [X: a] :
( ( member_a @ X @ A5 )
=> ! [Y: a] :
( ( member_a @ Y @ ( B5 @ X ) )
=> ( C
!= ( product_Pair_a_a @ X @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_649_times__eq__iff,axiom,
! [A5: set_a,B5: set_a,C3: set_a,D: set_a] :
( ( ( product_Sigma_a_a @ A5
@ ^ [Uu: a] : B5 )
= ( product_Sigma_a_a @ C3
@ ^ [Uu: a] : D ) )
= ( ( ( A5 = C3 )
& ( B5 = D ) )
| ( ( ( A5 = bot_bot_set_a )
| ( B5 = bot_bot_set_a ) )
& ( ( C3 = bot_bot_set_a )
| ( D = bot_bot_set_a ) ) ) ) ) ).
% times_eq_iff
thf(fact_650_Times__Int__Times,axiom,
! [A5: set_a,B5: set_a,C3: set_a,D: set_a] :
( ( inf_in8905007599844390133od_a_a
@ ( product_Sigma_a_a @ A5
@ ^ [Uu: a] : B5 )
@ ( product_Sigma_a_a @ C3
@ ^ [Uu: a] : D ) )
= ( product_Sigma_a_a @ ( inf_inf_set_a @ A5 @ C3 )
@ ^ [Uu: a] : ( inf_inf_set_a @ B5 @ D ) ) ) ).
% Times_Int_Times
thf(fact_651_Collect__case__prod__Sigma,axiom,
! [P: a > $o,Q: a > a > $o] :
( ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X3: a,Y2: a] :
( ( P @ X3 )
& ( Q @ X3 @ Y2 ) ) ) )
= ( product_Sigma_a_a @ ( collect_a @ P )
@ ^ [X3: a] : ( collect_a @ ( Q @ X3 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_652_curryE,axiom,
! [F: product_prod_a_a > $o,A3: a,B3: a] :
( ( product_curry_a_a_o @ F @ A3 @ B3 )
=> ( F @ ( product_Pair_a_a @ A3 @ B3 ) ) ) ).
% curryE
thf(fact_653_curryD,axiom,
! [F: product_prod_a_a > $o,A3: a,B3: a] :
( ( product_curry_a_a_o @ F @ A3 @ B3 )
=> ( F @ ( product_Pair_a_a @ A3 @ B3 ) ) ) ).
% curryD
thf(fact_654_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_655_is__empty__set,axiom,
! [Xs: list_P1396940483166286381od_a_a] :
( ( is_emp2937470224744679417od_a_a @ ( set_Product_prod_a_a2 @ Xs ) )
= ( null_P9109340212867140917od_a_a @ Xs ) ) ).
% is_empty_set
thf(fact_656_null__rec_I2_J,axiom,
null_P9109340212867140917od_a_a @ nil_Product_prod_a_a ).
% null_rec(2)
thf(fact_657_eq__Nil__null,axiom,
! [Xs: list_P1396940483166286381od_a_a] :
( ( Xs = nil_Product_prod_a_a )
= ( null_P9109340212867140917od_a_a @ Xs ) ) ).
% eq_Nil_null
thf(fact_658_lex__prod__def,axiom,
( lex_prod_a_a
= ( ^ [Ra: set_Product_prod_a_a,Rb: set_Product_prod_a_a] :
( collec10116633892588882od_a_a
@ ( produc4261069933152462124_a_a_o
@ ( produc5089340801457792962_a_a_o
@ ^ [A2: a,B: a] :
( produc6436628058953941356_a_a_o
@ ^ [A7: a,B7: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ A7 ) @ Ra )
| ( ( A2 = A7 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ B7 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_659_member__product,axiom,
! [X4: product_prod_a_a,A5: set_a,B5: set_a] :
( ( member1426531477525435216od_a_a @ X4 @ ( product_product_a_a @ A5 @ B5 ) )
= ( member1426531477525435216od_a_a @ X4
@ ( product_Sigma_a_a @ A5
@ ^ [Uu: a] : B5 ) ) ) ).
% member_product
thf(fact_660_class__def,axiom,
( class_6315440051758036822od_a_a
= ( ^ [X3: product_prod_a_a,Qeq2: set_Pr8600417178894128327od_a_a] :
( if_opt7251041032425086355od_a_a
@ ? [Y2: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ Y2 @ ( classe2514596149140685704od_a_a @ Qeq2 ) )
& ( member1426531477525435216od_a_a @ X3 @ Y2 ) )
@ ( some_s3837607710808129772od_a_a
@ ( the_se8866575142567302670od_a_a
@ ^ [X10: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X10 @ ( classe2514596149140685704od_a_a @ Qeq2 ) )
& ( member1426531477525435216od_a_a @ X3 @ X10 ) ) ) )
@ none_s5714488769912451432od_a_a ) ) ) ).
% class_def
thf(fact_661_class__def,axiom,
( class_a
= ( ^ [X3: a,Qeq2: set_Product_prod_a_a] :
( if_option_set_a
@ ? [Y2: set_a] :
( ( member_set_a @ Y2 @ ( classes_a @ Qeq2 ) )
& ( member_a @ X3 @ Y2 ) )
@ ( some_set_a
@ ( the_set_a
@ ^ [X10: set_a] :
( ( member_set_a @ X10 @ ( classes_a @ Qeq2 ) )
& ( member_a @ X3 @ X10 ) ) ) )
@ none_set_a ) ) ) ).
% class_def
thf(fact_662_bex__empty,axiom,
! [P: a > $o] :
~ ? [X5: a] :
( ( member_a @ X5 @ bot_bot_set_a )
& ( P @ X5 ) ) ).
% bex_empty
thf(fact_663_in__lex__prod,axiom,
! [A3: a,B3: a,A4: a,B4: a,R3: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ ( product_Pair_a_a @ A4 @ B4 ) ) @ ( lex_prod_a_a @ R3 @ S2 ) )
= ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ A4 ) @ R3 )
| ( ( A3 = A4 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ B4 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_664_Bex__def,axiom,
( bex_a
= ( ^ [A6: set_a,P3: a > $o] :
? [X3: a] :
( ( member_a @ X3 @ A6 )
& ( P3 @ X3 ) ) ) ) ).
% Bex_def
thf(fact_665_Bex__def,axiom,
( bex_Product_prod_a_a
= ( ^ [A6: set_Product_prod_a_a,P3: product_prod_a_a > $o] :
? [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A6 )
& ( P3 @ X3 ) ) ) ) ).
% Bex_def
thf(fact_666_Id__on__empty,axiom,
( ( id_on_a @ bot_bot_set_a )
= bot_bo3357376287454694259od_a_a ) ).
% Id_on_empty
thf(fact_667_class__None__eq,axiom,
! [X4: product_prod_a_a,Qeq: set_Pr8600417178894128327od_a_a] :
( ( ( class_6315440051758036822od_a_a @ X4 @ Qeq )
= none_s5714488769912451432od_a_a )
= ( ~ ( member1426531477525435216od_a_a @ X4 @ ( field_1126092520709947252od_a_a @ Qeq ) ) ) ) ).
% class_None_eq
thf(fact_668_class__None__eq,axiom,
! [X4: a,Qeq: set_Product_prod_a_a] :
( ( ( class_a @ X4 @ Qeq )
= none_set_a )
= ( ~ ( member_a @ X4 @ ( field_a @ Qeq ) ) ) ) ).
% class_None_eq
thf(fact_669_relation__of__def,axiom,
( order_relation_of_a
= ( ^ [P3: a > a > $o,A6: set_a] :
( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [A2: a,B: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B )
@ ( product_Sigma_a_a @ A6
@ ^ [Uu: a] : A6 ) )
& ( P3 @ A2 @ B ) ) ) ) ) ) ).
% relation_of_def
thf(fact_670_Collect__const__case__prod,axiom,
! [P: $o] :
( ( P
=> ( ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [A2: a,B: a] : P ) )
= top_to8063371432257647191od_a_a ) )
& ( ~ P
=> ( ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [A2: a,B: a] : P ) )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% Collect_const_case_prod
thf(fact_671_UNIV__I,axiom,
! [X4: a] : ( member_a @ X4 @ top_top_set_a ) ).
% UNIV_I
thf(fact_672_UNIV__I,axiom,
! [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ top_to8063371432257647191od_a_a ) ).
% UNIV_I
thf(fact_673_inf__top__left,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ X4 )
= X4 ) ).
% inf_top_left
thf(fact_674_inf__top__right,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ top_top_set_a )
= X4 ) ).
% inf_top_right
thf(fact_675_inf__eq__top__iff,axiom,
! [X4: set_a,Y3: set_a] :
( ( ( inf_inf_set_a @ X4 @ Y3 )
= top_top_set_a )
= ( ( X4 = top_top_set_a )
& ( Y3 = top_top_set_a ) ) ) ).
% inf_eq_top_iff
thf(fact_676_top__eq__inf__iff,axiom,
! [X4: set_a,Y3: set_a] :
( ( top_top_set_a
= ( inf_inf_set_a @ X4 @ Y3 ) )
= ( ( X4 = top_top_set_a )
& ( Y3 = top_top_set_a ) ) ) ).
% top_eq_inf_iff
thf(fact_677_inf__top_Oeq__neutr__iff,axiom,
! [A3: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A3 @ B3 )
= top_top_set_a )
= ( ( A3 = top_top_set_a )
& ( B3 = top_top_set_a ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_678_inf__top_Oleft__neutral,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ A3 )
= A3 ) ).
% inf_top.left_neutral
thf(fact_679_inf__top_Oneutr__eq__iff,axiom,
! [A3: set_a,B3: set_a] :
( ( top_top_set_a
= ( inf_inf_set_a @ A3 @ B3 ) )
= ( ( A3 = top_top_set_a )
& ( B3 = top_top_set_a ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_680_inf__top_Oright__neutral,axiom,
! [A3: set_a] :
( ( inf_inf_set_a @ A3 @ top_top_set_a )
= A3 ) ).
% inf_top.right_neutral
thf(fact_681_List_Ofinite__set,axiom,
! [Xs: list_P1396940483166286381od_a_a] : ( finite6544458595007987280od_a_a @ ( set_Product_prod_a_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_682_Int__UNIV,axiom,
! [A5: set_a,B5: set_a] :
( ( ( inf_inf_set_a @ A5 @ B5 )
= top_top_set_a )
= ( ( A5 = top_top_set_a )
& ( B5 = top_top_set_a ) ) ) ).
% Int_UNIV
thf(fact_683_Id__onI,axiom,
! [A3: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A3 @ A5 )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A3 @ A3 ) @ ( id_on_7193410965977890500od_a_a @ A5 ) ) ) ).
% Id_onI
thf(fact_684_Id__onI,axiom,
! [A3: a,A5: set_a] :
( ( member_a @ A3 @ A5 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ A3 ) @ ( id_on_a @ A5 ) ) ) ).
% Id_onI
thf(fact_685_Field__transymcl,axiom,
! [R3: set_Product_prod_a_a] :
( ( field_a @ ( transymcl_a @ R3 ) )
= ( field_a @ R3 ) ) ).
% Field_transymcl
thf(fact_686_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_a
@ ^ [S3: a] : P )
= top_top_set_a ) )
& ( ~ P
=> ( ( collect_a
@ ^ [S3: a] : P )
= bot_bot_set_a ) ) ) ).
% Collect_const
thf(fact_687_Field__symcl,axiom,
! [R3: set_Product_prod_a_a] :
( ( field_a @ ( symcl_a @ R3 ) )
= ( field_a @ R3 ) ) ).
% Field_symcl
thf(fact_688_Field__empty,axiom,
( ( field_a @ bot_bo3357376287454694259od_a_a )
= bot_bot_set_a ) ).
% Field_empty
thf(fact_689_UNIV__witness,axiom,
? [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_witness
thf(fact_690_UNIV__witness,axiom,
? [X: product_prod_a_a] : ( member1426531477525435216od_a_a @ X @ top_to8063371432257647191od_a_a ) ).
% UNIV_witness
thf(fact_691_UNIV__eq__I,axiom,
! [A5: set_a] :
( ! [X: a] : ( member_a @ X @ A5 )
=> ( top_top_set_a = A5 ) ) ).
% UNIV_eq_I
thf(fact_692_UNIV__eq__I,axiom,
! [A5: set_Product_prod_a_a] :
( ! [X: product_prod_a_a] : ( member1426531477525435216od_a_a @ X @ A5 )
=> ( top_to8063371432257647191od_a_a = A5 ) ) ).
% UNIV_eq_I
thf(fact_693_boolean__algebra_Oconj__one__right,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ top_top_set_a )
= X4 ) ).
% boolean_algebra.conj_one_right
thf(fact_694_FieldI1,axiom,
! [I: product_prod_a_a,J: product_prod_a_a,R2: set_Pr8600417178894128327od_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ I @ J ) @ R2 )
=> ( member1426531477525435216od_a_a @ I @ ( field_1126092520709947252od_a_a @ R2 ) ) ) ).
% FieldI1
thf(fact_695_FieldI1,axiom,
! [I: a,J: a,R2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R2 )
=> ( member_a @ I @ ( field_a @ R2 ) ) ) ).
% FieldI1
thf(fact_696_FieldI2,axiom,
! [I: product_prod_a_a,J: product_prod_a_a,R2: set_Pr8600417178894128327od_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ I @ J ) @ R2 )
=> ( member1426531477525435216od_a_a @ J @ ( field_1126092520709947252od_a_a @ R2 ) ) ) ).
% FieldI2
thf(fact_697_FieldI2,axiom,
! [I: a,J: a,R2: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R2 )
=> ( member_a @ J @ ( field_a @ R2 ) ) ) ).
% FieldI2
thf(fact_698_finite__list,axiom,
! [A5: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A5 )
=> ? [Xs3: list_P1396940483166286381od_a_a] :
( ( set_Product_prod_a_a2 @ Xs3 )
= A5 ) ) ).
% finite_list
thf(fact_699_empty__not__UNIV,axiom,
bot_bot_set_a != top_top_set_a ).
% empty_not_UNIV
thf(fact_700_Int__UNIV__left,axiom,
! [B5: set_a] :
( ( inf_inf_set_a @ top_top_set_a @ B5 )
= B5 ) ).
% Int_UNIV_left
thf(fact_701_Int__UNIV__right,axiom,
! [A5: set_a] :
( ( inf_inf_set_a @ A5 @ top_top_set_a )
= A5 ) ).
% Int_UNIV_right
thf(fact_702_Id__onE,axiom,
! [C: produc3498347346309940967od_a_a,A5: set_Product_prod_a_a] :
( ( member6330455413206600464od_a_a @ C @ ( id_on_7193410965977890500od_a_a @ A5 ) )
=> ~ ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A5 )
=> ( C
!= ( produc7886510207707329367od_a_a @ X @ X ) ) ) ) ).
% Id_onE
thf(fact_703_Id__onE,axiom,
! [C: product_prod_a_a,A5: set_a] :
( ( member1426531477525435216od_a_a @ C @ ( id_on_a @ A5 ) )
=> ~ ! [X: a] :
( ( member_a @ X @ A5 )
=> ( C
!= ( product_Pair_a_a @ X @ X ) ) ) ) ).
% Id_onE
thf(fact_704_Id__on__eqI,axiom,
! [A3: product_prod_a_a,B3: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( A3 = B3 )
=> ( ( member1426531477525435216od_a_a @ A3 @ A5 )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A3 @ B3 ) @ ( id_on_7193410965977890500od_a_a @ A5 ) ) ) ) ).
% Id_on_eqI
thf(fact_705_Id__on__eqI,axiom,
! [A3: a,B3: a,A5: set_a] :
( ( A3 = B3 )
=> ( ( member_a @ A3 @ A5 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ ( id_on_a @ A5 ) ) ) ) ).
% Id_on_eqI
thf(fact_706_Id__on__iff,axiom,
! [X4: product_prod_a_a,Y3: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X4 @ Y3 ) @ ( id_on_7193410965977890500od_a_a @ A5 ) )
= ( ( X4 = Y3 )
& ( member1426531477525435216od_a_a @ X4 @ A5 ) ) ) ).
% Id_on_iff
thf(fact_707_Id__on__iff,axiom,
! [X4: a,Y3: a,A5: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ ( id_on_a @ A5 ) )
= ( ( X4 = Y3 )
& ( member_a @ X4 @ A5 ) ) ) ).
% Id_on_iff
thf(fact_708_Id__on__def_H,axiom,
! [A5: a > $o] :
( ( id_on_a @ ( collect_a @ A5 ) )
= ( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X3: a,Y2: a] :
( ( X3 = Y2 )
& ( A5 @ X3 ) ) ) ) ) ).
% Id_on_def'
thf(fact_709_Field__transymcl__self,axiom,
! [A3: product_prod_a_a,Qeq: set_Pr8600417178894128327od_a_a] :
( ( member1426531477525435216od_a_a @ A3 @ ( field_1126092520709947252od_a_a @ Qeq ) )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A3 @ A3 ) @ ( transy2380204104860363337od_a_a @ Qeq ) ) ) ).
% Field_transymcl_self
thf(fact_710_Field__transymcl__self,axiom,
! [A3: a,Qeq: set_Product_prod_a_a] :
( ( member_a @ A3 @ ( field_a @ Qeq ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ A3 ) @ ( transymcl_a @ Qeq ) ) ) ).
% Field_transymcl_self
thf(fact_711_ex__class,axiom,
! [X4: product_prod_a_a,Qeq: set_Pr8600417178894128327od_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ ( field_1126092520709947252od_a_a @ Qeq ) )
=> ? [X6: set_Product_prod_a_a] :
( ( ( class_6315440051758036822od_a_a @ X4 @ Qeq )
= ( some_s3837607710808129772od_a_a @ X6 ) )
& ( member1426531477525435216od_a_a @ X4 @ X6 ) ) ) ).
% ex_class
thf(fact_712_ex__class,axiom,
! [X4: a,Qeq: set_Product_prod_a_a] :
( ( member_a @ X4 @ ( field_a @ Qeq ) )
=> ? [X6: set_a] :
( ( ( class_a @ X4 @ Qeq )
= ( some_set_a @ X6 ) )
& ( member_a @ X4 @ X6 ) ) ) ).
% ex_class
thf(fact_713_finite__Int,axiom,
! [F3: set_a,G3: set_a] :
( ( ( finite_finite_a @ F3 )
| ( finite_finite_a @ G3 ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F3 @ G3 ) ) ) ).
% finite_Int
thf(fact_714_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_715_top__empty__eq,axiom,
( top_to8687885267596698950_a_a_o
= ( ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ top_to8063371432257647191od_a_a ) ) ) ).
% top_empty_eq
thf(fact_716_top__empty__eq2,axiom,
( top_top_a_a_o
= ( ^ [X3: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ top_to8063371432257647191od_a_a ) ) ) ).
% top_empty_eq2
thf(fact_717_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_718_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_719_finite__cartesian__product__iff,axiom,
! [A5: set_a,B5: set_a] :
( ( finite6544458595007987280od_a_a
@ ( product_Sigma_a_a @ A5
@ ^ [Uu: a] : B5 ) )
= ( ( A5 = bot_bot_set_a )
| ( B5 = bot_bot_set_a )
| ( ( finite_finite_a @ A5 )
& ( finite_finite_a @ B5 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_720_finite__SigmaI2,axiom,
! [A5: set_a,B5: a > set_a] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A5 )
& ( ( B5 @ X3 )
!= bot_bot_set_a ) ) ) )
=> ( ! [A: a] :
( ( member_a @ A @ A5 )
=> ( finite_finite_a @ ( B5 @ A ) ) )
=> ( finite6544458595007987280od_a_a @ ( product_Sigma_a_a @ A5 @ B5 ) ) ) ) ).
% finite_SigmaI2
thf(fact_721_finite__SigmaI2,axiom,
! [A5: set_Product_prod_a_a,B5: product_prod_a_a > set_a] :
( ( finite6544458595007987280od_a_a
@ ( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A5 )
& ( ( B5 @ X3 )
!= bot_bot_set_a ) ) ) )
=> ( ! [A: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A @ A5 )
=> ( finite_finite_a @ ( B5 @ A ) ) )
=> ( finite5607752495362350695_a_a_a @ ( produc2379640491490746847_a_a_a @ A5 @ B5 ) ) ) ) ).
% finite_SigmaI2
thf(fact_722_max__extp_Omax__extI,axiom,
! [X7: set_a,Y4: set_a,R2: a > a > $o] :
( ( finite_finite_a @ X7 )
=> ( ( finite_finite_a @ Y4 )
=> ( ( Y4
!= ( collect_a @ bot_bot_a_o ) )
=> ( ! [X: a] :
( ( member_a @ X @ X7 )
=> ? [Xa: a] :
( ( member_a @ Xa @ Y4 )
& ( R2 @ X @ Xa ) ) )
=> ( max_extp_a @ R2 @ X7 @ Y4 ) ) ) ) ) ).
% max_extp.max_extI
thf(fact_723_max__extp_Omax__extI,axiom,
! [X7: set_Product_prod_a_a,Y4: set_Product_prod_a_a,R2: product_prod_a_a > product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ X7 )
=> ( ( finite6544458595007987280od_a_a @ Y4 )
=> ( ( Y4
!= ( collec3336397797384452498od_a_a @ bot_bo4160289986317612842_a_a_o ) )
=> ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ X7 )
=> ? [Xa: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Xa @ Y4 )
& ( R2 @ X @ Xa ) ) )
=> ( max_ex6057718940173624378od_a_a @ R2 @ X7 @ Y4 ) ) ) ) ) ).
% max_extp.max_extI
thf(fact_724_max__extp_Osimps,axiom,
( max_extp_a
= ( ^ [R4: a > a > $o,A1: set_a,A22: set_a] :
( ( finite_finite_a @ A1 )
& ( finite_finite_a @ A22 )
& ( A22
!= ( collect_a @ bot_bot_a_o ) )
& ! [X3: a] :
( ( member_a @ X3 @ A1 )
=> ? [Y2: a] :
( ( member_a @ Y2 @ A22 )
& ( R4 @ X3 @ Y2 ) ) ) ) ) ) ).
% max_extp.simps
thf(fact_725_max__extp_Osimps,axiom,
( max_ex6057718940173624378od_a_a
= ( ^ [R4: product_prod_a_a > product_prod_a_a > $o,A1: set_Product_prod_a_a,A22: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A1 )
& ( finite6544458595007987280od_a_a @ A22 )
& ( A22
!= ( collec3336397797384452498od_a_a @ bot_bo4160289986317612842_a_a_o ) )
& ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A1 )
=> ? [Y2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y2 @ A22 )
& ( R4 @ X3 @ Y2 ) ) ) ) ) ) ).
% max_extp.simps
thf(fact_726_max__extp_Ocases,axiom,
! [R2: a > a > $o,A12: set_a,A23: set_a] :
( ( max_extp_a @ R2 @ A12 @ A23 )
=> ~ ( ( finite_finite_a @ A12 )
=> ( ( finite_finite_a @ A23 )
=> ( ( A23
!= ( collect_a @ bot_bot_a_o ) )
=> ~ ! [X5: a] :
( ( member_a @ X5 @ A12 )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ A23 )
& ( R2 @ X5 @ Xa2 ) ) ) ) ) ) ) ).
% max_extp.cases
thf(fact_727_max__extp_Ocases,axiom,
! [R2: product_prod_a_a > product_prod_a_a > $o,A12: set_Product_prod_a_a,A23: set_Product_prod_a_a] :
( ( max_ex6057718940173624378od_a_a @ R2 @ A12 @ A23 )
=> ~ ( ( finite6544458595007987280od_a_a @ A12 )
=> ( ( finite6544458595007987280od_a_a @ A23 )
=> ( ( A23
!= ( collec3336397797384452498od_a_a @ bot_bo4160289986317612842_a_a_o ) )
=> ~ ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A12 )
=> ? [Xa2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Xa2 @ A23 )
& ( R2 @ X5 @ Xa2 ) ) ) ) ) ) ) ).
% max_extp.cases
thf(fact_728_cofinal__def,axiom,
( bNF_Ca4386975739854426319inal_a
= ( ^ [A6: set_a,R: set_Product_prod_a_a] :
! [X3: a] :
( ( member_a @ X3 @ ( field_a @ R ) )
=> ? [Y2: a] :
( ( member_a @ Y2 @ A6 )
& ( X3 != Y2 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R ) ) ) ) ) ).
% cofinal_def
thf(fact_729_max__ext_Omax__extI,axiom,
! [X7: set_Product_prod_a_a,Y4: set_Product_prod_a_a,R2: set_Pr8600417178894128327od_a_a] :
( ( finite6544458595007987280od_a_a @ X7 )
=> ( ( finite6544458595007987280od_a_a @ Y4 )
=> ( ( Y4 != bot_bo3357376287454694259od_a_a )
=> ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ X7 )
=> ? [Xa: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Xa @ Y4 )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Xa ) @ R2 ) ) )
=> ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ X7 @ Y4 ) @ ( max_ex7229726517752475106od_a_a @ R2 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_730_max__ext_Omax__extI,axiom,
! [X7: set_a,Y4: set_a,R2: set_Product_prod_a_a] :
( ( finite_finite_a @ X7 )
=> ( ( finite_finite_a @ Y4 )
=> ( ( Y4 != bot_bot_set_a )
=> ( ! [X: a] :
( ( member_a @ X @ X7 )
=> ? [Xa: a] :
( ( member_a @ Xa @ Y4 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Xa ) @ R2 ) ) )
=> ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X7 @ Y4 ) @ ( max_ext_a @ R2 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_731_max__ext_Osimps,axiom,
! [A12: set_Product_prod_a_a,A23: set_Product_prod_a_a,R2: set_Pr8600417178894128327od_a_a] :
( ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ A12 @ A23 ) @ ( max_ex7229726517752475106od_a_a @ R2 ) )
= ( ( finite6544458595007987280od_a_a @ A12 )
& ( finite6544458595007987280od_a_a @ A23 )
& ( A23 != bot_bo3357376287454694259od_a_a )
& ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A12 )
=> ? [Y2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y2 @ A23 )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ Y2 ) @ R2 ) ) ) ) ) ).
% max_ext.simps
thf(fact_732_max__ext_Osimps,axiom,
! [A12: set_a,A23: set_a,R2: set_Product_prod_a_a] :
( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A12 @ A23 ) @ ( max_ext_a @ R2 ) )
= ( ( finite_finite_a @ A12 )
& ( finite_finite_a @ A23 )
& ( A23 != bot_bot_set_a )
& ! [X3: a] :
( ( member_a @ X3 @ A12 )
=> ? [Y2: a] :
( ( member_a @ Y2 @ A23 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R2 ) ) ) ) ) ).
% max_ext.simps
thf(fact_733_max__ext_Ocases,axiom,
! [A12: set_Product_prod_a_a,A23: set_Product_prod_a_a,R2: set_Pr8600417178894128327od_a_a] :
( ( member4903802553882211088od_a_a @ ( produc3372137660326521687od_a_a @ A12 @ A23 ) @ ( max_ex7229726517752475106od_a_a @ R2 ) )
=> ~ ( ( finite6544458595007987280od_a_a @ A12 )
=> ( ( finite6544458595007987280od_a_a @ A23 )
=> ( ( A23 != bot_bo3357376287454694259od_a_a )
=> ~ ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A12 )
=> ? [Xa2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Xa2 @ A23 )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X5 @ Xa2 ) @ R2 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_734_max__ext_Ocases,axiom,
! [A12: set_a,A23: set_a,R2: set_Product_prod_a_a] :
( ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ A12 @ A23 ) @ ( max_ext_a @ R2 ) )
=> ~ ( ( finite_finite_a @ A12 )
=> ( ( finite_finite_a @ A23 )
=> ( ( A23 != bot_bot_set_a )
=> ~ ! [X5: a] :
( ( member_a @ X5 @ A12 )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ A23 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Xa2 ) @ R2 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_735_preorder__on__empty,axiom,
order_preorder_on_a @ bot_bot_set_a @ bot_bo3357376287454694259od_a_a ).
% preorder_on_empty
thf(fact_736_max__ext__def,axiom,
( max_ext_a
= ( ^ [R4: set_Product_prod_a_a] :
( collec8259436133773553042_set_a
@ ( produc2941226113218366828et_a_o
@ ( max_extp_a
@ ^ [X3: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R4 ) ) ) ) ) ) ).
% max_ext_def
thf(fact_737_max__extp__max__ext__eq,axiom,
! [R2: set_Product_prod_a_a] :
( ( max_extp_a
@ ^ [X3: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R2 ) )
= ( ^ [X3: set_a,Y2: set_a] : ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X3 @ Y2 ) @ ( max_ext_a @ R2 ) ) ) ) ).
% max_extp_max_ext_eq
thf(fact_738_max__extp__eq,axiom,
( max_extp_a
= ( ^ [R: a > a > $o,X3: set_a,Y2: set_a] : ( member7983343339038529360_set_a @ ( produc9088192753505129239_set_a @ X3 @ Y2 ) @ ( max_ext_a @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ R ) ) ) ) ) ) ).
% max_extp_eq
thf(fact_739_Refl__Field__Restr,axiom,
! [R3: set_Product_prod_a_a,A5: set_a] :
( ( refl_on_a @ ( field_a @ R3 ) @ R3 )
=> ( ( field_a
@ ( inf_in8905007599844390133od_a_a @ R3
@ ( product_Sigma_a_a @ A5
@ ^ [Uu: a] : A5 ) ) )
= ( inf_inf_set_a @ ( field_a @ R3 ) @ A5 ) ) ) ).
% Refl_Field_Restr
thf(fact_740_aboveS__def,axiom,
( order_aboveS_a
= ( ^ [R: set_Product_prod_a_a,A2: a] :
( collect_a
@ ^ [B: a] :
( ( B != A2 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ R ) ) ) ) ) ).
% aboveS_def
thf(fact_741_inter__Set__filter,axiom,
! [B5: set_Product_prod_a_a,A5: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ B5 )
=> ( ( inf_in8905007599844390133od_a_a @ A5 @ B5 )
= ( filter4380578791250724440od_a_a
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A5 )
@ B5 ) ) ) ).
% inter_Set_filter
thf(fact_742_inter__Set__filter,axiom,
! [B5: set_a,A5: set_a] :
( ( finite_finite_a @ B5 )
=> ( ( inf_inf_set_a @ A5 @ B5 )
= ( filter_a
@ ^ [X3: a] : ( member_a @ X3 @ A5 )
@ B5 ) ) ) ).
% inter_Set_filter
thf(fact_743_member__filter,axiom,
! [X4: a,P: a > $o,A5: set_a] :
( ( member_a @ X4 @ ( filter_a @ P @ A5 ) )
= ( ( member_a @ X4 @ A5 )
& ( P @ X4 ) ) ) ).
% member_filter
thf(fact_744_member__filter,axiom,
! [X4: product_prod_a_a,P: product_prod_a_a > $o,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ ( filter4380578791250724440od_a_a @ P @ A5 ) )
= ( ( member1426531477525435216od_a_a @ X4 @ A5 )
& ( P @ X4 ) ) ) ).
% member_filter
thf(fact_745_Set_Ofilter__def,axiom,
( filter_a
= ( ^ [P3: a > $o,A6: set_a] :
( collect_a
@ ^ [A2: a] :
( ( member_a @ A2 @ A6 )
& ( P3 @ A2 ) ) ) ) ) ).
% Set.filter_def
thf(fact_746_Set_Ofilter__def,axiom,
( filter4380578791250724440od_a_a
= ( ^ [P3: product_prod_a_a > $o,A6: set_Product_prod_a_a] :
( collec3336397797384452498od_a_a
@ ^ [A2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ A6 )
& ( P3 @ A2 ) ) ) ) ) ).
% Set.filter_def
thf(fact_747_refl__on__domain,axiom,
! [A5: set_Product_prod_a_a,R3: set_Pr8600417178894128327od_a_a,A3: product_prod_a_a,B3: product_prod_a_a] :
( ( refl_o7745108929832855590od_a_a @ A5 @ R3 )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A3 @ B3 ) @ R3 )
=> ( ( member1426531477525435216od_a_a @ A3 @ A5 )
& ( member1426531477525435216od_a_a @ B3 @ A5 ) ) ) ) ).
% refl_on_domain
thf(fact_748_refl__on__domain,axiom,
! [A5: set_a,R3: set_Product_prod_a_a,A3: a,B3: a] :
( ( refl_on_a @ A5 @ R3 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ R3 )
=> ( ( member_a @ A3 @ A5 )
& ( member_a @ B3 @ A5 ) ) ) ) ).
% refl_on_domain
thf(fact_749_refl__onD,axiom,
! [A5: set_Product_prod_a_a,R3: set_Pr8600417178894128327od_a_a,A3: product_prod_a_a] :
( ( refl_o7745108929832855590od_a_a @ A5 @ R3 )
=> ( ( member1426531477525435216od_a_a @ A3 @ A5 )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A3 @ A3 ) @ R3 ) ) ) ).
% refl_onD
thf(fact_750_refl__onD,axiom,
! [A5: set_a,R3: set_Product_prod_a_a,A3: a] :
( ( refl_on_a @ A5 @ R3 )
=> ( ( member_a @ A3 @ A5 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ A3 ) @ R3 ) ) ) ).
% refl_onD
thf(fact_751_refl__onD1,axiom,
! [A5: set_Product_prod_a_a,R3: set_Pr8600417178894128327od_a_a,X4: product_prod_a_a,Y3: product_prod_a_a] :
( ( refl_o7745108929832855590od_a_a @ A5 @ R3 )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X4 @ Y3 ) @ R3 )
=> ( member1426531477525435216od_a_a @ X4 @ A5 ) ) ) ).
% refl_onD1
thf(fact_752_refl__onD1,axiom,
! [A5: set_a,R3: set_Product_prod_a_a,X4: a,Y3: a] :
( ( refl_on_a @ A5 @ R3 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R3 )
=> ( member_a @ X4 @ A5 ) ) ) ).
% refl_onD1
thf(fact_753_refl__onD2,axiom,
! [A5: set_Product_prod_a_a,R3: set_Pr8600417178894128327od_a_a,X4: product_prod_a_a,Y3: product_prod_a_a] :
( ( refl_o7745108929832855590od_a_a @ A5 @ R3 )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X4 @ Y3 ) @ R3 )
=> ( member1426531477525435216od_a_a @ Y3 @ A5 ) ) ) ).
% refl_onD2
thf(fact_754_refl__onD2,axiom,
! [A5: set_a,R3: set_Product_prod_a_a,X4: a,Y3: a] :
( ( refl_on_a @ A5 @ R3 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R3 )
=> ( member_a @ Y3 @ A5 ) ) ) ).
% refl_onD2
thf(fact_755_reflI,axiom,
! [R3: set_Product_prod_a_a] :
( ! [X: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ X ) @ R3 )
=> ( refl_on_a @ top_top_set_a @ R3 ) ) ).
% reflI
thf(fact_756_reflD,axiom,
! [R3: set_Product_prod_a_a,A3: a] :
( ( refl_on_a @ top_top_set_a @ R3 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ A3 ) @ R3 ) ) ).
% reflD
thf(fact_757_refl__on__empty,axiom,
refl_on_a @ bot_bot_set_a @ bot_bo3357376287454694259od_a_a ).
% refl_on_empty
thf(fact_758_refl__on__Int,axiom,
! [A5: set_a,R3: set_Product_prod_a_a,B5: set_a,S2: set_Product_prod_a_a] :
( ( refl_on_a @ A5 @ R3 )
=> ( ( refl_on_a @ B5 @ S2 )
=> ( refl_on_a @ ( inf_inf_set_a @ A5 @ B5 ) @ ( inf_in8905007599844390133od_a_a @ R3 @ S2 ) ) ) ) ).
% refl_on_Int
thf(fact_759_lnear__order__on__empty,axiom,
order_8768733634509060147r_on_a @ bot_bot_set_a @ bot_bo3357376287454694259od_a_a ).
% lnear_order_on_empty
thf(fact_760_partial__order__on__empty,axiom,
order_5272072345360262643r_on_a @ bot_bot_set_a @ bot_bo3357376287454694259od_a_a ).
% partial_order_on_empty
thf(fact_761_linear__order__on__Restr,axiom,
! [A5: set_a,R3: set_Product_prod_a_a,X4: a] :
( ( order_8768733634509060147r_on_a @ A5 @ R3 )
=> ( order_8768733634509060147r_on_a @ ( inf_inf_set_a @ A5 @ ( order_above_a @ R3 @ X4 ) )
@ ( inf_in8905007599844390133od_a_a @ R3
@ ( product_Sigma_a_a @ ( order_above_a @ R3 @ X4 )
@ ^ [Uu: a] : ( order_above_a @ R3 @ X4 ) ) ) ) ) ).
% linear_order_on_Restr
thf(fact_762_max__ext__eq,axiom,
( max_ex7229726517752475106od_a_a
= ( ^ [R4: set_Pr8600417178894128327od_a_a] :
( collec3859266087184377170od_a_a
@ ( produc4361984678274043180_a_a_o
@ ^ [X10: set_Product_prod_a_a,Y7: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ X10 )
& ( finite6544458595007987280od_a_a @ Y7 )
& ( Y7 != bot_bo3357376287454694259od_a_a )
& ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ X10 )
=> ? [Y2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y2 @ Y7 )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ Y2 ) @ R4 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_763_max__ext__eq,axiom,
( max_ext_a
= ( ^ [R4: set_Product_prod_a_a] :
( collec8259436133773553042_set_a
@ ( produc2941226113218366828et_a_o
@ ^ [X10: set_a,Y7: set_a] :
( ( finite_finite_a @ X10 )
& ( finite_finite_a @ Y7 )
& ( Y7 != bot_bot_set_a )
& ! [X3: a] :
( ( member_a @ X3 @ X10 )
=> ? [Y2: a] :
( ( member_a @ Y2 @ Y7 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R4 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_764_finite__range__Some,axiom,
( ( finite3831083272032232269_set_a @ ( image_2983976770393306955_set_a @ some_set_a @ top_top_set_set_a ) )
= ( finite_finite_set_a @ top_top_set_set_a ) ) ).
% finite_range_Some
thf(fact_765_Id__on__set,axiom,
! [Xs: list_P1396940483166286381od_a_a] :
( ( id_on_7193410965977890500od_a_a @ ( set_Product_prod_a_a2 @ Xs ) )
= ( set_Pr6493056779096584572od_a_a
@ ( map_Pr5477722341457123780od_a_a
@ ^ [X3: product_prod_a_a] : ( produc7886510207707329367od_a_a @ X3 @ X3 )
@ Xs ) ) ) ).
% Id_on_set
thf(fact_766_Id__on__set,axiom,
! [Xs: list_a] :
( ( id_on_a @ ( set_a2 @ Xs ) )
= ( set_Product_prod_a_a2
@ ( map_a_7860052162900579309od_a_a
@ ^ [X3: a] : ( product_Pair_a_a @ X3 @ X3 )
@ Xs ) ) ) ).
% Id_on_set
thf(fact_767_image__eqI,axiom,
! [B3: a,F: a > a,X4: a,A5: set_a] :
( ( B3
= ( F @ X4 ) )
=> ( ( member_a @ X4 @ A5 )
=> ( member_a @ B3 @ ( image_a_a2 @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_768_image__eqI,axiom,
! [B3: product_prod_a_a,F: a > product_prod_a_a,X4: a,A5: set_a] :
( ( B3
= ( F @ X4 ) )
=> ( ( member_a @ X4 @ A5 )
=> ( member1426531477525435216od_a_a @ B3 @ ( image_7400625782589995694od_a_a @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_769_image__eqI,axiom,
! [B3: a,F: product_prod_a_a > a,X4: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( B3
= ( F @ X4 ) )
=> ( ( member1426531477525435216od_a_a @ X4 @ A5 )
=> ( member_a @ B3 @ ( image_3437945252899457948_a_a_a @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_770_image__eqI,axiom,
! [B3: product_prod_a_a,F: product_prod_a_a > product_prod_a_a,X4: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( B3
= ( F @ X4 ) )
=> ( ( member1426531477525435216od_a_a @ X4 @ A5 )
=> ( member1426531477525435216od_a_a @ B3 @ ( image_4636654165204879301od_a_a @ F @ A5 ) ) ) ) ).
% image_eqI
thf(fact_771_subsetI,axiom,
! [A5: set_a,B5: set_a] :
( ! [X: a] :
( ( member_a @ X @ A5 )
=> ( member_a @ X @ B5 ) )
=> ( ord_less_eq_set_a @ A5 @ B5 ) ) ).
% subsetI
thf(fact_772_subsetI,axiom,
! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A5 )
=> ( member1426531477525435216od_a_a @ X @ B5 ) )
=> ( ord_le746702958409616551od_a_a @ A5 @ B5 ) ) ).
% subsetI
thf(fact_773_inf_Obounded__iff,axiom,
! [A3: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C ) )
= ( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ A3 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_774_le__inf__iff,axiom,
! [X4: set_a,Y3: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z ) )
= ( ( ord_less_eq_set_a @ X4 @ Y3 )
& ( ord_less_eq_set_a @ X4 @ Z ) ) ) ).
% le_inf_iff
thf(fact_775_image__is__empty,axiom,
! [F: a > a,A5: set_a] :
( ( ( image_a_a2 @ F @ A5 )
= bot_bot_set_a )
= ( A5 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_776_empty__is__image,axiom,
! [F: a > a,A5: set_a] :
( ( bot_bot_set_a
= ( image_a_a2 @ F @ A5 ) )
= ( A5 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_777_image__empty,axiom,
! [F: a > a] :
( ( image_a_a2 @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_778_empty__subsetI,axiom,
! [A5: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A5 ) ).
% empty_subsetI
thf(fact_779_subset__empty,axiom,
! [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ bot_bot_set_a )
= ( A5 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_780_Int__subset__iff,axiom,
! [C3: set_a,A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A5 @ B5 ) )
= ( ( ord_less_eq_set_a @ C3 @ A5 )
& ( ord_less_eq_set_a @ C3 @ B5 ) ) ) ).
% Int_subset_iff
thf(fact_781_map__is__Nil__conv,axiom,
! [F: product_prod_a_a > product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( ( map_Pr7904243085458786820od_a_a @ F @ Xs )
= nil_Product_prod_a_a )
= ( Xs = nil_Product_prod_a_a ) ) ).
% map_is_Nil_conv
thf(fact_782_Nil__is__map__conv,axiom,
! [F: product_prod_a_a > product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( nil_Product_prod_a_a
= ( map_Pr7904243085458786820od_a_a @ F @ Xs ) )
= ( Xs = nil_Product_prod_a_a ) ) ).
% Nil_is_map_conv
thf(fact_783_list_Omap__disc__iff,axiom,
! [F: product_prod_a_a > product_prod_a_a,A3: list_P1396940483166286381od_a_a] :
( ( ( map_Pr7904243085458786820od_a_a @ F @ A3 )
= nil_Product_prod_a_a )
= ( A3 = nil_Product_prod_a_a ) ) ).
% list.map_disc_iff
thf(fact_784_list_Oset__map,axiom,
! [F: product_prod_a_a > product_prod_a_a,V: list_P1396940483166286381od_a_a] :
( ( set_Product_prod_a_a2 @ ( map_Pr7904243085458786820od_a_a @ F @ V ) )
= ( image_4636654165204879301od_a_a @ F @ ( set_Product_prod_a_a2 @ V ) ) ) ).
% list.set_map
thf(fact_785_Compr__image__eq,axiom,
! [F: a > a,A5: set_a,P: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( image_a_a2 @ F @ A5 ) )
& ( P @ X3 ) ) )
= ( image_a_a2 @ F
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A5 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_786_Compr__image__eq,axiom,
! [F: product_prod_a_a > a,A5: set_Product_prod_a_a,P: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( image_3437945252899457948_a_a_a @ F @ A5 ) )
& ( P @ X3 ) ) )
= ( image_3437945252899457948_a_a_a @ F
@ ( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A5 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_787_Compr__image__eq,axiom,
! [F: a > product_prod_a_a,A5: set_a,P: product_prod_a_a > $o] :
( ( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( image_7400625782589995694od_a_a @ F @ A5 ) )
& ( P @ X3 ) ) )
= ( image_7400625782589995694od_a_a @ F
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A5 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_788_Compr__image__eq,axiom,
! [F: product_prod_a_a > product_prod_a_a,A5: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( image_4636654165204879301od_a_a @ F @ A5 ) )
& ( P @ X3 ) ) )
= ( image_4636654165204879301od_a_a @ F
@ ( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A5 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_789_imageE,axiom,
! [B3: a,F: a > a,A5: set_a] :
( ( member_a @ B3 @ ( image_a_a2 @ F @ A5 ) )
=> ~ ! [X: a] :
( ( B3
= ( F @ X ) )
=> ~ ( member_a @ X @ A5 ) ) ) ).
% imageE
thf(fact_790_imageE,axiom,
! [B3: a,F: product_prod_a_a > a,A5: set_Product_prod_a_a] :
( ( member_a @ B3 @ ( image_3437945252899457948_a_a_a @ F @ A5 ) )
=> ~ ! [X: product_prod_a_a] :
( ( B3
= ( F @ X ) )
=> ~ ( member1426531477525435216od_a_a @ X @ A5 ) ) ) ).
% imageE
thf(fact_791_imageE,axiom,
! [B3: product_prod_a_a,F: a > product_prod_a_a,A5: set_a] :
( ( member1426531477525435216od_a_a @ B3 @ ( image_7400625782589995694od_a_a @ F @ A5 ) )
=> ~ ! [X: a] :
( ( B3
= ( F @ X ) )
=> ~ ( member_a @ X @ A5 ) ) ) ).
% imageE
thf(fact_792_imageE,axiom,
! [B3: product_prod_a_a,F: product_prod_a_a > product_prod_a_a,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B3 @ ( image_4636654165204879301od_a_a @ F @ A5 ) )
=> ~ ! [X: product_prod_a_a] :
( ( B3
= ( F @ X ) )
=> ~ ( member1426531477525435216od_a_a @ X @ A5 ) ) ) ).
% imageE
thf(fact_793_Collect__subset,axiom,
! [A5: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A5 )
& ( P @ X3 ) ) )
@ A5 ) ).
% Collect_subset
thf(fact_794_Collect__subset,axiom,
! [A5: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ord_le746702958409616551od_a_a
@ ( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A5 )
& ( P @ X3 ) ) )
@ A5 ) ).
% Collect_subset
thf(fact_795_imageI,axiom,
! [X4: a,A5: set_a,F: a > a] :
( ( member_a @ X4 @ A5 )
=> ( member_a @ ( F @ X4 ) @ ( image_a_a2 @ F @ A5 ) ) ) ).
% imageI
thf(fact_796_imageI,axiom,
! [X4: a,A5: set_a,F: a > product_prod_a_a] :
( ( member_a @ X4 @ A5 )
=> ( member1426531477525435216od_a_a @ ( F @ X4 ) @ ( image_7400625782589995694od_a_a @ F @ A5 ) ) ) ).
% imageI
thf(fact_797_imageI,axiom,
! [X4: product_prod_a_a,A5: set_Product_prod_a_a,F: product_prod_a_a > a] :
( ( member1426531477525435216od_a_a @ X4 @ A5 )
=> ( member_a @ ( F @ X4 ) @ ( image_3437945252899457948_a_a_a @ F @ A5 ) ) ) ).
% imageI
thf(fact_798_imageI,axiom,
! [X4: product_prod_a_a,A5: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A5 )
=> ( member1426531477525435216od_a_a @ ( F @ X4 ) @ ( image_4636654165204879301od_a_a @ F @ A5 ) ) ) ).
% imageI
thf(fact_799_in__mono,axiom,
! [A5: set_a,B5: set_a,X4: a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
=> ( ( member_a @ X4 @ A5 )
=> ( member_a @ X4 @ B5 ) ) ) ).
% in_mono
thf(fact_800_in__mono,axiom,
! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a,X4: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
=> ( ( member1426531477525435216od_a_a @ X4 @ A5 )
=> ( member1426531477525435216od_a_a @ X4 @ B5 ) ) ) ).
% in_mono
thf(fact_801_subsetD,axiom,
! [A5: set_a,B5: set_a,C: a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
=> ( ( member_a @ C @ A5 )
=> ( member_a @ C @ B5 ) ) ) ).
% subsetD
thf(fact_802_subsetD,axiom,
! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
=> ( ( member1426531477525435216od_a_a @ C @ A5 )
=> ( member1426531477525435216od_a_a @ C @ B5 ) ) ) ).
% subsetD
thf(fact_803_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A6 )
=> ( member_a @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_804_subset__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A6 )
=> ( member1426531477525435216od_a_a @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_805_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
! [T: a] :
( ( member_a @ T @ A6 )
=> ( member_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_806_subset__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
! [T: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T @ A6 )
=> ( member1426531477525435216od_a_a @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_807_image__subsetI,axiom,
! [A5: set_a,F: a > a,B5: set_a] :
( ! [X: a] :
( ( member_a @ X @ A5 )
=> ( member_a @ ( F @ X ) @ B5 ) )
=> ( ord_less_eq_set_a @ ( image_a_a2 @ F @ A5 ) @ B5 ) ) ).
% image_subsetI
thf(fact_808_image__subsetI,axiom,
! [A5: set_a,F: a > product_prod_a_a,B5: set_Product_prod_a_a] :
( ! [X: a] :
( ( member_a @ X @ A5 )
=> ( member1426531477525435216od_a_a @ ( F @ X ) @ B5 ) )
=> ( ord_le746702958409616551od_a_a @ ( image_7400625782589995694od_a_a @ F @ A5 ) @ B5 ) ) ).
% image_subsetI
thf(fact_809_image__subsetI,axiom,
! [A5: set_Product_prod_a_a,F: product_prod_a_a > a,B5: set_a] :
( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A5 )
=> ( member_a @ ( F @ X ) @ B5 ) )
=> ( ord_less_eq_set_a @ ( image_3437945252899457948_a_a_a @ F @ A5 ) @ B5 ) ) ).
% image_subsetI
thf(fact_810_image__subsetI,axiom,
! [A5: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a,B5: set_Product_prod_a_a] :
( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A5 )
=> ( member1426531477525435216od_a_a @ ( F @ X ) @ B5 ) )
=> ( ord_le746702958409616551od_a_a @ ( image_4636654165204879301od_a_a @ F @ A5 ) @ B5 ) ) ).
% image_subsetI
thf(fact_811_rev__image__eqI,axiom,
! [X4: a,A5: set_a,B3: a,F: a > a] :
( ( member_a @ X4 @ A5 )
=> ( ( B3
= ( F @ X4 ) )
=> ( member_a @ B3 @ ( image_a_a2 @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_812_rev__image__eqI,axiom,
! [X4: a,A5: set_a,B3: product_prod_a_a,F: a > product_prod_a_a] :
( ( member_a @ X4 @ A5 )
=> ( ( B3
= ( F @ X4 ) )
=> ( member1426531477525435216od_a_a @ B3 @ ( image_7400625782589995694od_a_a @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_813_rev__image__eqI,axiom,
! [X4: product_prod_a_a,A5: set_Product_prod_a_a,B3: a,F: product_prod_a_a > a] :
( ( member1426531477525435216od_a_a @ X4 @ A5 )
=> ( ( B3
= ( F @ X4 ) )
=> ( member_a @ B3 @ ( image_3437945252899457948_a_a_a @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_814_rev__image__eqI,axiom,
! [X4: product_prod_a_a,A5: set_Product_prod_a_a,B3: product_prod_a_a,F: product_prod_a_a > product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A5 )
=> ( ( B3
= ( F @ X4 ) )
=> ( member1426531477525435216od_a_a @ B3 @ ( image_4636654165204879301od_a_a @ F @ A5 ) ) ) ) ).
% rev_image_eqI
thf(fact_815_image__set,axiom,
! [F: product_prod_a_a > product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( image_4636654165204879301od_a_a @ F @ ( set_Product_prod_a_a2 @ Xs ) )
= ( set_Product_prod_a_a2 @ ( map_Pr7904243085458786820od_a_a @ F @ Xs ) ) ) ).
% image_set
thf(fact_816_image__Int__subset,axiom,
! [F: a > a,A5: set_a,B5: set_a] : ( ord_less_eq_set_a @ ( image_a_a2 @ F @ ( inf_inf_set_a @ A5 @ B5 ) ) @ ( inf_inf_set_a @ ( image_a_a2 @ F @ A5 ) @ ( image_a_a2 @ F @ B5 ) ) ) ).
% image_Int_subset
thf(fact_817_list_Osimps_I8_J,axiom,
! [F: product_prod_a_a > product_prod_a_a] :
( ( map_Pr7904243085458786820od_a_a @ F @ nil_Product_prod_a_a )
= nil_Product_prod_a_a ) ).
% list.simps(8)
thf(fact_818_map__idI,axiom,
! [Xs: list_a,F: a > a] :
( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( F @ X )
= X ) )
=> ( ( map_a_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_819_map__idI,axiom,
! [Xs: list_P1396940483166286381od_a_a,F: product_prod_a_a > product_prod_a_a] :
( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ( F @ X )
= X ) )
=> ( ( map_Pr7904243085458786820od_a_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_820_list_Omap__ident__strong,axiom,
! [T2: list_a,F: a > a] :
( ! [Z2: a] :
( ( member_a @ Z2 @ ( set_a2 @ T2 ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_a_a @ F @ T2 )
= T2 ) ) ).
% list.map_ident_strong
thf(fact_821_list_Omap__ident__strong,axiom,
! [T2: list_P1396940483166286381od_a_a,F: product_prod_a_a > product_prod_a_a] :
( ! [Z2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Z2 @ ( set_Product_prod_a_a2 @ T2 ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_Pr7904243085458786820od_a_a @ F @ T2 )
= T2 ) ) ).
% list.map_ident_strong
thf(fact_822_bot_Oextremum__uniqueI,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
=> ( A3 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_823_bot_Oextremum__unique,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_824_bot_Oextremum,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% bot.extremum
thf(fact_825_subrelI,axiom,
! [R3: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
( ! [X: a,Y: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R3 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ S2 ) )
=> ( ord_le746702958409616551od_a_a @ R3 @ S2 ) ) ).
% subrelI
thf(fact_826_inf_OcoboundedI2,axiom,
! [B3: set_a,C: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_827_inf_OcoboundedI1,axiom,
! [A3: set_a,C: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_828_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B: set_a,A2: set_a] :
( ( inf_inf_set_a @ A2 @ B )
= B ) ) ) ).
% inf.absorb_iff2
thf(fact_829_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ A2 @ B )
= A2 ) ) ) ).
% inf.absorb_iff1
thf(fact_830_inf_Ocobounded2,axiom,
! [A3: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_831_inf_Ocobounded1,axiom,
! [A3: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ A3 ) ).
% inf.cobounded1
thf(fact_832_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B: set_a] :
( A2
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% inf.order_iff
thf(fact_833_inf__greatest,axiom,
! [X4: set_a,Y3: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( ord_less_eq_set_a @ X4 @ Z )
=> ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_834_inf_OboundedI,axiom,
! [A3: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ A3 @ C )
=> ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_835_inf_OboundedE,axiom,
! [A3: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A3 @ ( inf_inf_set_a @ B3 @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B3 )
=> ~ ( ord_less_eq_set_a @ A3 @ C ) ) ) ).
% inf.boundedE
thf(fact_836_inf__absorb2,axiom,
! [Y3: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X4 )
=> ( ( inf_inf_set_a @ X4 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_837_inf__absorb1,axiom,
! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( inf_inf_set_a @ X4 @ Y3 )
= X4 ) ) ).
% inf_absorb1
thf(fact_838_inf_Oabsorb2,axiom,
! [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( ( inf_inf_set_a @ A3 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_839_inf_Oabsorb1,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( inf_inf_set_a @ A3 @ B3 )
= A3 ) ) ).
% inf.absorb1
thf(fact_840_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X3 @ Y2 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_841_inf__unique,axiom,
! [F: set_a > set_a > set_a,X4: set_a,Y3: set_a] :
( ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( F @ X @ Y ) @ X )
=> ( ! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( F @ X @ Y ) @ Y )
=> ( ! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z2 )
=> ( ord_less_eq_set_a @ X @ ( F @ Y @ Z2 ) ) ) )
=> ( ( inf_inf_set_a @ X4 @ Y3 )
= ( F @ X4 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_842_inf_OorderI,axiom,
! [A3: set_a,B3: set_a] :
( ( A3
= ( inf_inf_set_a @ A3 @ B3 ) )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% inf.orderI
thf(fact_843_inf_OorderE,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( A3
= ( inf_inf_set_a @ A3 @ B3 ) ) ) ).
% inf.orderE
thf(fact_844_le__infI2,axiom,
! [B3: set_a,X4: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ X4 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ X4 ) ) ).
% le_infI2
thf(fact_845_le__infI1,axiom,
! [A3: set_a,X4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ X4 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ X4 ) ) ).
% le_infI1
thf(fact_846_inf__mono,axiom,
! [A3: set_a,C: set_a,B3: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A3 @ C )
=> ( ( ord_less_eq_set_a @ B3 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A3 @ B3 ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_847_le__infI,axiom,
! [X4: set_a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ X4 @ A3 )
=> ( ( ord_less_eq_set_a @ X4 @ B3 )
=> ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ A3 @ B3 ) ) ) ) ).
% le_infI
thf(fact_848_le__infE,axiom,
! [X4: set_a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ A3 @ B3 ) )
=> ~ ( ( ord_less_eq_set_a @ X4 @ A3 )
=> ~ ( ord_less_eq_set_a @ X4 @ B3 ) ) ) ).
% le_infE
thf(fact_849_inf__le2,axiom,
! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_850_inf__le1,axiom,
! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ X4 ) ).
% inf_le1
thf(fact_851_inf__sup__ord_I1_J,axiom,
! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ X4 ) ).
% inf_sup_ord(1)
thf(fact_852_inf__sup__ord_I2_J,axiom,
! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_853_subset__code_I1_J,axiom,
! [Xs: list_a,B5: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B5 )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( member_a @ X3 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_854_subset__code_I1_J,axiom,
! [Xs: list_P1396940483166286381od_a_a,B5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ B5 )
= ( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( member1426531477525435216od_a_a @ X3 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_855_times__subset__iff,axiom,
! [A5: set_a,C3: set_a,B5: set_a,D: set_a] :
( ( ord_le746702958409616551od_a_a
@ ( product_Sigma_a_a @ A5
@ ^ [Uu: a] : C3 )
@ ( product_Sigma_a_a @ B5
@ ^ [Uu: a] : D ) )
= ( ( A5 = bot_bot_set_a )
| ( C3 = bot_bot_set_a )
| ( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ C3 @ D ) ) ) ) ).
% times_subset_iff
thf(fact_856_Int__Collect__mono,axiom,
! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a,P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ A5 @ B5 )
=> ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A5 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A5 @ ( collec3336397797384452498od_a_a @ P ) ) @ ( inf_in8905007599844390133od_a_a @ B5 @ ( collec3336397797384452498od_a_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_857_Int__Collect__mono,axiom,
! [A5: set_a,B5: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A5 @ B5 )
=> ( ! [X: a] :
( ( member_a @ X @ A5 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A5 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B5 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_858_Int__greatest,axiom,
! [C3: set_a,A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ C3 @ A5 )
=> ( ( ord_less_eq_set_a @ C3 @ B5 )
=> ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A5 @ B5 ) ) ) ) ).
% Int_greatest
thf(fact_859_Int__absorb2,axiom,
! [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
=> ( ( inf_inf_set_a @ A5 @ B5 )
= A5 ) ) ).
% Int_absorb2
thf(fact_860_Int__absorb1,axiom,
! [B5: set_a,A5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A5 )
=> ( ( inf_inf_set_a @ A5 @ B5 )
= B5 ) ) ).
% Int_absorb1
thf(fact_861_Int__lower2,axiom,
! [A5: set_a,B5: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ B5 ) ).
% Int_lower2
thf(fact_862_Int__lower1,axiom,
! [A5: set_a,B5: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ A5 ) ).
% Int_lower1
thf(fact_863_Int__mono,axiom,
! [A5: set_a,C3: set_a,B5: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A5 @ C3 )
=> ( ( ord_less_eq_set_a @ B5 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ ( inf_inf_set_a @ C3 @ D ) ) ) ) ).
% Int_mono
thf(fact_864_finite__set__of__finite__funs,axiom,
! [A5: set_a,B5: set_a,D2: a] :
( ( finite_finite_a @ A5 )
=> ( ( finite_finite_a @ B5 )
=> ( finite_finite_a_a
@ ( collect_a_a
@ ^ [F2: a > a] :
! [X3: a] :
( ( ( member_a @ X3 @ A5 )
=> ( member_a @ ( F2 @ X3 ) @ B5 ) )
& ( ~ ( member_a @ X3 @ A5 )
=> ( ( F2 @ X3 )
= D2 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_865_finite__set__of__finite__funs,axiom,
! [A5: set_a,B5: set_Product_prod_a_a,D2: product_prod_a_a] :
( ( finite_finite_a @ A5 )
=> ( ( finite6544458595007987280od_a_a @ B5 )
=> ( finite8821664692192630949od_a_a
@ ( collec3127584877517960419od_a_a
@ ^ [F2: a > product_prod_a_a] :
! [X3: a] :
( ( ( member_a @ X3 @ A5 )
=> ( member1426531477525435216od_a_a @ ( F2 @ X3 ) @ B5 ) )
& ( ~ ( member_a @ X3 @ A5 )
=> ( ( F2 @ X3 )
= D2 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_866_finite__set__of__finite__funs,axiom,
! [A5: set_Product_prod_a_a,B5: set_a,D2: a] :
( ( finite6544458595007987280od_a_a @ A5 )
=> ( ( finite_finite_a @ B5 )
=> ( finite8580459156145615379_a_a_a
@ ( collec2886379341470944849_a_a_a
@ ^ [F2: product_prod_a_a > a] :
! [X3: product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ X3 @ A5 )
=> ( member_a @ ( F2 @ X3 ) @ B5 ) )
& ( ~ ( member1426531477525435216od_a_a @ X3 @ A5 )
=> ( ( F2 @ X3 )
= D2 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_867_finite__set__of__finite__funs,axiom,
! [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a,D2: product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A5 )
=> ( ( finite6544458595007987280od_a_a @ B5 )
=> ( finite2714571839071174076od_a_a
@ ( collec8125451137695935482od_a_a
@ ^ [F2: product_prod_a_a > product_prod_a_a] :
! [X3: product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ X3 @ A5 )
=> ( member1426531477525435216od_a_a @ ( F2 @ X3 ) @ B5 ) )
& ( ~ ( member1426531477525435216od_a_a @ X3 @ A5 )
=> ( ( F2 @ X3 )
= D2 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_868_None__notin__image__Some,axiom,
! [A5: set_Product_prod_a_a] :
~ ( member8183384484874023062od_a_a @ none_P1914705783780380040od_a_a @ ( image_588318201180639627od_a_a @ some_P3592067295195376908od_a_a @ A5 ) ) ).
% None_notin_image_Some
thf(fact_869_None__notin__image__Some,axiom,
! [A5: set_set_a] :
~ ( member_option_set_a @ none_set_a @ ( image_2983976770393306955_set_a @ some_set_a @ A5 ) ) ).
% None_notin_image_Some
thf(fact_870_notin__range__Some,axiom,
! [X4: option5210160422955383789od_a_a] :
( ( ~ ( member8183384484874023062od_a_a @ X4 @ ( image_588318201180639627od_a_a @ some_P3592067295195376908od_a_a @ top_to8063371432257647191od_a_a ) ) )
= ( X4 = none_P1914705783780380040od_a_a ) ) ).
% notin_range_Some
thf(fact_871_notin__range__Some,axiom,
! [X4: option_set_a] :
( ( ~ ( member_option_set_a @ X4 @ ( image_2983976770393306955_set_a @ some_set_a @ top_top_set_set_a ) ) )
= ( X4 = none_set_a ) ) ).
% notin_range_Some
thf(fact_872_refl__on__def,axiom,
( refl_on_a
= ( ^ [A6: set_a,R: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R
@ ( product_Sigma_a_a @ A6
@ ^ [Uu: a] : A6 ) )
& ! [X3: a] :
( ( member_a @ X3 @ A6 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ X3 ) @ R ) ) ) ) ) ).
% refl_on_def
thf(fact_873_refl__onI,axiom,
! [R3: set_Pr8600417178894128327od_a_a,A5: set_Product_prod_a_a] :
( ( ord_le3469131294019144807od_a_a @ R3
@ ( produc5899993699339346696od_a_a @ A5
@ ^ [Uu: product_prod_a_a] : A5 ) )
=> ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A5 )
=> ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ X ) @ R3 ) )
=> ( refl_o7745108929832855590od_a_a @ A5 @ R3 ) ) ) ).
% refl_onI
thf(fact_874_refl__onI,axiom,
! [R3: set_Product_prod_a_a,A5: set_a] :
( ( ord_le746702958409616551od_a_a @ R3
@ ( product_Sigma_a_a @ A5
@ ^ [Uu: a] : A5 ) )
=> ( ! [X: a] :
( ( member_a @ X @ A5 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ X ) @ R3 ) )
=> ( refl_on_a @ A5 @ R3 ) ) ) ).
% refl_onI
thf(fact_875_above__def,axiom,
( order_above_a
= ( ^ [R: set_Product_prod_a_a,A2: a] :
( collect_a
@ ^ [B: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ R ) ) ) ) ).
% above_def
thf(fact_876_subset__emptyI,axiom,
! [A5: set_Product_prod_a_a] :
( ! [X: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X @ A5 )
=> ( ord_le746702958409616551od_a_a @ A5 @ bot_bo3357376287454694259od_a_a ) ) ).
% subset_emptyI
thf(fact_877_subset__emptyI,axiom,
! [A5: set_a] :
( ! [X: a] :
~ ( member_a @ X @ A5 )
=> ( ord_less_eq_set_a @ A5 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_878_Some__image__these__eq,axiom,
! [A5: set_op7160277562814721357od_a_a] :
( ( image_588318201180639627od_a_a @ some_P3592067295195376908od_a_a @ ( these_5100388957577570148od_a_a @ A5 ) )
= ( collec431113534271933400od_a_a
@ ^ [X3: option5210160422955383789od_a_a] :
( ( member8183384484874023062od_a_a @ X3 @ A5 )
& ( X3 != none_P1914705783780380040od_a_a ) ) ) ) ).
% Some_image_these_eq
thf(fact_879_Some__image__these__eq,axiom,
! [A5: set_option_set_a] :
( ( image_2983976770393306955_set_a @ some_set_a @ ( these_set_a @ A5 ) )
= ( collect_option_set_a
@ ^ [X3: option_set_a] :
( ( member_option_set_a @ X3 @ A5 )
& ( X3 != none_set_a ) ) ) ) ).
% Some_image_these_eq
thf(fact_880_these__empty,axiom,
( ( these_a @ bot_bot_set_option_a )
= bot_bot_set_a ) ).
% these_empty
thf(fact_881_pair__imageI,axiom,
! [A3: a,B3: a,A5: set_Product_prod_a_a,F: a > a > a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ A5 )
=> ( member_a @ ( F @ A3 @ B3 ) @ ( image_3437945252899457948_a_a_a @ ( produc8815886927560695506_a_a_a @ F ) @ A5 ) ) ) ).
% pair_imageI
thf(fact_882_pair__imageI,axiom,
! [A3: a,B3: a,A5: set_Product_prod_a_a,F: a > a > product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ A5 )
=> ( member1426531477525435216od_a_a @ ( F @ A3 @ B3 ) @ ( image_4636654165204879301od_a_a @ ( produc408267641121961211od_a_a @ F ) @ A5 ) ) ) ).
% pair_imageI
thf(fact_883_pair__imageI,axiom,
! [A3: a,B3: a,A5: set_Product_prod_a_a,F: a > a > $o] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ A5 )
=> ( member_o @ ( F @ A3 @ B3 ) @ ( image_9022731552424948534_a_a_o @ ( produc6436628058953941356_a_a_o @ F ) @ A5 ) ) ) ).
% pair_imageI
thf(fact_884_these__image__Some__eq,axiom,
! [A5: set_set_a] :
( ( these_set_a @ ( image_2983976770393306955_set_a @ some_set_a @ A5 ) )
= A5 ) ).
% these_image_Some_eq
thf(fact_885_Collect__case__prod__mono,axiom,
! [A5: a > a > $o,B5: a > a > $o] :
( ( ord_less_eq_a_a_o @ A5 @ B5 )
=> ( ord_le746702958409616551od_a_a @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ A5 ) ) @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ B5 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_886_pred__subset__eq2,axiom,
! [R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_less_eq_a_a_o
@ ^ [X3: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R2 )
@ ^ [X3: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ S ) )
= ( ord_le746702958409616551od_a_a @ R2 @ S ) ) ).
% pred_subset_eq2
thf(fact_887_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ord_less_eq_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A6 )
@ ^ [X3: a] : ( member_a @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_888_less__eq__set__def,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( ord_le1591150415168442102_a_a_o
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A6 )
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_889_pred__subset__eq,axiom,
! [R2: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X3: a] : ( member_a @ X3 @ R2 )
@ ^ [X3: a] : ( member_a @ X3 @ S ) )
= ( ord_less_eq_set_a @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_890_pred__subset__eq,axiom,
! [R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_le1591150415168442102_a_a_o
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ R2 )
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ S ) )
= ( ord_le746702958409616551od_a_a @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_891_in__these__eq,axiom,
! [X4: a,A5: set_option_a] :
( ( member_a @ X4 @ ( these_a @ A5 ) )
= ( member_option_a @ ( some_a @ X4 ) @ A5 ) ) ).
% in_these_eq
thf(fact_892_in__these__eq,axiom,
! [X4: product_prod_a_a,A5: set_op7160277562814721357od_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ ( these_5100388957577570148od_a_a @ A5 ) )
= ( member8183384484874023062od_a_a @ ( some_P3592067295195376908od_a_a @ X4 ) @ A5 ) ) ).
% in_these_eq
thf(fact_893_in__these__eq,axiom,
! [X4: set_a,A5: set_option_set_a] :
( ( member_set_a @ X4 @ ( these_set_a @ A5 ) )
= ( member_option_set_a @ ( some_set_a @ X4 ) @ A5 ) ) ).
% in_these_eq
thf(fact_894_ssubst__Pair__rhs,axiom,
! [R3: a,S2: a,R2: set_Product_prod_a_a,S4: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ R3 @ S2 ) @ R2 )
=> ( ( S4 = S2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ R3 @ S4 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_895_swap__product,axiom,
! [A5: set_a,B5: set_a] :
( ( image_4636654165204879301od_a_a
@ ( produc408267641121961211od_a_a
@ ^ [I2: a,J2: a] : ( product_Pair_a_a @ J2 @ I2 ) )
@ ( product_Sigma_a_a @ A5
@ ^ [Uu: a] : B5 ) )
= ( product_Sigma_a_a @ B5
@ ^ [Uu: a] : A5 ) ) ).
% swap_product
thf(fact_896_Collect__restrict,axiom,
! [X7: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ X7 )
& ( P @ X3 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_897_Collect__restrict,axiom,
! [X7: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ord_le746702958409616551od_a_a
@ ( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ X7 )
& ( P @ X3 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_898_prop__restrict,axiom,
! [X4: a,Z3: set_a,X7: set_a,P: a > $o] :
( ( member_a @ X4 @ Z3 )
=> ( ( ord_less_eq_set_a @ Z3
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ X7 )
& ( P @ X3 ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_899_prop__restrict,axiom,
! [X4: product_prod_a_a,Z3: set_Product_prod_a_a,X7: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ X4 @ Z3 )
=> ( ( ord_le746702958409616551od_a_a @ Z3
@ ( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ X7 )
& ( P @ X3 ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_900_product__code,axiom,
! [Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a] :
( ( produc6719770313124183570od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ ( set_Product_prod_a_a2 @ Ys ) )
= ( set_Pr6493056779096584572od_a_a
@ ( concat8362408604789070120od_a_a
@ ( map_Pr2879994515260393034od_a_a
@ ^ [X3: product_prod_a_a] : ( map_Pr5477722341457123780od_a_a @ ( produc7886510207707329367od_a_a @ X3 ) @ Ys )
@ Xs ) ) ) ) ).
% product_code
thf(fact_901_product__code,axiom,
! [Xs: list_a,Ys: list_a] :
( ( product_product_a_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
= ( set_Product_prod_a_a2
@ ( concat6071998529708183912od_a_a
@ ( map_a_6618942900564114931od_a_a
@ ^ [X3: a] : ( map_a_7860052162900579309od_a_a @ ( product_Pair_a_a @ X3 ) @ Ys )
@ Xs ) ) ) ) ).
% product_code
thf(fact_902_UNIV__option__conv,axiom,
( top_to5085949387790111389od_a_a
= ( insert8185611903228756413od_a_a @ none_P1914705783780380040od_a_a @ ( image_588318201180639627od_a_a @ some_P3592067295195376908od_a_a @ top_to8063371432257647191od_a_a ) ) ) ).
% UNIV_option_conv
thf(fact_903_UNIV__option__conv,axiom,
( top_to3949272007228979924_set_a
= ( insert_option_set_a @ none_set_a @ ( image_2983976770393306955_set_a @ some_set_a @ top_top_set_set_a ) ) ) ).
% UNIV_option_conv
thf(fact_904_insertCI,axiom,
! [A3: a,B5: set_a,B3: a] :
( ( ~ ( member_a @ A3 @ B5 )
=> ( A3 = B3 ) )
=> ( member_a @ A3 @ ( insert_a @ B3 @ B5 ) ) ) ).
% insertCI
thf(fact_905_insertCI,axiom,
! [A3: product_prod_a_a,B5: set_Product_prod_a_a,B3: product_prod_a_a] :
( ( ~ ( member1426531477525435216od_a_a @ A3 @ B5 )
=> ( A3 = B3 ) )
=> ( member1426531477525435216od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ B3 @ B5 ) ) ) ).
% insertCI
thf(fact_906_insert__iff,axiom,
! [A3: a,B3: a,A5: set_a] :
( ( member_a @ A3 @ ( insert_a @ B3 @ A5 ) )
= ( ( A3 = B3 )
| ( member_a @ A3 @ A5 ) ) ) ).
% insert_iff
thf(fact_907_insert__iff,axiom,
! [A3: product_prod_a_a,B3: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ B3 @ A5 ) )
= ( ( A3 = B3 )
| ( member1426531477525435216od_a_a @ A3 @ A5 ) ) ) ).
% insert_iff
thf(fact_908_singletonI,axiom,
! [A3: product_prod_a_a] : ( member1426531477525435216od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ A3 @ bot_bo3357376287454694259od_a_a ) ) ).
% singletonI
thf(fact_909_singletonI,axiom,
! [A3: a] : ( member_a @ A3 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_910_insert__subset,axiom,
! [X4: a,A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X4 @ A5 ) @ B5 )
= ( ( member_a @ X4 @ B5 )
& ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ).
% insert_subset
thf(fact_911_insert__subset,axiom,
! [X4: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( insert4534936382041156343od_a_a @ X4 @ A5 ) @ B5 )
= ( ( member1426531477525435216od_a_a @ X4 @ B5 )
& ( ord_le746702958409616551od_a_a @ A5 @ B5 ) ) ) ).
% insert_subset
thf(fact_912_Int__insert__left__if0,axiom,
! [A3: product_prod_a_a,C3: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A3 @ C3 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A3 @ B5 ) @ C3 )
= ( inf_in8905007599844390133od_a_a @ B5 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_913_Int__insert__left__if0,axiom,
! [A3: a,C3: set_a,B5: set_a] :
( ~ ( member_a @ A3 @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A3 @ B5 ) @ C3 )
= ( inf_inf_set_a @ B5 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_914_Int__insert__left__if1,axiom,
! [A3: product_prod_a_a,C3: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A3 @ C3 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A3 @ B5 ) @ C3 )
= ( insert4534936382041156343od_a_a @ A3 @ ( inf_in8905007599844390133od_a_a @ B5 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_915_Int__insert__left__if1,axiom,
! [A3: a,C3: set_a,B5: set_a] :
( ( member_a @ A3 @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A3 @ B5 ) @ C3 )
= ( insert_a @ A3 @ ( inf_inf_set_a @ B5 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_916_insert__inter__insert,axiom,
! [A3: a,A5: set_a,B5: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A3 @ A5 ) @ ( insert_a @ A3 @ B5 ) )
= ( insert_a @ A3 @ ( inf_inf_set_a @ A5 @ B5 ) ) ) ).
% insert_inter_insert
thf(fact_917_Int__insert__right__if0,axiom,
! [A3: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A3 @ A5 )
=> ( ( inf_in8905007599844390133od_a_a @ A5 @ ( insert4534936382041156343od_a_a @ A3 @ B5 ) )
= ( inf_in8905007599844390133od_a_a @ A5 @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_918_Int__insert__right__if0,axiom,
! [A3: a,A5: set_a,B5: set_a] :
( ~ ( member_a @ A3 @ A5 )
=> ( ( inf_inf_set_a @ A5 @ ( insert_a @ A3 @ B5 ) )
= ( inf_inf_set_a @ A5 @ B5 ) ) ) ).
% Int_insert_right_if0
thf(fact_919_Int__insert__right__if1,axiom,
! [A3: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A3 @ A5 )
=> ( ( inf_in8905007599844390133od_a_a @ A5 @ ( insert4534936382041156343od_a_a @ A3 @ B5 ) )
= ( insert4534936382041156343od_a_a @ A3 @ ( inf_in8905007599844390133od_a_a @ A5 @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_920_Int__insert__right__if1,axiom,
! [A3: a,A5: set_a,B5: set_a] :
( ( member_a @ A3 @ A5 )
=> ( ( inf_inf_set_a @ A5 @ ( insert_a @ A3 @ B5 ) )
= ( insert_a @ A3 @ ( inf_inf_set_a @ A5 @ B5 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_921_concat__eq__Nil__conv,axiom,
! [Xss: list_l2076173926988081843od_a_a] :
( ( ( concat6071998529708183912od_a_a @ Xss )
= nil_Product_prod_a_a )
= ( ! [X3: list_P1396940483166286381od_a_a] :
( ( member6824001069763096534od_a_a @ X3 @ ( set_li4974249543031949378od_a_a @ Xss ) )
=> ( X3 = nil_Product_prod_a_a ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_922_Nil__eq__concat__conv,axiom,
! [Xss: list_l2076173926988081843od_a_a] :
( ( nil_Product_prod_a_a
= ( concat6071998529708183912od_a_a @ Xss ) )
= ( ! [X3: list_P1396940483166286381od_a_a] :
( ( member6824001069763096534od_a_a @ X3 @ ( set_li4974249543031949378od_a_a @ Xss ) )
=> ( X3 = nil_Product_prod_a_a ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_923_singleton__conv2,axiom,
! [A3: a] :
( ( collect_a
@ ( ^ [Y8: a,Z4: a] : ( Y8 = Z4 )
@ A3 ) )
= ( insert_a @ A3 @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_924_singleton__conv,axiom,
! [A3: a] :
( ( collect_a
@ ^ [X3: a] : ( X3 = A3 ) )
= ( insert_a @ A3 @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_925_singleton__insert__inj__eq,axiom,
! [B3: a,A3: a,A5: set_a] :
( ( ( insert_a @ B3 @ bot_bot_set_a )
= ( insert_a @ A3 @ A5 ) )
= ( ( A3 = B3 )
& ( ord_less_eq_set_a @ A5 @ ( insert_a @ B3 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_926_singleton__insert__inj__eq_H,axiom,
! [A3: a,A5: set_a,B3: a] :
( ( ( insert_a @ A3 @ A5 )
= ( insert_a @ B3 @ bot_bot_set_a ) )
= ( ( A3 = B3 )
& ( ord_less_eq_set_a @ A5 @ ( insert_a @ B3 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_927_insert__disjoint_I1_J,axiom,
! [A3: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A3 @ A5 ) @ B5 )
= bot_bo3357376287454694259od_a_a )
= ( ~ ( member1426531477525435216od_a_a @ A3 @ B5 )
& ( ( inf_in8905007599844390133od_a_a @ A5 @ B5 )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% insert_disjoint(1)
thf(fact_928_insert__disjoint_I1_J,axiom,
! [A3: a,A5: set_a,B5: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A3 @ A5 ) @ B5 )
= bot_bot_set_a )
= ( ~ ( member_a @ A3 @ B5 )
& ( ( inf_inf_set_a @ A5 @ B5 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_929_insert__disjoint_I2_J,axiom,
! [A3: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A3 @ A5 ) @ B5 ) )
= ( ~ ( member1426531477525435216od_a_a @ A3 @ B5 )
& ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A5 @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_930_insert__disjoint_I2_J,axiom,
! [A3: a,A5: set_a,B5: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A3 @ A5 ) @ B5 ) )
= ( ~ ( member_a @ A3 @ B5 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A5 @ B5 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_931_disjoint__insert_I1_J,axiom,
! [B5: set_Product_prod_a_a,A3: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ B5 @ ( insert4534936382041156343od_a_a @ A3 @ A5 ) )
= bot_bo3357376287454694259od_a_a )
= ( ~ ( member1426531477525435216od_a_a @ A3 @ B5 )
& ( ( inf_in8905007599844390133od_a_a @ B5 @ A5 )
= bot_bo3357376287454694259od_a_a ) ) ) ).
% disjoint_insert(1)
thf(fact_932_disjoint__insert_I1_J,axiom,
! [B5: set_a,A3: a,A5: set_a] :
( ( ( inf_inf_set_a @ B5 @ ( insert_a @ A3 @ A5 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A3 @ B5 )
& ( ( inf_inf_set_a @ B5 @ A5 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_933_disjoint__insert_I2_J,axiom,
! [A5: set_Product_prod_a_a,B3: product_prod_a_a,B5: set_Product_prod_a_a] :
( ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A5 @ ( insert4534936382041156343od_a_a @ B3 @ B5 ) ) )
= ( ~ ( member1426531477525435216od_a_a @ B3 @ A5 )
& ( bot_bo3357376287454694259od_a_a
= ( inf_in8905007599844390133od_a_a @ A5 @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_934_disjoint__insert_I2_J,axiom,
! [A5: set_a,B3: a,B5: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A5 @ ( insert_a @ B3 @ B5 ) ) )
= ( ~ ( member_a @ B3 @ A5 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A5 @ B5 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_935_these__insert__None,axiom,
! [A5: set_op7160277562814721357od_a_a] :
( ( these_5100388957577570148od_a_a @ ( insert8185611903228756413od_a_a @ none_P1914705783780380040od_a_a @ A5 ) )
= ( these_5100388957577570148od_a_a @ A5 ) ) ).
% these_insert_None
thf(fact_936_these__insert__Some,axiom,
! [X4: set_a,A5: set_option_set_a] :
( ( these_set_a @ ( insert_option_set_a @ ( some_set_a @ X4 ) @ A5 ) )
= ( insert_set_a @ X4 @ ( these_set_a @ A5 ) ) ) ).
% these_insert_Some
thf(fact_937_singletonD,axiom,
! [B3: product_prod_a_a,A3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B3 @ ( insert4534936382041156343od_a_a @ A3 @ bot_bo3357376287454694259od_a_a ) )
=> ( B3 = A3 ) ) ).
% singletonD
thf(fact_938_singletonD,axiom,
! [B3: a,A3: a] :
( ( member_a @ B3 @ ( insert_a @ A3 @ bot_bot_set_a ) )
=> ( B3 = A3 ) ) ).
% singletonD
thf(fact_939_singleton__iff,axiom,
! [B3: product_prod_a_a,A3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B3 @ ( insert4534936382041156343od_a_a @ A3 @ bot_bo3357376287454694259od_a_a ) )
= ( B3 = A3 ) ) ).
% singleton_iff
thf(fact_940_singleton__iff,axiom,
! [B3: a,A3: a] :
( ( member_a @ B3 @ ( insert_a @ A3 @ bot_bot_set_a ) )
= ( B3 = A3 ) ) ).
% singleton_iff
thf(fact_941_doubleton__eq__iff,axiom,
! [A3: a,B3: a,C: a,D2: a] :
( ( ( insert_a @ A3 @ ( insert_a @ B3 @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A3 = C )
& ( B3 = D2 ) )
| ( ( A3 = D2 )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_942_insert__not__empty,axiom,
! [A3: a,A5: set_a] :
( ( insert_a @ A3 @ A5 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_943_singleton__inject,axiom,
! [A3: a,B3: a] :
( ( ( insert_a @ A3 @ bot_bot_set_a )
= ( insert_a @ B3 @ bot_bot_set_a ) )
=> ( A3 = B3 ) ) ).
% singleton_inject
thf(fact_944_subset__insert,axiom,
! [X4: a,A5: set_a,B5: set_a] :
( ~ ( member_a @ X4 @ A5 )
=> ( ( ord_less_eq_set_a @ A5 @ ( insert_a @ X4 @ B5 ) )
= ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ).
% subset_insert
thf(fact_945_subset__insert,axiom,
! [X4: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X4 @ A5 )
=> ( ( ord_le746702958409616551od_a_a @ A5 @ ( insert4534936382041156343od_a_a @ X4 @ B5 ) )
= ( ord_le746702958409616551od_a_a @ A5 @ B5 ) ) ) ).
% subset_insert
thf(fact_946_Int__insert__left,axiom,
! [A3: product_prod_a_a,C3: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ A3 @ C3 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A3 @ B5 ) @ C3 )
= ( insert4534936382041156343od_a_a @ A3 @ ( inf_in8905007599844390133od_a_a @ B5 @ C3 ) ) ) )
& ( ~ ( member1426531477525435216od_a_a @ A3 @ C3 )
=> ( ( inf_in8905007599844390133od_a_a @ ( insert4534936382041156343od_a_a @ A3 @ B5 ) @ C3 )
= ( inf_in8905007599844390133od_a_a @ B5 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_947_Int__insert__left,axiom,
! [A3: a,C3: set_a,B5: set_a] :
( ( ( member_a @ A3 @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A3 @ B5 ) @ C3 )
= ( insert_a @ A3 @ ( inf_inf_set_a @ B5 @ C3 ) ) ) )
& ( ~ ( member_a @ A3 @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A3 @ B5 ) @ C3 )
= ( inf_inf_set_a @ B5 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_948_Int__insert__right,axiom,
! [A3: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ A3 @ A5 )
=> ( ( inf_in8905007599844390133od_a_a @ A5 @ ( insert4534936382041156343od_a_a @ A3 @ B5 ) )
= ( insert4534936382041156343od_a_a @ A3 @ ( inf_in8905007599844390133od_a_a @ A5 @ B5 ) ) ) )
& ( ~ ( member1426531477525435216od_a_a @ A3 @ A5 )
=> ( ( inf_in8905007599844390133od_a_a @ A5 @ ( insert4534936382041156343od_a_a @ A3 @ B5 ) )
= ( inf_in8905007599844390133od_a_a @ A5 @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_949_Int__insert__right,axiom,
! [A3: a,A5: set_a,B5: set_a] :
( ( ( member_a @ A3 @ A5 )
=> ( ( inf_inf_set_a @ A5 @ ( insert_a @ A3 @ B5 ) )
= ( insert_a @ A3 @ ( inf_inf_set_a @ A5 @ B5 ) ) ) )
& ( ~ ( member_a @ A3 @ A5 )
=> ( ( inf_inf_set_a @ A5 @ ( insert_a @ A3 @ B5 ) )
= ( inf_inf_set_a @ A5 @ B5 ) ) ) ) ).
% Int_insert_right
thf(fact_950_concat_Osimps_I1_J,axiom,
( ( concat6071998529708183912od_a_a @ nil_li8564743906319921459od_a_a )
= nil_Product_prod_a_a ) ).
% concat.simps(1)
thf(fact_951_insert__compr,axiom,
( insert_a
= ( ^ [A2: a,B6: set_a] :
( collect_a
@ ^ [X3: a] :
( ( X3 = A2 )
| ( member_a @ X3 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_952_insert__compr,axiom,
( insert4534936382041156343od_a_a
= ( ^ [A2: product_prod_a_a,B6: set_Product_prod_a_a] :
( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] :
( ( X3 = A2 )
| ( member1426531477525435216od_a_a @ X3 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_953_insertE,axiom,
! [A3: a,B3: a,A5: set_a] :
( ( member_a @ A3 @ ( insert_a @ B3 @ A5 ) )
=> ( ( A3 != B3 )
=> ( member_a @ A3 @ A5 ) ) ) ).
% insertE
thf(fact_954_insertE,axiom,
! [A3: product_prod_a_a,B3: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ B3 @ A5 ) )
=> ( ( A3 != B3 )
=> ( member1426531477525435216od_a_a @ A3 @ A5 ) ) ) ).
% insertE
thf(fact_955_insertI1,axiom,
! [A3: a,B5: set_a] : ( member_a @ A3 @ ( insert_a @ A3 @ B5 ) ) ).
% insertI1
thf(fact_956_insertI1,axiom,
! [A3: product_prod_a_a,B5: set_Product_prod_a_a] : ( member1426531477525435216od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ A3 @ B5 ) ) ).
% insertI1
thf(fact_957_insertI2,axiom,
! [A3: a,B5: set_a,B3: a] :
( ( member_a @ A3 @ B5 )
=> ( member_a @ A3 @ ( insert_a @ B3 @ B5 ) ) ) ).
% insertI2
thf(fact_958_insertI2,axiom,
! [A3: product_prod_a_a,B5: set_Product_prod_a_a,B3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A3 @ B5 )
=> ( member1426531477525435216od_a_a @ A3 @ ( insert4534936382041156343od_a_a @ B3 @ B5 ) ) ) ).
% insertI2
thf(fact_959_Set_Oset__insert,axiom,
! [X4: a,A5: set_a] :
( ( member_a @ X4 @ A5 )
=> ~ ! [B8: set_a] :
( ( A5
= ( insert_a @ X4 @ B8 ) )
=> ( member_a @ X4 @ B8 ) ) ) ).
% Set.set_insert
thf(fact_960_Set_Oset__insert,axiom,
! [X4: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A5 )
=> ~ ! [B8: set_Product_prod_a_a] :
( ( A5
= ( insert4534936382041156343od_a_a @ X4 @ B8 ) )
=> ( member1426531477525435216od_a_a @ X4 @ B8 ) ) ) ).
% Set.set_insert
thf(fact_961_insert__ident,axiom,
! [X4: a,A5: set_a,B5: set_a] :
( ~ ( member_a @ X4 @ A5 )
=> ( ~ ( member_a @ X4 @ B5 )
=> ( ( ( insert_a @ X4 @ A5 )
= ( insert_a @ X4 @ B5 ) )
= ( A5 = B5 ) ) ) ) ).
% insert_ident
thf(fact_962_insert__ident,axiom,
! [X4: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ X4 @ A5 )
=> ( ~ ( member1426531477525435216od_a_a @ X4 @ B5 )
=> ( ( ( insert4534936382041156343od_a_a @ X4 @ A5 )
= ( insert4534936382041156343od_a_a @ X4 @ B5 ) )
= ( A5 = B5 ) ) ) ) ).
% insert_ident
thf(fact_963_insert__absorb,axiom,
! [A3: a,A5: set_a] :
( ( member_a @ A3 @ A5 )
=> ( ( insert_a @ A3 @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_964_insert__absorb,axiom,
! [A3: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A3 @ A5 )
=> ( ( insert4534936382041156343od_a_a @ A3 @ A5 )
= A5 ) ) ).
% insert_absorb
thf(fact_965_insert__eq__iff,axiom,
! [A3: a,A5: set_a,B3: a,B5: set_a] :
( ~ ( member_a @ A3 @ A5 )
=> ( ~ ( member_a @ B3 @ B5 )
=> ( ( ( insert_a @ A3 @ A5 )
= ( insert_a @ B3 @ B5 ) )
= ( ( ( A3 = B3 )
=> ( A5 = B5 ) )
& ( ( A3 != B3 )
=> ? [C4: set_a] :
( ( A5
= ( insert_a @ B3 @ C4 ) )
& ~ ( member_a @ B3 @ C4 )
& ( B5
= ( insert_a @ A3 @ C4 ) )
& ~ ( member_a @ A3 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_966_insert__eq__iff,axiom,
! [A3: product_prod_a_a,A5: set_Product_prod_a_a,B3: product_prod_a_a,B5: set_Product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A3 @ A5 )
=> ( ~ ( member1426531477525435216od_a_a @ B3 @ B5 )
=> ( ( ( insert4534936382041156343od_a_a @ A3 @ A5 )
= ( insert4534936382041156343od_a_a @ B3 @ B5 ) )
= ( ( ( A3 = B3 )
=> ( A5 = B5 ) )
& ( ( A3 != B3 )
=> ? [C4: set_Product_prod_a_a] :
( ( A5
= ( insert4534936382041156343od_a_a @ B3 @ C4 ) )
& ~ ( member1426531477525435216od_a_a @ B3 @ C4 )
& ( B5
= ( insert4534936382041156343od_a_a @ A3 @ C4 ) )
& ~ ( member1426531477525435216od_a_a @ A3 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_967_mk__disjoint__insert,axiom,
! [A3: a,A5: set_a] :
( ( member_a @ A3 @ A5 )
=> ? [B8: set_a] :
( ( A5
= ( insert_a @ A3 @ B8 ) )
& ~ ( member_a @ A3 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_968_mk__disjoint__insert,axiom,
! [A3: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A3 @ A5 )
=> ? [B8: set_Product_prod_a_a] :
( ( A5
= ( insert4534936382041156343od_a_a @ A3 @ B8 ) )
& ~ ( member1426531477525435216od_a_a @ A3 @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_969_Collect__conv__if2,axiom,
! [P: a > $o,A3: a] :
( ( ( P @ A3 )
=> ( ( collect_a
@ ^ [X3: a] :
( ( A3 = X3 )
& ( P @ X3 ) ) )
= ( insert_a @ A3 @ bot_bot_set_a ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect_a
@ ^ [X3: a] :
( ( A3 = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_970_Collect__conv__if,axiom,
! [P: a > $o,A3: a] :
( ( ( P @ A3 )
=> ( ( collect_a
@ ^ [X3: a] :
( ( X3 = A3 )
& ( P @ X3 ) ) )
= ( insert_a @ A3 @ bot_bot_set_a ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect_a
@ ^ [X3: a] :
( ( X3 = A3 )
& ( P @ X3 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_971_refl__on__singleton,axiom,
! [X4: a] : ( refl_on_a @ ( insert_a @ X4 @ bot_bot_set_a ) @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X4 @ X4 ) @ bot_bo3357376287454694259od_a_a ) ) ).
% refl_on_singleton
thf(fact_972_finite_Ocases,axiom,
! [A3: set_a] :
( ( finite_finite_a @ A3 )
=> ( ( A3 != bot_bot_set_a )
=> ~ ! [A8: set_a] :
( ? [A: a] :
( A3
= ( insert_a @ A @ A8 ) )
=> ~ ( finite_finite_a @ A8 ) ) ) ) ).
% finite.cases
thf(fact_973_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A2: set_a] :
( ( A2 = bot_bot_set_a )
| ? [A6: set_a,B: a] :
( ( A2
= ( insert_a @ B @ A6 ) )
& ( finite_finite_a @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_974_finite__induct,axiom,
! [F3: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [X: product_prod_a_a,F4: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F4 )
=> ( ~ ( member1426531477525435216od_a_a @ X @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert4534936382041156343od_a_a @ X @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_975_finite__induct,axiom,
! [F3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ~ ( member_a @ X @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_976_finite__ne__induct,axiom,
! [F3: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ( F3 != bot_bo3357376287454694259od_a_a )
=> ( ! [X: product_prod_a_a] : ( P @ ( insert4534936382041156343od_a_a @ X @ bot_bo3357376287454694259od_a_a ) )
=> ( ! [X: product_prod_a_a,F4: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F4 )
=> ( ( F4 != bot_bo3357376287454694259od_a_a )
=> ( ~ ( member1426531477525435216od_a_a @ X @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert4534936382041156343od_a_a @ X @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_977_finite__ne__induct,axiom,
! [F3: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ! [X: a] : ( P @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ! [X: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( F4 != bot_bot_set_a )
=> ( ~ ( member_a @ X @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_978_infinite__finite__induct,axiom,
! [P: set_Product_prod_a_a > $o,A5: set_Product_prod_a_a] :
( ! [A8: set_Product_prod_a_a] :
( ~ ( finite6544458595007987280od_a_a @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [X: product_prod_a_a,F4: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F4 )
=> ( ~ ( member1426531477525435216od_a_a @ X @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert4534936382041156343od_a_a @ X @ F4 ) ) ) ) )
=> ( P @ A5 ) ) ) ) ).
% infinite_finite_induct
thf(fact_979_infinite__finite__induct,axiom,
! [P: set_a > $o,A5: set_a] :
( ! [A8: set_a] :
( ~ ( finite_finite_a @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ~ ( member_a @ X @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X @ F4 ) ) ) ) )
=> ( P @ A5 ) ) ) ) ).
% infinite_finite_induct
thf(fact_980_linear__order__on__singleton,axiom,
! [X4: a] : ( order_8768733634509060147r_on_a @ ( insert_a @ X4 @ bot_bot_set_a ) @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X4 @ X4 ) @ bot_bo3357376287454694259od_a_a ) ) ).
% linear_order_on_singleton
thf(fact_981_subset__singleton__iff,axiom,
! [X7: set_a,A3: a] :
( ( ord_less_eq_set_a @ X7 @ ( insert_a @ A3 @ bot_bot_set_a ) )
= ( ( X7 = bot_bot_set_a )
| ( X7
= ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_982_subset__singletonD,axiom,
! [A5: set_a,X4: a] :
( ( ord_less_eq_set_a @ A5 @ ( insert_a @ X4 @ bot_bot_set_a ) )
=> ( ( A5 = bot_bot_set_a )
| ( A5
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_983_image__constant,axiom,
! [X4: a,A5: set_a,C: a] :
( ( member_a @ X4 @ A5 )
=> ( ( image_a_a2
@ ^ [X3: a] : C
@ A5 )
= ( insert_a @ C @ bot_bot_set_a ) ) ) ).
% image_constant
thf(fact_984_image__constant,axiom,
! [X4: product_prod_a_a,A5: set_Product_prod_a_a,C: a] :
( ( member1426531477525435216od_a_a @ X4 @ A5 )
=> ( ( image_3437945252899457948_a_a_a
@ ^ [X3: product_prod_a_a] : C
@ A5 )
= ( insert_a @ C @ bot_bot_set_a ) ) ) ).
% image_constant
thf(fact_985_image__constant__conv,axiom,
! [A5: set_a,C: a] :
( ( ( A5 = bot_bot_set_a )
=> ( ( image_a_a2
@ ^ [X3: a] : C
@ A5 )
= bot_bot_set_a ) )
& ( ( A5 != bot_bot_set_a )
=> ( ( image_a_a2
@ ^ [X3: a] : C
@ A5 )
= ( insert_a @ C @ bot_bot_set_a ) ) ) ) ).
% image_constant_conv
thf(fact_986_finite__subset__induct,axiom,
! [F3: set_Product_prod_a_a,A5: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ( ord_le746702958409616551od_a_a @ F3 @ A5 )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [A: product_prod_a_a,F4: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F4 )
=> ( ( member1426531477525435216od_a_a @ A @ A5 )
=> ( ~ ( member1426531477525435216od_a_a @ A @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert4534936382041156343od_a_a @ A @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_987_finite__subset__induct,axiom,
! [F3: set_a,A5: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A5 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A @ A5 )
=> ( ~ ( member_a @ A @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_988_finite__subset__induct_H,axiom,
! [F3: set_Product_prod_a_a,A5: set_Product_prod_a_a,P: set_Product_prod_a_a > $o] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ( ord_le746702958409616551od_a_a @ F3 @ A5 )
=> ( ( P @ bot_bo3357376287454694259od_a_a )
=> ( ! [A: product_prod_a_a,F4: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ F4 )
=> ( ( member1426531477525435216od_a_a @ A @ A5 )
=> ( ( ord_le746702958409616551od_a_a @ F4 @ A5 )
=> ( ~ ( member1426531477525435216od_a_a @ A @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert4534936382041156343od_a_a @ A @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_989_finite__subset__induct_H,axiom,
! [F3: set_a,A5: set_a,P: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A5 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A @ A5 )
=> ( ( ord_less_eq_set_a @ F4 @ A5 )
=> ( ~ ( member_a @ A @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_990_these__empty__eq,axiom,
! [B5: set_op7160277562814721357od_a_a] :
( ( ( these_5100388957577570148od_a_a @ B5 )
= bot_bo3357376287454694259od_a_a )
= ( ( B5 = bot_bo441534785844220345od_a_a )
| ( B5
= ( insert8185611903228756413od_a_a @ none_P1914705783780380040od_a_a @ bot_bo441534785844220345od_a_a ) ) ) ) ).
% these_empty_eq
thf(fact_991_these__empty__eq,axiom,
! [B5: set_option_a] :
( ( ( these_a @ B5 )
= bot_bot_set_a )
= ( ( B5 = bot_bot_set_option_a )
| ( B5
= ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).
% these_empty_eq
thf(fact_992_these__not__empty__eq,axiom,
! [B5: set_op7160277562814721357od_a_a] :
( ( ( these_5100388957577570148od_a_a @ B5 )
!= bot_bo3357376287454694259od_a_a )
= ( ( B5 != bot_bo441534785844220345od_a_a )
& ( B5
!= ( insert8185611903228756413od_a_a @ none_P1914705783780380040od_a_a @ bot_bo441534785844220345od_a_a ) ) ) ) ).
% these_not_empty_eq
thf(fact_993_these__not__empty__eq,axiom,
! [B5: set_option_a] :
( ( ( these_a @ B5 )
!= bot_bot_set_a )
= ( ( B5 != bot_bot_set_option_a )
& ( B5
!= ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).
% these_not_empty_eq
thf(fact_994_product__concat__map,axiom,
( product_a_a
= ( ^ [Xs2: list_a,Ys2: list_a] :
( concat6071998529708183912od_a_a
@ ( map_a_6618942900564114931od_a_a
@ ^ [X3: a] : ( map_a_7860052162900579309od_a_a @ ( product_Pair_a_a @ X3 ) @ Ys2 )
@ Xs2 ) ) ) ) ).
% product_concat_map
thf(fact_995_the__elem__def,axiom,
( the_elem_a
= ( ^ [X10: set_a] :
( the_a
@ ^ [X3: a] :
( X10
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ).
% the_elem_def
thf(fact_996_the__elem__eq,axiom,
! [X4: a] :
( ( the_elem_a @ ( insert_a @ X4 @ bot_bot_set_a ) )
= X4 ) ).
% the_elem_eq
thf(fact_997_is__singletonI,axiom,
! [X4: a] : ( is_singleton_a @ ( insert_a @ X4 @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_998_transymcl__insert__same__class,axiom,
! [X4: a,Y3: a,Qeq: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ ( transymcl_a @ Qeq ) )
=> ( ( transymcl_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ Qeq ) )
= ( transymcl_a @ Qeq ) ) ) ).
% transymcl_insert_same_class
thf(fact_999_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A6: set_a] :
( A6
= ( insert_a @ ( the_elem_a @ A6 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1000_symcl__insert,axiom,
! [X4: a,Y3: a,Qeq: set_Product_prod_a_a] :
( ( symcl_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ Qeq ) )
= ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ Y3 @ X4 ) @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ ( symcl_a @ Qeq ) ) ) ) ).
% symcl_insert
thf(fact_1001_is__singletonI_H,axiom,
! [A5: set_Product_prod_a_a] :
( ( A5 != bot_bo3357376287454694259od_a_a )
=> ( ! [X: product_prod_a_a,Y: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A5 )
=> ( ( member1426531477525435216od_a_a @ Y @ A5 )
=> ( X = Y ) ) )
=> ( is_sin3171834905898671131od_a_a @ A5 ) ) ) ).
% is_singletonI'
thf(fact_1002_is__singletonI_H,axiom,
! [A5: set_a] :
( ( A5 != bot_bot_set_a )
=> ( ! [X: a,Y: a] :
( ( member_a @ X @ A5 )
=> ( ( member_a @ Y @ A5 )
=> ( X = Y ) ) )
=> ( is_singleton_a @ A5 ) ) ) ).
% is_singletonI'
thf(fact_1003_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A6: set_a] :
? [X3: a] :
( A6
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_1004_is__singletonE,axiom,
! [A5: set_a] :
( ( is_singleton_a @ A5 )
=> ~ ! [X: a] :
( A5
!= ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_1005_image__split__eq__Sigma,axiom,
! [F: a > a,G: a > a,A5: set_a] :
( ( image_7400625782589995694od_a_a
@ ^ [X3: a] : ( product_Pair_a_a @ ( F @ X3 ) @ ( G @ X3 ) )
@ A5 )
= ( product_Sigma_a_a @ ( image_a_a2 @ F @ A5 )
@ ^ [X3: a] : ( image_a_a2 @ G @ ( inf_inf_set_a @ ( vimage_a_a @ F @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ A5 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_1006_Field__insert,axiom,
! [A3: a,B3: a,R3: set_Product_prod_a_a] :
( ( field_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ R3 ) )
= ( sup_sup_set_a @ ( insert_a @ A3 @ ( insert_a @ B3 @ bot_bot_set_a ) ) @ ( field_a @ R3 ) ) ) ).
% Field_insert
thf(fact_1007_UnCI,axiom,
! [C: a,B5: set_a,A5: set_a] :
( ( ~ ( member_a @ C @ B5 )
=> ( member_a @ C @ A5 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A5 @ B5 ) ) ) ).
% UnCI
thf(fact_1008_UnCI,axiom,
! [C: product_prod_a_a,B5: set_Product_prod_a_a,A5: set_Product_prod_a_a] :
( ( ~ ( member1426531477525435216od_a_a @ C @ B5 )
=> ( member1426531477525435216od_a_a @ C @ A5 ) )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A5 @ B5 ) ) ) ).
% UnCI
thf(fact_1009_Un__iff,axiom,
! [C: a,A5: set_a,B5: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A5 @ B5 ) )
= ( ( member_a @ C @ A5 )
| ( member_a @ C @ B5 ) ) ) ).
% Un_iff
thf(fact_1010_Un__iff,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A5 @ B5 ) )
= ( ( member1426531477525435216od_a_a @ C @ A5 )
| ( member1426531477525435216od_a_a @ C @ B5 ) ) ) ).
% Un_iff
thf(fact_1011_vimageI,axiom,
! [F: a > a,A3: a,B3: a,B5: set_a] :
( ( ( F @ A3 )
= B3 )
=> ( ( member_a @ B3 @ B5 )
=> ( member_a @ A3 @ ( vimage_a_a @ F @ B5 ) ) ) ) ).
% vimageI
thf(fact_1012_vimageI,axiom,
! [F: product_prod_a_a > a,A3: product_prod_a_a,B3: a,B5: set_a] :
( ( ( F @ A3 )
= B3 )
=> ( ( member_a @ B3 @ B5 )
=> ( member1426531477525435216od_a_a @ A3 @ ( vimage1181244665347930962_a_a_a @ F @ B5 ) ) ) ) ).
% vimageI
thf(fact_1013_vimageI,axiom,
! [F: a > product_prod_a_a,A3: a,B3: product_prod_a_a,B5: set_Product_prod_a_a] :
( ( ( F @ A3 )
= B3 )
=> ( ( member1426531477525435216od_a_a @ B3 @ B5 )
=> ( member_a @ A3 @ ( vimage5143925195038468708od_a_a @ F @ B5 ) ) ) ) ).
% vimageI
thf(fact_1014_vimageI,axiom,
! [F: product_prod_a_a > product_prod_a_a,A3: product_prod_a_a,B3: product_prod_a_a,B5: set_Product_prod_a_a] :
( ( ( F @ A3 )
= B3 )
=> ( ( member1426531477525435216od_a_a @ B3 @ B5 )
=> ( member1426531477525435216od_a_a @ A3 @ ( vimage889120284960592251od_a_a @ F @ B5 ) ) ) ) ).
% vimageI
thf(fact_1015_vimage__eq,axiom,
! [A3: a,F: a > a,B5: set_a] :
( ( member_a @ A3 @ ( vimage_a_a @ F @ B5 ) )
= ( member_a @ ( F @ A3 ) @ B5 ) ) ).
% vimage_eq
thf(fact_1016_vimage__eq,axiom,
! [A3: a,F: a > product_prod_a_a,B5: set_Product_prod_a_a] :
( ( member_a @ A3 @ ( vimage5143925195038468708od_a_a @ F @ B5 ) )
= ( member1426531477525435216od_a_a @ ( F @ A3 ) @ B5 ) ) ).
% vimage_eq
thf(fact_1017_vimage__eq,axiom,
! [A3: product_prod_a_a,F: product_prod_a_a > a,B5: set_a] :
( ( member1426531477525435216od_a_a @ A3 @ ( vimage1181244665347930962_a_a_a @ F @ B5 ) )
= ( member_a @ ( F @ A3 ) @ B5 ) ) ).
% vimage_eq
thf(fact_1018_vimage__eq,axiom,
! [A3: product_prod_a_a,F: product_prod_a_a > product_prod_a_a,B5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A3 @ ( vimage889120284960592251od_a_a @ F @ B5 ) )
= ( member1426531477525435216od_a_a @ ( F @ A3 ) @ B5 ) ) ).
% vimage_eq
thf(fact_1019_sup__bot_Oright__neutral,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ A3 @ bot_bot_set_a )
= A3 ) ).
% sup_bot.right_neutral
thf(fact_1020_sup__bot_Oneutr__eq__iff,axiom,
! [A3: set_a,B3: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A3 @ B3 ) )
= ( ( A3 = bot_bot_set_a )
& ( B3 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_1021_sup__bot_Oleft__neutral,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A3 )
= A3 ) ).
% sup_bot.left_neutral
thf(fact_1022_sup__bot_Oeq__neutr__iff,axiom,
! [A3: set_a,B3: set_a] :
( ( ( sup_sup_set_a @ A3 @ B3 )
= bot_bot_set_a )
= ( ( A3 = bot_bot_set_a )
& ( B3 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_1023_sup__eq__bot__iff,axiom,
! [X4: set_a,Y3: set_a] :
( ( ( sup_sup_set_a @ X4 @ Y3 )
= bot_bot_set_a )
= ( ( X4 = bot_bot_set_a )
& ( Y3 = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_1024_bot__eq__sup__iff,axiom,
! [X4: set_a,Y3: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X4 @ Y3 ) )
= ( ( X4 = bot_bot_set_a )
& ( Y3 = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_1025_sup__bot__right,axiom,
! [X4: set_a] :
( ( sup_sup_set_a @ X4 @ bot_bot_set_a )
= X4 ) ).
% sup_bot_right
thf(fact_1026_sup__bot__left,axiom,
! [X4: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_1027_inf__sup__absorb,axiom,
! [X4: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X4 @ ( sup_sup_set_a @ X4 @ Y3 ) )
= X4 ) ).
% inf_sup_absorb
thf(fact_1028_sup__inf__absorb,axiom,
! [X4: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X4 @ ( inf_inf_set_a @ X4 @ Y3 ) )
= X4 ) ).
% sup_inf_absorb
thf(fact_1029_Un__empty,axiom,
! [A5: set_a,B5: set_a] :
( ( ( sup_sup_set_a @ A5 @ B5 )
= bot_bot_set_a )
= ( ( A5 = bot_bot_set_a )
& ( B5 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_1030_Int__Un__eq_I4_J,axiom,
! [T3: set_a,S: set_a] :
( ( sup_sup_set_a @ T3 @ ( inf_inf_set_a @ S @ T3 ) )
= T3 ) ).
% Int_Un_eq(4)
thf(fact_1031_Int__Un__eq_I3_J,axiom,
! [S: set_a,T3: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T3 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_1032_Int__Un__eq_I2_J,axiom,
! [S: set_a,T3: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T3 ) @ T3 )
= T3 ) ).
% Int_Un_eq(2)
thf(fact_1033_Int__Un__eq_I1_J,axiom,
! [S: set_a,T3: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T3 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_1034_Un__Int__eq_I4_J,axiom,
! [T3: set_a,S: set_a] :
( ( inf_inf_set_a @ T3 @ ( sup_sup_set_a @ S @ T3 ) )
= T3 ) ).
% Un_Int_eq(4)
thf(fact_1035_Un__Int__eq_I3_J,axiom,
! [S: set_a,T3: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T3 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_1036_Un__Int__eq_I2_J,axiom,
! [S: set_a,T3: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T3 ) @ T3 )
= T3 ) ).
% Un_Int_eq(2)
thf(fact_1037_Un__Int__eq_I1_J,axiom,
! [S: set_a,T3: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T3 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_1038_vimage__empty,axiom,
! [F: a > a] :
( ( vimage_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% vimage_empty
thf(fact_1039_vimage__Int,axiom,
! [F: a > a,A5: set_a,B5: set_a] :
( ( vimage_a_a @ F @ ( inf_inf_set_a @ A5 @ B5 ) )
= ( inf_inf_set_a @ ( vimage_a_a @ F @ A5 ) @ ( vimage_a_a @ F @ B5 ) ) ) ).
% vimage_Int
thf(fact_1040_vimage__const,axiom,
! [C: a,A5: set_a] :
( ( ( member_a @ C @ A5 )
=> ( ( vimage_a_a
@ ^ [X3: a] : C
@ A5 )
= top_top_set_a ) )
& ( ~ ( member_a @ C @ A5 )
=> ( ( vimage_a_a
@ ^ [X3: a] : C
@ A5 )
= bot_bot_set_a ) ) ) ).
% vimage_const
thf(fact_1041_vimage__const,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( ( member1426531477525435216od_a_a @ C @ A5 )
=> ( ( vimage5143925195038468708od_a_a
@ ^ [X3: a] : C
@ A5 )
= top_top_set_a ) )
& ( ~ ( member1426531477525435216od_a_a @ C @ A5 )
=> ( ( vimage5143925195038468708od_a_a
@ ^ [X3: a] : C
@ A5 )
= bot_bot_set_a ) ) ) ).
% vimage_const
thf(fact_1042_insert__Times__insert,axiom,
! [A3: a,A5: set_a,B3: a,B5: set_a] :
( ( product_Sigma_a_a @ ( insert_a @ A3 @ A5 )
@ ^ [Uu: a] : ( insert_a @ B3 @ B5 ) )
= ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A3 @ B3 )
@ ( sup_su3048258781599657691od_a_a
@ ( product_Sigma_a_a @ A5
@ ^ [Uu: a] : ( insert_a @ B3 @ B5 ) )
@ ( product_Sigma_a_a @ ( insert_a @ A3 @ A5 )
@ ^ [Uu: a] : B5 ) ) ) ) ).
% insert_Times_insert
thf(fact_1043_distrib__imp1,axiom,
! [X4: set_a,Y3: set_a,Z: set_a] :
( ! [X: set_a,Y: set_a,Z2: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z2 ) ) )
=> ( ( sup_sup_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X4 @ Y3 ) @ ( sup_sup_set_a @ X4 @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_1044_distrib__imp2,axiom,
! [X4: set_a,Y3: set_a,Z: set_a] :
( ! [X: set_a,Y: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z2 ) ) )
=> ( ( inf_inf_set_a @ X4 @ ( sup_sup_set_a @ Y3 @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ ( inf_inf_set_a @ X4 @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_1045_inf__sup__distrib1,axiom,
! [X4: set_a,Y3: set_a,Z: set_a] :
( ( inf_inf_set_a @ X4 @ ( sup_sup_set_a @ Y3 @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ ( inf_inf_set_a @ X4 @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_1046_inf__sup__distrib2,axiom,
! [Y3: set_a,Z: set_a,X4: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y3 @ Z ) @ X4 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y3 @ X4 ) @ ( inf_inf_set_a @ Z @ X4 ) ) ) ).
% inf_sup_distrib2
thf(fact_1047_sup__inf__distrib1,axiom,
! [X4: set_a,Y3: set_a,Z: set_a] :
( ( sup_sup_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X4 @ Y3 ) @ ( sup_sup_set_a @ X4 @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_1048_sup__inf__distrib2,axiom,
! [Y3: set_a,Z: set_a,X4: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y3 @ Z ) @ X4 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y3 @ X4 ) @ ( sup_sup_set_a @ Z @ X4 ) ) ) ).
% sup_inf_distrib2
thf(fact_1049_boolean__algebra_Oconj__disj__distrib,axiom,
! [X4: set_a,Y3: set_a,Z: set_a] :
( ( inf_inf_set_a @ X4 @ ( sup_sup_set_a @ Y3 @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ ( inf_inf_set_a @ X4 @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1050_boolean__algebra_Odisj__conj__distrib,axiom,
! [X4: set_a,Y3: set_a,Z: set_a] :
( ( sup_sup_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X4 @ Y3 ) @ ( sup_sup_set_a @ X4 @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1051_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y3: set_a,Z: set_a,X4: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y3 @ Z ) @ X4 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y3 @ X4 ) @ ( inf_inf_set_a @ Z @ X4 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_1052_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y3: set_a,Z: set_a,X4: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y3 @ Z ) @ X4 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y3 @ X4 ) @ ( sup_sup_set_a @ Z @ X4 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_1053_Un__Int__crazy,axiom,
! [A5: set_a,B5: set_a,C3: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ ( inf_inf_set_a @ B5 @ C3 ) ) @ ( inf_inf_set_a @ C3 @ A5 ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A5 @ B5 ) @ ( sup_sup_set_a @ B5 @ C3 ) ) @ ( sup_sup_set_a @ C3 @ A5 ) ) ) ).
% Un_Int_crazy
thf(fact_1054_Int__Un__distrib,axiom,
! [A5: set_a,B5: set_a,C3: set_a] :
( ( inf_inf_set_a @ A5 @ ( sup_sup_set_a @ B5 @ C3 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ ( inf_inf_set_a @ A5 @ C3 ) ) ) ).
% Int_Un_distrib
thf(fact_1055_Un__Int__distrib,axiom,
! [A5: set_a,B5: set_a,C3: set_a] :
( ( sup_sup_set_a @ A5 @ ( inf_inf_set_a @ B5 @ C3 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A5 @ B5 ) @ ( sup_sup_set_a @ A5 @ C3 ) ) ) ).
% Un_Int_distrib
thf(fact_1056_Int__Un__distrib2,axiom,
! [B5: set_a,C3: set_a,A5: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B5 @ C3 ) @ A5 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B5 @ A5 ) @ ( inf_inf_set_a @ C3 @ A5 ) ) ) ).
% Int_Un_distrib2
thf(fact_1057_Un__Int__distrib2,axiom,
! [B5: set_a,C3: set_a,A5: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B5 @ C3 ) @ A5 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B5 @ A5 ) @ ( sup_sup_set_a @ C3 @ A5 ) ) ) ).
% Un_Int_distrib2
thf(fact_1058_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A6 )
| ( member_a @ X3 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_1059_Un__def,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A6 )
| ( member1426531477525435216od_a_a @ X3 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_1060_UnE,axiom,
! [C: a,A5: set_a,B5: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A5 @ B5 ) )
=> ( ~ ( member_a @ C @ A5 )
=> ( member_a @ C @ B5 ) ) ) ).
% UnE
thf(fact_1061_UnE,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A5 @ B5 ) )
=> ( ~ ( member1426531477525435216od_a_a @ C @ A5 )
=> ( member1426531477525435216od_a_a @ C @ B5 ) ) ) ).
% UnE
thf(fact_1062_UnI1,axiom,
! [C: a,A5: set_a,B5: set_a] :
( ( member_a @ C @ A5 )
=> ( member_a @ C @ ( sup_sup_set_a @ A5 @ B5 ) ) ) ).
% UnI1
thf(fact_1063_UnI1,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A5 )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A5 @ B5 ) ) ) ).
% UnI1
thf(fact_1064_UnI2,axiom,
! [C: a,B5: set_a,A5: set_a] :
( ( member_a @ C @ B5 )
=> ( member_a @ C @ ( sup_sup_set_a @ A5 @ B5 ) ) ) ).
% UnI2
thf(fact_1065_UnI2,axiom,
! [C: product_prod_a_a,B5: set_Product_prod_a_a,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ B5 )
=> ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A5 @ B5 ) ) ) ).
% UnI2
thf(fact_1066_vimageD,axiom,
! [A3: a,F: a > a,A5: set_a] :
( ( member_a @ A3 @ ( vimage_a_a @ F @ A5 ) )
=> ( member_a @ ( F @ A3 ) @ A5 ) ) ).
% vimageD
thf(fact_1067_vimageD,axiom,
! [A3: a,F: a > product_prod_a_a,A5: set_Product_prod_a_a] :
( ( member_a @ A3 @ ( vimage5143925195038468708od_a_a @ F @ A5 ) )
=> ( member1426531477525435216od_a_a @ ( F @ A3 ) @ A5 ) ) ).
% vimageD
thf(fact_1068_vimageD,axiom,
! [A3: product_prod_a_a,F: product_prod_a_a > a,A5: set_a] :
( ( member1426531477525435216od_a_a @ A3 @ ( vimage1181244665347930962_a_a_a @ F @ A5 ) )
=> ( member_a @ ( F @ A3 ) @ A5 ) ) ).
% vimageD
thf(fact_1069_vimageD,axiom,
! [A3: product_prod_a_a,F: product_prod_a_a > product_prod_a_a,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A3 @ ( vimage889120284960592251od_a_a @ F @ A5 ) )
=> ( member1426531477525435216od_a_a @ ( F @ A3 ) @ A5 ) ) ).
% vimageD
thf(fact_1070_vimageE,axiom,
! [A3: a,F: a > a,B5: set_a] :
( ( member_a @ A3 @ ( vimage_a_a @ F @ B5 ) )
=> ( member_a @ ( F @ A3 ) @ B5 ) ) ).
% vimageE
thf(fact_1071_vimageE,axiom,
! [A3: a,F: a > product_prod_a_a,B5: set_Product_prod_a_a] :
( ( member_a @ A3 @ ( vimage5143925195038468708od_a_a @ F @ B5 ) )
=> ( member1426531477525435216od_a_a @ ( F @ A3 ) @ B5 ) ) ).
% vimageE
thf(fact_1072_vimageE,axiom,
! [A3: product_prod_a_a,F: product_prod_a_a > a,B5: set_a] :
( ( member1426531477525435216od_a_a @ A3 @ ( vimage1181244665347930962_a_a_a @ F @ B5 ) )
=> ( member_a @ ( F @ A3 ) @ B5 ) ) ).
% vimageE
thf(fact_1073_vimageE,axiom,
! [A3: product_prod_a_a,F: product_prod_a_a > product_prod_a_a,B5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A3 @ ( vimage889120284960592251od_a_a @ F @ B5 ) )
=> ( member1426531477525435216od_a_a @ ( F @ A3 ) @ B5 ) ) ).
% vimageE
thf(fact_1074_vimageI2,axiom,
! [F: a > a,A3: a,A5: set_a] :
( ( member_a @ ( F @ A3 ) @ A5 )
=> ( member_a @ A3 @ ( vimage_a_a @ F @ A5 ) ) ) ).
% vimageI2
thf(fact_1075_vimageI2,axiom,
! [F: product_prod_a_a > a,A3: product_prod_a_a,A5: set_a] :
( ( member_a @ ( F @ A3 ) @ A5 )
=> ( member1426531477525435216od_a_a @ A3 @ ( vimage1181244665347930962_a_a_a @ F @ A5 ) ) ) ).
% vimageI2
thf(fact_1076_vimageI2,axiom,
! [F: a > product_prod_a_a,A3: a,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( F @ A3 ) @ A5 )
=> ( member_a @ A3 @ ( vimage5143925195038468708od_a_a @ F @ A5 ) ) ) ).
% vimageI2
thf(fact_1077_vimageI2,axiom,
! [F: product_prod_a_a > product_prod_a_a,A3: product_prod_a_a,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( F @ A3 ) @ A5 )
=> ( member1426531477525435216od_a_a @ A3 @ ( vimage889120284960592251od_a_a @ F @ A5 ) ) ) ).
% vimageI2
thf(fact_1078_Un__empty__right,axiom,
! [A5: set_a] :
( ( sup_sup_set_a @ A5 @ bot_bot_set_a )
= A5 ) ).
% Un_empty_right
thf(fact_1079_Un__empty__left,axiom,
! [B5: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B5 )
= B5 ) ).
% Un_empty_left
thf(fact_1080_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_a] :
( ( sup_sup_set_a @ X4 @ bot_bot_set_a )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_1081_vimage__singleton__eq,axiom,
! [A3: a,F: a > a,B3: a] :
( ( member_a @ A3 @ ( vimage_a_a @ F @ ( insert_a @ B3 @ bot_bot_set_a ) ) )
= ( ( F @ A3 )
= B3 ) ) ).
% vimage_singleton_eq
thf(fact_1082_vimage__singleton__eq,axiom,
! [A3: product_prod_a_a,F: product_prod_a_a > a,B3: a] :
( ( member1426531477525435216od_a_a @ A3 @ ( vimage1181244665347930962_a_a_a @ F @ ( insert_a @ B3 @ bot_bot_set_a ) ) )
= ( ( F @ A3 )
= B3 ) ) ).
% vimage_singleton_eq
thf(fact_1083_distrib__inf__le,axiom,
! [X4: set_a,Y3: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X4 @ Y3 ) @ ( inf_inf_set_a @ X4 @ Z ) ) @ ( inf_inf_set_a @ X4 @ ( sup_sup_set_a @ Y3 @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1084_distrib__sup__le,axiom,
! [X4: set_a,Y3: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X4 @ ( inf_inf_set_a @ Y3 @ Z ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X4 @ Y3 ) @ ( sup_sup_set_a @ X4 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1085_singleton__Un__iff,axiom,
! [X4: a,A5: set_a,B5: set_a] :
( ( ( insert_a @ X4 @ bot_bot_set_a )
= ( sup_sup_set_a @ A5 @ B5 ) )
= ( ( ( A5 = bot_bot_set_a )
& ( B5
= ( insert_a @ X4 @ bot_bot_set_a ) ) )
| ( ( A5
= ( insert_a @ X4 @ bot_bot_set_a ) )
& ( B5 = bot_bot_set_a ) )
| ( ( A5
= ( insert_a @ X4 @ bot_bot_set_a ) )
& ( B5
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1086_Un__singleton__iff,axiom,
! [A5: set_a,B5: set_a,X4: a] :
( ( ( sup_sup_set_a @ A5 @ B5 )
= ( insert_a @ X4 @ bot_bot_set_a ) )
= ( ( ( A5 = bot_bot_set_a )
& ( B5
= ( insert_a @ X4 @ bot_bot_set_a ) ) )
| ( ( A5
= ( insert_a @ X4 @ bot_bot_set_a ) )
& ( B5 = bot_bot_set_a ) )
| ( ( A5
= ( insert_a @ X4 @ bot_bot_set_a ) )
& ( B5
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1087_insert__is__Un,axiom,
( insert_a
= ( ^ [A2: a] : ( sup_sup_set_a @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_1088_Un__Int__assoc__eq,axiom,
! [A5: set_a,B5: set_a,C3: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A5 @ B5 ) @ C3 )
= ( inf_inf_set_a @ A5 @ ( sup_sup_set_a @ B5 @ C3 ) ) )
= ( ord_less_eq_set_a @ C3 @ A5 ) ) ).
% Un_Int_assoc_eq
thf(fact_1089_Pair__vimage__Sigma,axiom,
! [X4: product_prod_a_a,A5: set_Product_prod_a_a,F: product_prod_a_a > set_a] :
( ( ( member1426531477525435216od_a_a @ X4 @ A5 )
=> ( ( vimage434944399782712955_a_a_a @ ( produc5692536848587512110_a_a_a @ X4 ) @ ( produc2379640491490746847_a_a_a @ A5 @ F ) )
= ( F @ X4 ) ) )
& ( ~ ( member1426531477525435216od_a_a @ X4 @ A5 )
=> ( ( vimage434944399782712955_a_a_a @ ( produc5692536848587512110_a_a_a @ X4 ) @ ( produc2379640491490746847_a_a_a @ A5 @ F ) )
= bot_bot_set_a ) ) ) ).
% Pair_vimage_Sigma
thf(fact_1090_Pair__vimage__Sigma,axiom,
! [X4: a,A5: set_a,F: a > set_a] :
( ( ( member_a @ X4 @ A5 )
=> ( ( vimage5143925195038468708od_a_a @ ( product_Pair_a_a @ X4 ) @ ( product_Sigma_a_a @ A5 @ F ) )
= ( F @ X4 ) ) )
& ( ~ ( member_a @ X4 @ A5 )
=> ( ( vimage5143925195038468708od_a_a @ ( product_Pair_a_a @ X4 ) @ ( product_Sigma_a_a @ A5 @ F ) )
= bot_bot_set_a ) ) ) ).
% Pair_vimage_Sigma
thf(fact_1091_boolean__algebra_Ocomplement__unique,axiom,
! [A3: set_a,X4: set_a,Y3: set_a] :
( ( ( inf_inf_set_a @ A3 @ X4 )
= bot_bot_set_a )
=> ( ( ( sup_sup_set_a @ A3 @ X4 )
= top_top_set_a )
=> ( ( ( inf_inf_set_a @ A3 @ Y3 )
= bot_bot_set_a )
=> ( ( ( sup_sup_set_a @ A3 @ Y3 )
= top_top_set_a )
=> ( X4 = Y3 ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_1092_finite__finite__vimage__IntI,axiom,
! [F3: set_Product_prod_a_a,H: a > product_prod_a_a,A5: set_a] :
( ( finite6544458595007987280od_a_a @ F3 )
=> ( ! [Y: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y @ F3 )
=> ( finite_finite_a @ ( inf_inf_set_a @ ( vimage5143925195038468708od_a_a @ H @ ( insert4534936382041156343od_a_a @ Y @ bot_bo3357376287454694259od_a_a ) ) @ A5 ) ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ ( vimage5143925195038468708od_a_a @ H @ F3 ) @ A5 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_1093_finite__finite__vimage__IntI,axiom,
! [F3: set_a,H: a > a,A5: set_a] :
( ( finite_finite_a @ F3 )
=> ( ! [Y: a] :
( ( member_a @ Y @ F3 )
=> ( finite_finite_a @ ( inf_inf_set_a @ ( vimage_a_a @ H @ ( insert_a @ Y @ bot_bot_set_a ) ) @ A5 ) ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ ( vimage_a_a @ H @ F3 ) @ A5 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_1094_inf__img__fin__domE_H,axiom,
! [F: a > product_prod_a_a,A5: set_a] :
( ( finite6544458595007987280od_a_a @ ( image_7400625782589995694od_a_a @ F @ A5 ) )
=> ( ~ ( finite_finite_a @ A5 )
=> ~ ! [Y: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y @ ( image_7400625782589995694od_a_a @ F @ A5 ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ ( vimage5143925195038468708od_a_a @ F @ ( insert4534936382041156343od_a_a @ Y @ bot_bo3357376287454694259od_a_a ) ) @ A5 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_1095_inf__img__fin__domE_H,axiom,
! [F: a > a,A5: set_a] :
( ( finite_finite_a @ ( image_a_a2 @ F @ A5 ) )
=> ( ~ ( finite_finite_a @ A5 )
=> ~ ! [Y: a] :
( ( member_a @ Y @ ( image_a_a2 @ F @ A5 ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ ( vimage_a_a @ F @ ( insert_a @ Y @ bot_bot_set_a ) ) @ A5 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_1096_inf__img__fin__dom_H,axiom,
! [F: a > a,A5: set_a] :
( ( finite_finite_a @ ( image_a_a2 @ F @ A5 ) )
=> ( ~ ( finite_finite_a @ A5 )
=> ? [X: a] :
( ( member_a @ X @ ( image_a_a2 @ F @ A5 ) )
& ~ ( finite_finite_a @ ( inf_inf_set_a @ ( vimage_a_a @ F @ ( insert_a @ X @ bot_bot_set_a ) ) @ A5 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_1097_transymcl__insert__both__new,axiom,
! [A3: product_prod_a_a,Qeq: set_Pr8600417178894128327od_a_a,B3: product_prod_a_a] :
( ~ ( member1426531477525435216od_a_a @ A3 @ ( field_1126092520709947252od_a_a @ Qeq ) )
=> ( ~ ( member1426531477525435216od_a_a @ B3 @ ( field_1126092520709947252od_a_a @ Qeq ) )
=> ( ( transy2380204104860363337od_a_a @ ( insert8933621553020740791od_a_a @ ( produc7886510207707329367od_a_a @ A3 @ B3 ) @ Qeq ) )
= ( sup_su8193676293155882651od_a_a @ ( transy2380204104860363337od_a_a @ Qeq ) @ ( insert8933621553020740791od_a_a @ ( produc7886510207707329367od_a_a @ A3 @ A3 ) @ ( insert8933621553020740791od_a_a @ ( produc7886510207707329367od_a_a @ B3 @ B3 ) @ ( insert8933621553020740791od_a_a @ ( produc7886510207707329367od_a_a @ A3 @ B3 ) @ ( insert8933621553020740791od_a_a @ ( produc7886510207707329367od_a_a @ B3 @ A3 ) @ bot_bo510284599550014259od_a_a ) ) ) ) ) ) ) ) ).
% transymcl_insert_both_new
thf(fact_1098_transymcl__insert__both__new,axiom,
! [A3: a,Qeq: set_Product_prod_a_a,B3: a] :
( ~ ( member_a @ A3 @ ( field_a @ Qeq ) )
=> ( ~ ( member_a @ B3 @ ( field_a @ Qeq ) )
=> ( ( transymcl_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ Qeq ) )
= ( sup_su3048258781599657691od_a_a @ ( transymcl_a @ Qeq ) @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A3 @ A3 ) @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ B3 @ B3 ) @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ B3 @ A3 ) @ bot_bo3357376287454694259od_a_a ) ) ) ) ) ) ) ) ).
% transymcl_insert_both_new
thf(fact_1099_surj__vimage__empty,axiom,
! [F: a > a,A5: set_a] :
( ( ( image_a_a2 @ F @ top_top_set_a )
= top_top_set_a )
=> ( ( ( vimage_a_a @ F @ A5 )
= bot_bot_set_a )
= ( A5 = bot_bot_set_a ) ) ) ).
% surj_vimage_empty
thf(fact_1100_set__union,axiom,
! [Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a] :
( ( set_Product_prod_a_a2 @ ( union_7798659288537573153od_a_a @ Xs @ Ys ) )
= ( sup_su3048258781599657691od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ ( set_Product_prod_a_a2 @ Ys ) ) ) ).
% set_union
thf(fact_1101_set__rec,axiom,
( set_Product_prod_a_a2
= ( rec_li798439049087562902od_a_a @ bot_bo3357376287454694259od_a_a
@ ^ [X3: product_prod_a_a,Uu: list_P1396940483166286381od_a_a] : ( insert4534936382041156343od_a_a @ X3 ) ) ) ).
% set_rec
thf(fact_1102_set__rec,axiom,
( set_a2
= ( rec_list_set_a_a @ bot_bot_set_a
@ ^ [X3: a,Uu: list_a] : ( insert_a @ X3 ) ) ) ).
% set_rec
thf(fact_1103_sup__Un__eq,axiom,
! [R2: set_a,S: set_a] :
( ( sup_sup_a_o
@ ^ [X3: a] : ( member_a @ X3 @ R2 )
@ ^ [X3: a] : ( member_a @ X3 @ S ) )
= ( ^ [X3: a] : ( member_a @ X3 @ ( sup_sup_set_a @ R2 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1104_sup__Un__eq,axiom,
! [R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( sup_su1640154476453168578_a_a_o
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ R2 )
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ S ) )
= ( ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ ( sup_su3048258781599657691od_a_a @ R2 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1105_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B6: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A6 )
@ ^ [X3: a] : ( member_a @ X3 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_1106_sup__set__def,axiom,
( sup_su3048258781599657691od_a_a
= ( ^ [A6: set_Product_prod_a_a,B6: set_Product_prod_a_a] :
( collec3336397797384452498od_a_a
@ ( sup_su1640154476453168578_a_a_o
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A6 )
@ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_1107_sup__Un__eq2,axiom,
! [R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( sup_sup_a_a_o
@ ^ [X3: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R2 )
@ ^ [X3: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ S ) )
= ( ^ [X3: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ ( sup_su3048258781599657691od_a_a @ R2 @ S ) ) ) ) ).
% sup_Un_eq2
thf(fact_1108_inj__on__vimage__singleton,axiom,
! [F: product_prod_a_a > a,A5: set_Product_prod_a_a,A3: a] :
( ( inj_on4978979553551044360_a_a_a @ F @ A5 )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ ( vimage1181244665347930962_a_a_a @ F @ ( insert_a @ A3 @ bot_bot_set_a ) ) @ A5 )
@ ( insert4534936382041156343od_a_a
@ ( the_Product_prod_a_a
@ ^ [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A5 )
& ( ( F @ X3 )
= A3 ) ) )
@ bot_bo3357376287454694259od_a_a ) ) ) ).
% inj_on_vimage_singleton
thf(fact_1109_inj__on__vimage__singleton,axiom,
! [F: a > a,A5: set_a,A3: a] :
( ( inj_on_a_a @ F @ A5 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ ( vimage_a_a @ F @ ( insert_a @ A3 @ bot_bot_set_a ) ) @ A5 )
@ ( insert_a
@ ( the_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A5 )
& ( ( F @ X3 )
= A3 ) ) )
@ bot_bot_set_a ) ) ) ).
% inj_on_vimage_singleton
thf(fact_1110_inj__singleton,axiom,
! [A5: set_a] :
( inj_on_a_set_a
@ ^ [X3: a] : ( insert_a @ X3 @ bot_bot_set_a )
@ A5 ) ).
% inj_singleton
thf(fact_1111_swap__inj__on,axiom,
! [A5: set_Product_prod_a_a] :
( inj_on2566144670800592689od_a_a
@ ( produc408267641121961211od_a_a
@ ^ [I2: a,J2: a] : ( product_Pair_a_a @ J2 @ I2 ) )
@ A5 ) ).
% swap_inj_on
thf(fact_1112_inj__graph,axiom,
( inj_on8154521630074256687od_a_a
@ ^ [F2: a > a] :
( collec3336397797384452498od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X3: a,Y2: a] :
( Y2
= ( F2 @ X3 ) ) ) )
@ top_top_set_a_a ) ).
% inj_graph
thf(fact_1113_inj__Some,axiom,
! [A5: set_set_a] : ( inj_on1261829562225431735_set_a @ some_set_a @ A5 ) ).
% inj_Some
thf(fact_1114_inj__on__convol__ident,axiom,
! [F: a > a,X7: set_a] :
( inj_on8941660083241582106od_a_a
@ ^ [X3: a] : ( product_Pair_a_a @ X3 @ ( F @ X3 ) )
@ X7 ) ).
% inj_on_convol_ident
thf(fact_1115_finite__inverse__image__gen,axiom,
! [A5: set_a,F: a > a,D: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( inj_on_a_a @ F @ D )
=> ( finite_finite_a
@ ( collect_a
@ ^ [J2: a] :
( ( member_a @ J2 @ D )
& ( member_a @ ( F @ J2 ) @ A5 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_1116_finite__inverse__image__gen,axiom,
! [A5: set_Product_prod_a_a,F: a > product_prod_a_a,D: set_a] :
( ( finite6544458595007987280od_a_a @ A5 )
=> ( ( inj_on8941660083241582106od_a_a @ F @ D )
=> ( finite_finite_a
@ ( collect_a
@ ^ [J2: a] :
( ( member_a @ J2 @ D )
& ( member1426531477525435216od_a_a @ ( F @ J2 ) @ A5 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_1117_finite__inverse__image__gen,axiom,
! [A5: set_a,F: product_prod_a_a > a,D: set_Product_prod_a_a] :
( ( finite_finite_a @ A5 )
=> ( ( inj_on4978979553551044360_a_a_a @ F @ D )
=> ( finite6544458595007987280od_a_a
@ ( collec3336397797384452498od_a_a
@ ^ [J2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ J2 @ D )
& ( member_a @ ( F @ J2 ) @ A5 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_1118_finite__inverse__image__gen,axiom,
! [A5: set_Product_prod_a_a,F: product_prod_a_a > product_prod_a_a,D: set_Product_prod_a_a] :
( ( finite6544458595007987280od_a_a @ A5 )
=> ( ( inj_on2566144670800592689od_a_a @ F @ D )
=> ( finite6544458595007987280od_a_a
@ ( collec3336397797384452498od_a_a
@ ^ [J2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ J2 @ D )
& ( member1426531477525435216od_a_a @ ( F @ J2 ) @ A5 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_1119_image__Int,axiom,
! [F: a > a,A5: set_a,B5: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( image_a_a2 @ F @ ( inf_inf_set_a @ A5 @ B5 ) )
= ( inf_inf_set_a @ ( image_a_a2 @ F @ A5 ) @ ( image_a_a2 @ F @ B5 ) ) ) ) ).
% image_Int
thf(fact_1120_inj__on__image__Int,axiom,
! [F: a > a,C3: set_a,A5: set_a,B5: set_a] :
( ( inj_on_a_a @ F @ C3 )
=> ( ( ord_less_eq_set_a @ A5 @ C3 )
=> ( ( ord_less_eq_set_a @ B5 @ C3 )
=> ( ( image_a_a2 @ F @ ( inf_inf_set_a @ A5 @ B5 ) )
= ( inf_inf_set_a @ ( image_a_a2 @ F @ A5 ) @ ( image_a_a2 @ F @ B5 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_1121_inj__on__disjoint__Un,axiom,
! [F: a > a,A5: set_a,G: a > a,B5: set_a] :
( ( inj_on_a_a @ F @ A5 )
=> ( ( inj_on_a_a @ G @ B5 )
=> ( ( ( inf_inf_set_a @ ( image_a_a2 @ F @ A5 ) @ ( image_a_a2 @ G @ B5 ) )
= bot_bot_set_a )
=> ( inj_on_a_a
@ ^ [X3: a] : ( if_a @ ( member_a @ X3 @ A5 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( sup_sup_set_a @ A5 @ B5 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_1122_inj__on__disjoint__Un,axiom,
! [F: product_prod_a_a > a,A5: set_Product_prod_a_a,G: product_prod_a_a > a,B5: set_Product_prod_a_a] :
( ( inj_on4978979553551044360_a_a_a @ F @ A5 )
=> ( ( inj_on4978979553551044360_a_a_a @ G @ B5 )
=> ( ( ( inf_inf_set_a @ ( image_3437945252899457948_a_a_a @ F @ A5 ) @ ( image_3437945252899457948_a_a_a @ G @ B5 ) )
= bot_bot_set_a )
=> ( inj_on4978979553551044360_a_a_a
@ ^ [X3: product_prod_a_a] : ( if_a @ ( member1426531477525435216od_a_a @ X3 @ A5 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( sup_su3048258781599657691od_a_a @ A5 @ B5 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_1123_subset__CollectI,axiom,
! [B5: set_a,A5: set_a,Q: a > $o,P: a > $o] :
( ( ord_less_eq_set_a @ B5 @ A5 )
=> ( ! [X: a] :
( ( member_a @ X @ B5 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ B5 )
& ( Q @ X3 ) ) )
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A5 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1124_subset__CollectI,axiom,
! [B5: set_Product_prod_a_a,A5: set_Product_prod_a_a,Q: product_prod_a_a > $o,P: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ B5 @ A5 )
=> ( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ B5 )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le746702958409616551od_a_a
@ ( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ B5 )
& ( Q @ X3 ) ) )
@ ( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A5 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_1125_subset__Collect__iff,axiom,
! [B5: set_a,A5: set_a,P: a > $o] :
( ( ord_less_eq_set_a @ B5 @ A5 )
=> ( ( ord_less_eq_set_a @ B5
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A5 )
& ( P @ X3 ) ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ B5 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1126_subset__Collect__iff,axiom,
! [B5: set_Product_prod_a_a,A5: set_Product_prod_a_a,P: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ B5 @ A5 )
=> ( ( ord_le746702958409616551od_a_a @ B5
@ ( collec3336397797384452498od_a_a
@ ^ [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ A5 )
& ( P @ X3 ) ) ) )
= ( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ B5 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_1127_inj__vimage__singleton,axiom,
! [F: a > a,A3: a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ord_less_eq_set_a @ ( vimage_a_a @ F @ ( insert_a @ A3 @ bot_bot_set_a ) )
@ ( insert_a
@ ( the_a
@ ^ [X3: a] :
( ( F @ X3 )
= A3 ) )
@ bot_bot_set_a ) ) ) ).
% inj_vimage_singleton
thf(fact_1128_GrD2,axiom,
! [X4: a,Fx: a,A5: set_a,F: a > a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Fx ) @ ( bNF_Gr_a_a @ A5 @ F ) )
=> ( ( F @ X4 )
= Fx ) ) ).
% GrD2
thf(fact_1129_GrD1,axiom,
! [X4: a,Fx: a,A5: set_a,F: a > a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Fx ) @ ( bNF_Gr_a_a @ A5 @ F ) )
=> ( member_a @ X4 @ A5 ) ) ).
% GrD1
thf(fact_1130_apfst__conv,axiom,
! [F: a > a,X4: a,Y3: a] :
( ( product_apfst_a_a_a @ F @ ( product_Pair_a_a @ X4 @ Y3 ) )
= ( product_Pair_a_a @ ( F @ X4 ) @ Y3 ) ) ).
% apfst_conv
thf(fact_1131_apsnd__conv,axiom,
! [F: a > a,X4: a,Y3: a] :
( ( product_apsnd_a_a_a @ F @ ( product_Pair_a_a @ X4 @ Y3 ) )
= ( product_Pair_a_a @ X4 @ ( F @ Y3 ) ) ) ).
% apsnd_conv
thf(fact_1132_list_Osimps_I15_J,axiom,
! [X21: product_prod_a_a,X222: list_P1396940483166286381od_a_a] :
( ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X21 @ X222 ) )
= ( insert4534936382041156343od_a_a @ X21 @ ( set_Product_prod_a_a2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_1133_list_Odistinct_I1_J,axiom,
! [X21: product_prod_a_a,X222: list_P1396940483166286381od_a_a] :
( nil_Product_prod_a_a
!= ( cons_P7316939126706565853od_a_a @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_1134_list_OdiscI,axiom,
! [List: list_P1396940483166286381od_a_a,X21: product_prod_a_a,X222: list_P1396940483166286381od_a_a] :
( ( List
= ( cons_P7316939126706565853od_a_a @ X21 @ X222 ) )
=> ( List != nil_Product_prod_a_a ) ) ).
% list.discI
thf(fact_1135_list_Oexhaust,axiom,
! [Y3: list_P1396940483166286381od_a_a] :
( ( Y3 != nil_Product_prod_a_a )
=> ~ ! [X212: product_prod_a_a,X223: list_P1396940483166286381od_a_a] :
( Y3
!= ( cons_P7316939126706565853od_a_a @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_1136_splice_Ocases,axiom,
! [X4: produc5715173734304154471od_a_a] :
( ! [Ys3: list_P1396940483166286381od_a_a] :
( X4
!= ( produc1014507885701610967od_a_a @ nil_Product_prod_a_a @ Ys3 ) )
=> ~ ! [X: product_prod_a_a,Xs3: list_P1396940483166286381od_a_a,Ys3: list_P1396940483166286381od_a_a] :
( X4
!= ( produc1014507885701610967od_a_a @ ( cons_P7316939126706565853od_a_a @ X @ Xs3 ) @ Ys3 ) ) ) ).
% splice.cases
thf(fact_1137_shuffles_Ocases,axiom,
! [X4: produc5715173734304154471od_a_a] :
( ! [Ys3: list_P1396940483166286381od_a_a] :
( X4
!= ( produc1014507885701610967od_a_a @ nil_Product_prod_a_a @ Ys3 ) )
=> ( ! [Xs3: list_P1396940483166286381od_a_a] :
( X4
!= ( produc1014507885701610967od_a_a @ Xs3 @ nil_Product_prod_a_a ) )
=> ~ ! [X: product_prod_a_a,Xs3: list_P1396940483166286381od_a_a,Y: product_prod_a_a,Ys3: list_P1396940483166286381od_a_a] :
( X4
!= ( produc1014507885701610967od_a_a @ ( cons_P7316939126706565853od_a_a @ X @ Xs3 ) @ ( cons_P7316939126706565853od_a_a @ Y @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_1138_sorted__wrt_Ocases,axiom,
! [X4: produc5831972324637314982od_a_a] :
( ! [P5: product_prod_a_a > product_prod_a_a > $o] :
( X4
!= ( produc7609767723203257750od_a_a @ P5 @ nil_Product_prod_a_a ) )
=> ~ ! [P5: product_prod_a_a > product_prod_a_a > $o,X: product_prod_a_a,Ys3: list_P1396940483166286381od_a_a] :
( X4
!= ( produc7609767723203257750od_a_a @ P5 @ ( cons_P7316939126706565853od_a_a @ X @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_1139_remdups__adj_Ocases,axiom,
! [X4: list_P1396940483166286381od_a_a] :
( ( X4 != nil_Product_prod_a_a )
=> ( ! [X: product_prod_a_a] :
( X4
!= ( cons_P7316939126706565853od_a_a @ X @ nil_Product_prod_a_a ) )
=> ~ ! [X: product_prod_a_a,Y: product_prod_a_a,Xs3: list_P1396940483166286381od_a_a] :
( X4
!= ( cons_P7316939126706565853od_a_a @ X @ ( cons_P7316939126706565853od_a_a @ Y @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_1140_successively_Ocases,axiom,
! [X4: produc5831972324637314982od_a_a] :
( ! [P5: product_prod_a_a > product_prod_a_a > $o] :
( X4
!= ( produc7609767723203257750od_a_a @ P5 @ nil_Product_prod_a_a ) )
=> ( ! [P5: product_prod_a_a > product_prod_a_a > $o,X: product_prod_a_a] :
( X4
!= ( produc7609767723203257750od_a_a @ P5 @ ( cons_P7316939126706565853od_a_a @ X @ nil_Product_prod_a_a ) ) )
=> ~ ! [P5: product_prod_a_a > product_prod_a_a > $o,X: product_prod_a_a,Y: product_prod_a_a,Xs3: list_P1396940483166286381od_a_a] :
( X4
!= ( produc7609767723203257750od_a_a @ P5 @ ( cons_P7316939126706565853od_a_a @ X @ ( cons_P7316939126706565853od_a_a @ Y @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_1141_lookup_Ocases,axiom,
! [X4: produc3196479822238948891od_a_a] :
( ! [X: product_prod_a_a,Xs3: list_P1396940483166286381od_a_a,Y: product_prod_a_a,Ys3: list_P1396940483166286381od_a_a,Z2: product_prod_a_a] :
( X4
!= ( produc1537927547703807115od_a_a @ ( cons_P7316939126706565853od_a_a @ X @ Xs3 ) @ ( produc8307971631241958865od_a_a @ ( cons_P7316939126706565853od_a_a @ Y @ Ys3 ) @ Z2 ) ) )
=> ( ! [Uv2: list_P1396940483166286381od_a_a,Uw: product_prod_a_a] :
( X4
!= ( produc1537927547703807115od_a_a @ nil_Product_prod_a_a @ ( produc8307971631241958865od_a_a @ Uv2 @ Uw ) ) )
=> ~ ! [Uu3: list_P1396940483166286381od_a_a,Uw: product_prod_a_a] :
( X4
!= ( produc1537927547703807115od_a_a @ Uu3 @ ( produc8307971631241958865od_a_a @ nil_Product_prod_a_a @ Uw ) ) ) ) ) ).
% lookup.cases
thf(fact_1142_restrict_Ocases,axiom,
! [X4: produc6759108136775012935od_a_a] :
( ! [A8: set_Product_prod_a_a,X: product_prod_a_a,Xs3: list_P1396940483166286381od_a_a,Y: product_prod_a_a,Ys3: list_P1396940483166286381od_a_a] :
( X4
!= ( produc9141123965022120375od_a_a @ A8 @ ( produc1014507885701610967od_a_a @ ( cons_P7316939126706565853od_a_a @ X @ Xs3 ) @ ( cons_P7316939126706565853od_a_a @ Y @ Ys3 ) ) ) )
=> ( ! [A8: set_Product_prod_a_a,Uv2: list_P1396940483166286381od_a_a] :
( X4
!= ( produc9141123965022120375od_a_a @ A8 @ ( produc1014507885701610967od_a_a @ nil_Product_prod_a_a @ Uv2 ) ) )
=> ~ ! [A8: set_Product_prod_a_a,Uu3: list_P1396940483166286381od_a_a] :
( X4
!= ( produc9141123965022120375od_a_a @ A8 @ ( produc1014507885701610967od_a_a @ Uu3 @ nil_Product_prod_a_a ) ) ) ) ) ).
% restrict.cases
thf(fact_1143_neq__Nil__conv,axiom,
! [Xs: list_P1396940483166286381od_a_a] :
( ( Xs != nil_Product_prod_a_a )
= ( ? [Y2: product_prod_a_a,Ys2: list_P1396940483166286381od_a_a] :
( Xs
= ( cons_P7316939126706565853od_a_a @ Y2 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_1144_list__induct2_H,axiom,
! [P: list_P1396940483166286381od_a_a > list_P1396940483166286381od_a_a > $o,Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a] :
( ( P @ nil_Product_prod_a_a @ nil_Product_prod_a_a )
=> ( ! [X: product_prod_a_a,Xs3: list_P1396940483166286381od_a_a] : ( P @ ( cons_P7316939126706565853od_a_a @ X @ Xs3 ) @ nil_Product_prod_a_a )
=> ( ! [Y: product_prod_a_a,Ys3: list_P1396940483166286381od_a_a] : ( P @ nil_Product_prod_a_a @ ( cons_P7316939126706565853od_a_a @ Y @ Ys3 ) )
=> ( ! [X: product_prod_a_a,Xs3: list_P1396940483166286381od_a_a,Y: product_prod_a_a,Ys3: list_P1396940483166286381od_a_a] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_P7316939126706565853od_a_a @ X @ Xs3 ) @ ( cons_P7316939126706565853od_a_a @ Y @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1145_list__nonempty__induct,axiom,
! [Xs: list_P1396940483166286381od_a_a,P: list_P1396940483166286381od_a_a > $o] :
( ( Xs != nil_Product_prod_a_a )
=> ( ! [X: product_prod_a_a] : ( P @ ( cons_P7316939126706565853od_a_a @ X @ nil_Product_prod_a_a ) )
=> ( ! [X: product_prod_a_a,Xs3: list_P1396940483166286381od_a_a] :
( ( Xs3 != nil_Product_prod_a_a )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_P7316939126706565853od_a_a @ X @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_1146_list_Oset__intros_I2_J,axiom,
! [Y3: a,X222: list_a,X21: a] :
( ( member_a @ Y3 @ ( set_a2 @ X222 ) )
=> ( member_a @ Y3 @ ( set_a2 @ ( cons_a @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1147_list_Oset__intros_I2_J,axiom,
! [Y3: product_prod_a_a,X222: list_P1396940483166286381od_a_a,X21: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_Product_prod_a_a2 @ X222 ) )
=> ( member1426531477525435216od_a_a @ Y3 @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1148_list_Oset__intros_I1_J,axiom,
! [X21: a,X222: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_1149_list_Oset__intros_I1_J,axiom,
! [X21: product_prod_a_a,X222: list_P1396940483166286381od_a_a] : ( member1426531477525435216od_a_a @ X21 @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_1150_list_Oset__cases,axiom,
! [E: a,A3: list_a] :
( ( member_a @ E @ ( set_a2 @ A3 ) )
=> ( ! [Z22: list_a] :
( A3
!= ( cons_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: list_a] :
( ( A3
= ( cons_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1151_list_Oset__cases,axiom,
! [E: product_prod_a_a,A3: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ E @ ( set_Product_prod_a_a2 @ A3 ) )
=> ( ! [Z22: list_P1396940483166286381od_a_a] :
( A3
!= ( cons_P7316939126706565853od_a_a @ E @ Z22 ) )
=> ~ ! [Z1: product_prod_a_a,Z22: list_P1396940483166286381od_a_a] :
( ( A3
= ( cons_P7316939126706565853od_a_a @ Z1 @ Z22 ) )
=> ~ ( member1426531477525435216od_a_a @ E @ ( set_Product_prod_a_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1152_set__ConsD,axiom,
! [Y3: a,X4: a,Xs: list_a] :
( ( member_a @ Y3 @ ( set_a2 @ ( cons_a @ X4 @ Xs ) ) )
=> ( ( Y3 = X4 )
| ( member_a @ Y3 @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1153_set__ConsD,axiom,
! [Y3: product_prod_a_a,X4: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X4 @ Xs ) ) )
=> ( ( Y3 = X4 )
| ( member1426531477525435216od_a_a @ Y3 @ ( set_Product_prod_a_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1154_transpose_Ocases,axiom,
! [X4: list_l2076173926988081843od_a_a] :
( ( X4 != nil_li8564743906319921459od_a_a )
=> ( ! [Xss2: list_l2076173926988081843od_a_a] :
( X4
!= ( cons_l836352187635934435od_a_a @ nil_Product_prod_a_a @ Xss2 ) )
=> ~ ! [X: product_prod_a_a,Xs3: list_P1396940483166286381od_a_a,Xss2: list_l2076173926988081843od_a_a] :
( X4
!= ( cons_l836352187635934435od_a_a @ ( cons_P7316939126706565853od_a_a @ X @ Xs3 ) @ Xss2 ) ) ) ) ).
% transpose.cases
thf(fact_1155_set__subset__Cons,axiom,
! [Xs: list_P1396940483166286381od_a_a,X4: product_prod_a_a] : ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X4 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1156_find_Osimps_I2_J,axiom,
! [P: product_prod_a_a > $o,X4: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( ( P @ X4 )
=> ( ( find_P2248954792982286147od_a_a @ P @ ( cons_P7316939126706565853od_a_a @ X4 @ Xs ) )
= ( some_P3592067295195376908od_a_a @ X4 ) ) )
& ( ~ ( P @ X4 )
=> ( ( find_P2248954792982286147od_a_a @ P @ ( cons_P7316939126706565853od_a_a @ X4 @ Xs ) )
= ( find_P2248954792982286147od_a_a @ P @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_1157_find_Osimps_I2_J,axiom,
! [P: set_a > $o,X4: set_a,Xs: list_set_a] :
( ( ( P @ X4 )
=> ( ( find_set_a @ P @ ( cons_set_a @ X4 @ Xs ) )
= ( some_set_a @ X4 ) ) )
& ( ~ ( P @ X4 )
=> ( ( find_set_a @ P @ ( cons_set_a @ X4 @ Xs ) )
= ( find_set_a @ P @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_1158_the__elem__set,axiom,
! [X4: product_prod_a_a] :
( ( the_el8589169208993665564od_a_a @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X4 @ nil_Product_prod_a_a ) ) )
= X4 ) ).
% the_elem_set
thf(fact_1159_set__Cons__sing__Nil,axiom,
! [A5: set_Product_prod_a_a] :
( ( set_Co1270446422447248349od_a_a @ A5 @ ( insert6721513254612355069od_a_a @ nil_Product_prod_a_a @ bot_bo6558180029120801017od_a_a ) )
= ( image_5211047369904514507od_a_a
@ ^ [X3: product_prod_a_a] : ( cons_P7316939126706565853od_a_a @ X3 @ nil_Product_prod_a_a )
@ A5 ) ) ).
% set_Cons_sing_Nil
thf(fact_1160_subset__Image1__Image1__iff,axiom,
! [R3: set_Pr8600417178894128327od_a_a,A3: product_prod_a_a,B3: product_prod_a_a] :
( ( order_3202267349275844158od_a_a @ ( field_1126092520709947252od_a_a @ R3 ) @ R3 )
=> ( ( member1426531477525435216od_a_a @ A3 @ ( field_1126092520709947252od_a_a @ R3 ) )
=> ( ( member1426531477525435216od_a_a @ B3 @ ( field_1126092520709947252od_a_a @ R3 ) )
=> ( ( ord_le746702958409616551od_a_a @ ( image_9076584400576816019od_a_a @ R3 @ ( insert4534936382041156343od_a_a @ A3 @ bot_bo3357376287454694259od_a_a ) ) @ ( image_9076584400576816019od_a_a @ R3 @ ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) ) )
= ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ B3 @ A3 ) @ R3 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1161_subset__Image1__Image1__iff,axiom,
! [R3: set_Product_prod_a_a,A3: a,B3: a] :
( ( order_preorder_on_a @ ( field_a @ R3 ) @ R3 )
=> ( ( member_a @ A3 @ ( field_a @ R3 ) )
=> ( ( member_a @ B3 @ ( field_a @ R3 ) )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ R3 @ ( insert_a @ A3 @ bot_bot_set_a ) ) @ ( image_a_a @ R3 @ ( insert_a @ B3 @ bot_bot_set_a ) ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B3 @ A3 ) @ R3 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1162_ImageI,axiom,
! [A3: a,B3: product_prod_a_a,R3: set_Pr5530083903271594800od_a_a,A5: set_a] :
( ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ A3 @ B3 ) @ R3 )
=> ( ( member_a @ A3 @ A5 )
=> ( member1426531477525435216od_a_a @ B3 @ ( image_2799180466780705916od_a_a @ R3 @ A5 ) ) ) ) ).
% ImageI
thf(fact_1163_ImageI,axiom,
! [A3: product_prod_a_a,B3: a,R3: set_Pr8876520727511657886_a_a_a,A5: set_Product_prod_a_a] :
( ( member2829916517802586983_a_a_a @ ( produc5692536848587512110_a_a_a @ A3 @ B3 ) @ R3 )
=> ( ( member1426531477525435216od_a_a @ A3 @ A5 )
=> ( member_a @ B3 @ ( image_8059871973944943978_a_a_a @ R3 @ A5 ) ) ) ) ).
% ImageI
thf(fact_1164_ImageI,axiom,
! [A3: product_prod_a_a,B3: product_prod_a_a,R3: set_Pr8600417178894128327od_a_a,A5: set_Product_prod_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A3 @ B3 ) @ R3 )
=> ( ( member1426531477525435216od_a_a @ A3 @ A5 )
=> ( member1426531477525435216od_a_a @ B3 @ ( image_9076584400576816019od_a_a @ R3 @ A5 ) ) ) ) ).
% ImageI
thf(fact_1165_ImageI,axiom,
! [A3: a,B3: a,R3: set_Product_prod_a_a,A5: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ R3 )
=> ( ( member_a @ A3 @ A5 )
=> ( member_a @ B3 @ ( image_a_a @ R3 @ A5 ) ) ) ) ).
% ImageI
thf(fact_1166_Image__empty2,axiom,
! [R2: set_Product_prod_a_a] :
( ( image_a_a @ R2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Image_empty2
thf(fact_1167_Image__Id__on,axiom,
! [A5: set_a,B5: set_a] :
( ( image_a_a @ ( id_on_a @ A5 ) @ B5 )
= ( inf_inf_set_a @ A5 @ B5 ) ) ).
% Image_Id_on
thf(fact_1168_Image__Collect__case__prod,axiom,
! [P: a > a > $o,A5: set_a] :
( ( image_a_a @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ P ) ) @ A5 )
= ( collect_a
@ ^ [Y2: a] :
? [X3: a] :
( ( member_a @ X3 @ A5 )
& ( P @ X3 @ Y2 ) ) ) ) ).
% Image_Collect_case_prod
thf(fact_1169_Image__singleton__iff,axiom,
! [B3: product_prod_a_a,R3: set_Pr5530083903271594800od_a_a,A3: a] :
( ( member1426531477525435216od_a_a @ B3 @ ( image_2799180466780705916od_a_a @ R3 @ ( insert_a @ A3 @ bot_bot_set_a ) ) )
= ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ A3 @ B3 ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_1170_Image__singleton__iff,axiom,
! [B3: a,R3: set_Product_prod_a_a,A3: a] :
( ( member_a @ B3 @ ( image_a_a @ R3 @ ( insert_a @ A3 @ bot_bot_set_a ) ) )
= ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ R3 ) ) ).
% Image_singleton_iff
thf(fact_1171_Image__Int__subset,axiom,
! [R2: set_Product_prod_a_a,A5: set_a,B5: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ R2 @ ( inf_inf_set_a @ A5 @ B5 ) ) @ ( inf_inf_set_a @ ( image_a_a @ R2 @ A5 ) @ ( image_a_a @ R2 @ B5 ) ) ) ).
% Image_Int_subset
thf(fact_1172_insert__Image__triv,axiom,
! [X4: a,R3: set_a,Y3: a,Qeq: set_Product_prod_a_a] :
( ~ ( member_a @ X4 @ R3 )
=> ( ( image_a_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ Qeq ) @ R3 )
= ( image_a_a @ Qeq @ R3 ) ) ) ).
% insert_Image_triv
thf(fact_1173_Image__def,axiom,
( image_a_a
= ( ^ [R: set_Product_prod_a_a,S3: set_a] :
( collect_a
@ ^ [Y2: a] :
? [X3: a] :
( ( member_a @ X3 @ S3 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R ) ) ) ) ) ).
% Image_def
thf(fact_1174_ImageE,axiom,
! [B3: a,R3: set_Pr8876520727511657886_a_a_a,A5: set_Product_prod_a_a] :
( ( member_a @ B3 @ ( image_8059871973944943978_a_a_a @ R3 @ A5 ) )
=> ~ ! [X: product_prod_a_a] :
( ( member2829916517802586983_a_a_a @ ( produc5692536848587512110_a_a_a @ X @ B3 ) @ R3 )
=> ~ ( member1426531477525435216od_a_a @ X @ A5 ) ) ) ).
% ImageE
thf(fact_1175_ImageE,axiom,
! [B3: product_prod_a_a,R3: set_Pr5530083903271594800od_a_a,A5: set_a] :
( ( member1426531477525435216od_a_a @ B3 @ ( image_2799180466780705916od_a_a @ R3 @ A5 ) )
=> ~ ! [X: a] :
( ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ X @ B3 ) @ R3 )
=> ~ ( member_a @ X @ A5 ) ) ) ).
% ImageE
thf(fact_1176_ImageE,axiom,
! [B3: product_prod_a_a,R3: set_Pr8600417178894128327od_a_a,A5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ B3 @ ( image_9076584400576816019od_a_a @ R3 @ A5 ) )
=> ~ ! [X: product_prod_a_a] :
( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ B3 ) @ R3 )
=> ~ ( member1426531477525435216od_a_a @ X @ A5 ) ) ) ).
% ImageE
thf(fact_1177_ImageE,axiom,
! [B3: a,R3: set_Product_prod_a_a,A5: set_a] :
( ( member_a @ B3 @ ( image_a_a @ R3 @ A5 ) )
=> ~ ! [X: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ B3 ) @ R3 )
=> ~ ( member_a @ X @ A5 ) ) ) ).
% ImageE
thf(fact_1178_Image__iff,axiom,
! [B3: a,R3: set_Product_prod_a_a,A5: set_a] :
( ( member_a @ B3 @ ( image_a_a @ R3 @ A5 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A5 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ B3 ) @ R3 ) ) ) ) ).
% Image_iff
thf(fact_1179_rev__ImageI,axiom,
! [A3: a,A5: set_a,B3: product_prod_a_a,R3: set_Pr5530083903271594800od_a_a] :
( ( member_a @ A3 @ A5 )
=> ( ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ A3 @ B3 ) @ R3 )
=> ( member1426531477525435216od_a_a @ B3 @ ( image_2799180466780705916od_a_a @ R3 @ A5 ) ) ) ) ).
% rev_ImageI
thf(fact_1180_rev__ImageI,axiom,
! [A3: product_prod_a_a,A5: set_Product_prod_a_a,B3: a,R3: set_Pr8876520727511657886_a_a_a] :
( ( member1426531477525435216od_a_a @ A3 @ A5 )
=> ( ( member2829916517802586983_a_a_a @ ( produc5692536848587512110_a_a_a @ A3 @ B3 ) @ R3 )
=> ( member_a @ B3 @ ( image_8059871973944943978_a_a_a @ R3 @ A5 ) ) ) ) ).
% rev_ImageI
thf(fact_1181_rev__ImageI,axiom,
! [A3: product_prod_a_a,A5: set_Product_prod_a_a,B3: product_prod_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ( member1426531477525435216od_a_a @ A3 @ A5 )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A3 @ B3 ) @ R3 )
=> ( member1426531477525435216od_a_a @ B3 @ ( image_9076584400576816019od_a_a @ R3 @ A5 ) ) ) ) ).
% rev_ImageI
thf(fact_1182_rev__ImageI,axiom,
! [A3: a,A5: set_a,B3: a,R3: set_Product_prod_a_a] :
( ( member_a @ A3 @ A5 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B3 ) @ R3 )
=> ( member_a @ B3 @ ( image_a_a @ R3 @ A5 ) ) ) ) ).
% rev_ImageI
thf(fact_1183_Image__singleton,axiom,
! [R3: set_Product_prod_a_a,A3: a] :
( ( image_a_a @ R3 @ ( insert_a @ A3 @ bot_bot_set_a ) )
= ( collect_a
@ ^ [B: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B ) @ R3 ) ) ) ).
% Image_singleton
thf(fact_1184_Partial__order__eq__Image1__Image1__iff,axiom,
! [R3: set_Pr8600417178894128327od_a_a,A3: product_prod_a_a,B3: product_prod_a_a] :
( ( order_7408868903334687516od_a_a @ ( field_1126092520709947252od_a_a @ R3 ) @ R3 )
=> ( ( member1426531477525435216od_a_a @ A3 @ ( field_1126092520709947252od_a_a @ R3 ) )
=> ( ( member1426531477525435216od_a_a @ B3 @ ( field_1126092520709947252od_a_a @ R3 ) )
=> ( ( ( image_9076584400576816019od_a_a @ R3 @ ( insert4534936382041156343od_a_a @ A3 @ bot_bo3357376287454694259od_a_a ) )
= ( image_9076584400576816019od_a_a @ R3 @ ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) ) )
= ( A3 = B3 ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_1185_Partial__order__eq__Image1__Image1__iff,axiom,
! [R3: set_Product_prod_a_a,A3: a,B3: a] :
( ( order_5272072345360262643r_on_a @ ( field_a @ R3 ) @ R3 )
=> ( ( member_a @ A3 @ ( field_a @ R3 ) )
=> ( ( member_a @ B3 @ ( field_a @ R3 ) )
=> ( ( ( image_a_a @ R3 @ ( insert_a @ A3 @ bot_bot_set_a ) )
= ( image_a_a @ R3 @ ( insert_a @ B3 @ bot_bot_set_a ) ) )
= ( A3 = B3 ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_1186_subset__Image__Image__iff,axiom,
! [R3: set_Product_prod_a_a,A5: set_a,B5: set_a] :
( ( order_preorder_on_a @ ( field_a @ R3 ) @ R3 )
=> ( ( ord_less_eq_set_a @ A5 @ ( field_a @ R3 ) )
=> ( ( ord_less_eq_set_a @ B5 @ ( field_a @ R3 ) )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ R3 @ A5 ) @ ( image_a_a @ R3 @ B5 ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ? [Y2: a] :
( ( member_a @ Y2 @ B5 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y2 @ X3 ) @ R3 ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_1187_listset_Osimps_I1_J,axiom,
( ( listse8494732006900235788od_a_a @ nil_se7151937861063948045od_a_a )
= ( insert6721513254612355069od_a_a @ nil_Product_prod_a_a @ bot_bo6558180029120801017od_a_a ) ) ).
% listset.simps(1)
thf(fact_1188_listrel__Nil,axiom,
! [R3: set_Pr8600417178894128327od_a_a] :
( ( image_8944511214546096659od_a_a @ ( listre6710817562338460593od_a_a @ R3 ) @ ( insert6721513254612355069od_a_a @ nil_Product_prod_a_a @ bot_bo6558180029120801017od_a_a ) )
= ( insert6721513254612355069od_a_a @ nil_Product_prod_a_a @ bot_bo6558180029120801017od_a_a ) ) ).
% listrel_Nil
thf(fact_1189_listrel__Nil2,axiom,
! [Xs: list_P1396940483166286381od_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ( member1370157277785437840od_a_a @ ( produc1014507885701610967od_a_a @ Xs @ nil_Product_prod_a_a ) @ ( listre6710817562338460593od_a_a @ R3 ) )
=> ( Xs = nil_Product_prod_a_a ) ) ).
% listrel_Nil2
thf(fact_1190_listrel__Nil1,axiom,
! [Xs: list_P1396940483166286381od_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ( member1370157277785437840od_a_a @ ( produc1014507885701610967od_a_a @ nil_Product_prod_a_a @ Xs ) @ ( listre6710817562338460593od_a_a @ R3 ) )
=> ( Xs = nil_Product_prod_a_a ) ) ).
% listrel_Nil1
thf(fact_1191_listrel_ONil,axiom,
! [R3: set_Pr8600417178894128327od_a_a] : ( member1370157277785437840od_a_a @ ( produc1014507885701610967od_a_a @ nil_Product_prod_a_a @ nil_Product_prod_a_a ) @ ( listre6710817562338460593od_a_a @ R3 ) ) ).
% listrel.Nil
thf(fact_1192_listrel_OCons,axiom,
! [X4: a,Y3: a,R3: set_Product_prod_a_a,Xs: list_a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R3 )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel_a_a @ R3 ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X4 @ Xs ) @ ( cons_a @ Y3 @ Ys ) ) @ ( listrel_a_a @ R3 ) ) ) ) ).
% listrel.Cons
thf(fact_1193_listrel__Cons1,axiom,
! [Y3: a,Ys: list_a,Xs: list_a,R3: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ Y3 @ Ys ) @ Xs ) @ ( listrel_a_a @ R3 ) )
=> ~ ! [Y: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ Y @ Ys3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ Y ) @ R3 )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Ys3 ) @ ( listrel_a_a @ R3 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_1194_listrel__Cons2,axiom,
! [Xs: list_a,Y3: a,Ys: list_a,R3: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ ( cons_a @ Y3 @ Ys ) ) @ ( listrel_a_a @ R3 ) )
=> ~ ! [X: a,Xs3: list_a] :
( ( Xs
= ( cons_a @ X @ Xs3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R3 )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs3 @ Ys ) @ ( listrel_a_a @ R3 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_1195_listrel_Ocases,axiom,
! [A12: list_P1396940483166286381od_a_a,A23: list_P1396940483166286381od_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ( member1370157277785437840od_a_a @ ( produc1014507885701610967od_a_a @ A12 @ A23 ) @ ( listre6710817562338460593od_a_a @ R3 ) )
=> ( ( ( A12 = nil_Product_prod_a_a )
=> ( A23 != nil_Product_prod_a_a ) )
=> ~ ! [X: product_prod_a_a,Y: product_prod_a_a,Xs3: list_P1396940483166286381od_a_a] :
( ( A12
= ( cons_P7316939126706565853od_a_a @ X @ Xs3 ) )
=> ! [Ys3: list_P1396940483166286381od_a_a] :
( ( A23
= ( cons_P7316939126706565853od_a_a @ Y @ Ys3 ) )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y ) @ R3 )
=> ~ ( member1370157277785437840od_a_a @ ( produc1014507885701610967od_a_a @ Xs3 @ Ys3 ) @ ( listre6710817562338460593od_a_a @ R3 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_1196_listrel_Ocases,axiom,
! [A12: list_a,A23: list_a,R3: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A12 @ A23 ) @ ( listrel_a_a @ R3 ) )
=> ( ( ( A12 = nil_a )
=> ( A23 != nil_a ) )
=> ~ ! [X: a,Y: a,Xs3: list_a] :
( ( A12
= ( cons_a @ X @ Xs3 ) )
=> ! [Ys3: list_a] :
( ( A23
= ( cons_a @ Y @ Ys3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R3 )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs3 @ Ys3 ) @ ( listrel_a_a @ R3 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_1197_listrel_Osimps,axiom,
! [A12: list_P1396940483166286381od_a_a,A23: list_P1396940483166286381od_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ( member1370157277785437840od_a_a @ ( produc1014507885701610967od_a_a @ A12 @ A23 ) @ ( listre6710817562338460593od_a_a @ R3 ) )
= ( ( ( A12 = nil_Product_prod_a_a )
& ( A23 = nil_Product_prod_a_a ) )
| ? [X3: product_prod_a_a,Y2: product_prod_a_a,Xs2: list_P1396940483166286381od_a_a,Ys2: list_P1396940483166286381od_a_a] :
( ( A12
= ( cons_P7316939126706565853od_a_a @ X3 @ Xs2 ) )
& ( A23
= ( cons_P7316939126706565853od_a_a @ Y2 @ Ys2 ) )
& ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X3 @ Y2 ) @ R3 )
& ( member1370157277785437840od_a_a @ ( produc1014507885701610967od_a_a @ Xs2 @ Ys2 ) @ ( listre6710817562338460593od_a_a @ R3 ) ) ) ) ) ).
% listrel.simps
thf(fact_1198_listrel_Osimps,axiom,
! [A12: list_a,A23: list_a,R3: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A12 @ A23 ) @ ( listrel_a_a @ R3 ) )
= ( ( ( A12 = nil_a )
& ( A23 = nil_a ) )
| ? [X3: a,Y2: a,Xs2: list_a,Ys2: list_a] :
( ( A12
= ( cons_a @ X3 @ Xs2 ) )
& ( A23
= ( cons_a @ Y2 @ Ys2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R3 )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys2 ) @ ( listrel_a_a @ R3 ) ) ) ) ) ).
% listrel.simps
thf(fact_1199_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [R3: set_Pr8600417178894128327od_a_a,A3: product_prod_a_a,B3: product_prod_a_a] :
( ( refl_o7745108929832855590od_a_a @ ( field_1126092520709947252od_a_a @ R3 ) @ R3 )
=> ( ( antisy9008168496540133130od_a_a @ top_to8063371432257647191od_a_a @ R3 )
=> ( ( member1426531477525435216od_a_a @ A3 @ ( field_1126092520709947252od_a_a @ R3 ) )
=> ( ( member1426531477525435216od_a_a @ B3 @ ( field_1126092520709947252od_a_a @ R3 ) )
=> ( ( ( image_9076584400576816019od_a_a @ R3 @ ( insert4534936382041156343od_a_a @ A3 @ bot_bo3357376287454694259od_a_a ) )
= ( image_9076584400576816019od_a_a @ R3 @ ( insert4534936382041156343od_a_a @ B3 @ bot_bo3357376287454694259od_a_a ) ) )
= ( A3 = B3 ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_1200_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [R3: set_Product_prod_a_a,A3: a,B3: a] :
( ( refl_on_a @ ( field_a @ R3 ) @ R3 )
=> ( ( antisym_on_a @ top_top_set_a @ R3 )
=> ( ( member_a @ A3 @ ( field_a @ R3 ) )
=> ( ( member_a @ B3 @ ( field_a @ R3 ) )
=> ( ( ( image_a_a @ R3 @ ( insert_a @ A3 @ bot_bot_set_a ) )
= ( image_a_a @ R3 @ ( insert_a @ B3 @ bot_bot_set_a ) ) )
= ( A3 = B3 ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_1201_antisym__on__def,axiom,
( antisym_on_a
= ( ^ [A6: set_a,R: set_Product_prod_a_a] :
! [X3: a] :
( ( member_a @ X3 @ A6 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ A6 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y2 @ X3 ) @ R )
=> ( X3 = Y2 ) ) ) ) ) ) ) ).
% antisym_on_def
thf(fact_1202_antisym__onI,axiom,
! [A5: set_Product_prod_a_a,R3: set_Pr8600417178894128327od_a_a] :
( ! [X: product_prod_a_a,Y: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A5 )
=> ( ( member1426531477525435216od_a_a @ Y @ A5 )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X @ Y ) @ R3 )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y @ X ) @ R3 )
=> ( X = Y ) ) ) ) )
=> ( antisy9008168496540133130od_a_a @ A5 @ R3 ) ) ).
% antisym_onI
thf(fact_1203_antisym__onI,axiom,
! [A5: set_a,R3: set_Product_prod_a_a] :
( ! [X: a,Y: a] :
( ( member_a @ X @ A5 )
=> ( ( member_a @ Y @ A5 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R3 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X ) @ R3 )
=> ( X = Y ) ) ) ) )
=> ( antisym_on_a @ A5 @ R3 ) ) ).
% antisym_onI
thf(fact_1204_antisym__onD,axiom,
! [A5: set_Product_prod_a_a,R3: set_Pr8600417178894128327od_a_a,X4: product_prod_a_a,Y3: product_prod_a_a] :
( ( antisy9008168496540133130od_a_a @ A5 @ R3 )
=> ( ( member1426531477525435216od_a_a @ X4 @ A5 )
=> ( ( member1426531477525435216od_a_a @ Y3 @ A5 )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X4 @ Y3 ) @ R3 )
=> ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ Y3 @ X4 ) @ R3 )
=> ( X4 = Y3 ) ) ) ) ) ) ).
% antisym_onD
thf(fact_1205_antisym__onD,axiom,
! [A5: set_a,R3: set_Product_prod_a_a,X4: a,Y3: a] :
( ( antisym_on_a @ A5 @ R3 )
=> ( ( member_a @ X4 @ A5 )
=> ( ( member_a @ Y3 @ A5 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R3 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X4 ) @ R3 )
=> ( X4 = Y3 ) ) ) ) ) ) ).
% antisym_onD
thf(fact_1206_antisymD,axiom,
! [R3: set_Product_prod_a_a,X4: a,Y3: a] :
( ( antisym_on_a @ top_top_set_a @ R3 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R3 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X4 ) @ R3 )
=> ( X4 = Y3 ) ) ) ) ).
% antisymD
thf(fact_1207_antisymI,axiom,
! [R3: set_Product_prod_a_a] :
( ! [X: a,Y: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R3 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X ) @ R3 )
=> ( X = Y ) ) )
=> ( antisym_on_a @ top_top_set_a @ R3 ) ) ).
% antisymI
thf(fact_1208_product__lists_Osimps_I1_J,axiom,
( ( produc5891818802274520598od_a_a @ nil_li8564743906319921459od_a_a )
= ( cons_l836352187635934435od_a_a @ nil_Product_prod_a_a @ nil_li8564743906319921459od_a_a ) ) ).
% product_lists.simps(1)
thf(fact_1209_extend_Osimps_I2_J,axiom,
! [A5: set_nat,Uv3: list_P1396940483166286381od_a_a] :
( ( extend8942801199121607652od_a_a @ A5 @ nil_nat @ Uv3 )
= ( insert6721513254612355069od_a_a @ nil_Product_prod_a_a @ bot_bo6558180029120801017od_a_a ) ) ).
% extend.simps(2)
thf(fact_1210_transpose_Osimps_I3_J,axiom,
! [X4: product_prod_a_a,Xs: list_P1396940483166286381od_a_a,Xss: list_l2076173926988081843od_a_a] :
( ( transp4276400492052872375od_a_a @ ( cons_l836352187635934435od_a_a @ ( cons_P7316939126706565853od_a_a @ X4 @ Xs ) @ Xss ) )
= ( cons_l836352187635934435od_a_a
@ ( cons_P7316939126706565853od_a_a @ X4
@ ( concat6071998529708183912od_a_a
@ ( map_li6019632663516159876od_a_a
@ ( case_l3432923752981944992od_a_a @ nil_Product_prod_a_a
@ ^ [H2: product_prod_a_a,T: list_P1396940483166286381od_a_a] : ( cons_P7316939126706565853od_a_a @ H2 @ nil_Product_prod_a_a ) )
@ Xss ) ) )
@ ( transp4276400492052872375od_a_a
@ ( cons_l836352187635934435od_a_a @ Xs
@ ( concat8629597595045315054od_a_a
@ ( map_li711184933725290634od_a_a
@ ( case_l8205109941394486810od_a_a @ nil_li8564743906319921459od_a_a
@ ^ [H2: product_prod_a_a,T: list_P1396940483166286381od_a_a] : ( cons_l836352187635934435od_a_a @ T @ nil_li8564743906319921459od_a_a ) )
@ Xss ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_1211_transpose_Oelims,axiom,
! [X4: list_l2076173926988081843od_a_a,Y3: list_l2076173926988081843od_a_a] :
( ( ( transp4276400492052872375od_a_a @ X4 )
= Y3 )
=> ( ( ( X4 = nil_li8564743906319921459od_a_a )
=> ( Y3 != nil_li8564743906319921459od_a_a ) )
=> ( ! [Xss2: list_l2076173926988081843od_a_a] :
( ( X4
= ( cons_l836352187635934435od_a_a @ nil_Product_prod_a_a @ Xss2 ) )
=> ( Y3
!= ( transp4276400492052872375od_a_a @ Xss2 ) ) )
=> ~ ! [X: product_prod_a_a,Xs3: list_P1396940483166286381od_a_a,Xss2: list_l2076173926988081843od_a_a] :
( ( X4
= ( cons_l836352187635934435od_a_a @ ( cons_P7316939126706565853od_a_a @ X @ Xs3 ) @ Xss2 ) )
=> ( Y3
!= ( cons_l836352187635934435od_a_a
@ ( cons_P7316939126706565853od_a_a @ X
@ ( concat6071998529708183912od_a_a
@ ( map_li6019632663516159876od_a_a
@ ( case_l3432923752981944992od_a_a @ nil_Product_prod_a_a
@ ^ [H2: product_prod_a_a,T: list_P1396940483166286381od_a_a] : ( cons_P7316939126706565853od_a_a @ H2 @ nil_Product_prod_a_a ) )
@ Xss2 ) ) )
@ ( transp4276400492052872375od_a_a
@ ( cons_l836352187635934435od_a_a @ Xs3
@ ( concat8629597595045315054od_a_a
@ ( map_li711184933725290634od_a_a
@ ( case_l8205109941394486810od_a_a @ nil_li8564743906319921459od_a_a
@ ^ [H2: product_prod_a_a,T: list_P1396940483166286381od_a_a] : ( cons_l836352187635934435od_a_a @ T @ nil_li8564743906319921459od_a_a ) )
@ Xss2 ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_1212_transpose_Osimps_I2_J,axiom,
! [Xss: list_l2076173926988081843od_a_a] :
( ( transp4276400492052872375od_a_a @ ( cons_l836352187635934435od_a_a @ nil_Product_prod_a_a @ Xss ) )
= ( transp4276400492052872375od_a_a @ Xss ) ) ).
% transpose.simps(2)
thf(fact_1213_list_Odisc__eq__case_I2_J,axiom,
! [List: list_P1396940483166286381od_a_a] :
( ( List != nil_Product_prod_a_a )
= ( case_l5046764540854462197od_a_a @ $false
@ ^ [Uu: product_prod_a_a,Uv: list_P1396940483166286381od_a_a] : $true
@ List ) ) ).
% list.disc_eq_case(2)
thf(fact_1214_list_Odisc__eq__case_I1_J,axiom,
! [List: list_P1396940483166286381od_a_a] :
( ( List = nil_Product_prod_a_a )
= ( case_l5046764540854462197od_a_a @ $true
@ ^ [Uu: product_prod_a_a,Uv: list_P1396940483166286381od_a_a] : $false
@ List ) ) ).
% list.disc_eq_case(1)
thf(fact_1215_transpose__empty,axiom,
! [Xs: list_l2076173926988081843od_a_a] :
( ( ( transp4276400492052872375od_a_a @ Xs )
= nil_li8564743906319921459od_a_a )
= ( ! [X3: list_P1396940483166286381od_a_a] :
( ( member6824001069763096534od_a_a @ X3 @ ( set_li4974249543031949378od_a_a @ Xs ) )
=> ( X3 = nil_Product_prod_a_a ) ) ) ) ).
% transpose_empty
thf(fact_1216_transpose_Opsimps_I3_J,axiom,
! [X4: product_prod_a_a,Xs: list_P1396940483166286381od_a_a,Xss: list_l2076173926988081843od_a_a] :
( ( accp_l288472794138438844od_a_a @ transp70083455532304476od_a_a @ ( cons_l836352187635934435od_a_a @ ( cons_P7316939126706565853od_a_a @ X4 @ Xs ) @ Xss ) )
=> ( ( transp4276400492052872375od_a_a @ ( cons_l836352187635934435od_a_a @ ( cons_P7316939126706565853od_a_a @ X4 @ Xs ) @ Xss ) )
= ( cons_l836352187635934435od_a_a
@ ( cons_P7316939126706565853od_a_a @ X4
@ ( concat6071998529708183912od_a_a
@ ( map_li6019632663516159876od_a_a
@ ( case_l3432923752981944992od_a_a @ nil_Product_prod_a_a
@ ^ [H2: product_prod_a_a,T: list_P1396940483166286381od_a_a] : ( cons_P7316939126706565853od_a_a @ H2 @ nil_Product_prod_a_a ) )
@ Xss ) ) )
@ ( transp4276400492052872375od_a_a
@ ( cons_l836352187635934435od_a_a @ Xs
@ ( concat8629597595045315054od_a_a
@ ( map_li711184933725290634od_a_a
@ ( case_l8205109941394486810od_a_a @ nil_li8564743906319921459od_a_a
@ ^ [H2: product_prod_a_a,T: list_P1396940483166286381od_a_a] : ( cons_l836352187635934435od_a_a @ T @ nil_li8564743906319921459od_a_a ) )
@ Xss ) ) ) ) ) ) ) ).
% transpose.psimps(3)
thf(fact_1217_Cons__listrel1__Cons,axiom,
! [X4: a,Xs: list_a,Y3: a,Ys: list_a,R3: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X4 @ Xs ) @ ( cons_a @ Y3 @ Ys ) ) @ ( listrel1_a @ R3 ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R3 )
& ( Xs = Ys ) )
| ( ( X4 = Y3 )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R3 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_1218_transpose_Opsimps_I2_J,axiom,
! [Xss: list_l2076173926988081843od_a_a] :
( ( accp_l288472794138438844od_a_a @ transp70083455532304476od_a_a @ ( cons_l836352187635934435od_a_a @ nil_Product_prod_a_a @ Xss ) )
=> ( ( transp4276400492052872375od_a_a @ ( cons_l836352187635934435od_a_a @ nil_Product_prod_a_a @ Xss ) )
= ( transp4276400492052872375od_a_a @ Xss ) ) ) ).
% transpose.psimps(2)
thf(fact_1219_not__Nil__listrel1,axiom,
! [Xs: list_P1396940483166286381od_a_a,R3: set_Pr8600417178894128327od_a_a] :
~ ( member1370157277785437840od_a_a @ ( produc1014507885701610967od_a_a @ nil_Product_prod_a_a @ Xs ) @ ( listre2955263628197028358od_a_a @ R3 ) ) ).
% not_Nil_listrel1
thf(fact_1220_not__listrel1__Nil,axiom,
! [Xs: list_P1396940483166286381od_a_a,R3: set_Pr8600417178894128327od_a_a] :
~ ( member1370157277785437840od_a_a @ ( produc1014507885701610967od_a_a @ Xs @ nil_Product_prod_a_a ) @ ( listre2955263628197028358od_a_a @ R3 ) ) ).
% not_listrel1_Nil
thf(fact_1221_listrel1I1,axiom,
! [X4: a,Y3: a,R3: set_Product_prod_a_a,Xs: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ R3 )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X4 @ Xs ) @ ( cons_a @ Y3 @ Xs ) ) @ ( listrel1_a @ R3 ) ) ) ).
% listrel1I1
thf(fact_1222_Cons__listrel1E1,axiom,
! [X4: a,Xs: list_a,Ys: list_a,R3: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X4 @ Xs ) @ Ys ) @ ( listrel1_a @ R3 ) )
=> ( ! [Y: a] :
( ( Ys
= ( cons_a @ Y @ Xs ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y ) @ R3 ) )
=> ~ ! [Zs: list_a] :
( ( Ys
= ( cons_a @ X4 @ Zs ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Zs ) @ ( listrel1_a @ R3 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_1223_Cons__listrel1E2,axiom,
! [Xs: list_a,Y3: a,Ys: list_a,R3: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ ( cons_a @ Y3 @ Ys ) ) @ ( listrel1_a @ R3 ) )
=> ( ! [X: a] :
( ( Xs
= ( cons_a @ X @ Ys ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R3 ) )
=> ~ ! [Zs: list_a] :
( ( Xs
= ( cons_a @ Y3 @ Zs ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Zs @ Ys ) @ ( listrel1_a @ R3 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_1224_transpose_Opinduct,axiom,
! [A0: list_l2076173926988081843od_a_a,P: list_l2076173926988081843od_a_a > $o] :
( ( accp_l288472794138438844od_a_a @ transp70083455532304476od_a_a @ A0 )
=> ( ( ( accp_l288472794138438844od_a_a @ transp70083455532304476od_a_a @ nil_li8564743906319921459od_a_a )
=> ( P @ nil_li8564743906319921459od_a_a ) )
=> ( ! [Xss2: list_l2076173926988081843od_a_a] :
( ( accp_l288472794138438844od_a_a @ transp70083455532304476od_a_a @ ( cons_l836352187635934435od_a_a @ nil_Product_prod_a_a @ Xss2 ) )
=> ( ( P @ Xss2 )
=> ( P @ ( cons_l836352187635934435od_a_a @ nil_Product_prod_a_a @ Xss2 ) ) ) )
=> ( ! [X: product_prod_a_a,Xs3: list_P1396940483166286381od_a_a,Xss2: list_l2076173926988081843od_a_a] :
( ( accp_l288472794138438844od_a_a @ transp70083455532304476od_a_a @ ( cons_l836352187635934435od_a_a @ ( cons_P7316939126706565853od_a_a @ X @ Xs3 ) @ Xss2 ) )
=> ( ( P
@ ( cons_l836352187635934435od_a_a @ Xs3
@ ( concat8629597595045315054od_a_a
@ ( map_li711184933725290634od_a_a
@ ( case_l8205109941394486810od_a_a @ nil_li8564743906319921459od_a_a
@ ^ [H2: product_prod_a_a,T: list_P1396940483166286381od_a_a] : ( cons_l836352187635934435od_a_a @ T @ nil_li8564743906319921459od_a_a ) )
@ Xss2 ) ) ) )
=> ( P @ ( cons_l836352187635934435od_a_a @ ( cons_P7316939126706565853od_a_a @ X @ Xs3 ) @ Xss2 ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% transpose.pinduct
thf(fact_1225_listrel1p__def,axiom,
( listrel1p_a
= ( ^ [R: a > a > $o,Xs2: list_a,Ys2: list_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys2 ) @ ( listrel1_a @ ( collec3336397797384452498od_a_a @ ( produc6436628058953941356_a_a_o @ R ) ) ) ) ) ) ).
% listrel1p_def
thf(fact_1226_lists__empty,axiom,
( ( lists_9062004080735213851od_a_a @ bot_bo3357376287454694259od_a_a )
= ( insert6721513254612355069od_a_a @ nil_Product_prod_a_a @ bot_bo6558180029120801017od_a_a ) ) ).
% lists_empty
thf(fact_1227_lists__empty,axiom,
( ( lists_a @ bot_bot_set_a )
= ( insert_list_a @ nil_a @ bot_bot_set_list_a ) ) ).
% lists_empty
thf(fact_1228_Cons__in__lists__iff,axiom,
! [X4: a,Xs: list_a,A5: set_a] :
( ( member_list_a @ ( cons_a @ X4 @ Xs ) @ ( lists_a @ A5 ) )
= ( ( member_a @ X4 @ A5 )
& ( member_list_a @ Xs @ ( lists_a @ A5 ) ) ) ) ).
% Cons_in_lists_iff
thf(fact_1229_Cons__in__lists__iff,axiom,
! [X4: product_prod_a_a,Xs: list_P1396940483166286381od_a_a,A5: set_Product_prod_a_a] :
( ( member6824001069763096534od_a_a @ ( cons_P7316939126706565853od_a_a @ X4 @ Xs ) @ ( lists_9062004080735213851od_a_a @ A5 ) )
= ( ( member1426531477525435216od_a_a @ X4 @ A5 )
& ( member6824001069763096534od_a_a @ Xs @ ( lists_9062004080735213851od_a_a @ A5 ) ) ) ) ).
% Cons_in_lists_iff
thf(fact_1230_in__listsI,axiom,
! [Xs: list_a,A5: set_a] :
( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( member_a @ X @ A5 ) )
=> ( member_list_a @ Xs @ ( lists_a @ A5 ) ) ) ).
% in_listsI
thf(fact_1231_in__listsI,axiom,
! [Xs: list_P1396940483166286381od_a_a,A5: set_Product_prod_a_a] :
( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( member1426531477525435216od_a_a @ X @ A5 ) )
=> ( member6824001069763096534od_a_a @ Xs @ ( lists_9062004080735213851od_a_a @ A5 ) ) ) ).
% in_listsI
thf(fact_1232_lists__Int__eq,axiom,
! [A5: set_a,B5: set_a] :
( ( lists_a @ ( inf_inf_set_a @ A5 @ B5 ) )
= ( inf_inf_set_list_a @ ( lists_a @ A5 ) @ ( lists_a @ B5 ) ) ) ).
% lists_Int_eq
thf(fact_1233_lists_ONil,axiom,
! [A5: set_Product_prod_a_a] : ( member6824001069763096534od_a_a @ nil_Product_prod_a_a @ ( lists_9062004080735213851od_a_a @ A5 ) ) ).
% lists.Nil
thf(fact_1234_lists__IntI,axiom,
! [L2: list_a,A5: set_a,B5: set_a] :
( ( member_list_a @ L2 @ ( lists_a @ A5 ) )
=> ( ( member_list_a @ L2 @ ( lists_a @ B5 ) )
=> ( member_list_a @ L2 @ ( lists_a @ ( inf_inf_set_a @ A5 @ B5 ) ) ) ) ) ).
% lists_IntI
thf(fact_1235_lists__eq__set,axiom,
( lists_9062004080735213851od_a_a
= ( ^ [A6: set_Product_prod_a_a] :
( collec2774578310585404696od_a_a
@ ^ [Xs2: list_P1396940483166286381od_a_a] : ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs2 ) @ A6 ) ) ) ) ).
% lists_eq_set
thf(fact_1236_lists_OCons,axiom,
! [A3: a,A5: set_a,L2: list_a] :
( ( member_a @ A3 @ A5 )
=> ( ( member_list_a @ L2 @ ( lists_a @ A5 ) )
=> ( member_list_a @ ( cons_a @ A3 @ L2 ) @ ( lists_a @ A5 ) ) ) ) ).
% lists.Cons
thf(fact_1237_lists_OCons,axiom,
! [A3: product_prod_a_a,A5: set_Product_prod_a_a,L2: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ A3 @ A5 )
=> ( ( member6824001069763096534od_a_a @ L2 @ ( lists_9062004080735213851od_a_a @ A5 ) )
=> ( member6824001069763096534od_a_a @ ( cons_P7316939126706565853od_a_a @ A3 @ L2 ) @ ( lists_9062004080735213851od_a_a @ A5 ) ) ) ) ).
% lists.Cons
thf(fact_1238_listsE,axiom,
! [X4: a,L2: list_a,A5: set_a] :
( ( member_list_a @ ( cons_a @ X4 @ L2 ) @ ( lists_a @ A5 ) )
=> ~ ( ( member_a @ X4 @ A5 )
=> ~ ( member_list_a @ L2 @ ( lists_a @ A5 ) ) ) ) ).
% listsE
thf(fact_1239_listsE,axiom,
! [X4: product_prod_a_a,L2: list_P1396940483166286381od_a_a,A5: set_Product_prod_a_a] :
( ( member6824001069763096534od_a_a @ ( cons_P7316939126706565853od_a_a @ X4 @ L2 ) @ ( lists_9062004080735213851od_a_a @ A5 ) )
=> ~ ( ( member1426531477525435216od_a_a @ X4 @ A5 )
=> ~ ( member6824001069763096534od_a_a @ L2 @ ( lists_9062004080735213851od_a_a @ A5 ) ) ) ) ).
% listsE
thf(fact_1240_in__listsD,axiom,
! [Xs: list_a,A5: set_a] :
( ( member_list_a @ Xs @ ( lists_a @ A5 ) )
=> ! [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
=> ( member_a @ X5 @ A5 ) ) ) ).
% in_listsD
thf(fact_1241_in__listsD,axiom,
! [Xs: list_P1396940483166286381od_a_a,A5: set_Product_prod_a_a] :
( ( member6824001069763096534od_a_a @ Xs @ ( lists_9062004080735213851od_a_a @ A5 ) )
=> ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( member1426531477525435216od_a_a @ X5 @ A5 ) ) ) ).
% in_listsD
thf(fact_1242_in__lists__conv__set,axiom,
! [Xs: list_a,A5: set_a] :
( ( member_list_a @ Xs @ ( lists_a @ A5 ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( member_a @ X3 @ A5 ) ) ) ) ).
% in_lists_conv_set
thf(fact_1243_in__lists__conv__set,axiom,
! [Xs: list_P1396940483166286381od_a_a,A5: set_Product_prod_a_a] :
( ( member6824001069763096534od_a_a @ Xs @ ( lists_9062004080735213851od_a_a @ A5 ) )
= ( ! [X3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( member1426531477525435216od_a_a @ X3 @ A5 ) ) ) ) ).
% in_lists_conv_set
thf(fact_1244_lists_Osimps,axiom,
! [A3: list_a,A5: set_a] :
( ( member_list_a @ A3 @ ( lists_a @ A5 ) )
= ( ( A3 = nil_a )
| ? [A2: a,L: list_a] :
( ( A3
= ( cons_a @ A2 @ L ) )
& ( member_a @ A2 @ A5 )
& ( member_list_a @ L @ ( lists_a @ A5 ) ) ) ) ) ).
% lists.simps
thf(fact_1245_lists_Osimps,axiom,
! [A3: list_P1396940483166286381od_a_a,A5: set_Product_prod_a_a] :
( ( member6824001069763096534od_a_a @ A3 @ ( lists_9062004080735213851od_a_a @ A5 ) )
= ( ( A3 = nil_Product_prod_a_a )
| ? [A2: product_prod_a_a,L: list_P1396940483166286381od_a_a] :
( ( A3
= ( cons_P7316939126706565853od_a_a @ A2 @ L ) )
& ( member1426531477525435216od_a_a @ A2 @ A5 )
& ( member6824001069763096534od_a_a @ L @ ( lists_9062004080735213851od_a_a @ A5 ) ) ) ) ) ).
% lists.simps
thf(fact_1246_lists_Ocases,axiom,
! [A3: list_a,A5: set_a] :
( ( member_list_a @ A3 @ ( lists_a @ A5 ) )
=> ( ( A3 != nil_a )
=> ~ ! [A: a,L3: list_a] :
( ( A3
= ( cons_a @ A @ L3 ) )
=> ( ( member_a @ A @ A5 )
=> ~ ( member_list_a @ L3 @ ( lists_a @ A5 ) ) ) ) ) ) ).
% lists.cases
thf(fact_1247_lists_Ocases,axiom,
! [A3: list_P1396940483166286381od_a_a,A5: set_Product_prod_a_a] :
( ( member6824001069763096534od_a_a @ A3 @ ( lists_9062004080735213851od_a_a @ A5 ) )
=> ( ( A3 != nil_Product_prod_a_a )
=> ~ ! [A: product_prod_a_a,L3: list_P1396940483166286381od_a_a] :
( ( A3
= ( cons_P7316939126706565853od_a_a @ A @ L3 ) )
=> ( ( member1426531477525435216od_a_a @ A @ A5 )
=> ~ ( member6824001069763096534od_a_a @ L3 @ ( lists_9062004080735213851od_a_a @ A5 ) ) ) ) ) ) ).
% lists.cases
thf(fact_1248_listrelp__listrel__eq,axiom,
! [R3: set_Product_prod_a_a] :
( ( listrelp_a_a
@ ^ [X3: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R3 ) )
= ( ^ [X3: list_a,Y2: list_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X3 @ Y2 ) @ ( listrel_a_a @ R3 ) ) ) ) ).
% listrelp_listrel_eq
thf(fact_1249_maps__simps_I2_J,axiom,
! [F: product_prod_a_a > list_P1396940483166286381od_a_a] :
( ( maps_P6194213432232786177od_a_a @ F @ nil_Product_prod_a_a )
= nil_Product_prod_a_a ) ).
% maps_simps(2)
thf(fact_1250_listrelp_ONil,axiom,
! [R3: product_prod_a_a > product_prod_a_a > $o] : ( listre3943852969021380607od_a_a @ R3 @ nil_Product_prod_a_a @ nil_Product_prod_a_a ) ).
% listrelp.Nil
thf(fact_1251_listrelp_Ocases,axiom,
! [R3: product_prod_a_a > product_prod_a_a > $o,A12: list_P1396940483166286381od_a_a,A23: list_P1396940483166286381od_a_a] :
( ( listre3943852969021380607od_a_a @ R3 @ A12 @ A23 )
=> ( ( ( A12 = nil_Product_prod_a_a )
=> ( A23 != nil_Product_prod_a_a ) )
=> ~ ! [X: product_prod_a_a,Y: product_prod_a_a,Xs3: list_P1396940483166286381od_a_a] :
( ( A12
= ( cons_P7316939126706565853od_a_a @ X @ Xs3 ) )
=> ! [Ys3: list_P1396940483166286381od_a_a] :
( ( A23
= ( cons_P7316939126706565853od_a_a @ Y @ Ys3 ) )
=> ( ( R3 @ X @ Y )
=> ~ ( listre3943852969021380607od_a_a @ R3 @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_1252_listrelp_Osimps,axiom,
( listre3943852969021380607od_a_a
= ( ^ [R: product_prod_a_a > product_prod_a_a > $o,A1: list_P1396940483166286381od_a_a,A22: list_P1396940483166286381od_a_a] :
( ( ( A1 = nil_Product_prod_a_a )
& ( A22 = nil_Product_prod_a_a ) )
| ? [X3: product_prod_a_a,Y2: product_prod_a_a,Xs2: list_P1396940483166286381od_a_a,Ys2: list_P1396940483166286381od_a_a] :
( ( A1
= ( cons_P7316939126706565853od_a_a @ X3 @ Xs2 ) )
& ( A22
= ( cons_P7316939126706565853od_a_a @ Y2 @ Ys2 ) )
& ( R @ X3 @ Y2 )
& ( listre3943852969021380607od_a_a @ R @ Xs2 @ Ys2 ) ) ) ) ) ).
% listrelp.simps
thf(fact_1253_listrel__def,axiom,
( listrel_a_a
= ( ^ [R: set_Product_prod_a_a] :
( collec943055143889122450list_a
@ ( produc8172378796822260076st_a_o
@ ( listrelp_a_a
@ ^ [X3: a,Y2: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R ) ) ) ) ) ) ).
% listrel_def
thf(fact_1254_shuffles_Oelims,axiom,
! [X4: list_P1396940483166286381od_a_a,Xa3: list_P1396940483166286381od_a_a,Y3: set_li8827807065578854541od_a_a] :
( ( ( shuffl3267877308853796546od_a_a @ X4 @ Xa3 )
= Y3 )
=> ( ( ( X4 = nil_Product_prod_a_a )
=> ( Y3
!= ( insert6721513254612355069od_a_a @ Xa3 @ bot_bo6558180029120801017od_a_a ) ) )
=> ( ( ( Xa3 = nil_Product_prod_a_a )
=> ( Y3
!= ( insert6721513254612355069od_a_a @ X4 @ bot_bo6558180029120801017od_a_a ) ) )
=> ~ ! [X: product_prod_a_a,Xs3: list_P1396940483166286381od_a_a] :
( ( X4
= ( cons_P7316939126706565853od_a_a @ X @ Xs3 ) )
=> ! [Y: product_prod_a_a,Ys3: list_P1396940483166286381od_a_a] :
( ( Xa3
= ( cons_P7316939126706565853od_a_a @ Y @ Ys3 ) )
=> ( Y3
!= ( sup_su6214745143002101857od_a_a @ ( image_788595945937515589od_a_a @ ( cons_P7316939126706565853od_a_a @ X ) @ ( shuffl3267877308853796546od_a_a @ Xs3 @ ( cons_P7316939126706565853od_a_a @ Y @ Ys3 ) ) ) @ ( image_788595945937515589od_a_a @ ( cons_P7316939126706565853od_a_a @ Y ) @ ( shuffl3267877308853796546od_a_a @ ( cons_P7316939126706565853od_a_a @ X @ Xs3 ) @ Ys3 ) ) ) ) ) ) ) ) ) ).
% shuffles.elims
thf(fact_1255_Id__on__fold,axiom,
! [A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( id_on_a @ A5 )
= ( finite8041173845075148881od_a_a
@ ^ [X3: a] : ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X3 @ X3 ) )
@ bot_bo3357376287454694259od_a_a
@ A5 ) ) ) ).
% Id_on_fold
thf(fact_1256_Nil__in__shuffles,axiom,
! [Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a] :
( ( member6824001069763096534od_a_a @ nil_Product_prod_a_a @ ( shuffl3267877308853796546od_a_a @ Xs @ Ys ) )
= ( ( Xs = nil_Product_prod_a_a )
& ( Ys = nil_Product_prod_a_a ) ) ) ).
% Nil_in_shuffles
thf(fact_1257_Nil__in__shufflesI,axiom,
! [Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a] :
( ( Xs = nil_Product_prod_a_a )
=> ( ( Ys = nil_Product_prod_a_a )
=> ( member6824001069763096534od_a_a @ nil_Product_prod_a_a @ ( shuffl3267877308853796546od_a_a @ Xs @ Ys ) ) ) ) ).
% Nil_in_shufflesI
thf(fact_1258_set__shuffles,axiom,
! [Zs2: list_P1396940483166286381od_a_a,Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a] :
( ( member6824001069763096534od_a_a @ Zs2 @ ( shuffl3267877308853796546od_a_a @ Xs @ Ys ) )
=> ( ( set_Product_prod_a_a2 @ Zs2 )
= ( sup_su3048258781599657691od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ ( set_Product_prod_a_a2 @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_1259_shufflesE,axiom,
! [Zs2: list_P1396940483166286381od_a_a,Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a] :
( ( member6824001069763096534od_a_a @ Zs2 @ ( shuffl3267877308853796546od_a_a @ Xs @ Ys ) )
=> ( ( ( Zs2 = Xs )
=> ( Ys != nil_Product_prod_a_a ) )
=> ( ( ( Zs2 = Ys )
=> ( Xs != nil_Product_prod_a_a ) )
=> ( ! [X: product_prod_a_a,Xs4: list_P1396940483166286381od_a_a] :
( ( Xs
= ( cons_P7316939126706565853od_a_a @ X @ Xs4 ) )
=> ! [Z2: product_prod_a_a,Zs3: list_P1396940483166286381od_a_a] :
( ( Zs2
= ( cons_P7316939126706565853od_a_a @ Z2 @ Zs3 ) )
=> ( ( X = Z2 )
=> ~ ( member6824001069763096534od_a_a @ Zs3 @ ( shuffl3267877308853796546od_a_a @ Xs4 @ Ys ) ) ) ) )
=> ~ ! [Y: product_prod_a_a,Ys4: list_P1396940483166286381od_a_a] :
( ( Ys
= ( cons_P7316939126706565853od_a_a @ Y @ Ys4 ) )
=> ! [Z2: product_prod_a_a,Zs3: list_P1396940483166286381od_a_a] :
( ( Zs2
= ( cons_P7316939126706565853od_a_a @ Z2 @ Zs3 ) )
=> ( ( Y = Z2 )
=> ~ ( member6824001069763096534od_a_a @ Zs3 @ ( shuffl3267877308853796546od_a_a @ Xs @ Ys4 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_1260_shuffles_Osimps_I2_J,axiom,
! [Xs: list_P1396940483166286381od_a_a] :
( ( shuffl3267877308853796546od_a_a @ Xs @ nil_Product_prod_a_a )
= ( insert6721513254612355069od_a_a @ Xs @ bot_bo6558180029120801017od_a_a ) ) ).
% shuffles.simps(2)
thf(fact_1261_shuffles_Osimps_I1_J,axiom,
! [Ys: list_P1396940483166286381od_a_a] :
( ( shuffl3267877308853796546od_a_a @ nil_Product_prod_a_a @ Ys )
= ( insert6721513254612355069od_a_a @ Ys @ bot_bo6558180029120801017od_a_a ) ) ).
% shuffles.simps(1)
thf(fact_1262_Set__filter__fold,axiom,
! [A5: set_a,P: a > $o] :
( ( finite_finite_a @ A5 )
=> ( ( filter_a @ P @ A5 )
= ( finite_fold_a_set_a
@ ^ [X3: a,A9: set_a] : ( if_set_a @ ( P @ X3 ) @ ( insert_a @ X3 @ A9 ) @ A9 )
@ bot_bot_set_a
@ A5 ) ) ) ).
% Set_filter_fold
thf(fact_1263_product__fold,axiom,
! [A5: set_a,B5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( finite_finite_a @ B5 )
=> ( ( product_Sigma_a_a @ A5
@ ^ [Uu: a] : B5 )
= ( finite8041173845075148881od_a_a
@ ^ [X3: a,Z5: set_Product_prod_a_a] :
( finite8041173845075148881od_a_a
@ ^ [Y2: a] : ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) )
@ Z5
@ B5 )
@ bot_bo3357376287454694259od_a_a
@ A5 ) ) ) ) ).
% product_fold
thf(fact_1264_comp__fun__commute__product__fold,axiom,
! [B5: set_a] :
( ( finite_finite_a @ B5 )
=> ( finite9169933705475362010od_a_a
@ ^ [X3: a,Z5: set_Product_prod_a_a] :
( finite8041173845075148881od_a_a
@ ^ [Y2: a] : ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) )
@ Z5
@ B5 ) ) ) ).
% comp_fun_commute_product_fold
thf(fact_1265_Image__fold,axiom,
! [R2: set_Product_prod_a_a,S: set_a] :
( ( finite6544458595007987280od_a_a @ R2 )
=> ( ( image_a_a @ R2 @ S )
= ( finite3448380600287114943_set_a
@ ( produc2520556867391656337_set_a
@ ^ [X3: a,Y2: a,A6: set_a] : ( if_set_a @ ( member_a @ X3 @ S ) @ ( insert_a @ Y2 @ A6 ) @ A6 ) )
@ bot_bot_set_a
@ R2 ) ) ) ).
% Image_fold
thf(fact_1266_Image__fold,axiom,
! [R2: set_Pr8876520727511657886_a_a_a,S: set_Product_prod_a_a] :
( ( finite5607752495362350695_a_a_a @ R2 )
=> ( ( image_8059871973944943978_a_a_a @ R2 @ S )
= ( finite443596670980949096_set_a
@ ( produc7523845443869562280_set_a
@ ^ [X3: product_prod_a_a,Y2: a,A6: set_a] : ( if_set_a @ ( member1426531477525435216od_a_a @ X3 @ S ) @ ( insert_a @ Y2 @ A6 ) @ A6 ) )
@ bot_bot_set_a
@ R2 ) ) ) ).
% Image_fold
thf(fact_1267_fold__union__pair,axiom,
! [B5: set_a,X4: a,A5: set_Product_prod_a_a] :
( ( finite_finite_a @ B5 )
=> ( ( sup_su3048258781599657691od_a_a
@ ( comple8421679170691845492od_a_a
@ ( image_4421510592991446670od_a_a
@ ^ [Y2: a] : ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X4 @ Y2 ) @ bot_bo3357376287454694259od_a_a )
@ B5 ) )
@ A5 )
= ( finite8041173845075148881od_a_a
@ ^ [Y2: a] : ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X4 @ Y2 ) )
@ A5
@ B5 ) ) ) ).
% fold_union_pair
thf(fact_1268_DiffI,axiom,
! [C: a,A5: set_a,B5: set_a] :
( ( member_a @ C @ A5 )
=> ( ~ ( member_a @ C @ B5 )
=> ( member_a @ C @ ( minus_minus_set_a @ A5 @ B5 ) ) ) ) ).
% DiffI
thf(fact_1269_DiffI,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A5 )
=> ( ~ ( member1426531477525435216od_a_a @ C @ B5 )
=> ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A5 @ B5 ) ) ) ) ).
% DiffI
thf(fact_1270_Diff__iff,axiom,
! [C: a,A5: set_a,B5: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A5 @ B5 ) )
= ( ( member_a @ C @ A5 )
& ~ ( member_a @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_1271_Diff__iff,axiom,
! [C: product_prod_a_a,A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A5 @ B5 ) )
= ( ( member1426531477525435216od_a_a @ C @ A5 )
& ~ ( member1426531477525435216od_a_a @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_1272_Diff__empty,axiom,
! [A5: set_a] :
( ( minus_minus_set_a @ A5 @ bot_bot_set_a )
= A5 ) ).
% Diff_empty
% Helper facts (9)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X4: a,Y3: a] :
( ( if_a @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X4: a,Y3: a] :
( ( if_a @ $true @ X4 @ Y3 )
= X4 ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X4: set_a,Y3: set_a] :
( ( if_set_a @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__a_J_T,axiom,
! [X4: set_a,Y3: set_a] :
( ( if_set_a @ $true @ X4 @ Y3 )
= X4 ) ).
thf(help_If_2_1_If_001t__Option__Ooption_It__Set__Oset_Itf__a_J_J_T,axiom,
! [X4: option_set_a,Y3: option_set_a] :
( ( if_option_set_a @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__Set__Oset_Itf__a_J_J_T,axiom,
! [X4: option_set_a,Y3: option_set_a] :
( ( if_option_set_a @ $true @ X4 @ Y3 )
= X4 ) ).
thf(help_If_3_1_If_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_T,axiom,
! [X4: option3362143566709901261od_a_a,Y3: option3362143566709901261od_a_a] :
( ( if_opt7251041032425086355od_a_a @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J_J_T,axiom,
! [X4: option3362143566709901261od_a_a,Y3: option3362143566709901261od_a_a] :
( ( if_opt7251041032425086355od_a_a @ $true @ X4 @ Y3 )
= X4 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( find_P2248954792982286147od_a_a
@ ( produc6436628058953941356_a_a_o
@ ^ [X3: a,Y2: a] :
( ( member_a @ X3 @ a2 )
| ( member_a @ Y2 @ a2 ) ) )
@ xys )
!= none_P1914705783780380040od_a_a ) ).
%------------------------------------------------------------------------------