TPTP Problem File: SLH0862^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% 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 : SCC_Bloemen_Sequential/0000_SCC_Bloemen_Sequential/prob_03062_105445__6592276_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1597 ( 769 unt; 317 typ; 0 def)
% Number of atoms : 3457 (1590 equ; 0 cnn)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 10678 ( 446 ~; 66 |; 363 &;8689 @)
% ( 0 <=>;1114 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 5 avg)
% Number of types : 52 ( 51 usr)
% Number of type conns : 644 ( 644 >; 0 *; 0 +; 0 <<)
% Number of symbols : 269 ( 266 usr; 55 con; 0-9 aty)
% Number of variables : 3264 ( 171 ^;2947 !; 146 ?;3264 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:54:24.684
%------------------------------------------------------------------------------
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sCC_Bl349061681862590396t_unit: ( list_v > list_v ) > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit ).
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sCC_Bl5498988629518860705t_unit: sCC_Bl7326425374436813197t_unit > set_Product_prod_v_v ).
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sCC_Bl4645233313691564917t_unit: sCC_Bl1394983891496994913t_unit > set_v ).
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sCC_Bl7870604408699998558t_unit: ( set_v > set_v ) > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit ).
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sCC_Bl3878977043676959280t_unit: sCC_Bl7326425374436813197t_unit > product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ovsuccs_001tf__v_001t__Product____Type__Ounit,type,
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thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ovsuccs__update_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl48393358579903213t_unit: ( ( v > set_v ) > v > set_v ) > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_001_Eo,type,
sCC_Bloemen_graph_o: set_o > ( $o > set_o ) > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_001t__Nat__Onat,type,
sCC_Bl8035451632035226289ph_nat: set_nat > ( nat > set_nat ) > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl8307124943676871238od_v_v: set_Product_prod_v_v > ( product_prod_v_v > set_Product_prod_v_v ) > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_001t__Product____Type__Ounit,type,
sCC_Bl1875605551932356204t_unit: set_Product_unit > ( product_unit > set_Product_unit ) > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_001t__Set__Oset_Itf__v_J,type,
sCC_Bl5810666556806954322_set_v: set_set_v > ( set_v > set_set_v ) > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_001tf__v,type,
sCC_Bloemen_graph_v: set_v > ( v > set_v ) > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Odfs_001tf__v,type,
sCC_Bloemen_dfs_v: ( v > set_v ) > v > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Odfs__dfss__rel_001tf__v,type,
sCC_Bl907557413677168252_rel_v: ( v > set_v ) > sum_su8181647976486975269t_unit > sum_su8181647976486975269t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Odfss_001tf__v,type,
sCC_Bloemen_dfss_v: ( v > set_v ) > v > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__scc_001_Eo,type,
sCC_Bloemen_is_scc_o: ( $o > set_o ) > set_o > $o ).
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sCC_Bl6242042402218619277od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > set_Product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__scc_001t__Product____Type__Ounit,type,
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thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__scc_001t__Set__Oset_Itf__v_J,type,
sCC_Bl1515522642333523865_set_v: ( set_v > set_set_v ) > set_set_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__scc_001tf__v,type,
sCC_Bloemen_is_scc_v: ( v > set_v ) > set_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__subscc_001_Eo,type,
sCC_Bl6048899608199235562bscc_o: ( $o > set_o ) > set_o > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__subscc_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl2301996248249672505od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > set_Product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__subscc_001t__Product____Type__Ounit,type,
sCC_Bl5104783218331194207t_unit: ( product_unit > set_Product_unit ) > set_Product_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__subscc_001t__Set__Oset_Itf__v_J,type,
sCC_Bl7907073126578335045_set_v: ( set_v > set_set_v ) > set_set_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ois__subscc_001tf__v,type,
sCC_Bl5398416737448265317bscc_v: ( v > set_v ) > set_v > $o ).
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thf(sy_c_SCC__Bloemen__Sequential_Ograph_Opost__dfss_001tf__v_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
sCC_Bl6082031138996704384t_unit: ( v > set_v ) > v > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Opre__dfs_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl36166008131615352t_unit: ( v > set_v ) > v > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Opre__dfss_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit,type,
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thf(sy_c_SCC__Bloemen__Sequential_Ograph_Opre__dfss_001tf__v_001t__Product____Type__Ounit,type,
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thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable_001_Eo,type,
sCC_Bl1688025099652275002able_o: ( $o > set_o ) > $o > $o > $o ).
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sCC_Bl4981926079593201289od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > product_prod_v_v > product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable_001t__Product____Type__Ounit,type,
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thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable_001t__Set__Oset_Itf__v_J,type,
sCC_Bl7354734129683093653_set_v: ( set_v > set_set_v ) > set_v > set_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable_001tf__v,type,
sCC_Bl649662514949026229able_v: ( v > set_v ) > v > v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__avoiding_001tf__v,type,
sCC_Bl4291963740693775144ding_v: ( v > set_v ) > v > v > set_Product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__end_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl4714988717384592488od_v_v: ( product_prod_v_v > set_Product_prod_v_v ) > product_prod_v_v > product_prod_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable__end_001tf__v,type,
sCC_Bl770211535891879572_end_v: ( v > set_v ) > v > v > $o ).
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sCC_Bl7963838319573962697t_unit: sCC_Bl7326425374436813197t_unit > sCC_Bl7326425374436813197t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Osub__env_001tf__v_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
sCC_Bl5768913643336123637t_unit: sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ounite_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl4702006153222411093od_v_v: product_prod_v_v > product_prod_v_v > sCC_Bl7326425374436813197t_unit > sCC_Bl7326425374436813197t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Ounite_001tf__v,type,
sCC_Bloemen_unite_v: v > v > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Owf__env_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit,type,
sCC_Bl7798947040364291444t_unit: ( product_prod_v_v > set_Product_prod_v_v ) > sCC_Bl7326425374436813197t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Owf__env_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl9196236973127232072t_unit: ( v > set_v ) > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Oinit__env_001tf__v,type,
sCC_Bl7693227186847904995_env_v: v > sCC_Bl1394983891496994913t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Oprecedes_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
sCC_Bl2026170059108282219od_v_v: product_prod_v_v > product_prod_v_v > list_P7986770385144383213od_v_v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Oprecedes_001tf__v,type,
sCC_Bl4022239298816431255edes_v: v > v > list_v > $o ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
collec140062887454715474od_v_v: ( product_prod_v_v > $o ) > set_Product_prod_v_v ).
thf(sy_c_Set_OCollect_001t__Product____Type__Ounit,type,
collect_Product_unit: ( product_unit > $o ) > set_Product_unit ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__v_J,type,
collect_set_v: ( set_v > $o ) > set_set_v ).
thf(sy_c_Set_OCollect_001tf__v,type,
collect_v: ( v > $o ) > set_v ).
thf(sy_c_Set_Oinsert_001_Eo,type,
insert_o: $o > set_o > set_o ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
insert1338601472111419319od_v_v: product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Ounit,type,
insert_Product_unit: product_unit > set_Product_unit > set_Product_unit ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__v_J,type,
insert_set_v: set_v > set_set_v > set_set_v ).
thf(sy_c_Set_Oinsert_001tf__v,type,
insert_v: v > set_v > set_v ).
thf(sy_c_Set_Othe__elem_001_Eo,type,
the_elem_o: set_o > $o ).
thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
the_el5392834299063928540od_v_v: set_Product_prod_v_v > product_prod_v_v ).
thf(sy_c_Set_Othe__elem_001t__Product____Type__Ounit,type,
the_el608902216710161154t_unit: set_Product_unit > product_unit ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_Itf__v_J,type,
the_elem_set_v: set_set_v > set_v ).
thf(sy_c_Set_Othe__elem_001tf__v,type,
the_elem_v: set_v > v ).
thf(sy_c_Sum__Type_OInl_001t__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_001t__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J,type,
sum_In526841707622398774t_unit: produc5741669702376414499t_unit > sum_su8181647976486975269t_unit ).
thf(sy_c_Sum__Type_OInr_001t__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_001t__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J,type,
sum_In5289330923152326972t_unit: produc5741669702376414499t_unit > sum_su8181647976486975269t_unit ).
thf(sy_c_Wellfounded_Oaccp_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_Mt__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J_J,type,
accp_S2303753412255344476t_unit: ( sum_su8181647976486975269t_unit > sum_su8181647976486975269t_unit > $o ) > sum_su8181647976486975269t_unit > $o ).
thf(sy_c_fChoice_001tf__v,type,
fChoice_v: ( v > $o ) > v ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__v_Mt__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_J,type,
member7924940910754673978t_unit: produc5741669702376414499t_unit > set_Pr6425124735969554649t_unit > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
member7453568604450474000od_v_v: product_prod_v_v > set_Product_prod_v_v > $o ).
thf(sy_c_member_001t__Product____Type__Ounit,type,
member_Product_unit: product_unit > set_Product_unit > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__v_J,type,
member_set_v: set_v > set_set_v > $o ).
thf(sy_c_member_001tf__v,type,
member_v2: v > set_v > $o ).
thf(sy_v_e,type,
e: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_successors,type,
successors: v > set_v ).
thf(sy_v_v,type,
v2: v ).
thf(sy_v_vertices,type,
vertices: set_v ).
% Relevant facts (1273)
thf(fact_0__092_060open_062stack_Ae_A_092_060noteq_062_A_091_093_092_060close_062,axiom,
( ( sCC_Bl8828226123343373779t_unit @ e )
!= nil_v ) ).
% \<open>stack e \<noteq> []\<close>
thf(fact_1_sub__env__trans,axiom,
! [E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
=> ( ( sCC_Bl5768913643336123637t_unit @ E2 @ E3 )
=> ( sCC_Bl5768913643336123637t_unit @ E @ E3 ) ) ) ).
% sub_env_trans
thf(fact_2_dfs__dfss__rel_Ocong,axiom,
sCC_Bl907557413677168252_rel_v = sCC_Bl907557413677168252_rel_v ).
% dfs_dfss_rel.cong
thf(fact_3_graph_Opost__dfss_Ocong,axiom,
sCC_Bl6082031138996704384t_unit = sCC_Bl6082031138996704384t_unit ).
% graph.post_dfss.cong
thf(fact_4_dfss_Ocases,axiom,
! [X: produc5741669702376414499t_unit] :
~ ! [V: v,E4: sCC_Bl1394983891496994913t_unit] :
( X
!= ( produc3862955338007567901t_unit @ V @ E4 ) ) ).
% dfss.cases
thf(fact_5_graph_Opre__dfs_Ocong,axiom,
sCC_Bl36166008131615352t_unit = sCC_Bl36166008131615352t_unit ).
% graph.pre_dfs.cong
thf(fact_6_graph_Ois__scc_Ocong,axiom,
sCC_Bloemen_is_scc_v = sCC_Bloemen_is_scc_v ).
% graph.is_scc.cong
thf(fact_7_graph_Odfss_Ocong,axiom,
sCC_Bloemen_dfss_v = sCC_Bloemen_dfss_v ).
% graph.dfss.cong
thf(fact_8_graph_Opost__dfs_Ocong,axiom,
sCC_Bl8953792750115413617t_unit = sCC_Bl8953792750115413617t_unit ).
% graph.post_dfs.cong
thf(fact_9_graph_Opre__dfss_Ocong,axiom,
sCC_Bl1748261141445803503t_unit = sCC_Bl1748261141445803503t_unit ).
% graph.pre_dfss.cong
thf(fact_10_graph_Ois__subscc_Ocong,axiom,
sCC_Bl5398416737448265317bscc_v = sCC_Bl5398416737448265317bscc_v ).
% graph.is_subscc.cong
thf(fact_11_graph_Odfs_Ocong,axiom,
sCC_Bloemen_dfs_v = sCC_Bloemen_dfs_v ).
% graph.dfs.cong
thf(fact_12__092_060open_062cstack_Ae_A_061_A_091_093_092_060close_062,axiom,
( ( sCC_Bl9201514103433284750t_unit @ e )
= nil_v ) ).
% \<open>cstack e = []\<close>
thf(fact_13__092_060open_062hd_A_Istack_Ae_J_A_092_060preceq_062_Ahd_A_Istack_Ae_J_Ain_Astack_Ae_092_060close_062,axiom,
sCC_Bl4022239298816431255edes_v @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e ) ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e ) ) @ ( sCC_Bl8828226123343373779t_unit @ e ) ).
% \<open>hd (stack e) \<preceq> hd (stack e) in stack e\<close>
thf(fact_14_prod_Oinject,axiom,
! [X1: v,X2: sCC_Bl1394983891496994913t_unit,Y1: v,Y2: sCC_Bl1394983891496994913t_unit] :
( ( ( produc3862955338007567901t_unit @ X1 @ X2 )
= ( produc3862955338007567901t_unit @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_15_prod_Oinject,axiom,
! [X1: v,X2: v,Y1: v,Y2: v] :
( ( ( product_Pair_v_v @ X1 @ X2 )
= ( product_Pair_v_v @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_16_old_Oprod_Oinject,axiom,
! [A: v,B: sCC_Bl1394983891496994913t_unit,A2: v,B2: sCC_Bl1394983891496994913t_unit] :
( ( ( produc3862955338007567901t_unit @ A @ B )
= ( produc3862955338007567901t_unit @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_17_old_Oprod_Oinject,axiom,
! [A: v,B: v,A2: v,B2: v] :
( ( ( product_Pair_v_v @ A @ B )
= ( product_Pair_v_v @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_18_old_Oprod_Oexhaust,axiom,
! [Y: produc5741669702376414499t_unit] :
~ ! [A3: v,B3: sCC_Bl1394983891496994913t_unit] :
( Y
!= ( produc3862955338007567901t_unit @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_19_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_v_v] :
~ ! [A3: v,B3: v] :
( Y
!= ( product_Pair_v_v @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_20_surj__pair,axiom,
! [P: produc5741669702376414499t_unit] :
? [X3: v,Y3: sCC_Bl1394983891496994913t_unit] :
( P
= ( produc3862955338007567901t_unit @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_21_surj__pair,axiom,
! [P: product_prod_v_v] :
? [X3: v,Y3: v] :
( P
= ( product_Pair_v_v @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_22_prod__cases,axiom,
! [P2: produc5741669702376414499t_unit > $o,P: produc5741669702376414499t_unit] :
( ! [A3: v,B3: sCC_Bl1394983891496994913t_unit] : ( P2 @ ( produc3862955338007567901t_unit @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_23_prod__cases,axiom,
! [P2: product_prod_v_v > $o,P: product_prod_v_v] :
( ! [A3: v,B3: v] : ( P2 @ ( product_Pair_v_v @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_24_Pair__inject,axiom,
! [A: v,B: sCC_Bl1394983891496994913t_unit,A2: v,B2: sCC_Bl1394983891496994913t_unit] :
( ( ( produc3862955338007567901t_unit @ A @ B )
= ( produc3862955338007567901t_unit @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_25_Pair__inject,axiom,
! [A: v,B: v,A2: v,B2: v] :
( ( ( product_Pair_v_v @ A @ B )
= ( product_Pair_v_v @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_26_list__ex1__simps_I1_J,axiom,
! [P2: v > $o] :
~ ( list_ex1_v @ P2 @ nil_v ) ).
% list_ex1_simps(1)
thf(fact_27_bind__simps_I1_J,axiom,
! [F: v > list_v] :
( ( bind_v_v @ nil_v @ F )
= nil_v ) ).
% bind_simps(1)
thf(fact_28_member__rec_I2_J,axiom,
! [Y: v] :
~ ( member_v @ nil_v @ Y ) ).
% member_rec(2)
thf(fact_29_ssubst__Pair__rhs,axiom,
! [R: v,S: sCC_Bl1394983891496994913t_unit,R2: set_Pr6425124735969554649t_unit,S2: sCC_Bl1394983891496994913t_unit] :
( ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_30_ssubst__Pair__rhs,axiom,
! [R: v,S: v,R2: set_Product_prod_v_v,S2: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_31_in__measures_I1_J,axiom,
! [X: v,Y: v] :
~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ ( measures_v @ nil_v_nat ) ) ).
% in_measures(1)
thf(fact_32_mem__Collect__eq,axiom,
! [A: v,P2: v > $o] :
( ( member_v2 @ A @ ( collect_v @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_33_mem__Collect__eq,axiom,
! [A: product_prod_v_v,P2: product_prod_v_v > $o] :
( ( member7453568604450474000od_v_v @ A @ ( collec140062887454715474od_v_v @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_34_mem__Collect__eq,axiom,
! [A: set_v,P2: set_v > $o] :
( ( member_set_v @ A @ ( collect_set_v @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_35_Collect__mem__eq,axiom,
! [A4: set_v] :
( ( collect_v
@ ^ [X4: v] : ( member_v2 @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_36_Collect__mem__eq,axiom,
! [A4: set_Product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ^ [X4: product_prod_v_v] : ( member7453568604450474000od_v_v @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_37_Collect__mem__eq,axiom,
! [A4: set_set_v] :
( ( collect_set_v
@ ^ [X4: set_v] : ( member_set_v @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_38_Collect__cong,axiom,
! [P2: set_v > $o,Q: set_v > $o] :
( ! [X3: set_v] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_set_v @ P2 )
= ( collect_set_v @ Q ) ) ) ).
% Collect_cong
thf(fact_39_graph_Odfs__S__tl__stack_I1_J,axiom,
! [Vertices: set_v,Successors: v > set_v,V2: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V2 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ) ).
% graph.dfs_S_tl_stack(1)
thf(fact_40_fold__congs_I8_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R3: sCC_Bl1394983891496994913t_unit,V3: list_v,F: list_v > list_v,F2: list_v > list_v] :
( ( R = R3 )
=> ( ( ( sCC_Bl9201514103433284750t_unit @ R3 )
= V3 )
=> ( ! [V: list_v] :
( ( V3 = V )
=> ( ( F @ V )
= ( F2 @ V ) ) )
=> ( ( sCC_Bl7876664385711583351t_unit @ F @ R )
= ( sCC_Bl7876664385711583351t_unit @ F2 @ R3 ) ) ) ) ) ).
% fold_congs(8)
thf(fact_41_curryI,axiom,
! [F: produc5741669702376414499t_unit > $o,A: v,B: sCC_Bl1394983891496994913t_unit] :
( ( F @ ( produc3862955338007567901t_unit @ A @ B ) )
=> ( produc1221886420749542637unit_o @ F @ A @ B ) ) ).
% curryI
thf(fact_42_curryI,axiom,
! [F: product_prod_v_v > $o,A: v,B: v] :
( ( F @ ( product_Pair_v_v @ A @ B ) )
=> ( product_curry_v_v_o @ F @ A @ B ) ) ).
% curryI
thf(fact_43_graph_Odfss_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,X: produc5741669702376414499t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ~ ! [V: v,E4: sCC_Bl1394983891496994913t_unit] :
( X
!= ( produc3862955338007567901t_unit @ V @ E4 ) ) ) ).
% graph.dfss.cases
thf(fact_44_graph_Osub__env__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
=> ( ( sCC_Bl5768913643336123637t_unit @ E2 @ E3 )
=> ( sCC_Bl5768913643336123637t_unit @ E @ E3 ) ) ) ) ).
% graph.sub_env_trans
thf(fact_45_curryD,axiom,
! [F: produc5741669702376414499t_unit > $o,A: v,B: sCC_Bl1394983891496994913t_unit] :
( ( produc1221886420749542637unit_o @ F @ A @ B )
=> ( F @ ( produc3862955338007567901t_unit @ A @ B ) ) ) ).
% curryD
thf(fact_46_curryD,axiom,
! [F: product_prod_v_v > $o,A: v,B: v] :
( ( product_curry_v_v_o @ F @ A @ B )
=> ( F @ ( product_Pair_v_v @ A @ B ) ) ) ).
% curryD
thf(fact_47_curryE,axiom,
! [F: produc5741669702376414499t_unit > $o,A: v,B: sCC_Bl1394983891496994913t_unit] :
( ( produc1221886420749542637unit_o @ F @ A @ B )
=> ( F @ ( produc3862955338007567901t_unit @ A @ B ) ) ) ).
% curryE
thf(fact_48_curryE,axiom,
! [F: product_prod_v_v > $o,A: v,B: v] :
( ( product_curry_v_v_o @ F @ A @ B )
=> ( F @ ( product_Pair_v_v @ A @ B ) ) ) ).
% curryE
thf(fact_49_unfold__congs_I8_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R3: sCC_Bl1394983891496994913t_unit,V3: list_v,F: list_v > list_v,F2: list_v > list_v] :
( ( R = R3 )
=> ( ( ( sCC_Bl9201514103433284750t_unit @ R3 )
= V3 )
=> ( ! [V: list_v] :
( ( V = V3 )
=> ( ( F @ V )
= ( F2 @ V ) ) )
=> ( ( sCC_Bl7876664385711583351t_unit @ F @ R )
= ( sCC_Bl7876664385711583351t_unit @ F2 @ R3 ) ) ) ) ) ).
% unfold_congs(8)
thf(fact_50_UNIV__I,axiom,
! [X: v] : ( member_v2 @ X @ top_top_set_v ) ).
% UNIV_I
thf(fact_51_UNIV__I,axiom,
! [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ top_to5429829297380968215od_v_v ) ).
% UNIV_I
thf(fact_52_UNIV__I,axiom,
! [X: product_unit] : ( member_Product_unit @ X @ top_to1996260823553986621t_unit ) ).
% UNIV_I
thf(fact_53_UNIV__I,axiom,
! [X: $o] : ( member_o @ X @ top_top_set_o ) ).
% UNIV_I
thf(fact_54_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_55_iso__tuple__UNIV__I,axiom,
! [X: v] : ( member_v2 @ X @ top_top_set_v ) ).
% iso_tuple_UNIV_I
thf(fact_56_iso__tuple__UNIV__I,axiom,
! [X: product_prod_v_v] : ( member7453568604450474000od_v_v @ X @ top_to5429829297380968215od_v_v ) ).
% iso_tuple_UNIV_I
thf(fact_57_iso__tuple__UNIV__I,axiom,
! [X: product_unit] : ( member_Product_unit @ X @ top_to1996260823553986621t_unit ) ).
% iso_tuple_UNIV_I
thf(fact_58_iso__tuple__UNIV__I,axiom,
! [X: $o] : ( member_o @ X @ top_top_set_o ) ).
% iso_tuple_UNIV_I
thf(fact_59_iso__tuple__UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_60_graph_Oinit__env__pre__dfs,axiom,
! [Vertices: set_v,Successors: v > set_v,V2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl36166008131615352t_unit @ Successors @ V2 @ ( sCC_Bl7693227186847904995_env_v @ V2 ) ) ) ).
% graph.init_env_pre_dfs
thf(fact_61_graph_Opre__dfss__pre__dfs,axiom,
! [Vertices: set_v,Successors: v > set_v,V2: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V2 @ E )
=> ( ~ ( member_v2 @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( member_v2 @ W @ ( Successors @ V2 ) )
=> ( sCC_Bl36166008131615352t_unit @ Successors @ W @ E ) ) ) ) ) ).
% graph.pre_dfss_pre_dfs
thf(fact_62_graph_Odfs__S__hd__stack_I2_J,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V2: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V2 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v2 @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( member_v2 @ N @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ) ).
% graph.dfs_S_hd_stack(2)
thf(fact_63_graph_Odfs__S__hd__stack_I1_J,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V2: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V2 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v2 @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ) ) ) ).
% graph.dfs_S_hd_stack(1)
thf(fact_64_UNIV__coset,axiom,
( top_top_set_v
= ( coset_v @ nil_v ) ) ).
% UNIV_coset
thf(fact_65_UNIV__coset,axiom,
( top_to1996260823553986621t_unit
= ( coset_Product_unit @ nil_Product_unit ) ) ).
% UNIV_coset
thf(fact_66_UNIV__coset,axiom,
( top_top_set_o
= ( coset_o @ nil_o ) ) ).
% UNIV_coset
thf(fact_67_UNIV__coset,axiom,
( top_top_set_nat
= ( coset_nat @ nil_nat ) ) ).
% UNIV_coset
thf(fact_68_UNIV__eq__I,axiom,
! [A4: set_v] :
( ! [X3: v] : ( member_v2 @ X3 @ A4 )
=> ( top_top_set_v = A4 ) ) ).
% UNIV_eq_I
thf(fact_69_UNIV__eq__I,axiom,
! [A4: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ A4 )
=> ( top_to5429829297380968215od_v_v = A4 ) ) ).
% UNIV_eq_I
thf(fact_70_UNIV__eq__I,axiom,
! [A4: set_Product_unit] :
( ! [X3: product_unit] : ( member_Product_unit @ X3 @ A4 )
=> ( top_to1996260823553986621t_unit = A4 ) ) ).
% UNIV_eq_I
thf(fact_71_UNIV__eq__I,axiom,
! [A4: set_o] :
( ! [X3: $o] : ( member_o @ X3 @ A4 )
=> ( top_top_set_o = A4 ) ) ).
% UNIV_eq_I
thf(fact_72_UNIV__eq__I,axiom,
! [A4: set_nat] :
( ! [X3: nat] : ( member_nat @ X3 @ A4 )
=> ( top_top_set_nat = A4 ) ) ).
% UNIV_eq_I
thf(fact_73_UNIV__witness,axiom,
? [X3: v] : ( member_v2 @ X3 @ top_top_set_v ) ).
% UNIV_witness
thf(fact_74_UNIV__witness,axiom,
? [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ top_to5429829297380968215od_v_v ) ).
% UNIV_witness
thf(fact_75_UNIV__witness,axiom,
? [X3: product_unit] : ( member_Product_unit @ X3 @ top_to1996260823553986621t_unit ) ).
% UNIV_witness
thf(fact_76_UNIV__witness,axiom,
? [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% UNIV_witness
thf(fact_77_UNIV__witness,axiom,
? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_78_graph_Owf__env_Ocong,axiom,
sCC_Bl9196236973127232072t_unit = sCC_Bl9196236973127232072t_unit ).
% graph.wf_env.cong
thf(fact_79_top__set__def,axiom,
( top_top_set_set_v
= ( collect_set_v @ top_top_set_v_o ) ) ).
% top_set_def
thf(fact_80_top__set__def,axiom,
( top_to1996260823553986621t_unit
= ( collect_Product_unit @ top_to2465898995584390880unit_o ) ) ).
% top_set_def
thf(fact_81_top__set__def,axiom,
( top_top_set_o
= ( collect_o @ top_top_o_o ) ) ).
% top_set_def
thf(fact_82_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_83_graph_OS__reflexive,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( member_v2 @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ).
% graph.S_reflexive
thf(fact_84_graph_Ostack__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v2 @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( member_v2 @ N @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ).
% graph.stack_visited
thf(fact_85_dfs__S__hd__stack_I1_J,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V2: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ V2 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v2 @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ) ) ).
% dfs_S_hd_stack(1)
thf(fact_86_dfs__S__hd__stack_I2_J,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V2: v,E2: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ V2 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( member_v2 @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
=> ( member_v2 @ N @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ).
% dfs_S_hd_stack(2)
thf(fact_87_graph__axioms,axiom,
sCC_Bloemen_graph_v @ vertices @ successors ).
% graph_axioms
thf(fact_88_init__env__pre__dfs,axiom,
! [V2: v] : ( sCC_Bl36166008131615352t_unit @ successors @ V2 @ ( sCC_Bl7693227186847904995_env_v @ V2 ) ) ).
% init_env_pre_dfs
thf(fact_89_pre__dfss__pre__dfs,axiom,
! [V2: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V2 @ E )
=> ( ~ ( member_v2 @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( member_v2 @ W @ ( successors @ V2 ) )
=> ( sCC_Bl36166008131615352t_unit @ successors @ W @ E ) ) ) ) ).
% pre_dfss_pre_dfs
thf(fact_90_local_Owf,axiom,
sCC_Bl9196236973127232072t_unit @ successors @ e ).
% local.wf
thf(fact_91_stack__visited,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v2 @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( member_v2 @ N @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ).
% stack_visited
thf(fact_92_S__reflexive,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( member_v2 @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ).
% S_reflexive
thf(fact_93_dfs__S__tl__stack_I1_J,axiom,
! [V2: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V2 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v ) ) ) ).
% dfs_S_tl_stack(1)
thf(fact_94_e__def,axiom,
( e
= ( sCC_Bloemen_dfs_v @ successors @ v2 @ ( sCC_Bl7693227186847904995_env_v @ v2 ) ) ) ).
% e_def
thf(fact_95_precedes__refl,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ X @ Xs )
= ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_refl
thf(fact_96_precedes__refl,axiom,
! [X: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ X @ Xs )
= ( member_v2 @ X @ ( set_v2 @ Xs ) ) ) ).
% precedes_refl
thf(fact_97_precedes__mem_I2_J,axiom,
! [X: product_prod_v_v,Y: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ Xs )
=> ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_mem(2)
thf(fact_98_precedes__mem_I2_J,axiom,
! [X: v,Y: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( member_v2 @ Y @ ( set_v2 @ Xs ) ) ) ).
% precedes_mem(2)
thf(fact_99_precedes__mem_I1_J,axiom,
! [X: product_prod_v_v,Y: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ Xs )
=> ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% precedes_mem(1)
thf(fact_100_precedes__mem_I1_J,axiom,
! [X: v,Y: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( member_v2 @ X @ ( set_v2 @ Xs ) ) ) ).
% precedes_mem(1)
thf(fact_101_list__ex1__iff,axiom,
( list_e4337557499903403568od_v_v
= ( ^ [P3: product_prod_v_v > $o,Xs2: list_P7986770385144383213od_v_v] :
? [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ ( set_Product_prod_v_v2 @ Xs2 ) )
& ( P3 @ X4 )
& ! [Y4: product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ Y4 @ ( set_Product_prod_v_v2 @ Xs2 ) )
& ( P3 @ Y4 ) )
=> ( Y4 = X4 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_102_list__ex1__iff,axiom,
( list_ex1_v
= ( ^ [P3: v > $o,Xs2: list_v] :
? [X4: v] :
( ( member_v2 @ X4 @ ( set_v2 @ Xs2 ) )
& ( P3 @ X4 )
& ! [Y4: v] :
( ( ( member_v2 @ Y4 @ ( set_v2 @ Xs2 ) )
& ( P3 @ Y4 ) )
=> ( Y4 = X4 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_103_in__set__member,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
= ( member6878703317195979394od_v_v @ Xs @ X ) ) ).
% in_set_member
thf(fact_104_in__set__member,axiom,
! [X: v,Xs: list_v] :
( ( member_v2 @ X @ ( set_v2 @ Xs ) )
= ( member_v @ Xs @ X ) ) ).
% in_set_member
thf(fact_105_list_Oset__sel_I1_J,axiom,
! [A: list_P7986770385144383213od_v_v] :
( ( A != nil_Product_prod_v_v )
=> ( member7453568604450474000od_v_v @ ( hd_Product_prod_v_v @ A ) @ ( set_Product_prod_v_v2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_106_list_Oset__sel_I1_J,axiom,
! [A: list_v] :
( ( A != nil_v )
=> ( member_v2 @ ( hd_v @ A ) @ ( set_v2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_107_hd__in__set,axiom,
! [Xs: list_P7986770385144383213od_v_v] :
( ( Xs != nil_Product_prod_v_v )
=> ( member7453568604450474000od_v_v @ ( hd_Product_prod_v_v @ Xs ) @ ( set_Product_prod_v_v2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_108_hd__in__set,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( member_v2 @ ( hd_v @ Xs ) @ ( set_v2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_109_post,axiom,
sCC_Bl8953792750115413617t_unit @ successors @ v2 @ ( sCC_Bl7693227186847904995_env_v @ v2 ) @ e ).
% post
thf(fact_110_visited__unexplored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,M: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v2 @ M @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v2 @ M @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ~ ! [N2: v] :
( ( member_v2 @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v2 @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ) ) ).
% visited_unexplored
thf(fact_111_stack__unexplored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v2 @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v2 @ N @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ).
% stack_unexplored
thf(fact_112_dfs__S__tl__stack_I2_J,axiom,
! [V2: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V2 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ! [X5: v] :
( ( member_v2 @ X5 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X5 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X5 ) ) ) ) ) ).
% dfs_S_tl_stack(2)
thf(fact_113__092_060open_062pre__dfs_Av_A_Iinit__env_Av_J_092_060close_062,axiom,
sCC_Bl36166008131615352t_unit @ successors @ v2 @ ( sCC_Bl7693227186847904995_env_v @ v2 ) ).
% \<open>pre_dfs v (init_env v)\<close>
thf(fact_114_can__select__set__list__ex1,axiom,
! [P2: v > $o,A4: list_v] :
( ( can_select_v @ P2 @ ( set_v2 @ A4 ) )
= ( list_ex1_v @ P2 @ A4 ) ) ).
% can_select_set_list_ex1
thf(fact_115_graph_Ovisited__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,M: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v2 @ M @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v2 @ M @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ~ ! [N2: v] :
( ( member_v2 @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v2 @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ) ) ) ).
% graph.visited_unexplored
thf(fact_116_top__empty__eq,axiom,
( top_top_v_o
= ( ^ [X4: v] : ( member_v2 @ X4 @ top_top_set_v ) ) ) ).
% top_empty_eq
thf(fact_117_top__empty__eq,axiom,
( top_to3765765064819114118_v_v_o
= ( ^ [X4: product_prod_v_v] : ( member7453568604450474000od_v_v @ X4 @ top_to5429829297380968215od_v_v ) ) ) ).
% top_empty_eq
thf(fact_118_top__empty__eq,axiom,
( top_to2465898995584390880unit_o
= ( ^ [X4: product_unit] : ( member_Product_unit @ X4 @ top_to1996260823553986621t_unit ) ) ) ).
% top_empty_eq
thf(fact_119_top__empty__eq,axiom,
( top_top_o_o
= ( ^ [X4: $o] : ( member_o @ X4 @ top_top_set_o ) ) ) ).
% top_empty_eq
thf(fact_120_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X4: nat] : ( member_nat @ X4 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_121_reachable__end_Ocases,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ Y3 )
=> ~ ( member_v2 @ A22 @ ( successors @ Y3 ) ) ) ) ) ).
% reachable_end.cases
thf(fact_122_succ__re,axiom,
! [Y: v,X: v,Z: v] :
( ( member_v2 @ Y @ ( successors @ X ) )
=> ( ( sCC_Bl770211535891879572_end_v @ successors @ Y @ Z )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Z ) ) ) ).
% succ_re
thf(fact_123_reachable__end_Osimps,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A22 )
= ( ? [X4: v] :
( ( A1 = X4 )
& ( A22 = X4 ) )
| ? [X4: v,Y4: v,Z2: v] :
( ( A1 = X4 )
& ( A22 = Z2 )
& ( sCC_Bl770211535891879572_end_v @ successors @ X4 @ Y4 )
& ( member_v2 @ Z2 @ ( successors @ Y4 ) ) ) ) ) ).
% reachable_end.simps
thf(fact_124_re__succ,axiom,
! [X: v,Y: v,Z: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y )
=> ( ( member_v2 @ Z @ ( successors @ Y ) )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Z ) ) ) ).
% re_succ
thf(fact_125_re__refl,axiom,
! [X: v] : ( sCC_Bl770211535891879572_end_v @ successors @ X @ X ) ).
% re_refl
thf(fact_126_graph_Oreachable__end_Ocong,axiom,
sCC_Bl770211535891879572_end_v = sCC_Bl770211535891879572_end_v ).
% graph.reachable_end.cong
thf(fact_127_list_Osel_I2_J,axiom,
( ( tl_v @ nil_v )
= nil_v ) ).
% list.sel(2)
thf(fact_128_can__select__def,axiom,
( can_select_v
= ( ^ [P3: v > $o,A5: set_v] :
? [X4: v] :
( ( member_v2 @ X4 @ A5 )
& ( P3 @ X4 )
& ! [Y4: v] :
( ( ( member_v2 @ Y4 @ A5 )
& ( P3 @ Y4 ) )
=> ( Y4 = X4 ) ) ) ) ) ).
% can_select_def
thf(fact_129_can__select__def,axiom,
( can_se8548074686306234212od_v_v
= ( ^ [P3: product_prod_v_v > $o,A5: set_Product_prod_v_v] :
? [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A5 )
& ( P3 @ X4 )
& ! [Y4: product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ Y4 @ A5 )
& ( P3 @ Y4 ) )
=> ( Y4 = X4 ) ) ) ) ) ).
% can_select_def
thf(fact_130_graph_Ore__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Y )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_131_graph_Ore__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y )
=> ( ( member_v2 @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_132_graph_Ore__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ X ) ) ).
% graph.re_refl
thf(fact_133_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A22 )
= ( ? [X4: product_prod_v_v] :
( ( A1 = X4 )
& ( A22 = X4 ) )
| ? [X4: product_prod_v_v,Y4: product_prod_v_v,Z2: product_prod_v_v] :
( ( A1 = X4 )
& ( A22 = Z2 )
& ( sCC_Bl4714988717384592488od_v_v @ Successors @ X4 @ Y4 )
& ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y4 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_134_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A22 )
= ( ? [X4: v] :
( ( A1 = X4 )
& ( A22 = X4 ) )
| ? [X4: v,Y4: v,Z2: v] :
( ( A1 = X4 )
& ( A22 = Z2 )
& ( sCC_Bl770211535891879572_end_v @ Successors @ X4 @ Y4 )
& ( member_v2 @ Z2 @ ( Successors @ Y4 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_135_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: product_prod_v_v] :
( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ Y3 )
=> ~ ( member7453568604450474000od_v_v @ A22 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_136_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: v] :
( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ Y3 )
=> ~ ( member_v2 @ A22 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_137_graph_Osucc__re,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y: product_prod_v_v,X: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ Y @ Z )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_138_graph_Osucc__re,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v2 @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ Y @ Z )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_139_list_Oset__sel_I2_J,axiom,
! [A: list_P7986770385144383213od_v_v,X: product_prod_v_v] :
( ( A != nil_Product_prod_v_v )
=> ( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ ( tl_Product_prod_v_v @ A ) ) )
=> ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_140_list_Oset__sel_I2_J,axiom,
! [A: list_v,X: v] :
( ( A != nil_v )
=> ( ( member_v2 @ X @ ( set_v2 @ ( tl_v @ A ) ) )
=> ( member_v2 @ X @ ( set_v2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_141_list_Oexpand,axiom,
! [List: list_v,List2: list_v] :
( ( ( List = nil_v )
= ( List2 = nil_v ) )
=> ( ( ( List != nil_v )
=> ( ( List2 != nil_v )
=> ( ( ( hd_v @ List )
= ( hd_v @ List2 ) )
& ( ( tl_v @ List )
= ( tl_v @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_142_graph_Ostack__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v2 @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v2 @ N @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ).
% graph.stack_unexplored
thf(fact_143_graph_Odfs__S__tl__stack_I2_J,axiom,
! [Vertices: set_v,Successors: v > set_v,V2: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl8953792750115413617t_unit @ Successors @ V2 @ E @ E2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
=> ! [X5: v] :
( ( member_v2 @ X5 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X5 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X5 ) ) ) ) ) ) ).
% graph.dfs_S_tl_stack(2)
thf(fact_144_stack__class,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v,M: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v2 @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( ( member_v2 @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) )
=> ( member_v2 @ M @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ).
% stack_class
thf(fact_145_sclosed,axiom,
! [X5: v] :
( ( member_v2 @ X5 @ vertices )
=> ( ord_less_eq_set_v @ ( successors @ X5 ) @ vertices ) ) ).
% sclosed
thf(fact_146_graph_Ostack__class,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v,M: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v2 @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( ( member_v2 @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) )
=> ( member_v2 @ M @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ) ).
% graph.stack_class
thf(fact_147_assms_I2_J,axiom,
accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ v2 @ ( sCC_Bl7693227186847904995_env_v @ v2 ) ) ) ).
% assms(2)
thf(fact_148_vfin,axiom,
finite_finite_v @ vertices ).
% vfin
thf(fact_149_reachable__re,axiom,
! [X: v,Y: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y ) ) ).
% reachable_re
thf(fact_150_re__reachable,axiom,
! [X: v,Y: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% re_reachable
thf(fact_151_post__dfs__def,axiom,
! [V2: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl8953792750115413617t_unit @ successors @ V2 @ E @ E2 )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
& ( member_v2 @ V2 @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
& ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
& ( ( sCC_Bl3795065053823578884t_unit @ E2 @ V2 )
= ( successors @ V2 ) )
& ! [X4: v] :
( ( member_v2 @ X4 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ E2 @ X4 )
= ( sCC_Bl3795065053823578884t_unit @ E @ X4 ) ) )
& ! [X4: v] :
( ( member_v2 @ X4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X4 @ V2 ) )
& ? [Ns: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ E )
= ( append_v @ Ns @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
& ( ( ( member_v2 @ V2 @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E2 )
= ( sCC_Bl8828226123343373779t_unit @ E ) )
& ! [X4: v] :
( ( member_v2 @ X4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X4 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X4 ) ) ) )
| ( ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
& ( member_v2 @ V2 @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
& ! [X4: v] :
( ( member_v2 @ X4 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X4 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X4 ) ) ) ) )
& ( ( sCC_Bl9201514103433284750t_unit @ E2 )
= ( sCC_Bl9201514103433284750t_unit @ E ) ) ) ) ).
% post_dfs_def
thf(fact_152_unite__S__tl,axiom,
! [E: sCC_Bl1394983891496994913t_unit,W: v,V2: v,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v2 @ W @ ( successors @ V2 ) )
=> ( ~ ( member_v2 @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) )
=> ( ( member_v2 @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v2 @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v2 @ N @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V2 @ W @ E ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V2 @ W @ E ) @ N )
= ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ) ) ) ) ).
% unite_S_tl
thf(fact_153_reachable_Ocases,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: v] :
( ( member_v2 @ Y3 @ ( successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Y3 @ A22 ) ) ) ) ).
% reachable.cases
thf(fact_154_reachable__refl,axiom,
! [X: v] : ( sCC_Bl649662514949026229able_v @ successors @ X @ X ) ).
% reachable_refl
thf(fact_155_reachable__succ,axiom,
! [Y: v,X: v,Z: v] :
( ( member_v2 @ Y @ ( successors @ X ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% reachable_succ
thf(fact_156_reachable_Osimps,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A22 )
= ( ? [X4: v] :
( ( A1 = X4 )
& ( A22 = X4 ) )
| ? [X4: v,Y4: v,Z2: v] :
( ( A1 = X4 )
& ( A22 = Z2 )
& ( member_v2 @ Y4 @ ( successors @ X4 ) )
& ( sCC_Bl649662514949026229able_v @ successors @ Y4 @ Z2 ) ) ) ) ).
% reachable.simps
thf(fact_157_reachable__edge,axiom,
! [Y: v,X: v] :
( ( member_v2 @ Y @ ( successors @ X ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% reachable_edge
thf(fact_158_reachable__end__induct,axiom,
! [X: v,Y: v,P2: v > v > $o] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ! [X3: v] : ( P2 @ X3 @ X3 )
=> ( ! [X3: v,Y3: v,Z3: v] :
( ( P2 @ X3 @ Y3 )
=> ( ( member_v2 @ Z3 @ ( successors @ Y3 ) )
=> ( P2 @ X3 @ Z3 ) ) )
=> ( P2 @ X @ Y ) ) ) ) ).
% reachable_end_induct
thf(fact_159_reachable__trans,axiom,
! [X: v,Y: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% reachable_trans
thf(fact_160_succ__reachable,axiom,
! [X: v,Y: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( member_v2 @ Z @ ( successors @ Y ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% succ_reachable
thf(fact_161_sccE,axiom,
! [S3: set_v,X: v,Y: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S3 )
=> ( ( member_v2 @ X @ S3 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ X )
=> ( member_v2 @ Y @ S3 ) ) ) ) ) ).
% sccE
thf(fact_162_is__subscc__def,axiom,
! [S3: set_v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S3 )
= ( ! [X4: v] :
( ( member_v2 @ X4 @ S3 )
=> ! [Y4: v] :
( ( member_v2 @ Y4 @ S3 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X4 @ Y4 ) ) ) ) ) ).
% is_subscc_def
thf(fact_163_dual__order_Orefl,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ A @ A ) ).
% dual_order.refl
thf(fact_164_dual__order_Orefl,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ A ) ).
% dual_order.refl
thf(fact_165_order__refl,axiom,
! [X: set_v] : ( ord_less_eq_set_v @ X @ X ) ).
% order_refl
thf(fact_166_order__refl,axiom,
! [X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ X ) ).
% order_refl
thf(fact_167_subset__antisym,axiom,
! [A4: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ A4 @ B4 )
=> ( ( ord_less_eq_set_v @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_168_subset__antisym,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B4 )
=> ( ( ord_le7336532860387713383od_v_v @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_169_subsetI,axiom,
! [A4: set_v,B4: set_v] :
( ! [X3: v] :
( ( member_v2 @ X3 @ A4 )
=> ( member_v2 @ X3 @ B4 ) )
=> ( ord_less_eq_set_v @ A4 @ B4 ) ) ).
% subsetI
thf(fact_170_subsetI,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A4 )
=> ( member7453568604450474000od_v_v @ X3 @ B4 ) )
=> ( ord_le7336532860387713383od_v_v @ A4 @ B4 ) ) ).
% subsetI
thf(fact_171_Diff__idemp,axiom,
! [A4: set_v,B4: set_v] :
( ( minus_minus_set_v @ ( minus_minus_set_v @ A4 @ B4 ) @ B4 )
= ( minus_minus_set_v @ A4 @ B4 ) ) ).
% Diff_idemp
thf(fact_172_Diff__iff,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A4 @ B4 ) )
= ( ( member7453568604450474000od_v_v @ C @ A4 )
& ~ ( member7453568604450474000od_v_v @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_173_Diff__iff,axiom,
! [C: v,A4: set_v,B4: set_v] :
( ( member_v2 @ C @ ( minus_minus_set_v @ A4 @ B4 ) )
= ( ( member_v2 @ C @ A4 )
& ~ ( member_v2 @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_174_DiffI,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A4 )
=> ( ~ ( member7453568604450474000od_v_v @ C @ B4 )
=> ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A4 @ B4 ) ) ) ) ).
% DiffI
thf(fact_175_DiffI,axiom,
! [C: v,A4: set_v,B4: set_v] :
( ( member_v2 @ C @ A4 )
=> ( ~ ( member_v2 @ C @ B4 )
=> ( member_v2 @ C @ ( minus_minus_set_v @ A4 @ B4 ) ) ) ) ).
% DiffI
thf(fact_176_unite__sub__env,axiom,
! [V2: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V2 @ E )
=> ( ( member_v2 @ W @ ( successors @ V2 ) )
=> ( ~ ( member_v2 @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) )
=> ( ( member_v2 @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v2 @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5768913643336123637t_unit @ E @ ( sCC_Bloemen_unite_v @ V2 @ W @ E ) ) ) ) ) ) ) ).
% unite_sub_env
thf(fact_177_reachable__visited,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V2: v,W: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v2 @ V2 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V2 @ W )
=> ( ! [X3: v] :
( ( member_v2 @ X3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa: v] :
( ( member_v2 @ Xa @ ( minus_minus_set_v @ ( successors @ X3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X3 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V2 @ X3 )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Xa @ W ) ) ) )
=> ( member_v2 @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ).
% reachable_visited
thf(fact_178_List_Ofinite__set,axiom,
! [Xs: list_v] : ( finite_finite_v @ ( set_v2 @ Xs ) ) ).
% List.finite_set
thf(fact_179_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_180_append_Oright__neutral,axiom,
! [A: list_v] :
( ( append_v @ A @ nil_v )
= A ) ).
% append.right_neutral
thf(fact_181_append__Nil2,axiom,
! [Xs: list_v] :
( ( append_v @ Xs @ nil_v )
= Xs ) ).
% append_Nil2
thf(fact_182_append__self__conv,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( append_v @ Xs @ Ys )
= Xs )
= ( Ys = nil_v ) ) ).
% append_self_conv
thf(fact_183_self__append__conv,axiom,
! [Y: list_v,Ys: list_v] :
( ( Y
= ( append_v @ Y @ Ys ) )
= ( Ys = nil_v ) ) ).
% self_append_conv
thf(fact_184_append__self__conv2,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( append_v @ Xs @ Ys )
= Ys )
= ( Xs = nil_v ) ) ).
% append_self_conv2
thf(fact_185_self__append__conv2,axiom,
! [Y: list_v,Xs: list_v] :
( ( Y
= ( append_v @ Xs @ Y ) )
= ( Xs = nil_v ) ) ).
% self_append_conv2
thf(fact_186_Nil__is__append__conv,axiom,
! [Xs: list_v,Ys: list_v] :
( ( nil_v
= ( append_v @ Xs @ Ys ) )
= ( ( Xs = nil_v )
& ( Ys = nil_v ) ) ) ).
% Nil_is_append_conv
thf(fact_187_append__is__Nil__conv,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( append_v @ Xs @ Ys )
= nil_v )
= ( ( Xs = nil_v )
& ( Ys = nil_v ) ) ) ).
% append_is_Nil_conv
thf(fact_188_unite__wf__env,axiom,
! [V2: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V2 @ E )
=> ( ( member_v2 @ W @ ( successors @ V2 ) )
=> ( ~ ( member_v2 @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) )
=> ( ( member_v2 @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v2 @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl9196236973127232072t_unit @ successors @ ( sCC_Bloemen_unite_v @ V2 @ W @ E ) ) ) ) ) ) ) ).
% unite_wf_env
thf(fact_189_unite__subscc,axiom,
! [V2: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V2 @ E )
=> ( ( member_v2 @ W @ ( successors @ V2 ) )
=> ( ~ ( member_v2 @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) )
=> ( ( member_v2 @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v2 @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V2 @ W @ E ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V2 @ W @ E ) ) ) ) ) ) ) ) ) ) ).
% unite_subscc
thf(fact_190_pre__post_I1_J,axiom,
! [V2: v,E: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V2 @ E ) ) )
=> ( ( sCC_Bl36166008131615352t_unit @ successors @ V2 @ E )
=> ( sCC_Bl8953792750115413617t_unit @ successors @ V2 @ E @ ( sCC_Bloemen_dfs_v @ successors @ V2 @ E ) ) ) ) ).
% pre_post(1)
thf(fact_191_hd__append2,axiom,
! [Xs: list_v,Ys: list_v] :
( ( Xs != nil_v )
=> ( ( hd_v @ ( append_v @ Xs @ Ys ) )
= ( hd_v @ Xs ) ) ) ).
% hd_append2
thf(fact_192_tl__append2,axiom,
! [Xs: list_v,Ys: list_v] :
( ( Xs != nil_v )
=> ( ( tl_v @ ( append_v @ Xs @ Ys ) )
= ( append_v @ ( tl_v @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_193_SCC__Bloemen__Sequential_Ograph__def,axiom,
( sCC_Bl8035451632035226289ph_nat
= ( ^ [Vertices2: set_nat,Successors2: nat > set_nat] :
( ( finite_finite_nat @ Vertices2 )
& ! [X4: nat] :
( ( member_nat @ X4 @ Vertices2 )
=> ( ord_less_eq_set_nat @ ( Successors2 @ X4 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_194_SCC__Bloemen__Sequential_Ograph__def,axiom,
( sCC_Bl8307124943676871238od_v_v
= ( ^ [Vertices2: set_Product_prod_v_v,Successors2: product_prod_v_v > set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ Vertices2 )
& ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ Vertices2 )
=> ( ord_le7336532860387713383od_v_v @ ( Successors2 @ X4 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_195_SCC__Bloemen__Sequential_Ograph__def,axiom,
( sCC_Bloemen_graph_v
= ( ^ [Vertices2: set_v,Successors2: v > set_v] :
( ( finite_finite_v @ Vertices2 )
& ! [X4: v] :
( ( member_v2 @ X4 @ Vertices2 )
=> ( ord_less_eq_set_v @ ( Successors2 @ X4 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_196_graph_Ointro,axiom,
! [Vertices: set_nat,Successors: nat > set_nat] :
( ( finite_finite_nat @ Vertices )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ Vertices )
=> ( ord_less_eq_set_nat @ ( Successors @ X3 ) @ Vertices ) )
=> ( sCC_Bl8035451632035226289ph_nat @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_197_graph_Ointro,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ Vertices )
=> ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ Vertices )
=> ( ord_le7336532860387713383od_v_v @ ( Successors @ X3 ) @ Vertices ) )
=> ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_198_graph_Ointro,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( finite_finite_v @ Vertices )
=> ( ! [X3: v] :
( ( member_v2 @ X3 @ Vertices )
=> ( ord_less_eq_set_v @ ( Successors @ X3 ) @ Vertices ) )
=> ( sCC_Bloemen_graph_v @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_199_order__antisym__conv,axiom,
! [Y: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y @ X )
=> ( ( ord_less_eq_set_v @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_200_order__antisym__conv,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( ( ord_le7336532860387713383od_v_v @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_201_ord__le__eq__subst,axiom,
! [A: set_v,B: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_202_ord__le__eq__subst,axiom,
! [A: set_v,B: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_203_ord__le__eq__subst,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_204_ord__le__eq__subst,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_205_ord__eq__le__subst,axiom,
! [A: set_v,F: set_v > set_v,B: set_v,C: set_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ! [X3: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_206_ord__eq__le__subst,axiom,
! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B: set_v,C: set_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ! [X3: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_207_ord__eq__le__subst,axiom,
! [A: set_v,F: set_Product_prod_v_v > set_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ! [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_208_ord__eq__le__subst,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ! [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_209_order__eq__refl,axiom,
! [X: set_v,Y: set_v] :
( ( X = Y )
=> ( ord_less_eq_set_v @ X @ Y ) ) ).
% order_eq_refl
thf(fact_210_order__eq__refl,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( X = Y )
=> ( ord_le7336532860387713383od_v_v @ X @ Y ) ) ).
% order_eq_refl
thf(fact_211_order__subst2,axiom,
! [A: set_v,B: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ ( F @ B ) @ C )
=> ( ! [X3: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_212_order__subst2,axiom,
! [A: set_v,B: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B ) @ C )
=> ( ! [X3: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_213_order__subst2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_less_eq_set_v @ ( F @ B ) @ C )
=> ( ! [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_214_order__subst2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B ) @ C )
=> ( ! [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_215_order__subst1,axiom,
! [A: set_v,F: set_v > set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ! [X3: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_216_order__subst1,axiom,
! [A: set_v,F: set_Product_prod_v_v > set_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ! [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y3 )
=> ( ord_less_eq_set_v @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_217_order__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B: set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ! [X3: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_218_order__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ! [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_219_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_v,Z4: set_v] : ( Y5 = Z4 ) )
= ( ^ [A6: set_v,B5: set_v] :
( ( ord_less_eq_set_v @ A6 @ B5 )
& ( ord_less_eq_set_v @ B5 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_220_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y5 = Z4 ) )
= ( ^ [A6: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A6 @ B5 )
& ( ord_le7336532860387713383od_v_v @ B5 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_221_Collect__mono__iff,axiom,
! [P2: set_v > $o,Q: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P2 ) @ ( collect_set_v @ Q ) )
= ( ! [X4: set_v] :
( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_222_Collect__mono__iff,axiom,
! [P2: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ ( collect_v @ P2 ) @ ( collect_v @ Q ) )
= ( ! [X4: v] :
( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_223_Collect__mono__iff,axiom,
! [P2: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P2 ) @ ( collec140062887454715474od_v_v @ Q ) )
= ( ! [X4: product_prod_v_v] :
( ( P2 @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_224_set__eq__subset,axiom,
( ( ^ [Y5: set_v,Z4: set_v] : ( Y5 = Z4 ) )
= ( ^ [A5: set_v,B6: set_v] :
( ( ord_less_eq_set_v @ A5 @ B6 )
& ( ord_less_eq_set_v @ B6 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_225_set__eq__subset,axiom,
( ( ^ [Y5: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y5 = Z4 ) )
= ( ^ [A5: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A5 @ B6 )
& ( ord_le7336532860387713383od_v_v @ B6 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_226_antisym,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_227_antisym,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_228_subset__trans,axiom,
! [A4: set_v,B4: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A4 @ B4 )
=> ( ( ord_less_eq_set_v @ B4 @ C2 )
=> ( ord_less_eq_set_v @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_229_subset__trans,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B4 )
=> ( ( ord_le7336532860387713383od_v_v @ B4 @ C2 )
=> ( ord_le7336532860387713383od_v_v @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_230_Collect__mono,axiom,
! [P2: set_v > $o,Q: set_v > $o] :
( ! [X3: set_v] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P2 ) @ ( collect_set_v @ Q ) ) ) ).
% Collect_mono
thf(fact_231_Collect__mono,axiom,
! [P2: v > $o,Q: v > $o] :
( ! [X3: v] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_v @ ( collect_v @ P2 ) @ ( collect_v @ Q ) ) ) ).
% Collect_mono
thf(fact_232_Collect__mono,axiom,
! [P2: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ! [X3: product_prod_v_v] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P2 ) @ ( collec140062887454715474od_v_v @ Q ) ) ) ).
% Collect_mono
thf(fact_233_subset__refl,axiom,
! [A4: set_v] : ( ord_less_eq_set_v @ A4 @ A4 ) ).
% subset_refl
thf(fact_234_subset__refl,axiom,
! [A4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A4 @ A4 ) ).
% subset_refl
thf(fact_235_double__diff,axiom,
! [A4: set_v,B4: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A4 @ B4 )
=> ( ( ord_less_eq_set_v @ B4 @ C2 )
=> ( ( minus_minus_set_v @ B4 @ ( minus_minus_set_v @ C2 @ A4 ) )
= A4 ) ) ) ).
% double_diff
thf(fact_236_double__diff,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B4 )
=> ( ( ord_le7336532860387713383od_v_v @ B4 @ C2 )
=> ( ( minus_4183494784930505774od_v_v @ B4 @ ( minus_4183494784930505774od_v_v @ C2 @ A4 ) )
= A4 ) ) ) ).
% double_diff
thf(fact_237_Diff__subset,axiom,
! [A4: set_v,B4: set_v] : ( ord_less_eq_set_v @ ( minus_minus_set_v @ A4 @ B4 ) @ A4 ) ).
% Diff_subset
thf(fact_238_Diff__subset,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A4 @ B4 ) @ A4 ) ).
% Diff_subset
thf(fact_239_subset__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B6: set_v] :
! [T: v] :
( ( member_v2 @ T @ A5 )
=> ( member_v2 @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_240_subset__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
! [T: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ T @ A5 )
=> ( member7453568604450474000od_v_v @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_241_equalityD2,axiom,
! [A4: set_v,B4: set_v] :
( ( A4 = B4 )
=> ( ord_less_eq_set_v @ B4 @ A4 ) ) ).
% equalityD2
thf(fact_242_equalityD2,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( A4 = B4 )
=> ( ord_le7336532860387713383od_v_v @ B4 @ A4 ) ) ).
% equalityD2
thf(fact_243_equalityD1,axiom,
! [A4: set_v,B4: set_v] :
( ( A4 = B4 )
=> ( ord_less_eq_set_v @ A4 @ B4 ) ) ).
% equalityD1
thf(fact_244_equalityD1,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( A4 = B4 )
=> ( ord_le7336532860387713383od_v_v @ A4 @ B4 ) ) ).
% equalityD1
thf(fact_245_subset__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B6: set_v] :
! [X4: v] :
( ( member_v2 @ X4 @ A5 )
=> ( member_v2 @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_246_subset__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A5 )
=> ( member7453568604450474000od_v_v @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_247_equalityE,axiom,
! [A4: set_v,B4: set_v] :
( ( A4 = B4 )
=> ~ ( ( ord_less_eq_set_v @ A4 @ B4 )
=> ~ ( ord_less_eq_set_v @ B4 @ A4 ) ) ) ).
% equalityE
thf(fact_248_equalityE,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( A4 = B4 )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A4 @ B4 )
=> ~ ( ord_le7336532860387713383od_v_v @ B4 @ A4 ) ) ) ).
% equalityE
thf(fact_249_Diff__mono,axiom,
! [A4: set_v,C2: set_v,D: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ A4 @ C2 )
=> ( ( ord_less_eq_set_v @ D @ B4 )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A4 @ B4 ) @ ( minus_minus_set_v @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_250_Diff__mono,axiom,
! [A4: set_Product_prod_v_v,C2: set_Product_prod_v_v,D: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ D @ B4 )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A4 @ B4 ) @ ( minus_4183494784930505774od_v_v @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_251_subsetD,axiom,
! [A4: set_v,B4: set_v,C: v] :
( ( ord_less_eq_set_v @ A4 @ B4 )
=> ( ( member_v2 @ C @ A4 )
=> ( member_v2 @ C @ B4 ) ) ) ).
% subsetD
thf(fact_252_subsetD,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v,C: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B4 )
=> ( ( member7453568604450474000od_v_v @ C @ A4 )
=> ( member7453568604450474000od_v_v @ C @ B4 ) ) ) ).
% subsetD
thf(fact_253_in__mono,axiom,
! [A4: set_v,B4: set_v,X: v] :
( ( ord_less_eq_set_v @ A4 @ B4 )
=> ( ( member_v2 @ X @ A4 )
=> ( member_v2 @ X @ B4 ) ) ) ).
% in_mono
thf(fact_254_in__mono,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B4 )
=> ( ( member7453568604450474000od_v_v @ X @ A4 )
=> ( member7453568604450474000od_v_v @ X @ B4 ) ) ) ).
% in_mono
thf(fact_255_DiffD2,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A4 @ B4 ) )
=> ~ ( member7453568604450474000od_v_v @ C @ B4 ) ) ).
% DiffD2
thf(fact_256_DiffD2,axiom,
! [C: v,A4: set_v,B4: set_v] :
( ( member_v2 @ C @ ( minus_minus_set_v @ A4 @ B4 ) )
=> ~ ( member_v2 @ C @ B4 ) ) ).
% DiffD2
thf(fact_257_DiffD1,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A4 @ B4 ) )
=> ( member7453568604450474000od_v_v @ C @ A4 ) ) ).
% DiffD1
thf(fact_258_DiffD1,axiom,
! [C: v,A4: set_v,B4: set_v] :
( ( member_v2 @ C @ ( minus_minus_set_v @ A4 @ B4 ) )
=> ( member_v2 @ C @ A4 ) ) ).
% DiffD1
thf(fact_259_DiffE,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A4 @ B4 ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A4 )
=> ( member7453568604450474000od_v_v @ C @ B4 ) ) ) ).
% DiffE
thf(fact_260_DiffE,axiom,
! [C: v,A4: set_v,B4: set_v] :
( ( member_v2 @ C @ ( minus_minus_set_v @ A4 @ B4 ) )
=> ~ ( ( member_v2 @ C @ A4 )
=> ( member_v2 @ C @ B4 ) ) ) ).
% DiffE
thf(fact_261_dual__order_Otrans,axiom,
! [B: set_v,A: set_v,C: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( ord_less_eq_set_v @ C @ B )
=> ( ord_less_eq_set_v @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_262_dual__order_Otrans,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C @ B )
=> ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_263_dual__order_Oantisym,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( ord_less_eq_set_v @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_264_dual__order_Oantisym,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_265_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_v,Z4: set_v] : ( Y5 = Z4 ) )
= ( ^ [A6: set_v,B5: set_v] :
( ( ord_less_eq_set_v @ B5 @ A6 )
& ( ord_less_eq_set_v @ A6 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_266_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y5 = Z4 ) )
= ( ^ [A6: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B5 @ A6 )
& ( ord_le7336532860387713383od_v_v @ A6 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_267_order__trans,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( ord_less_eq_set_v @ Y @ Z )
=> ( ord_less_eq_set_v @ X @ Z ) ) ) ).
% order_trans
thf(fact_268_order__trans,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ Y @ Z )
=> ( ord_le7336532860387713383od_v_v @ X @ Z ) ) ) ).
% order_trans
thf(fact_269_order_Otrans,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% order.trans
thf(fact_270_order_Otrans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% order.trans
thf(fact_271_order__antisym,axiom,
! [X: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( ord_less_eq_set_v @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_272_order__antisym,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_273_ord__le__eq__trans,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_274_ord__le__eq__trans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( B = C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_275_ord__eq__le__trans,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( A = B )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_276_ord__eq__le__trans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A = B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_277_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_v,Z4: set_v] : ( Y5 = Z4 ) )
= ( ^ [X4: set_v,Y4: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y4 )
& ( ord_less_eq_set_v @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_278_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y5 = Z4 ) )
= ( ^ [X4: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y4 )
& ( ord_le7336532860387713383od_v_v @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_279_graph_Oreachable_Ocong,axiom,
sCC_Bl649662514949026229able_v = sCC_Bl649662514949026229able_v ).
% graph.reachable.cong
thf(fact_280_graph_Ounite__sub__env,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V2: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V2 @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V2 ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl7963838319573962697t_unit @ E @ ( sCC_Bl4702006153222411093od_v_v @ V2 @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_281_graph_Ounite__sub__env,axiom,
! [Vertices: set_v,Successors: v > set_v,V2: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V2 @ E )
=> ( ( member_v2 @ W @ ( Successors @ V2 ) )
=> ( ~ ( member_v2 @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) )
=> ( ( member_v2 @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v2 @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5768913643336123637t_unit @ E @ ( sCC_Bloemen_unite_v @ V2 @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_282_append__Nil,axiom,
! [Ys: list_v] :
( ( append_v @ nil_v @ Ys )
= Ys ) ).
% append_Nil
thf(fact_283_append_Oleft__neutral,axiom,
! [A: list_v] :
( ( append_v @ nil_v @ A )
= A ) ).
% append.left_neutral
thf(fact_284_eq__Nil__appendI,axiom,
! [Xs: list_v,Ys: list_v] :
( ( Xs = Ys )
=> ( Xs
= ( append_v @ nil_v @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_285_finite__list,axiom,
! [A4: set_v] :
( ( finite_finite_v @ A4 )
=> ? [Xs3: list_v] :
( ( set_v2 @ Xs3 )
= A4 ) ) ).
% finite_list
thf(fact_286_finite__list,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ? [Xs3: list_nat] :
( ( set_nat2 @ Xs3 )
= A4 ) ) ).
% finite_list
thf(fact_287_top__greatest,axiom,
! [A: product_unit] : ( ord_le3221252021190050221t_unit @ A @ top_top_Product_unit ) ).
% top_greatest
thf(fact_288_top__greatest,axiom,
! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A @ top_to1996260823553986621t_unit ) ).
% top_greatest
thf(fact_289_top__greatest,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ A @ top_top_set_o ) ).
% top_greatest
thf(fact_290_top__greatest,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% top_greatest
thf(fact_291_top__greatest,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ A @ top_top_set_v ) ).
% top_greatest
thf(fact_292_top__greatest,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ top_to5429829297380968215od_v_v ) ).
% top_greatest
thf(fact_293_top_Oextremum__unique,axiom,
! [A: product_unit] :
( ( ord_le3221252021190050221t_unit @ top_top_Product_unit @ A )
= ( A = top_top_Product_unit ) ) ).
% top.extremum_unique
thf(fact_294_top_Oextremum__unique,axiom,
! [A: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A )
= ( A = top_to1996260823553986621t_unit ) ) ).
% top.extremum_unique
thf(fact_295_top_Oextremum__unique,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A )
= ( A = top_top_set_o ) ) ).
% top.extremum_unique
thf(fact_296_top_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
= ( A = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_297_top_Oextremum__unique,axiom,
! [A: set_v] :
( ( ord_less_eq_set_v @ top_top_set_v @ A )
= ( A = top_top_set_v ) ) ).
% top.extremum_unique
thf(fact_298_top_Oextremum__unique,axiom,
! [A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ top_to5429829297380968215od_v_v @ A )
= ( A = top_to5429829297380968215od_v_v ) ) ).
% top.extremum_unique
thf(fact_299_top_Oextremum__uniqueI,axiom,
! [A: product_unit] :
( ( ord_le3221252021190050221t_unit @ top_top_Product_unit @ A )
=> ( A = top_top_Product_unit ) ) ).
% top.extremum_uniqueI
thf(fact_300_top_Oextremum__uniqueI,axiom,
! [A: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A )
=> ( A = top_to1996260823553986621t_unit ) ) ).
% top.extremum_uniqueI
thf(fact_301_top_Oextremum__uniqueI,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A )
=> ( A = top_top_set_o ) ) ).
% top.extremum_uniqueI
thf(fact_302_top_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
=> ( A = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_303_top_Oextremum__uniqueI,axiom,
! [A: set_v] :
( ( ord_less_eq_set_v @ top_top_set_v @ A )
=> ( A = top_top_set_v ) ) ).
% top.extremum_uniqueI
thf(fact_304_top_Oextremum__uniqueI,axiom,
! [A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ top_to5429829297380968215od_v_v @ A )
=> ( A = top_to5429829297380968215od_v_v ) ) ).
% top.extremum_uniqueI
thf(fact_305_subset__UNIV,axiom,
! [A4: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A4 @ top_to1996260823553986621t_unit ) ).
% subset_UNIV
thf(fact_306_subset__UNIV,axiom,
! [A4: set_o] : ( ord_less_eq_set_o @ A4 @ top_top_set_o ) ).
% subset_UNIV
thf(fact_307_subset__UNIV,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ A4 @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_308_subset__UNIV,axiom,
! [A4: set_v] : ( ord_less_eq_set_v @ A4 @ top_top_set_v ) ).
% subset_UNIV
thf(fact_309_subset__UNIV,axiom,
! [A4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A4 @ top_to5429829297380968215od_v_v ) ).
% subset_UNIV
thf(fact_310_subset__code_I1_J,axiom,
! [Xs: list_v,B4: set_v] :
( ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ B4 )
= ( ! [X4: v] :
( ( member_v2 @ X4 @ ( set_v2 @ Xs ) )
=> ( member_v2 @ X4 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_311_subset__code_I1_J,axiom,
! [Xs: list_P7986770385144383213od_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ B4 )
= ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( member7453568604450474000od_v_v @ X4 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_312_graph_Oreachable__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V2: v,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v2 @ V2 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V2 @ W )
=> ( ! [X3: v] :
( ( member_v2 @ X3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa: v] :
( ( member_v2 @ Xa @ ( minus_minus_set_v @ ( Successors @ X3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X3 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V2 @ X3 )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Xa @ W ) ) ) )
=> ( member_v2 @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ) ).
% graph.reachable_visited
thf(fact_313_graph_Ovfin,axiom,
! [Vertices: set_nat,Successors: nat > set_nat] :
( ( sCC_Bl8035451632035226289ph_nat @ Vertices @ Successors )
=> ( finite_finite_nat @ Vertices ) ) ).
% graph.vfin
thf(fact_314_graph_Ovfin,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( finite_finite_v @ Vertices ) ) ).
% graph.vfin
thf(fact_315_graph_Osclosed,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ! [X5: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X5 @ Vertices )
=> ( ord_le7336532860387713383od_v_v @ ( Successors @ X5 ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_316_graph_Osclosed,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ! [X5: v] :
( ( member_v2 @ X5 @ Vertices )
=> ( ord_less_eq_set_v @ ( Successors @ X5 ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_317_precedes__append__left,axiom,
! [X: v,Y: v,Xs: list_v,Ys: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( append_v @ Ys @ Xs ) ) ) ).
% precedes_append_left
thf(fact_318_precedes__append__right,axiom,
! [X: v,Y: v,Xs: list_v,Ys: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( append_v @ Xs @ Ys ) ) ) ).
% precedes_append_right
thf(fact_319_graph_Oreachable__edge,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y: product_prod_v_v,X: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y ) ) ) ).
% graph.reachable_edge
thf(fact_320_graph_Oreachable__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v2 @ Y @ ( Successors @ X ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y ) ) ) ).
% graph.reachable_edge
thf(fact_321_graph_Osucc__reachable,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_322_graph_Osucc__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( member_v2 @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_323_graph_Oreachable_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y3 @ A22 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_324_graph_Oreachable_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: v] :
( ( member_v2 @ Y3 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Y3 @ A22 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_325_graph_Oreachable_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ A1 @ A22 )
= ( ? [X4: product_prod_v_v] :
( ( A1 = X4 )
& ( A22 = X4 ) )
| ? [X4: product_prod_v_v,Y4: product_prod_v_v,Z2: product_prod_v_v] :
( ( A1 = X4 )
& ( A22 = Z2 )
& ( member7453568604450474000od_v_v @ Y4 @ ( Successors @ X4 ) )
& ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y4 @ Z2 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_326_graph_Oreachable_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ A1 @ A22 )
= ( ? [X4: v] :
( ( A1 = X4 )
& ( A22 = X4 ) )
| ? [X4: v,Y4: v,Z2: v] :
( ( A1 = X4 )
& ( A22 = Z2 )
& ( member_v2 @ Y4 @ ( Successors @ X4 ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ Y4 @ Z2 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_327_graph_Oreachable__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_trans
thf(fact_328_graph_Oreachable__end__induct,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,P2: product_prod_v_v > product_prod_v_v > $o] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ! [X3: product_prod_v_v] : ( P2 @ X3 @ X3 )
=> ( ! [X3: product_prod_v_v,Y3: product_prod_v_v,Z3: product_prod_v_v] :
( ( P2 @ X3 @ Y3 )
=> ( ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ Y3 ) )
=> ( P2 @ X3 @ Z3 ) ) )
=> ( P2 @ X @ Y ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_329_graph_Oreachable__end__induct,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,P2: v > v > $o] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ! [X3: v] : ( P2 @ X3 @ X3 )
=> ( ! [X3: v,Y3: v,Z3: v] :
( ( P2 @ X3 @ Y3 )
=> ( ( member_v2 @ Z3 @ ( Successors @ Y3 ) )
=> ( P2 @ X3 @ Z3 ) ) )
=> ( P2 @ X @ Y ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_330_graph_Oreachable__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ X ) ) ).
% graph.reachable_refl
thf(fact_331_graph_Oreachable__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y: product_prod_v_v,X: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ Z )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_332_graph_Oreachable__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v2 @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_333_graph_Ounite__wf__env,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V2: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V2 @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V2 ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl7798947040364291444t_unit @ Successors @ ( sCC_Bl4702006153222411093od_v_v @ V2 @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_wf_env
thf(fact_334_graph_Ounite__wf__env,axiom,
! [Vertices: set_v,Successors: v > set_v,V2: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V2 @ E )
=> ( ( member_v2 @ W @ ( Successors @ V2 ) )
=> ( ~ ( member_v2 @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) )
=> ( ( member_v2 @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v2 @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl9196236973127232072t_unit @ Successors @ ( sCC_Bloemen_unite_v @ V2 @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_wf_env
thf(fact_335_hd__append,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( Xs = nil_v )
=> ( ( hd_v @ ( append_v @ Xs @ Ys ) )
= ( hd_v @ Ys ) ) )
& ( ( Xs != nil_v )
=> ( ( hd_v @ ( append_v @ Xs @ Ys ) )
= ( hd_v @ Xs ) ) ) ) ).
% hd_append
thf(fact_336_longest__common__prefix,axiom,
! [Xs: list_v,Ys: list_v] :
? [Ps: list_v,Xs4: list_v,Ys2: list_v] :
( ( Xs
= ( append_v @ Ps @ Xs4 ) )
& ( Ys
= ( append_v @ Ps @ Ys2 ) )
& ( ( Xs4 = nil_v )
| ( Ys2 = nil_v )
| ( ( hd_v @ Xs4 )
!= ( hd_v @ Ys2 ) ) ) ) ).
% longest_common_prefix
thf(fact_337_tl__append__if,axiom,
! [Xs: list_v,Ys: list_v] :
( ( ( Xs = nil_v )
=> ( ( tl_v @ ( append_v @ Xs @ Ys ) )
= ( tl_v @ Ys ) ) )
& ( ( Xs != nil_v )
=> ( ( tl_v @ ( append_v @ Xs @ Ys ) )
= ( append_v @ ( tl_v @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_338_precedes__append__left__iff,axiom,
! [X: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,Y: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys ) )
=> ( ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ ( append2138873909117096322od_v_v @ Ys @ Xs ) )
= ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ Xs ) ) ) ).
% precedes_append_left_iff
thf(fact_339_precedes__append__left__iff,axiom,
! [X: v,Ys: list_v,Y: v,Xs: list_v] :
( ~ ( member_v2 @ X @ ( set_v2 @ Ys ) )
=> ( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( append_v @ Ys @ Xs ) )
= ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs ) ) ) ).
% precedes_append_left_iff
thf(fact_340_precedes__append__right__iff,axiom,
! [Y: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ Ys ) )
=> ( ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ Xs ) ) ) ).
% precedes_append_right_iff
thf(fact_341_precedes__append__right__iff,axiom,
! [Y: v,Ys: list_v,X: v,Xs: list_v] :
( ~ ( member_v2 @ Y @ ( set_v2 @ Ys ) )
=> ( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( append_v @ Xs @ Ys ) )
= ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs ) ) ) ).
% precedes_append_right_iff
thf(fact_342_subset__code_I2_J,axiom,
! [A4: set_v,Ys: list_v] :
( ( ord_less_eq_set_v @ A4 @ ( coset_v @ Ys ) )
= ( ! [X4: v] :
( ( member_v2 @ X4 @ ( set_v2 @ Ys ) )
=> ~ ( member_v2 @ X4 @ A4 ) ) ) ) ).
% subset_code(2)
thf(fact_343_subset__code_I2_J,axiom,
! [A4: set_Product_prod_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ ( coset_766761627116920666od_v_v @ Ys ) )
= ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ ( set_Product_prod_v_v2 @ Ys ) )
=> ~ ( member7453568604450474000od_v_v @ X4 @ A4 ) ) ) ) ).
% subset_code(2)
thf(fact_344_graph_Ore__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y ) ) ) ).
% graph.re_reachable
thf(fact_345_graph_Oreachable__re,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y ) ) ) ).
% graph.reachable_re
thf(fact_346_graph_Ois__subscc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S3: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S3 )
= ( ! [X4: v] :
( ( member_v2 @ X4 @ S3 )
=> ! [Y4: v] :
( ( member_v2 @ Y4 @ S3 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X4 @ Y4 ) ) ) ) ) ) ).
% graph.is_subscc_def
thf(fact_347_graph_OsccE,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S3: set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S3 )
=> ( ( member7453568604450474000od_v_v @ X @ S3 )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ X )
=> ( member7453568604450474000od_v_v @ Y @ S3 ) ) ) ) ) ) ).
% graph.sccE
thf(fact_348_graph_OsccE,axiom,
! [Vertices: set_v,Successors: v > set_v,S3: set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S3 )
=> ( ( member_v2 @ X @ S3 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ X )
=> ( member_v2 @ Y @ S3 ) ) ) ) ) ) ).
% graph.sccE
thf(fact_349_graph_Ounite__subscc,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V2: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V2 @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V2 ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V2 @ W @ E ) @ ( hd_Product_prod_v_v @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V2 @ W @ E ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_subscc
thf(fact_350_graph_Ounite__subscc,axiom,
! [Vertices: set_v,Successors: v > set_v,V2: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V2 @ E )
=> ( ( member_v2 @ W @ ( Successors @ V2 ) )
=> ( ~ ( member_v2 @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) )
=> ( ( member_v2 @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v2 @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V2 @ W @ E ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V2 @ W @ E ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_subscc
thf(fact_351_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_v @ ( coset_v @ nil_v ) @ ( set_v2 @ nil_v ) ) ).
% subset_code(3)
thf(fact_352_subset__code_I3_J,axiom,
~ ( ord_le7336532860387713383od_v_v @ ( coset_766761627116920666od_v_v @ nil_Product_prod_v_v ) @ ( set_Product_prod_v_v2 @ nil_Product_prod_v_v ) ) ).
% subset_code(3)
thf(fact_353_graph_Opre__post_I1_J,axiom,
! [Vertices: set_v,Successors: v > set_v,V2: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ Successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V2 @ E ) ) )
=> ( ( sCC_Bl36166008131615352t_unit @ Successors @ V2 @ E )
=> ( sCC_Bl8953792750115413617t_unit @ Successors @ V2 @ E @ ( sCC_Bloemen_dfs_v @ Successors @ V2 @ E ) ) ) ) ) ).
% graph.pre_post(1)
thf(fact_354_graph_Ounite__S__tl,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v,V2: product_prod_v_v,N: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl7798947040364291444t_unit @ Successors @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V2 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V2 ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( ( member7453568604450474000od_v_v @ N @ ( set_Product_prod_v_v2 @ ( tl_Product_prod_v_v @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V2 @ W @ E ) ) ) ) )
=> ( ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V2 @ W @ E ) @ N )
= ( sCC_Bl8440648026628373538t_unit @ E @ N ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_355_graph_Ounite__S__tl,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,W: v,V2: v,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v2 @ W @ ( Successors @ V2 ) )
=> ( ~ ( member_v2 @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) )
=> ( ( member_v2 @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v2 @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v2 @ N @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V2 @ W @ E ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V2 @ W @ E ) @ N )
= ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_356_finite__Plus__UNIV__iff,axiom,
( ( finite6078742274189123708um_v_v @ top_to6215363865728860875um_v_v )
= ( ( finite_finite_v @ top_top_set_v )
& ( finite_finite_v @ top_top_set_v ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_357_finite__Plus__UNIV__iff,axiom,
( ( finite2027638671530882166t_unit @ top_to1518419388207941061t_unit )
= ( ( finite_finite_v @ top_top_set_v )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_358_finite__Plus__UNIV__iff,axiom,
( ( finite7167594516733240705um_v_o @ top_to8752842168885503824um_v_o )
= ( ( finite_finite_v @ top_top_set_v )
& ( finite_finite_o @ top_top_set_o ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_359_finite__Plus__UNIV__iff,axiom,
( ( finite7999691852018159805_v_nat @ top_to7635744941665955908_v_nat )
= ( ( finite_finite_v @ top_top_set_v )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_360_finite__Plus__UNIV__iff,axiom,
( ( finite1276461646446175746unit_v @ top_to7050962321513869649unit_v )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite_finite_v @ top_top_set_v ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_361_finite__Plus__UNIV__iff,axiom,
( ( finite3146551501593861116t_unit @ top_to2771918933716375115t_unit )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_362_finite__Plus__UNIV__iff,axiom,
( ( finite5803017218550156807unit_o @ top_to8142889665740241110unit_o )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite_finite_o @ top_top_set_o ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_363_finite__Plus__UNIV__iff,axiom,
( ( finite4401952911629260215it_nat @ top_to2894617605782473790it_nat )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_364_finite__Plus__UNIV__iff,axiom,
( ( finite2208549381002230327um_o_v @ top_to2259075651460907270um_o_v )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite_finite_v @ top_top_set_v ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_365_finite__Plus__UNIV__iff,axiom,
( ( finite2105581108028245809t_unit @ top_to126344037801868544t_unit )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_366_finite__option__UNIV,axiom,
( ( finite1674126308695703426tion_v @ top_top_set_option_v )
= ( finite_finite_v @ top_top_set_v ) ) ).
% finite_option_UNIV
thf(fact_367_finite__option__UNIV,axiom,
( ( finite1445617369574913404t_unit @ top_to2690860209552263555t_unit )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% finite_option_UNIV
thf(fact_368_finite__option__UNIV,axiom,
( ( finite4093902646404507527tion_o @ top_top_set_option_o )
= ( finite_finite_o @ top_top_set_o ) ) ).
% finite_option_UNIV
thf(fact_369_finite__option__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_option_UNIV
thf(fact_370_pre__post_I2_J,axiom,
! [V2: v,E: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In5289330923152326972t_unit @ ( produc3862955338007567901t_unit @ V2 @ E ) ) )
=> ( ( sCC_Bl1748261141445803503t_unit @ successors @ V2 @ E )
=> ( sCC_Bl6082031138996704384t_unit @ successors @ V2 @ E @ ( sCC_Bloemen_dfss_v @ successors @ V2 @ E ) ) ) ) ).
% pre_post(2)
thf(fact_371_is__scc__def,axiom,
! [S3: set_v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S3 )
= ( ( S3 != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S3 )
& ! [S4: set_v] :
( ( ( ord_less_eq_set_v @ S3 @ S4 )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S4 ) )
=> ( S4 = S3 ) ) ) ) ).
% is_scc_def
thf(fact_372_pre__dfss__def,axiom,
! [V2: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V2 @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
& ( member_v2 @ V2 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
& ( member_v2 @ V2 @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
& ! [X4: v] :
( ( member_v2 @ X4 @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) )
=> ( member_v2 @ X4 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
& ! [X4: v] :
( ( member_v2 @ X4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X4 @ V2 ) )
& ? [Ns: list_v] :
( ( sCC_Bl9201514103433284750t_unit @ E )
= ( cons_v @ V2 @ Ns ) ) ) ) ).
% pre_dfss_def
thf(fact_373_pre__dfs__def,axiom,
! [V2: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl36166008131615352t_unit @ successors @ V2 @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
& ~ ( member_v2 @ V2 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( sCC_Bl649662514949026229able_v @ successors @ ( sCC_Bl1090238580953940555t_unit @ E ) @ V2 )
& ( ( sCC_Bl3795065053823578884t_unit @ E @ V2 )
= bot_bot_set_v )
& ! [X4: v] :
( ( member_v2 @ X4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X4 @ V2 ) ) ) ) ).
% pre_dfs_def
thf(fact_374_finite__prod,axiom,
( ( finite3348123685078250256od_v_v @ top_to5429829297380968215od_v_v )
= ( ( finite_finite_v @ top_top_set_v )
& ( finite_finite_v @ top_top_set_v ) ) ) ).
% finite_prod
thf(fact_375_finite__prod,axiom,
( ( finite1367261533680821002t_unit @ top_to6398825399363161617t_unit )
= ( ( finite_finite_v @ top_top_set_v )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_376_finite__prod,axiom,
( ( finite1845983951574860309od_v_o @ top_to3646780102395089308od_v_o )
= ( ( finite_finite_v @ top_top_set_v )
& ( finite_finite_o @ top_top_set_o ) ) ) ).
% finite_prod
thf(fact_377_finite__prod,axiom,
( ( finite4919113160666087721_v_nat @ top_to970446785217458424_v_nat )
= ( ( finite_finite_v @ top_top_set_v )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_378_finite__prod,axiom,
( ( finite616084508596114582unit_v @ top_to2707996295814314397unit_v )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite_finite_v @ top_top_set_v ) ) ) ).
% finite_prod
thf(fact_379_finite__prod,axiom,
( ( finite6816719414181127824t_unit @ top_to1835807148980544151t_unit )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_380_finite__prod,axiom,
( ( finite433095635221857947unit_o @ top_to422830600960239394unit_o )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite_finite_o @ top_top_set_o ) ) ) ).
% finite_prod
thf(fact_381_finite__prod,axiom,
( ( finite5187522816498166307it_nat @ top_to5974110478112770290it_nat )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_382_finite__prod,axiom,
( ( finite6110310852698625739od_o_v @ top_to6376385621825268562od_o_v )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite_finite_v @ top_top_set_v ) ) ) ).
% finite_prod
thf(fact_383_finite__prod,axiom,
( ( finite5959031561554722757t_unit @ top_to1629657009876642636t_unit )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_384_finite__Prod__UNIV,axiom,
( ( finite_finite_v @ top_top_set_v )
=> ( ( finite_finite_v @ top_top_set_v )
=> ( finite3348123685078250256od_v_v @ top_to5429829297380968215od_v_v ) ) ) ).
% finite_Prod_UNIV
thf(fact_385_finite__Prod__UNIV,axiom,
( ( finite_finite_v @ top_top_set_v )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite1367261533680821002t_unit @ top_to6398825399363161617t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_386_finite__Prod__UNIV,axiom,
( ( finite_finite_v @ top_top_set_v )
=> ( ( finite_finite_o @ top_top_set_o )
=> ( finite1845983951574860309od_v_o @ top_to3646780102395089308od_v_o ) ) ) ).
% finite_Prod_UNIV
thf(fact_387_finite__Prod__UNIV,axiom,
( ( finite_finite_v @ top_top_set_v )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite4919113160666087721_v_nat @ top_to970446785217458424_v_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_388_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite_finite_v @ top_top_set_v )
=> ( finite616084508596114582unit_v @ top_to2707996295814314397unit_v ) ) ) ).
% finite_Prod_UNIV
thf(fact_389_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite6816719414181127824t_unit @ top_to1835807148980544151t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_390_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite_finite_o @ top_top_set_o )
=> ( finite433095635221857947unit_o @ top_to422830600960239394unit_o ) ) ) ).
% finite_Prod_UNIV
thf(fact_391_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite5187522816498166307it_nat @ top_to5974110478112770290it_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_392_finite__Prod__UNIV,axiom,
( ( finite_finite_o @ top_top_set_o )
=> ( ( finite_finite_v @ top_top_set_v )
=> ( finite6110310852698625739od_o_v @ top_to6376385621825268562od_o_v ) ) ) ).
% finite_Prod_UNIV
thf(fact_393_finite__Prod__UNIV,axiom,
( ( finite_finite_o @ top_top_set_o )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite5959031561554722757t_unit @ top_to1629657009876642636t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_394_post__dfss__def,axiom,
! [V2: v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl6082031138996704384t_unit @ successors @ V2 @ E @ E2 )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E2 )
& ( ( sCC_Bl3795065053823578884t_unit @ E2 @ V2 )
= ( successors @ V2 ) )
& ! [X4: v] :
( ( member_v2 @ X4 @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V2 @ bot_bot_set_v ) ) )
=> ( ( sCC_Bl3795065053823578884t_unit @ E2 @ X4 )
= ( sCC_Bl3795065053823578884t_unit @ E @ X4 ) ) )
& ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
& ! [X4: v] :
( ( member_v2 @ X4 @ ( successors @ V2 ) )
=> ( member_v2 @ X4 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E2 ) @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) )
& ! [X4: v] :
( ( member_v2 @ X4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X4 @ V2 ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E2 )
!= nil_v )
& ? [Ns: list_v] :
( ( sCC_Bl8828226123343373779t_unit @ E )
= ( append_v @ Ns @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) )
& ( member_v2 @ V2 @ ( sCC_Bl1280885523602775798t_unit @ E2 @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
& ! [X4: v] :
( ( member_v2 @ X4 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ E2 @ X4 )
= ( sCC_Bl1280885523602775798t_unit @ E @ X4 ) ) )
& ( ( ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) )
= V2 )
=> ! [X4: v] :
( ( member_v2 @ X4 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ V2 @ X4 ) ) )
& ( ( sCC_Bl9201514103433284750t_unit @ E2 )
= ( sCC_Bl9201514103433284750t_unit @ E ) ) ) ) ).
% post_dfss_def
thf(fact_395_empty__iff,axiom,
! [C: product_unit] :
~ ( member_Product_unit @ C @ bot_bo3957492148770167129t_unit ) ).
% empty_iff
thf(fact_396_empty__iff,axiom,
! [C: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ C @ bot_bo723834152578015283od_v_v ) ).
% empty_iff
thf(fact_397_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_398_empty__iff,axiom,
! [C: set_v] :
~ ( member_set_v @ C @ bot_bot_set_set_v ) ).
% empty_iff
thf(fact_399_empty__iff,axiom,
! [C: v] :
~ ( member_v2 @ C @ bot_bot_set_v ) ).
% empty_iff
thf(fact_400_all__not__in__conv,axiom,
! [A4: set_Product_unit] :
( ( ! [X4: product_unit] :
~ ( member_Product_unit @ X4 @ A4 ) )
= ( A4 = bot_bo3957492148770167129t_unit ) ) ).
% all_not_in_conv
thf(fact_401_all__not__in__conv,axiom,
! [A4: set_Product_prod_v_v] :
( ( ! [X4: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ X4 @ A4 ) )
= ( A4 = bot_bo723834152578015283od_v_v ) ) ).
% all_not_in_conv
thf(fact_402_all__not__in__conv,axiom,
! [A4: set_o] :
( ( ! [X4: $o] :
~ ( member_o @ X4 @ A4 ) )
= ( A4 = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_403_all__not__in__conv,axiom,
! [A4: set_set_v] :
( ( ! [X4: set_v] :
~ ( member_set_v @ X4 @ A4 ) )
= ( A4 = bot_bot_set_set_v ) ) ).
% all_not_in_conv
thf(fact_404_all__not__in__conv,axiom,
! [A4: set_v] :
( ( ! [X4: v] :
~ ( member_v2 @ X4 @ A4 ) )
= ( A4 = bot_bot_set_v ) ) ).
% all_not_in_conv
thf(fact_405_Collect__empty__eq,axiom,
! [P2: product_unit > $o] :
( ( ( collect_Product_unit @ P2 )
= bot_bo3957492148770167129t_unit )
= ( ! [X4: product_unit] :
~ ( P2 @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_406_Collect__empty__eq,axiom,
! [P2: product_prod_v_v > $o] :
( ( ( collec140062887454715474od_v_v @ P2 )
= bot_bo723834152578015283od_v_v )
= ( ! [X4: product_prod_v_v] :
~ ( P2 @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_407_Collect__empty__eq,axiom,
! [P2: $o > $o] :
( ( ( collect_o @ P2 )
= bot_bot_set_o )
= ( ! [X4: $o] :
~ ( P2 @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_408_Collect__empty__eq,axiom,
! [P2: set_v > $o] :
( ( ( collect_set_v @ P2 )
= bot_bot_set_set_v )
= ( ! [X4: set_v] :
~ ( P2 @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_409_Collect__empty__eq,axiom,
! [P2: v > $o] :
( ( ( collect_v @ P2 )
= bot_bot_set_v )
= ( ! [X4: v] :
~ ( P2 @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_410_empty__Collect__eq,axiom,
! [P2: product_unit > $o] :
( ( bot_bo3957492148770167129t_unit
= ( collect_Product_unit @ P2 ) )
= ( ! [X4: product_unit] :
~ ( P2 @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_411_empty__Collect__eq,axiom,
! [P2: product_prod_v_v > $o] :
( ( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ P2 ) )
= ( ! [X4: product_prod_v_v] :
~ ( P2 @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_412_empty__Collect__eq,axiom,
! [P2: $o > $o] :
( ( bot_bot_set_o
= ( collect_o @ P2 ) )
= ( ! [X4: $o] :
~ ( P2 @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_413_empty__Collect__eq,axiom,
! [P2: set_v > $o] :
( ( bot_bot_set_set_v
= ( collect_set_v @ P2 ) )
= ( ! [X4: set_v] :
~ ( P2 @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_414_empty__Collect__eq,axiom,
! [P2: v > $o] :
( ( bot_bot_set_v
= ( collect_v @ P2 ) )
= ( ! [X4: v] :
~ ( P2 @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_415_list_Oinject,axiom,
! [X21: v,X22: list_v,Y21: v,Y22: list_v] :
( ( ( cons_v @ X21 @ X22 )
= ( cons_v @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_416_insertCI,axiom,
! [A: set_v,B4: set_set_v,B: set_v] :
( ( ~ ( member_set_v @ A @ B4 )
=> ( A = B ) )
=> ( member_set_v @ A @ ( insert_set_v @ B @ B4 ) ) ) ).
% insertCI
thf(fact_417_insertCI,axiom,
! [A: product_unit,B4: set_Product_unit,B: product_unit] :
( ( ~ ( member_Product_unit @ A @ B4 )
=> ( A = B ) )
=> ( member_Product_unit @ A @ ( insert_Product_unit @ B @ B4 ) ) ) ).
% insertCI
thf(fact_418_insertCI,axiom,
! [A: $o,B4: set_o,B: $o] :
( ( ~ ( member_o @ A @ B4 )
=> ( A = B ) )
=> ( member_o @ A @ ( insert_o @ B @ B4 ) ) ) ).
% insertCI
thf(fact_419_insertCI,axiom,
! [A: v,B4: set_v,B: v] :
( ( ~ ( member_v2 @ A @ B4 )
=> ( A = B ) )
=> ( member_v2 @ A @ ( insert_v @ B @ B4 ) ) ) ).
% insertCI
thf(fact_420_insertCI,axiom,
! [A: product_prod_v_v,B4: set_Product_prod_v_v,B: product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ A @ B4 )
=> ( A = B ) )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ B4 ) ) ) ).
% insertCI
thf(fact_421_insert__iff,axiom,
! [A: set_v,B: set_v,A4: set_set_v] :
( ( member_set_v @ A @ ( insert_set_v @ B @ A4 ) )
= ( ( A = B )
| ( member_set_v @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_422_insert__iff,axiom,
! [A: product_unit,B: product_unit,A4: set_Product_unit] :
( ( member_Product_unit @ A @ ( insert_Product_unit @ B @ A4 ) )
= ( ( A = B )
| ( member_Product_unit @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_423_insert__iff,axiom,
! [A: $o,B: $o,A4: set_o] :
( ( member_o @ A @ ( insert_o @ B @ A4 ) )
= ( ( A = B )
| ( member_o @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_424_insert__iff,axiom,
! [A: v,B: v,A4: set_v] :
( ( member_v2 @ A @ ( insert_v @ B @ A4 ) )
= ( ( A = B )
| ( member_v2 @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_425_insert__iff,axiom,
! [A: product_prod_v_v,B: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ A4 ) )
= ( ( A = B )
| ( member7453568604450474000od_v_v @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_426_insert__absorb2,axiom,
! [X: v,A4: set_v] :
( ( insert_v @ X @ ( insert_v @ X @ A4 ) )
= ( insert_v @ X @ A4 ) ) ).
% insert_absorb2
thf(fact_427_insert__absorb2,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( insert1338601472111419319od_v_v @ X @ A4 ) )
= ( insert1338601472111419319od_v_v @ X @ A4 ) ) ).
% insert_absorb2
thf(fact_428_insert__absorb2,axiom,
! [X: set_v,A4: set_set_v] :
( ( insert_set_v @ X @ ( insert_set_v @ X @ A4 ) )
= ( insert_set_v @ X @ A4 ) ) ).
% insert_absorb2
thf(fact_429_insert__absorb2,axiom,
! [X: product_unit,A4: set_Product_unit] :
( ( insert_Product_unit @ X @ ( insert_Product_unit @ X @ A4 ) )
= ( insert_Product_unit @ X @ A4 ) ) ).
% insert_absorb2
thf(fact_430_insert__absorb2,axiom,
! [X: $o,A4: set_o] :
( ( insert_o @ X @ ( insert_o @ X @ A4 ) )
= ( insert_o @ X @ A4 ) ) ).
% insert_absorb2
thf(fact_431_UnCI,axiom,
! [C: product_prod_v_v,B4: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ C @ B4 )
=> ( member7453568604450474000od_v_v @ C @ A4 ) )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A4 @ B4 ) ) ) ).
% UnCI
thf(fact_432_UnCI,axiom,
! [C: v,B4: set_v,A4: set_v] :
( ( ~ ( member_v2 @ C @ B4 )
=> ( member_v2 @ C @ A4 ) )
=> ( member_v2 @ C @ ( sup_sup_set_v @ A4 @ B4 ) ) ) ).
% UnCI
thf(fact_433_UnCI,axiom,
! [C: set_v,B4: set_set_v,A4: set_set_v] :
( ( ~ ( member_set_v @ C @ B4 )
=> ( member_set_v @ C @ A4 ) )
=> ( member_set_v @ C @ ( sup_sup_set_set_v @ A4 @ B4 ) ) ) ).
% UnCI
thf(fact_434_Un__iff,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A4 @ B4 ) )
= ( ( member7453568604450474000od_v_v @ C @ A4 )
| ( member7453568604450474000od_v_v @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_435_Un__iff,axiom,
! [C: v,A4: set_v,B4: set_v] :
( ( member_v2 @ C @ ( sup_sup_set_v @ A4 @ B4 ) )
= ( ( member_v2 @ C @ A4 )
| ( member_v2 @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_436_Un__iff,axiom,
! [C: set_v,A4: set_set_v,B4: set_set_v] :
( ( member_set_v @ C @ ( sup_sup_set_set_v @ A4 @ B4 ) )
= ( ( member_set_v @ C @ A4 )
| ( member_set_v @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_437_subscc__add,axiom,
! [S3: set_v,X: v,Y: v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S3 )
=> ( ( member_v2 @ X @ S3 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ X )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( insert_v @ Y @ S3 ) ) ) ) ) ) ).
% subscc_add
thf(fact_438_subset__empty,axiom,
! [A4: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A4 @ bot_bo3957492148770167129t_unit )
= ( A4 = bot_bo3957492148770167129t_unit ) ) ).
% subset_empty
thf(fact_439_subset__empty,axiom,
! [A4: set_o] :
( ( ord_less_eq_set_o @ A4 @ bot_bot_set_o )
= ( A4 = bot_bot_set_o ) ) ).
% subset_empty
thf(fact_440_subset__empty,axiom,
! [A4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A4 @ bot_bot_set_set_v )
= ( A4 = bot_bot_set_set_v ) ) ).
% subset_empty
thf(fact_441_subset__empty,axiom,
! [A4: set_v] :
( ( ord_less_eq_set_v @ A4 @ bot_bot_set_v )
= ( A4 = bot_bot_set_v ) ) ).
% subset_empty
thf(fact_442_subset__empty,axiom,
! [A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ bot_bo723834152578015283od_v_v )
= ( A4 = bot_bo723834152578015283od_v_v ) ) ).
% subset_empty
thf(fact_443_empty__subsetI,axiom,
! [A4: set_Product_unit] : ( ord_le3507040750410214029t_unit @ bot_bo3957492148770167129t_unit @ A4 ) ).
% empty_subsetI
thf(fact_444_empty__subsetI,axiom,
! [A4: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A4 ) ).
% empty_subsetI
thf(fact_445_empty__subsetI,axiom,
! [A4: set_set_v] : ( ord_le5216385588623774835_set_v @ bot_bot_set_set_v @ A4 ) ).
% empty_subsetI
thf(fact_446_empty__subsetI,axiom,
! [A4: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A4 ) ).
% empty_subsetI
thf(fact_447_empty__subsetI,axiom,
! [A4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A4 ) ).
% empty_subsetI
thf(fact_448_singletonI,axiom,
! [A: product_unit] : ( member_Product_unit @ A @ ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) ) ).
% singletonI
thf(fact_449_singletonI,axiom,
! [A: product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% singletonI
thf(fact_450_singletonI,axiom,
! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).
% singletonI
thf(fact_451_singletonI,axiom,
! [A: set_v] : ( member_set_v @ A @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) ).
% singletonI
thf(fact_452_singletonI,axiom,
! [A: v] : ( member_v2 @ A @ ( insert_v @ A @ bot_bot_set_v ) ) ).
% singletonI
thf(fact_453_insert__subset,axiom,
! [X: set_v,A4: set_set_v,B4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( insert_set_v @ X @ A4 ) @ B4 )
= ( ( member_set_v @ X @ B4 )
& ( ord_le5216385588623774835_set_v @ A4 @ B4 ) ) ) ).
% insert_subset
thf(fact_454_insert__subset,axiom,
! [X: product_unit,A4: set_Product_unit,B4: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ X @ A4 ) @ B4 )
= ( ( member_Product_unit @ X @ B4 )
& ( ord_le3507040750410214029t_unit @ A4 @ B4 ) ) ) ).
% insert_subset
thf(fact_455_insert__subset,axiom,
! [X: $o,A4: set_o,B4: set_o] :
( ( ord_less_eq_set_o @ ( insert_o @ X @ A4 ) @ B4 )
= ( ( member_o @ X @ B4 )
& ( ord_less_eq_set_o @ A4 @ B4 ) ) ) ).
% insert_subset
thf(fact_456_insert__subset,axiom,
! [X: v,A4: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ ( insert_v @ X @ A4 ) @ B4 )
= ( ( member_v2 @ X @ B4 )
& ( ord_less_eq_set_v @ A4 @ B4 ) ) ) ).
% insert_subset
thf(fact_457_insert__subset,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X @ A4 ) @ B4 )
= ( ( member7453568604450474000od_v_v @ X @ B4 )
& ( ord_le7336532860387713383od_v_v @ A4 @ B4 ) ) ) ).
% insert_subset
thf(fact_458_Un__empty,axiom,
! [A4: set_Product_unit,B4: set_Product_unit] :
( ( ( sup_su793286257634532545t_unit @ A4 @ B4 )
= bot_bo3957492148770167129t_unit )
= ( ( A4 = bot_bo3957492148770167129t_unit )
& ( B4 = bot_bo3957492148770167129t_unit ) ) ) ).
% Un_empty
thf(fact_459_Un__empty,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A4 @ B4 )
= bot_bo723834152578015283od_v_v )
= ( ( A4 = bot_bo723834152578015283od_v_v )
& ( B4 = bot_bo723834152578015283od_v_v ) ) ) ).
% Un_empty
thf(fact_460_Un__empty,axiom,
! [A4: set_o,B4: set_o] :
( ( ( sup_sup_set_o @ A4 @ B4 )
= bot_bot_set_o )
= ( ( A4 = bot_bot_set_o )
& ( B4 = bot_bot_set_o ) ) ) ).
% Un_empty
thf(fact_461_Un__empty,axiom,
! [A4: set_set_v,B4: set_set_v] :
( ( ( sup_sup_set_set_v @ A4 @ B4 )
= bot_bot_set_set_v )
= ( ( A4 = bot_bot_set_set_v )
& ( B4 = bot_bot_set_set_v ) ) ) ).
% Un_empty
thf(fact_462_Un__empty,axiom,
! [A4: set_v,B4: set_v] :
( ( ( sup_sup_set_v @ A4 @ B4 )
= bot_bot_set_v )
= ( ( A4 = bot_bot_set_v )
& ( B4 = bot_bot_set_v ) ) ) ).
% Un_empty
thf(fact_463_Diff__empty,axiom,
! [A4: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ A4 @ bot_bo3957492148770167129t_unit )
= A4 ) ).
% Diff_empty
thf(fact_464_Diff__empty,axiom,
! [A4: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A4 @ bot_bo723834152578015283od_v_v )
= A4 ) ).
% Diff_empty
thf(fact_465_Diff__empty,axiom,
! [A4: set_o] :
( ( minus_minus_set_o @ A4 @ bot_bot_set_o )
= A4 ) ).
% Diff_empty
thf(fact_466_Diff__empty,axiom,
! [A4: set_set_v] :
( ( minus_7228012346218142266_set_v @ A4 @ bot_bot_set_set_v )
= A4 ) ).
% Diff_empty
thf(fact_467_Diff__empty,axiom,
! [A4: set_v] :
( ( minus_minus_set_v @ A4 @ bot_bot_set_v )
= A4 ) ).
% Diff_empty
thf(fact_468_empty__Diff,axiom,
! [A4: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ bot_bo3957492148770167129t_unit @ A4 )
= bot_bo3957492148770167129t_unit ) ).
% empty_Diff
thf(fact_469_empty__Diff,axiom,
! [A4: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ bot_bo723834152578015283od_v_v @ A4 )
= bot_bo723834152578015283od_v_v ) ).
% empty_Diff
thf(fact_470_empty__Diff,axiom,
! [A4: set_o] :
( ( minus_minus_set_o @ bot_bot_set_o @ A4 )
= bot_bot_set_o ) ).
% empty_Diff
thf(fact_471_empty__Diff,axiom,
! [A4: set_set_v] :
( ( minus_7228012346218142266_set_v @ bot_bot_set_set_v @ A4 )
= bot_bot_set_set_v ) ).
% empty_Diff
thf(fact_472_empty__Diff,axiom,
! [A4: set_v] :
( ( minus_minus_set_v @ bot_bot_set_v @ A4 )
= bot_bot_set_v ) ).
% empty_Diff
thf(fact_473_Diff__cancel,axiom,
! [A4: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ A4 @ A4 )
= bot_bo3957492148770167129t_unit ) ).
% Diff_cancel
thf(fact_474_Diff__cancel,axiom,
! [A4: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A4 @ A4 )
= bot_bo723834152578015283od_v_v ) ).
% Diff_cancel
thf(fact_475_Diff__cancel,axiom,
! [A4: set_o] :
( ( minus_minus_set_o @ A4 @ A4 )
= bot_bot_set_o ) ).
% Diff_cancel
thf(fact_476_Diff__cancel,axiom,
! [A4: set_set_v] :
( ( minus_7228012346218142266_set_v @ A4 @ A4 )
= bot_bot_set_set_v ) ).
% Diff_cancel
thf(fact_477_Diff__cancel,axiom,
! [A4: set_v] :
( ( minus_minus_set_v @ A4 @ A4 )
= bot_bot_set_v ) ).
% Diff_cancel
thf(fact_478_Un__subset__iff,axiom,
! [A4: set_set_v,B4: set_set_v,C2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A4 @ B4 ) @ C2 )
= ( ( ord_le5216385588623774835_set_v @ A4 @ C2 )
& ( ord_le5216385588623774835_set_v @ B4 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_479_Un__subset__iff,axiom,
! [A4: set_v,B4: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A4 @ B4 ) @ C2 )
= ( ( ord_less_eq_set_v @ A4 @ C2 )
& ( ord_less_eq_set_v @ B4 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_480_Un__subset__iff,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B4 ) @ C2 )
= ( ( ord_le7336532860387713383od_v_v @ A4 @ C2 )
& ( ord_le7336532860387713383od_v_v @ B4 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_481_Un__insert__left,axiom,
! [A: product_unit,B4: set_Product_unit,C2: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ ( insert_Product_unit @ A @ B4 ) @ C2 )
= ( insert_Product_unit @ A @ ( sup_su793286257634532545t_unit @ B4 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_482_Un__insert__left,axiom,
! [A: $o,B4: set_o,C2: set_o] :
( ( sup_sup_set_o @ ( insert_o @ A @ B4 ) @ C2 )
= ( insert_o @ A @ ( sup_sup_set_o @ B4 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_483_Un__insert__left,axiom,
! [A: product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A @ B4 ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B4 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_484_Un__insert__left,axiom,
! [A: v,B4: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( insert_v @ A @ B4 ) @ C2 )
= ( insert_v @ A @ ( sup_sup_set_v @ B4 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_485_Un__insert__left,axiom,
! [A: set_v,B4: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ ( insert_set_v @ A @ B4 ) @ C2 )
= ( insert_set_v @ A @ ( sup_sup_set_set_v @ B4 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_486_Un__insert__right,axiom,
! [A4: set_Product_unit,A: product_unit,B4: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ A4 @ ( insert_Product_unit @ A @ B4 ) )
= ( insert_Product_unit @ A @ ( sup_su793286257634532545t_unit @ A4 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_487_Un__insert__right,axiom,
! [A4: set_o,A: $o,B4: set_o] :
( ( sup_sup_set_o @ A4 @ ( insert_o @ A @ B4 ) )
= ( insert_o @ A @ ( sup_sup_set_o @ A4 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_488_Un__insert__right,axiom,
! [A4: set_Product_prod_v_v,A: product_prod_v_v,B4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A @ B4 ) )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A4 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_489_Un__insert__right,axiom,
! [A4: set_v,A: v,B4: set_v] :
( ( sup_sup_set_v @ A4 @ ( insert_v @ A @ B4 ) )
= ( insert_v @ A @ ( sup_sup_set_v @ A4 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_490_Un__insert__right,axiom,
! [A4: set_set_v,A: set_v,B4: set_set_v] :
( ( sup_sup_set_set_v @ A4 @ ( insert_set_v @ A @ B4 ) )
= ( insert_set_v @ A @ ( sup_sup_set_set_v @ A4 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_491_Diff__insert0,axiom,
! [X: set_v,A4: set_set_v,B4: set_set_v] :
( ~ ( member_set_v @ X @ A4 )
=> ( ( minus_7228012346218142266_set_v @ A4 @ ( insert_set_v @ X @ B4 ) )
= ( minus_7228012346218142266_set_v @ A4 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_492_Diff__insert0,axiom,
! [X: product_unit,A4: set_Product_unit,B4: set_Product_unit] :
( ~ ( member_Product_unit @ X @ A4 )
=> ( ( minus_6452836326544984404t_unit @ A4 @ ( insert_Product_unit @ X @ B4 ) )
= ( minus_6452836326544984404t_unit @ A4 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_493_Diff__insert0,axiom,
! [X: $o,A4: set_o,B4: set_o] :
( ~ ( member_o @ X @ A4 )
=> ( ( minus_minus_set_o @ A4 @ ( insert_o @ X @ B4 ) )
= ( minus_minus_set_o @ A4 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_494_Diff__insert0,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A4 )
=> ( ( minus_4183494784930505774od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ X @ B4 ) )
= ( minus_4183494784930505774od_v_v @ A4 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_495_Diff__insert0,axiom,
! [X: v,A4: set_v,B4: set_v] :
( ~ ( member_v2 @ X @ A4 )
=> ( ( minus_minus_set_v @ A4 @ ( insert_v @ X @ B4 ) )
= ( minus_minus_set_v @ A4 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_496_insert__Diff1,axiom,
! [X: set_v,B4: set_set_v,A4: set_set_v] :
( ( member_set_v @ X @ B4 )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X @ A4 ) @ B4 )
= ( minus_7228012346218142266_set_v @ A4 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_497_insert__Diff1,axiom,
! [X: product_unit,B4: set_Product_unit,A4: set_Product_unit] :
( ( member_Product_unit @ X @ B4 )
=> ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit @ X @ A4 ) @ B4 )
= ( minus_6452836326544984404t_unit @ A4 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_498_insert__Diff1,axiom,
! [X: $o,B4: set_o,A4: set_o] :
( ( member_o @ X @ B4 )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A4 ) @ B4 )
= ( minus_minus_set_o @ A4 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_499_insert__Diff1,axiom,
! [X: product_prod_v_v,B4: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ B4 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A4 ) @ B4 )
= ( minus_4183494784930505774od_v_v @ A4 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_500_insert__Diff1,axiom,
! [X: v,B4: set_v,A4: set_v] :
( ( member_v2 @ X @ B4 )
=> ( ( minus_minus_set_v @ ( insert_v @ X @ A4 ) @ B4 )
= ( minus_minus_set_v @ A4 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_501_Un__Diff__cancel,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ ( minus_4183494784930505774od_v_v @ B4 @ A4 ) )
= ( sup_su414716646722978715od_v_v @ A4 @ B4 ) ) ).
% Un_Diff_cancel
thf(fact_502_Un__Diff__cancel,axiom,
! [A4: set_set_v,B4: set_set_v] :
( ( sup_sup_set_set_v @ A4 @ ( minus_7228012346218142266_set_v @ B4 @ A4 ) )
= ( sup_sup_set_set_v @ A4 @ B4 ) ) ).
% Un_Diff_cancel
thf(fact_503_Un__Diff__cancel,axiom,
! [A4: set_v,B4: set_v] :
( ( sup_sup_set_v @ A4 @ ( minus_minus_set_v @ B4 @ A4 ) )
= ( sup_sup_set_v @ A4 @ B4 ) ) ).
% Un_Diff_cancel
thf(fact_504_Un__Diff__cancel2,axiom,
! [B4: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ B4 @ A4 ) @ A4 )
= ( sup_su414716646722978715od_v_v @ B4 @ A4 ) ) ).
% Un_Diff_cancel2
thf(fact_505_Un__Diff__cancel2,axiom,
! [B4: set_set_v,A4: set_set_v] :
( ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ B4 @ A4 ) @ A4 )
= ( sup_sup_set_set_v @ B4 @ A4 ) ) ).
% Un_Diff_cancel2
thf(fact_506_Un__Diff__cancel2,axiom,
! [B4: set_v,A4: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ B4 @ A4 ) @ A4 )
= ( sup_sup_set_v @ B4 @ A4 ) ) ).
% Un_Diff_cancel2
thf(fact_507_Inl__Inr__False,axiom,
! [X: produc5741669702376414499t_unit,Y: produc5741669702376414499t_unit] :
( ( sum_In526841707622398774t_unit @ X )
!= ( sum_In5289330923152326972t_unit @ Y ) ) ).
% Inl_Inr_False
thf(fact_508_Inr__Inl__False,axiom,
! [X: produc5741669702376414499t_unit,Y: produc5741669702376414499t_unit] :
( ( sum_In5289330923152326972t_unit @ X )
!= ( sum_In526841707622398774t_unit @ Y ) ) ).
% Inr_Inl_False
thf(fact_509_singleton__insert__inj__eq_H,axiom,
! [A: product_unit,A4: set_Product_unit,B: product_unit] :
( ( ( insert_Product_unit @ A @ A4 )
= ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) )
= ( ( A = B )
& ( ord_le3507040750410214029t_unit @ A4 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_510_singleton__insert__inj__eq_H,axiom,
! [A: $o,A4: set_o,B: $o] :
( ( ( insert_o @ A @ A4 )
= ( insert_o @ B @ bot_bot_set_o ) )
= ( ( A = B )
& ( ord_less_eq_set_o @ A4 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_511_singleton__insert__inj__eq_H,axiom,
! [A: set_v,A4: set_set_v,B: set_v] :
( ( ( insert_set_v @ A @ A4 )
= ( insert_set_v @ B @ bot_bot_set_set_v ) )
= ( ( A = B )
& ( ord_le5216385588623774835_set_v @ A4 @ ( insert_set_v @ B @ bot_bot_set_set_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_512_singleton__insert__inj__eq_H,axiom,
! [A: v,A4: set_v,B: v] :
( ( ( insert_v @ A @ A4 )
= ( insert_v @ B @ bot_bot_set_v ) )
= ( ( A = B )
& ( ord_less_eq_set_v @ A4 @ ( insert_v @ B @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_513_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v,B: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ A4 )
= ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) )
= ( ( A = B )
& ( ord_le7336532860387713383od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_514_singleton__insert__inj__eq,axiom,
! [B: product_unit,A: product_unit,A4: set_Product_unit] :
( ( ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit )
= ( insert_Product_unit @ A @ A4 ) )
= ( ( A = B )
& ( ord_le3507040750410214029t_unit @ A4 @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_515_singleton__insert__inj__eq,axiom,
! [B: $o,A: $o,A4: set_o] :
( ( ( insert_o @ B @ bot_bot_set_o )
= ( insert_o @ A @ A4 ) )
= ( ( A = B )
& ( ord_less_eq_set_o @ A4 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_516_singleton__insert__inj__eq,axiom,
! [B: set_v,A: set_v,A4: set_set_v] :
( ( ( insert_set_v @ B @ bot_bot_set_set_v )
= ( insert_set_v @ A @ A4 ) )
= ( ( A = B )
& ( ord_le5216385588623774835_set_v @ A4 @ ( insert_set_v @ B @ bot_bot_set_set_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_517_singleton__insert__inj__eq,axiom,
! [B: v,A: v,A4: set_v] :
( ( ( insert_v @ B @ bot_bot_set_v )
= ( insert_v @ A @ A4 ) )
= ( ( A = B )
& ( ord_less_eq_set_v @ A4 @ ( insert_v @ B @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_518_singleton__insert__inj__eq,axiom,
! [B: product_prod_v_v,A: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ A @ A4 ) )
= ( ( A = B )
& ( ord_le7336532860387713383od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_519_set__empty,axiom,
! [Xs: list_Product_unit] :
( ( ( set_Product_unit2 @ Xs )
= bot_bo3957492148770167129t_unit )
= ( Xs = nil_Product_unit ) ) ).
% set_empty
thf(fact_520_set__empty,axiom,
! [Xs: list_P7986770385144383213od_v_v] :
( ( ( set_Product_prod_v_v2 @ Xs )
= bot_bo723834152578015283od_v_v )
= ( Xs = nil_Product_prod_v_v ) ) ).
% set_empty
thf(fact_521_set__empty,axiom,
! [Xs: list_o] :
( ( ( set_o2 @ Xs )
= bot_bot_set_o )
= ( Xs = nil_o ) ) ).
% set_empty
thf(fact_522_set__empty,axiom,
! [Xs: list_set_v] :
( ( ( set_set_v2 @ Xs )
= bot_bot_set_set_v )
= ( Xs = nil_set_v ) ) ).
% set_empty
thf(fact_523_set__empty,axiom,
! [Xs: list_v] :
( ( ( set_v2 @ Xs )
= bot_bot_set_v )
= ( Xs = nil_v ) ) ).
% set_empty
thf(fact_524_set__empty2,axiom,
! [Xs: list_Product_unit] :
( ( bot_bo3957492148770167129t_unit
= ( set_Product_unit2 @ Xs ) )
= ( Xs = nil_Product_unit ) ) ).
% set_empty2
thf(fact_525_set__empty2,axiom,
! [Xs: list_P7986770385144383213od_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( set_Product_prod_v_v2 @ Xs ) )
= ( Xs = nil_Product_prod_v_v ) ) ).
% set_empty2
thf(fact_526_set__empty2,axiom,
! [Xs: list_o] :
( ( bot_bot_set_o
= ( set_o2 @ Xs ) )
= ( Xs = nil_o ) ) ).
% set_empty2
thf(fact_527_set__empty2,axiom,
! [Xs: list_set_v] :
( ( bot_bot_set_set_v
= ( set_set_v2 @ Xs ) )
= ( Xs = nil_set_v ) ) ).
% set_empty2
thf(fact_528_set__empty2,axiom,
! [Xs: list_v] :
( ( bot_bot_set_v
= ( set_v2 @ Xs ) )
= ( Xs = nil_v ) ) ).
% set_empty2
thf(fact_529_list_Osimps_I15_J,axiom,
! [X21: product_prod_v_v,X22: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X22 ) )
= ( insert1338601472111419319od_v_v @ X21 @ ( set_Product_prod_v_v2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_530_list_Osimps_I15_J,axiom,
! [X21: set_v,X22: list_set_v] :
( ( set_set_v2 @ ( cons_set_v @ X21 @ X22 ) )
= ( insert_set_v @ X21 @ ( set_set_v2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_531_list_Osimps_I15_J,axiom,
! [X21: product_unit,X22: list_Product_unit] :
( ( set_Product_unit2 @ ( cons_Product_unit @ X21 @ X22 ) )
= ( insert_Product_unit @ X21 @ ( set_Product_unit2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_532_list_Osimps_I15_J,axiom,
! [X21: $o,X22: list_o] :
( ( set_o2 @ ( cons_o @ X21 @ X22 ) )
= ( insert_o @ X21 @ ( set_o2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_533_list_Osimps_I15_J,axiom,
! [X21: v,X22: list_v] :
( ( set_v2 @ ( cons_v @ X21 @ X22 ) )
= ( insert_v @ X21 @ ( set_v2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_534_Diff__UNIV,axiom,
! [A4: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A4 @ top_to5429829297380968215od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Diff_UNIV
thf(fact_535_Diff__UNIV,axiom,
! [A4: set_set_v] :
( ( minus_7228012346218142266_set_v @ A4 @ top_top_set_set_v )
= bot_bot_set_set_v ) ).
% Diff_UNIV
thf(fact_536_Diff__UNIV,axiom,
! [A4: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ A4 @ top_to1996260823553986621t_unit )
= bot_bo3957492148770167129t_unit ) ).
% Diff_UNIV
thf(fact_537_Diff__UNIV,axiom,
! [A4: set_o] :
( ( minus_minus_set_o @ A4 @ top_top_set_o )
= bot_bot_set_o ) ).
% Diff_UNIV
thf(fact_538_Diff__UNIV,axiom,
! [A4: set_nat] :
( ( minus_minus_set_nat @ A4 @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Diff_UNIV
thf(fact_539_Diff__UNIV,axiom,
! [A4: set_v] :
( ( minus_minus_set_v @ A4 @ top_top_set_v )
= bot_bot_set_v ) ).
% Diff_UNIV
thf(fact_540_Diff__eq__empty__iff,axiom,
! [A4: set_Product_unit,B4: set_Product_unit] :
( ( ( minus_6452836326544984404t_unit @ A4 @ B4 )
= bot_bo3957492148770167129t_unit )
= ( ord_le3507040750410214029t_unit @ A4 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_541_Diff__eq__empty__iff,axiom,
! [A4: set_o,B4: set_o] :
( ( ( minus_minus_set_o @ A4 @ B4 )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ A4 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_542_Diff__eq__empty__iff,axiom,
! [A4: set_set_v,B4: set_set_v] :
( ( ( minus_7228012346218142266_set_v @ A4 @ B4 )
= bot_bot_set_set_v )
= ( ord_le5216385588623774835_set_v @ A4 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_543_Diff__eq__empty__iff,axiom,
! [A4: set_v,B4: set_v] :
( ( ( minus_minus_set_v @ A4 @ B4 )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ A4 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_544_Diff__eq__empty__iff,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ A4 @ B4 )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ A4 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_545_insert__Diff__single,axiom,
! [A: product_unit,A4: set_Product_unit] :
( ( insert_Product_unit @ A @ ( minus_6452836326544984404t_unit @ A4 @ ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) ) )
= ( insert_Product_unit @ A @ A4 ) ) ).
% insert_Diff_single
thf(fact_546_insert__Diff__single,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= ( insert1338601472111419319od_v_v @ A @ A4 ) ) ).
% insert_Diff_single
thf(fact_547_insert__Diff__single,axiom,
! [A: $o,A4: set_o] :
( ( insert_o @ A @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( insert_o @ A @ A4 ) ) ).
% insert_Diff_single
thf(fact_548_insert__Diff__single,axiom,
! [A: set_v,A4: set_set_v] :
( ( insert_set_v @ A @ ( minus_7228012346218142266_set_v @ A4 @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) )
= ( insert_set_v @ A @ A4 ) ) ).
% insert_Diff_single
thf(fact_549_insert__Diff__single,axiom,
! [A: v,A4: set_v] :
( ( insert_v @ A @ ( minus_minus_set_v @ A4 @ ( insert_v @ A @ bot_bot_set_v ) ) )
= ( insert_v @ A @ A4 ) ) ).
% insert_Diff_single
thf(fact_550_append1__eq__conv,axiom,
! [Xs: list_v,X: v,Ys: list_v,Y: v] :
( ( ( append_v @ Xs @ ( cons_v @ X @ nil_v ) )
= ( append_v @ Ys @ ( cons_v @ Y @ nil_v ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_551_set__append,axiom,
! [Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( append2138873909117096322od_v_v @ Xs @ Ys ) )
= ( sup_su414716646722978715od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ Ys ) ) ) ).
% set_append
thf(fact_552_set__append,axiom,
! [Xs: list_v,Ys: list_v] :
( ( set_v2 @ ( append_v @ Xs @ Ys ) )
= ( sup_sup_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ Ys ) ) ) ).
% set_append
thf(fact_553_set__append,axiom,
! [Xs: list_set_v,Ys: list_set_v] :
( ( set_set_v2 @ ( append_set_v @ Xs @ Ys ) )
= ( sup_sup_set_set_v @ ( set_set_v2 @ Xs ) @ ( set_set_v2 @ Ys ) ) ) ).
% set_append
thf(fact_554_list_Ocollapse,axiom,
! [List: list_v] :
( ( List != nil_v )
=> ( ( cons_v @ ( hd_v @ List ) @ ( tl_v @ List ) )
= List ) ) ).
% list.collapse
thf(fact_555_hd__Cons__tl,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( ( cons_v @ ( hd_v @ Xs ) @ ( tl_v @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_556_subset__singleton__iff,axiom,
! [X6: set_Product_unit,A: product_unit] :
( ( ord_le3507040750410214029t_unit @ X6 @ ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) )
= ( ( X6 = bot_bo3957492148770167129t_unit )
| ( X6
= ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% subset_singleton_iff
thf(fact_557_subset__singleton__iff,axiom,
! [X6: set_o,A: $o] :
( ( ord_less_eq_set_o @ X6 @ ( insert_o @ A @ bot_bot_set_o ) )
= ( ( X6 = bot_bot_set_o )
| ( X6
= ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).
% subset_singleton_iff
thf(fact_558_subset__singleton__iff,axiom,
! [X6: set_set_v,A: set_v] :
( ( ord_le5216385588623774835_set_v @ X6 @ ( insert_set_v @ A @ bot_bot_set_set_v ) )
= ( ( X6 = bot_bot_set_set_v )
| ( X6
= ( insert_set_v @ A @ bot_bot_set_set_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_559_subset__singleton__iff,axiom,
! [X6: set_v,A: v] :
( ( ord_less_eq_set_v @ X6 @ ( insert_v @ A @ bot_bot_set_v ) )
= ( ( X6 = bot_bot_set_v )
| ( X6
= ( insert_v @ A @ bot_bot_set_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_560_subset__singleton__iff,axiom,
! [X6: set_Product_prod_v_v,A: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X6 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
= ( ( X6 = bot_bo723834152578015283od_v_v )
| ( X6
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_561_Diff__insert__absorb,axiom,
! [X: product_unit,A4: set_Product_unit] :
( ~ ( member_Product_unit @ X @ A4 )
=> ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit @ X @ A4 ) @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_562_Diff__insert__absorb,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A4 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A4 ) @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_563_Diff__insert__absorb,axiom,
! [X: $o,A4: set_o] :
( ~ ( member_o @ X @ A4 )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A4 ) @ ( insert_o @ X @ bot_bot_set_o ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_564_Diff__insert__absorb,axiom,
! [X: set_v,A4: set_set_v] :
( ~ ( member_set_v @ X @ A4 )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X @ A4 ) @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_565_Diff__insert__absorb,axiom,
! [X: v,A4: set_v] :
( ~ ( member_v2 @ X @ A4 )
=> ( ( minus_minus_set_v @ ( insert_v @ X @ A4 ) @ ( insert_v @ X @ bot_bot_set_v ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_566_subset__singletonD,axiom,
! [A4: set_Product_unit,X: product_unit] :
( ( ord_le3507040750410214029t_unit @ A4 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
=> ( ( A4 = bot_bo3957492148770167129t_unit )
| ( A4
= ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% subset_singletonD
thf(fact_567_subset__singletonD,axiom,
! [A4: set_o,X: $o] :
( ( ord_less_eq_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) )
=> ( ( A4 = bot_bot_set_o )
| ( A4
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% subset_singletonD
thf(fact_568_subset__singletonD,axiom,
! [A4: set_set_v,X: set_v] :
( ( ord_le5216385588623774835_set_v @ A4 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
=> ( ( A4 = bot_bot_set_set_v )
| ( A4
= ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ).
% subset_singletonD
thf(fact_569_subset__singletonD,axiom,
! [A4: set_v,X: v] :
( ( ord_less_eq_set_v @ A4 @ ( insert_v @ X @ bot_bot_set_v ) )
=> ( ( A4 = bot_bot_set_v )
| ( A4
= ( insert_v @ X @ bot_bot_set_v ) ) ) ) ).
% subset_singletonD
thf(fact_570_subset__singletonD,axiom,
! [A4: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
=> ( ( A4 = bot_bo723834152578015283od_v_v )
| ( A4
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singletonD
thf(fact_571_Diff__insert2,axiom,
! [A4: set_Product_unit,A: product_unit,B4: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ A4 @ ( insert_Product_unit @ A @ B4 ) )
= ( minus_6452836326544984404t_unit @ ( minus_6452836326544984404t_unit @ A4 @ ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_572_Diff__insert2,axiom,
! [A4: set_Product_prod_v_v,A: product_prod_v_v,B4: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A @ B4 ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_573_Diff__insert2,axiom,
! [A4: set_o,A: $o,B4: set_o] :
( ( minus_minus_set_o @ A4 @ ( insert_o @ A @ B4 ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_574_Diff__insert2,axiom,
! [A4: set_set_v,A: set_v,B4: set_set_v] :
( ( minus_7228012346218142266_set_v @ A4 @ ( insert_set_v @ A @ B4 ) )
= ( minus_7228012346218142266_set_v @ ( minus_7228012346218142266_set_v @ A4 @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_575_Diff__insert2,axiom,
! [A4: set_v,A: v,B4: set_v] :
( ( minus_minus_set_v @ A4 @ ( insert_v @ A @ B4 ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A4 @ ( insert_v @ A @ bot_bot_set_v ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_576_insert__Diff,axiom,
! [A: product_unit,A4: set_Product_unit] :
( ( member_Product_unit @ A @ A4 )
=> ( ( insert_Product_unit @ A @ ( minus_6452836326544984404t_unit @ A4 @ ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_577_insert__Diff,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A4 )
=> ( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_578_insert__Diff,axiom,
! [A: $o,A4: set_o] :
( ( member_o @ A @ A4 )
=> ( ( insert_o @ A @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_579_insert__Diff,axiom,
! [A: set_v,A4: set_set_v] :
( ( member_set_v @ A @ A4 )
=> ( ( insert_set_v @ A @ ( minus_7228012346218142266_set_v @ A4 @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_580_insert__Diff,axiom,
! [A: v,A4: set_v] :
( ( member_v2 @ A @ A4 )
=> ( ( insert_v @ A @ ( minus_minus_set_v @ A4 @ ( insert_v @ A @ bot_bot_set_v ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_581_Diff__insert,axiom,
! [A4: set_Product_unit,A: product_unit,B4: set_Product_unit] :
( ( minus_6452836326544984404t_unit @ A4 @ ( insert_Product_unit @ A @ B4 ) )
= ( minus_6452836326544984404t_unit @ ( minus_6452836326544984404t_unit @ A4 @ B4 ) @ ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) ) ) ).
% Diff_insert
thf(fact_582_Diff__insert,axiom,
! [A4: set_Product_prod_v_v,A: product_prod_v_v,B4: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A @ B4 ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A4 @ B4 ) @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ).
% Diff_insert
thf(fact_583_Diff__insert,axiom,
! [A4: set_o,A: $o,B4: set_o] :
( ( minus_minus_set_o @ A4 @ ( insert_o @ A @ B4 ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A4 @ B4 ) @ ( insert_o @ A @ bot_bot_set_o ) ) ) ).
% Diff_insert
thf(fact_584_Diff__insert,axiom,
! [A4: set_set_v,A: set_v,B4: set_set_v] :
( ( minus_7228012346218142266_set_v @ A4 @ ( insert_set_v @ A @ B4 ) )
= ( minus_7228012346218142266_set_v @ ( minus_7228012346218142266_set_v @ A4 @ B4 ) @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) ) ).
% Diff_insert
thf(fact_585_Diff__insert,axiom,
! [A4: set_v,A: v,B4: set_v] :
( ( minus_minus_set_v @ A4 @ ( insert_v @ A @ B4 ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A4 @ B4 ) @ ( insert_v @ A @ bot_bot_set_v ) ) ) ).
% Diff_insert
thf(fact_586_transpose_Ocases,axiom,
! [X: list_list_v] :
( ( X != nil_list_v )
=> ( ! [Xss: list_list_v] :
( X
!= ( cons_list_v @ nil_v @ Xss ) )
=> ~ ! [X3: v,Xs3: list_v,Xss: list_list_v] :
( X
!= ( cons_list_v @ ( cons_v @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_587_not__Cons__self2,axiom,
! [X: v,Xs: list_v] :
( ( cons_v @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_588_UnE,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A4 @ B4 ) )
=> ( ~ ( member7453568604450474000od_v_v @ C @ A4 )
=> ( member7453568604450474000od_v_v @ C @ B4 ) ) ) ).
% UnE
thf(fact_589_UnE,axiom,
! [C: v,A4: set_v,B4: set_v] :
( ( member_v2 @ C @ ( sup_sup_set_v @ A4 @ B4 ) )
=> ( ~ ( member_v2 @ C @ A4 )
=> ( member_v2 @ C @ B4 ) ) ) ).
% UnE
thf(fact_590_UnE,axiom,
! [C: set_v,A4: set_set_v,B4: set_set_v] :
( ( member_set_v @ C @ ( sup_sup_set_set_v @ A4 @ B4 ) )
=> ( ~ ( member_set_v @ C @ A4 )
=> ( member_set_v @ C @ B4 ) ) ) ).
% UnE
thf(fact_591_UnI1,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A4 )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A4 @ B4 ) ) ) ).
% UnI1
thf(fact_592_UnI1,axiom,
! [C: v,A4: set_v,B4: set_v] :
( ( member_v2 @ C @ A4 )
=> ( member_v2 @ C @ ( sup_sup_set_v @ A4 @ B4 ) ) ) ).
% UnI1
thf(fact_593_UnI1,axiom,
! [C: set_v,A4: set_set_v,B4: set_set_v] :
( ( member_set_v @ C @ A4 )
=> ( member_set_v @ C @ ( sup_sup_set_set_v @ A4 @ B4 ) ) ) ).
% UnI1
thf(fact_594_UnI2,axiom,
! [C: product_prod_v_v,B4: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ B4 )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A4 @ B4 ) ) ) ).
% UnI2
thf(fact_595_UnI2,axiom,
! [C: v,B4: set_v,A4: set_v] :
( ( member_v2 @ C @ B4 )
=> ( member_v2 @ C @ ( sup_sup_set_v @ A4 @ B4 ) ) ) ).
% UnI2
thf(fact_596_UnI2,axiom,
! [C: set_v,B4: set_set_v,A4: set_set_v] :
( ( member_set_v @ C @ B4 )
=> ( member_set_v @ C @ ( sup_sup_set_set_v @ A4 @ B4 ) ) ) ).
% UnI2
thf(fact_597_bex__Un,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v,P2: product_prod_v_v > $o] :
( ( ? [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ ( sup_su414716646722978715od_v_v @ A4 @ B4 ) )
& ( P2 @ X4 ) ) )
= ( ? [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A4 )
& ( P2 @ X4 ) )
| ? [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ B4 )
& ( P2 @ X4 ) ) ) ) ).
% bex_Un
thf(fact_598_bex__Un,axiom,
! [A4: set_v,B4: set_v,P2: v > $o] :
( ( ? [X4: v] :
( ( member_v2 @ X4 @ ( sup_sup_set_v @ A4 @ B4 ) )
& ( P2 @ X4 ) ) )
= ( ? [X4: v] :
( ( member_v2 @ X4 @ A4 )
& ( P2 @ X4 ) )
| ? [X4: v] :
( ( member_v2 @ X4 @ B4 )
& ( P2 @ X4 ) ) ) ) ).
% bex_Un
thf(fact_599_bex__Un,axiom,
! [A4: set_set_v,B4: set_set_v,P2: set_v > $o] :
( ( ? [X4: set_v] :
( ( member_set_v @ X4 @ ( sup_sup_set_set_v @ A4 @ B4 ) )
& ( P2 @ X4 ) ) )
= ( ? [X4: set_v] :
( ( member_set_v @ X4 @ A4 )
& ( P2 @ X4 ) )
| ? [X4: set_v] :
( ( member_set_v @ X4 @ B4 )
& ( P2 @ X4 ) ) ) ) ).
% bex_Un
thf(fact_600_emptyE,axiom,
! [A: product_unit] :
~ ( member_Product_unit @ A @ bot_bo3957492148770167129t_unit ) ).
% emptyE
thf(fact_601_emptyE,axiom,
! [A: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ A @ bot_bo723834152578015283od_v_v ) ).
% emptyE
thf(fact_602_emptyE,axiom,
! [A: $o] :
~ ( member_o @ A @ bot_bot_set_o ) ).
% emptyE
thf(fact_603_emptyE,axiom,
! [A: set_v] :
~ ( member_set_v @ A @ bot_bot_set_set_v ) ).
% emptyE
thf(fact_604_emptyE,axiom,
! [A: v] :
~ ( member_v2 @ A @ bot_bot_set_v ) ).
% emptyE
thf(fact_605_ball__Un,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v,P2: product_prod_v_v > $o] :
( ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ ( sup_su414716646722978715od_v_v @ A4 @ B4 ) )
=> ( P2 @ X4 ) ) )
= ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A4 )
=> ( P2 @ X4 ) )
& ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ B4 )
=> ( P2 @ X4 ) ) ) ) ).
% ball_Un
thf(fact_606_ball__Un,axiom,
! [A4: set_v,B4: set_v,P2: v > $o] :
( ( ! [X4: v] :
( ( member_v2 @ X4 @ ( sup_sup_set_v @ A4 @ B4 ) )
=> ( P2 @ X4 ) ) )
= ( ! [X4: v] :
( ( member_v2 @ X4 @ A4 )
=> ( P2 @ X4 ) )
& ! [X4: v] :
( ( member_v2 @ X4 @ B4 )
=> ( P2 @ X4 ) ) ) ) ).
% ball_Un
thf(fact_607_ball__Un,axiom,
! [A4: set_set_v,B4: set_set_v,P2: set_v > $o] :
( ( ! [X4: set_v] :
( ( member_set_v @ X4 @ ( sup_sup_set_set_v @ A4 @ B4 ) )
=> ( P2 @ X4 ) ) )
= ( ! [X4: set_v] :
( ( member_set_v @ X4 @ A4 )
=> ( P2 @ X4 ) )
& ! [X4: set_v] :
( ( member_set_v @ X4 @ B4 )
=> ( P2 @ X4 ) ) ) ) ).
% ball_Un
thf(fact_608_insertE,axiom,
! [A: set_v,B: set_v,A4: set_set_v] :
( ( member_set_v @ A @ ( insert_set_v @ B @ A4 ) )
=> ( ( A != B )
=> ( member_set_v @ A @ A4 ) ) ) ).
% insertE
thf(fact_609_insertE,axiom,
! [A: product_unit,B: product_unit,A4: set_Product_unit] :
( ( member_Product_unit @ A @ ( insert_Product_unit @ B @ A4 ) )
=> ( ( A != B )
=> ( member_Product_unit @ A @ A4 ) ) ) ).
% insertE
thf(fact_610_insertE,axiom,
! [A: $o,B: $o,A4: set_o] :
( ( member_o @ A @ ( insert_o @ B @ A4 ) )
=> ( ( A = (~ B) )
=> ( member_o @ A @ A4 ) ) ) ).
% insertE
thf(fact_611_insertE,axiom,
! [A: v,B: v,A4: set_v] :
( ( member_v2 @ A @ ( insert_v @ B @ A4 ) )
=> ( ( A != B )
=> ( member_v2 @ A @ A4 ) ) ) ).
% insertE
thf(fact_612_insertE,axiom,
! [A: product_prod_v_v,B: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ A4 ) )
=> ( ( A != B )
=> ( member7453568604450474000od_v_v @ A @ A4 ) ) ) ).
% insertE
thf(fact_613_Un__assoc,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B4 ) @ C2 )
= ( sup_su414716646722978715od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ B4 @ C2 ) ) ) ).
% Un_assoc
thf(fact_614_Un__assoc,axiom,
! [A4: set_v,B4: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A4 @ B4 ) @ C2 )
= ( sup_sup_set_v @ A4 @ ( sup_sup_set_v @ B4 @ C2 ) ) ) ).
% Un_assoc
thf(fact_615_Un__assoc,axiom,
! [A4: set_set_v,B4: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ A4 @ B4 ) @ C2 )
= ( sup_sup_set_set_v @ A4 @ ( sup_sup_set_set_v @ B4 @ C2 ) ) ) ).
% Un_assoc
thf(fact_616_equals0D,axiom,
! [A4: set_Product_unit,A: product_unit] :
( ( A4 = bot_bo3957492148770167129t_unit )
=> ~ ( member_Product_unit @ A @ A4 ) ) ).
% equals0D
thf(fact_617_equals0D,axiom,
! [A4: set_Product_prod_v_v,A: product_prod_v_v] :
( ( A4 = bot_bo723834152578015283od_v_v )
=> ~ ( member7453568604450474000od_v_v @ A @ A4 ) ) ).
% equals0D
thf(fact_618_equals0D,axiom,
! [A4: set_o,A: $o] :
( ( A4 = bot_bot_set_o )
=> ~ ( member_o @ A @ A4 ) ) ).
% equals0D
thf(fact_619_equals0D,axiom,
! [A4: set_set_v,A: set_v] :
( ( A4 = bot_bot_set_set_v )
=> ~ ( member_set_v @ A @ A4 ) ) ).
% equals0D
thf(fact_620_equals0D,axiom,
! [A4: set_v,A: v] :
( ( A4 = bot_bot_set_v )
=> ~ ( member_v2 @ A @ A4 ) ) ).
% equals0D
thf(fact_621_equals0I,axiom,
! [A4: set_Product_unit] :
( ! [Y3: product_unit] :
~ ( member_Product_unit @ Y3 @ A4 )
=> ( A4 = bot_bo3957492148770167129t_unit ) ) ).
% equals0I
thf(fact_622_equals0I,axiom,
! [A4: set_Product_prod_v_v] :
( ! [Y3: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ Y3 @ A4 )
=> ( A4 = bot_bo723834152578015283od_v_v ) ) ).
% equals0I
thf(fact_623_equals0I,axiom,
! [A4: set_o] :
( ! [Y3: $o] :
~ ( member_o @ Y3 @ A4 )
=> ( A4 = bot_bot_set_o ) ) ).
% equals0I
thf(fact_624_equals0I,axiom,
! [A4: set_set_v] :
( ! [Y3: set_v] :
~ ( member_set_v @ Y3 @ A4 )
=> ( A4 = bot_bot_set_set_v ) ) ).
% equals0I
thf(fact_625_equals0I,axiom,
! [A4: set_v] :
( ! [Y3: v] :
~ ( member_v2 @ Y3 @ A4 )
=> ( A4 = bot_bot_set_v ) ) ).
% equals0I
thf(fact_626_insertI1,axiom,
! [A: set_v,B4: set_set_v] : ( member_set_v @ A @ ( insert_set_v @ A @ B4 ) ) ).
% insertI1
thf(fact_627_insertI1,axiom,
! [A: product_unit,B4: set_Product_unit] : ( member_Product_unit @ A @ ( insert_Product_unit @ A @ B4 ) ) ).
% insertI1
thf(fact_628_insertI1,axiom,
! [A: $o,B4: set_o] : ( member_o @ A @ ( insert_o @ A @ B4 ) ) ).
% insertI1
thf(fact_629_insertI1,axiom,
! [A: v,B4: set_v] : ( member_v2 @ A @ ( insert_v @ A @ B4 ) ) ).
% insertI1
thf(fact_630_insertI1,axiom,
! [A: product_prod_v_v,B4: set_Product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ B4 ) ) ).
% insertI1
thf(fact_631_insertI2,axiom,
! [A: set_v,B4: set_set_v,B: set_v] :
( ( member_set_v @ A @ B4 )
=> ( member_set_v @ A @ ( insert_set_v @ B @ B4 ) ) ) ).
% insertI2
thf(fact_632_insertI2,axiom,
! [A: product_unit,B4: set_Product_unit,B: product_unit] :
( ( member_Product_unit @ A @ B4 )
=> ( member_Product_unit @ A @ ( insert_Product_unit @ B @ B4 ) ) ) ).
% insertI2
thf(fact_633_insertI2,axiom,
! [A: $o,B4: set_o,B: $o] :
( ( member_o @ A @ B4 )
=> ( member_o @ A @ ( insert_o @ B @ B4 ) ) ) ).
% insertI2
thf(fact_634_insertI2,axiom,
! [A: v,B4: set_v,B: v] :
( ( member_v2 @ A @ B4 )
=> ( member_v2 @ A @ ( insert_v @ B @ B4 ) ) ) ).
% insertI2
thf(fact_635_insertI2,axiom,
! [A: product_prod_v_v,B4: set_Product_prod_v_v,B: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ B4 )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ B4 ) ) ) ).
% insertI2
thf(fact_636_Un__absorb,axiom,
! [A4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_637_Un__absorb,axiom,
! [A4: set_v] :
( ( sup_sup_set_v @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_638_Un__absorb,axiom,
! [A4: set_set_v] :
( ( sup_sup_set_set_v @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_639_Un__commute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A5: set_Product_prod_v_v,B6: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B6 @ A5 ) ) ) ).
% Un_commute
thf(fact_640_Un__commute,axiom,
( sup_sup_set_v
= ( ^ [A5: set_v,B6: set_v] : ( sup_sup_set_v @ B6 @ A5 ) ) ) ).
% Un_commute
thf(fact_641_Un__commute,axiom,
( sup_sup_set_set_v
= ( ^ [A5: set_set_v,B6: set_set_v] : ( sup_sup_set_set_v @ B6 @ A5 ) ) ) ).
% Un_commute
thf(fact_642_ex__in__conv,axiom,
! [A4: set_Product_unit] :
( ( ? [X4: product_unit] : ( member_Product_unit @ X4 @ A4 ) )
= ( A4 != bot_bo3957492148770167129t_unit ) ) ).
% ex_in_conv
thf(fact_643_ex__in__conv,axiom,
! [A4: set_Product_prod_v_v] :
( ( ? [X4: product_prod_v_v] : ( member7453568604450474000od_v_v @ X4 @ A4 ) )
= ( A4 != bot_bo723834152578015283od_v_v ) ) ).
% ex_in_conv
thf(fact_644_ex__in__conv,axiom,
! [A4: set_o] :
( ( ? [X4: $o] : ( member_o @ X4 @ A4 ) )
= ( A4 != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_645_ex__in__conv,axiom,
! [A4: set_set_v] :
( ( ? [X4: set_v] : ( member_set_v @ X4 @ A4 ) )
= ( A4 != bot_bot_set_set_v ) ) ).
% ex_in_conv
thf(fact_646_ex__in__conv,axiom,
! [A4: set_v] :
( ( ? [X4: v] : ( member_v2 @ X4 @ A4 ) )
= ( A4 != bot_bot_set_v ) ) ).
% ex_in_conv
thf(fact_647_Set_Oset__insert,axiom,
! [X: set_v,A4: set_set_v] :
( ( member_set_v @ X @ A4 )
=> ~ ! [B7: set_set_v] :
( ( A4
= ( insert_set_v @ X @ B7 ) )
=> ( member_set_v @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_648_Set_Oset__insert,axiom,
! [X: product_unit,A4: set_Product_unit] :
( ( member_Product_unit @ X @ A4 )
=> ~ ! [B7: set_Product_unit] :
( ( A4
= ( insert_Product_unit @ X @ B7 ) )
=> ( member_Product_unit @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_649_Set_Oset__insert,axiom,
! [X: $o,A4: set_o] :
( ( member_o @ X @ A4 )
=> ~ ! [B7: set_o] :
( ( A4
= ( insert_o @ X @ B7 ) )
=> ( member_o @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_650_Set_Oset__insert,axiom,
! [X: v,A4: set_v] :
( ( member_v2 @ X @ A4 )
=> ~ ! [B7: set_v] :
( ( A4
= ( insert_v @ X @ B7 ) )
=> ( member_v2 @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_651_Set_Oset__insert,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A4 )
=> ~ ! [B7: set_Product_prod_v_v] :
( ( A4
= ( insert1338601472111419319od_v_v @ X @ B7 ) )
=> ( member7453568604450474000od_v_v @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_652_singletonD,axiom,
! [B: product_unit,A: product_unit] :
( ( member_Product_unit @ B @ ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_653_singletonD,axiom,
! [B: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_654_singletonD,axiom,
! [B: $o,A: $o] :
( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_655_singletonD,axiom,
! [B: set_v,A: set_v] :
( ( member_set_v @ B @ ( insert_set_v @ A @ bot_bot_set_set_v ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_656_singletonD,axiom,
! [B: v,A: v] :
( ( member_v2 @ B @ ( insert_v @ A @ bot_bot_set_v ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_657_insert__ident,axiom,
! [X: set_v,A4: set_set_v,B4: set_set_v] :
( ~ ( member_set_v @ X @ A4 )
=> ( ~ ( member_set_v @ X @ B4 )
=> ( ( ( insert_set_v @ X @ A4 )
= ( insert_set_v @ X @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% insert_ident
thf(fact_658_insert__ident,axiom,
! [X: product_unit,A4: set_Product_unit,B4: set_Product_unit] :
( ~ ( member_Product_unit @ X @ A4 )
=> ( ~ ( member_Product_unit @ X @ B4 )
=> ( ( ( insert_Product_unit @ X @ A4 )
= ( insert_Product_unit @ X @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% insert_ident
thf(fact_659_insert__ident,axiom,
! [X: $o,A4: set_o,B4: set_o] :
( ~ ( member_o @ X @ A4 )
=> ( ~ ( member_o @ X @ B4 )
=> ( ( ( insert_o @ X @ A4 )
= ( insert_o @ X @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% insert_ident
thf(fact_660_insert__ident,axiom,
! [X: v,A4: set_v,B4: set_v] :
( ~ ( member_v2 @ X @ A4 )
=> ( ~ ( member_v2 @ X @ B4 )
=> ( ( ( insert_v @ X @ A4 )
= ( insert_v @ X @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% insert_ident
thf(fact_661_insert__ident,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A4 )
=> ( ~ ( member7453568604450474000od_v_v @ X @ B4 )
=> ( ( ( insert1338601472111419319od_v_v @ X @ A4 )
= ( insert1338601472111419319od_v_v @ X @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% insert_ident
thf(fact_662_insert__is__Un,axiom,
( insert_Product_unit
= ( ^ [A6: product_unit] : ( sup_su793286257634532545t_unit @ ( insert_Product_unit @ A6 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% insert_is_Un
thf(fact_663_insert__is__Un,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A6: product_prod_v_v] : ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A6 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% insert_is_Un
thf(fact_664_insert__is__Un,axiom,
( insert_o
= ( ^ [A6: $o] : ( sup_sup_set_o @ ( insert_o @ A6 @ bot_bot_set_o ) ) ) ) ).
% insert_is_Un
thf(fact_665_insert__is__Un,axiom,
( insert_set_v
= ( ^ [A6: set_v] : ( sup_sup_set_set_v @ ( insert_set_v @ A6 @ bot_bot_set_set_v ) ) ) ) ).
% insert_is_Un
thf(fact_666_insert__is__Un,axiom,
( insert_v
= ( ^ [A6: v] : ( sup_sup_set_v @ ( insert_v @ A6 @ bot_bot_set_v ) ) ) ) ).
% insert_is_Un
thf(fact_667_Un__empty__left,axiom,
! [B4: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ bot_bo3957492148770167129t_unit @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_668_Un__empty__left,axiom,
! [B4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_669_Un__empty__left,axiom,
! [B4: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_670_Un__empty__left,axiom,
! [B4: set_set_v] :
( ( sup_sup_set_set_v @ bot_bot_set_set_v @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_671_Un__empty__left,axiom,
! [B4: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_672_insert__absorb,axiom,
! [A: set_v,A4: set_set_v] :
( ( member_set_v @ A @ A4 )
=> ( ( insert_set_v @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_673_insert__absorb,axiom,
! [A: product_unit,A4: set_Product_unit] :
( ( member_Product_unit @ A @ A4 )
=> ( ( insert_Product_unit @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_674_insert__absorb,axiom,
! [A: $o,A4: set_o] :
( ( member_o @ A @ A4 )
=> ( ( insert_o @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_675_insert__absorb,axiom,
! [A: v,A4: set_v] :
( ( member_v2 @ A @ A4 )
=> ( ( insert_v @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_676_insert__absorb,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A4 )
=> ( ( insert1338601472111419319od_v_v @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_677_insert__eq__iff,axiom,
! [A: set_v,A4: set_set_v,B: set_v,B4: set_set_v] :
( ~ ( member_set_v @ A @ A4 )
=> ( ~ ( member_set_v @ B @ B4 )
=> ( ( ( insert_set_v @ A @ A4 )
= ( insert_set_v @ B @ B4 ) )
= ( ( ( A = B )
=> ( A4 = B4 ) )
& ( ( A != B )
=> ? [C3: set_set_v] :
( ( A4
= ( insert_set_v @ B @ C3 ) )
& ~ ( member_set_v @ B @ C3 )
& ( B4
= ( insert_set_v @ A @ C3 ) )
& ~ ( member_set_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_678_insert__eq__iff,axiom,
! [A: product_unit,A4: set_Product_unit,B: product_unit,B4: set_Product_unit] :
( ~ ( member_Product_unit @ A @ A4 )
=> ( ~ ( member_Product_unit @ B @ B4 )
=> ( ( ( insert_Product_unit @ A @ A4 )
= ( insert_Product_unit @ B @ B4 ) )
= ( ( ( A = B )
=> ( A4 = B4 ) )
& ( ( A != B )
=> ? [C3: set_Product_unit] :
( ( A4
= ( insert_Product_unit @ B @ C3 ) )
& ~ ( member_Product_unit @ B @ C3 )
& ( B4
= ( insert_Product_unit @ A @ C3 ) )
& ~ ( member_Product_unit @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_679_insert__eq__iff,axiom,
! [A: $o,A4: set_o,B: $o,B4: set_o] :
( ~ ( member_o @ A @ A4 )
=> ( ~ ( member_o @ B @ B4 )
=> ( ( ( insert_o @ A @ A4 )
= ( insert_o @ B @ B4 ) )
= ( ( ( A = B )
=> ( A4 = B4 ) )
& ( ( A = (~ B) )
=> ? [C3: set_o] :
( ( A4
= ( insert_o @ B @ C3 ) )
& ~ ( member_o @ B @ C3 )
& ( B4
= ( insert_o @ A @ C3 ) )
& ~ ( member_o @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_680_insert__eq__iff,axiom,
! [A: v,A4: set_v,B: v,B4: set_v] :
( ~ ( member_v2 @ A @ A4 )
=> ( ~ ( member_v2 @ B @ B4 )
=> ( ( ( insert_v @ A @ A4 )
= ( insert_v @ B @ B4 ) )
= ( ( ( A = B )
=> ( A4 = B4 ) )
& ( ( A != B )
=> ? [C3: set_v] :
( ( A4
= ( insert_v @ B @ C3 ) )
& ~ ( member_v2 @ B @ C3 )
& ( B4
= ( insert_v @ A @ C3 ) )
& ~ ( member_v2 @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_681_insert__eq__iff,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v,B: product_prod_v_v,B4: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A4 )
=> ( ~ ( member7453568604450474000od_v_v @ B @ B4 )
=> ( ( ( insert1338601472111419319od_v_v @ A @ A4 )
= ( insert1338601472111419319od_v_v @ B @ B4 ) )
= ( ( ( A = B )
=> ( A4 = B4 ) )
& ( ( A != B )
=> ? [C3: set_Product_prod_v_v] :
( ( A4
= ( insert1338601472111419319od_v_v @ B @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ B @ C3 )
& ( B4
= ( insert1338601472111419319od_v_v @ A @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_682_singleton__iff,axiom,
! [B: product_unit,A: product_unit] :
( ( member_Product_unit @ B @ ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_683_singleton__iff,axiom,
! [B: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_684_singleton__iff,axiom,
! [B: $o,A: $o] :
( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_685_singleton__iff,axiom,
! [B: set_v,A: set_v] :
( ( member_set_v @ B @ ( insert_set_v @ A @ bot_bot_set_set_v ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_686_singleton__iff,axiom,
! [B: v,A: v] :
( ( member_v2 @ B @ ( insert_v @ A @ bot_bot_set_v ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_687_Un__empty__right,axiom,
! [A4: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ A4 @ bot_bo3957492148770167129t_unit )
= A4 ) ).
% Un_empty_right
thf(fact_688_Un__empty__right,axiom,
! [A4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ bot_bo723834152578015283od_v_v )
= A4 ) ).
% Un_empty_right
thf(fact_689_Un__empty__right,axiom,
! [A4: set_o] :
( ( sup_sup_set_o @ A4 @ bot_bot_set_o )
= A4 ) ).
% Un_empty_right
thf(fact_690_Un__empty__right,axiom,
! [A4: set_set_v] :
( ( sup_sup_set_set_v @ A4 @ bot_bot_set_set_v )
= A4 ) ).
% Un_empty_right
thf(fact_691_Un__empty__right,axiom,
! [A4: set_v] :
( ( sup_sup_set_v @ A4 @ bot_bot_set_v )
= A4 ) ).
% Un_empty_right
thf(fact_692_Un__left__absorb,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ A4 @ B4 ) )
= ( sup_su414716646722978715od_v_v @ A4 @ B4 ) ) ).
% Un_left_absorb
thf(fact_693_Un__left__absorb,axiom,
! [A4: set_v,B4: set_v] :
( ( sup_sup_set_v @ A4 @ ( sup_sup_set_v @ A4 @ B4 ) )
= ( sup_sup_set_v @ A4 @ B4 ) ) ).
% Un_left_absorb
thf(fact_694_Un__left__absorb,axiom,
! [A4: set_set_v,B4: set_set_v] :
( ( sup_sup_set_set_v @ A4 @ ( sup_sup_set_set_v @ A4 @ B4 ) )
= ( sup_sup_set_set_v @ A4 @ B4 ) ) ).
% Un_left_absorb
thf(fact_695_insert__commute,axiom,
! [X: v,Y: v,A4: set_v] :
( ( insert_v @ X @ ( insert_v @ Y @ A4 ) )
= ( insert_v @ Y @ ( insert_v @ X @ A4 ) ) ) ).
% insert_commute
thf(fact_696_insert__commute,axiom,
! [X: product_prod_v_v,Y: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( insert1338601472111419319od_v_v @ Y @ A4 ) )
= ( insert1338601472111419319od_v_v @ Y @ ( insert1338601472111419319od_v_v @ X @ A4 ) ) ) ).
% insert_commute
thf(fact_697_insert__commute,axiom,
! [X: set_v,Y: set_v,A4: set_set_v] :
( ( insert_set_v @ X @ ( insert_set_v @ Y @ A4 ) )
= ( insert_set_v @ Y @ ( insert_set_v @ X @ A4 ) ) ) ).
% insert_commute
thf(fact_698_insert__commute,axiom,
! [X: product_unit,Y: product_unit,A4: set_Product_unit] :
( ( insert_Product_unit @ X @ ( insert_Product_unit @ Y @ A4 ) )
= ( insert_Product_unit @ Y @ ( insert_Product_unit @ X @ A4 ) ) ) ).
% insert_commute
thf(fact_699_insert__commute,axiom,
! [X: $o,Y: $o,A4: set_o] :
( ( insert_o @ X @ ( insert_o @ Y @ A4 ) )
= ( insert_o @ Y @ ( insert_o @ X @ A4 ) ) ) ).
% insert_commute
thf(fact_700_Un__left__commute,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ B4 @ C2 ) )
= ( sup_su414716646722978715od_v_v @ B4 @ ( sup_su414716646722978715od_v_v @ A4 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_701_Un__left__commute,axiom,
! [A4: set_v,B4: set_v,C2: set_v] :
( ( sup_sup_set_v @ A4 @ ( sup_sup_set_v @ B4 @ C2 ) )
= ( sup_sup_set_v @ B4 @ ( sup_sup_set_v @ A4 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_702_Un__left__commute,axiom,
! [A4: set_set_v,B4: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ A4 @ ( sup_sup_set_set_v @ B4 @ C2 ) )
= ( sup_sup_set_set_v @ B4 @ ( sup_sup_set_set_v @ A4 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_703_Un__singleton__iff,axiom,
! [A4: set_Product_unit,B4: set_Product_unit,X: product_unit] :
( ( ( sup_su793286257634532545t_unit @ A4 @ B4 )
= ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= ( ( ( A4 = bot_bo3957492148770167129t_unit )
& ( B4
= ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
| ( ( A4
= ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
& ( B4 = bot_bo3957492148770167129t_unit ) )
| ( ( A4
= ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
& ( B4
= ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_704_Un__singleton__iff,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A4 @ B4 )
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= ( ( ( A4 = bot_bo723834152578015283od_v_v )
& ( B4
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A4
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B4 = bot_bo723834152578015283od_v_v ) )
| ( ( A4
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B4
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_705_Un__singleton__iff,axiom,
! [A4: set_o,B4: set_o,X: $o] :
( ( ( sup_sup_set_o @ A4 @ B4 )
= ( insert_o @ X @ bot_bot_set_o ) )
= ( ( ( A4 = bot_bot_set_o )
& ( B4
= ( insert_o @ X @ bot_bot_set_o ) ) )
| ( ( A4
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B4 = bot_bot_set_o ) )
| ( ( A4
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B4
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_706_Un__singleton__iff,axiom,
! [A4: set_set_v,B4: set_set_v,X: set_v] :
( ( ( sup_sup_set_set_v @ A4 @ B4 )
= ( insert_set_v @ X @ bot_bot_set_set_v ) )
= ( ( ( A4 = bot_bot_set_set_v )
& ( B4
= ( insert_set_v @ X @ bot_bot_set_set_v ) ) )
| ( ( A4
= ( insert_set_v @ X @ bot_bot_set_set_v ) )
& ( B4 = bot_bot_set_set_v ) )
| ( ( A4
= ( insert_set_v @ X @ bot_bot_set_set_v ) )
& ( B4
= ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_707_Un__singleton__iff,axiom,
! [A4: set_v,B4: set_v,X: v] :
( ( ( sup_sup_set_v @ A4 @ B4 )
= ( insert_v @ X @ bot_bot_set_v ) )
= ( ( ( A4 = bot_bot_set_v )
& ( B4
= ( insert_v @ X @ bot_bot_set_v ) ) )
| ( ( A4
= ( insert_v @ X @ bot_bot_set_v ) )
& ( B4 = bot_bot_set_v ) )
| ( ( A4
= ( insert_v @ X @ bot_bot_set_v ) )
& ( B4
= ( insert_v @ X @ bot_bot_set_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_708_doubleton__eq__iff,axiom,
! [A: product_unit,B: product_unit,C: product_unit,D2: product_unit] :
( ( ( insert_Product_unit @ A @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) )
= ( insert_Product_unit @ C @ ( insert_Product_unit @ D2 @ bot_bo3957492148770167129t_unit ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_709_doubleton__eq__iff,axiom,
! [A: product_prod_v_v,B: product_prod_v_v,C: product_prod_v_v,D2: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) )
= ( insert1338601472111419319od_v_v @ C @ ( insert1338601472111419319od_v_v @ D2 @ bot_bo723834152578015283od_v_v ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_710_doubleton__eq__iff,axiom,
! [A: $o,B: $o,C: $o,D2: $o] :
( ( ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) )
= ( insert_o @ C @ ( insert_o @ D2 @ bot_bot_set_o ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_711_doubleton__eq__iff,axiom,
! [A: set_v,B: set_v,C: set_v,D2: set_v] :
( ( ( insert_set_v @ A @ ( insert_set_v @ B @ bot_bot_set_set_v ) )
= ( insert_set_v @ C @ ( insert_set_v @ D2 @ bot_bot_set_set_v ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_712_doubleton__eq__iff,axiom,
! [A: v,B: v,C: v,D2: v] :
( ( ( insert_v @ A @ ( insert_v @ B @ bot_bot_set_v ) )
= ( insert_v @ C @ ( insert_v @ D2 @ bot_bot_set_v ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_713_insert__not__empty,axiom,
! [A: product_unit,A4: set_Product_unit] :
( ( insert_Product_unit @ A @ A4 )
!= bot_bo3957492148770167129t_unit ) ).
% insert_not_empty
thf(fact_714_insert__not__empty,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ A4 )
!= bot_bo723834152578015283od_v_v ) ).
% insert_not_empty
thf(fact_715_insert__not__empty,axiom,
! [A: $o,A4: set_o] :
( ( insert_o @ A @ A4 )
!= bot_bot_set_o ) ).
% insert_not_empty
thf(fact_716_insert__not__empty,axiom,
! [A: set_v,A4: set_set_v] :
( ( insert_set_v @ A @ A4 )
!= bot_bot_set_set_v ) ).
% insert_not_empty
thf(fact_717_insert__not__empty,axiom,
! [A: v,A4: set_v] :
( ( insert_v @ A @ A4 )
!= bot_bot_set_v ) ).
% insert_not_empty
thf(fact_718_singleton__Un__iff,axiom,
! [X: product_unit,A4: set_Product_unit,B4: set_Product_unit] :
( ( ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit )
= ( sup_su793286257634532545t_unit @ A4 @ B4 ) )
= ( ( ( A4 = bot_bo3957492148770167129t_unit )
& ( B4
= ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
| ( ( A4
= ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
& ( B4 = bot_bo3957492148770167129t_unit ) )
| ( ( A4
= ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
& ( B4
= ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_719_singleton__Un__iff,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v )
= ( sup_su414716646722978715od_v_v @ A4 @ B4 ) )
= ( ( ( A4 = bot_bo723834152578015283od_v_v )
& ( B4
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A4
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B4 = bot_bo723834152578015283od_v_v ) )
| ( ( A4
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B4
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_720_singleton__Un__iff,axiom,
! [X: $o,A4: set_o,B4: set_o] :
( ( ( insert_o @ X @ bot_bot_set_o )
= ( sup_sup_set_o @ A4 @ B4 ) )
= ( ( ( A4 = bot_bot_set_o )
& ( B4
= ( insert_o @ X @ bot_bot_set_o ) ) )
| ( ( A4
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B4 = bot_bot_set_o ) )
| ( ( A4
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B4
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_721_singleton__Un__iff,axiom,
! [X: set_v,A4: set_set_v,B4: set_set_v] :
( ( ( insert_set_v @ X @ bot_bot_set_set_v )
= ( sup_sup_set_set_v @ A4 @ B4 ) )
= ( ( ( A4 = bot_bot_set_set_v )
& ( B4
= ( insert_set_v @ X @ bot_bot_set_set_v ) ) )
| ( ( A4
= ( insert_set_v @ X @ bot_bot_set_set_v ) )
& ( B4 = bot_bot_set_set_v ) )
| ( ( A4
= ( insert_set_v @ X @ bot_bot_set_set_v ) )
& ( B4
= ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_722_singleton__Un__iff,axiom,
! [X: v,A4: set_v,B4: set_v] :
( ( ( insert_v @ X @ bot_bot_set_v )
= ( sup_sup_set_v @ A4 @ B4 ) )
= ( ( ( A4 = bot_bot_set_v )
& ( B4
= ( insert_v @ X @ bot_bot_set_v ) ) )
| ( ( A4
= ( insert_v @ X @ bot_bot_set_v ) )
& ( B4 = bot_bot_set_v ) )
| ( ( A4
= ( insert_v @ X @ bot_bot_set_v ) )
& ( B4
= ( insert_v @ X @ bot_bot_set_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_723_singleton__inject,axiom,
! [A: product_unit,B: product_unit] :
( ( ( insert_Product_unit @ A @ bot_bo3957492148770167129t_unit )
= ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_724_singleton__inject,axiom,
! [A: product_prod_v_v,B: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_725_singleton__inject,axiom,
! [A: $o,B: $o] :
( ( ( insert_o @ A @ bot_bot_set_o )
= ( insert_o @ B @ bot_bot_set_o ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_726_singleton__inject,axiom,
! [A: set_v,B: set_v] :
( ( ( insert_set_v @ A @ bot_bot_set_set_v )
= ( insert_set_v @ B @ bot_bot_set_set_v ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_727_singleton__inject,axiom,
! [A: v,B: v] :
( ( ( insert_v @ A @ bot_bot_set_v )
= ( insert_v @ B @ bot_bot_set_v ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_728_mk__disjoint__insert,axiom,
! [A: set_v,A4: set_set_v] :
( ( member_set_v @ A @ A4 )
=> ? [B7: set_set_v] :
( ( A4
= ( insert_set_v @ A @ B7 ) )
& ~ ( member_set_v @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_729_mk__disjoint__insert,axiom,
! [A: product_unit,A4: set_Product_unit] :
( ( member_Product_unit @ A @ A4 )
=> ? [B7: set_Product_unit] :
( ( A4
= ( insert_Product_unit @ A @ B7 ) )
& ~ ( member_Product_unit @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_730_mk__disjoint__insert,axiom,
! [A: $o,A4: set_o] :
( ( member_o @ A @ A4 )
=> ? [B7: set_o] :
( ( A4
= ( insert_o @ A @ B7 ) )
& ~ ( member_o @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_731_mk__disjoint__insert,axiom,
! [A: v,A4: set_v] :
( ( member_v2 @ A @ A4 )
=> ? [B7: set_v] :
( ( A4
= ( insert_v @ A @ B7 ) )
& ~ ( member_v2 @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_732_mk__disjoint__insert,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A4 )
=> ? [B7: set_Product_prod_v_v] :
( ( A4
= ( insert1338601472111419319od_v_v @ A @ B7 ) )
& ~ ( member7453568604450474000od_v_v @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_733_subset__insert__iff,axiom,
! [A4: set_Product_unit,X: product_unit,B4: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A4 @ ( insert_Product_unit @ X @ B4 ) )
= ( ( ( member_Product_unit @ X @ A4 )
=> ( ord_le3507040750410214029t_unit @ ( minus_6452836326544984404t_unit @ A4 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ B4 ) )
& ( ~ ( member_Product_unit @ X @ A4 )
=> ( ord_le3507040750410214029t_unit @ A4 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_734_subset__insert__iff,axiom,
! [A4: set_o,X: $o,B4: set_o] :
( ( ord_less_eq_set_o @ A4 @ ( insert_o @ X @ B4 ) )
= ( ( ( member_o @ X @ A4 )
=> ( ord_less_eq_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B4 ) )
& ( ~ ( member_o @ X @ A4 )
=> ( ord_less_eq_set_o @ A4 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_735_subset__insert__iff,axiom,
! [A4: set_set_v,X: set_v,B4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A4 @ ( insert_set_v @ X @ B4 ) )
= ( ( ( member_set_v @ X @ A4 )
=> ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A4 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) @ B4 ) )
& ( ~ ( member_set_v @ X @ A4 )
=> ( ord_le5216385588623774835_set_v @ A4 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_736_subset__insert__iff,axiom,
! [A4: set_v,X: v,B4: set_v] :
( ( ord_less_eq_set_v @ A4 @ ( insert_v @ X @ B4 ) )
= ( ( ( member_v2 @ X @ A4 )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A4 @ ( insert_v @ X @ bot_bot_set_v ) ) @ B4 ) )
& ( ~ ( member_v2 @ X @ A4 )
=> ( ord_less_eq_set_v @ A4 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_737_subset__insert__iff,axiom,
! [A4: set_Product_prod_v_v,X: product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ X @ B4 ) )
= ( ( ( member7453568604450474000od_v_v @ X @ A4 )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B4 ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ A4 )
=> ( ord_le7336532860387713383od_v_v @ A4 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_738_Diff__single__insert,axiom,
! [A4: set_Product_unit,X: product_unit,B4: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ ( minus_6452836326544984404t_unit @ A4 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) @ B4 )
=> ( ord_le3507040750410214029t_unit @ A4 @ ( insert_Product_unit @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_739_Diff__single__insert,axiom,
! [A4: set_o,X: $o,B4: set_o] :
( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B4 )
=> ( ord_less_eq_set_o @ A4 @ ( insert_o @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_740_Diff__single__insert,axiom,
! [A4: set_set_v,X: set_v,B4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A4 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) @ B4 )
=> ( ord_le5216385588623774835_set_v @ A4 @ ( insert_set_v @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_741_Diff__single__insert,axiom,
! [A4: set_v,X: v,B4: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A4 @ ( insert_v @ X @ bot_bot_set_v ) ) @ B4 )
=> ( ord_less_eq_set_v @ A4 @ ( insert_v @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_742_Diff__single__insert,axiom,
! [A4: set_Product_prod_v_v,X: product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B4 )
=> ( ord_le7336532860387713383od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_743_bot_Oextremum,axiom,
! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ bot_bo3957492148770167129t_unit @ A ) ).
% bot.extremum
thf(fact_744_bot_Oextremum,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% bot.extremum
thf(fact_745_bot_Oextremum,axiom,
! [A: set_set_v] : ( ord_le5216385588623774835_set_v @ bot_bot_set_set_v @ A ) ).
% bot.extremum
thf(fact_746_bot_Oextremum,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A ) ).
% bot.extremum
thf(fact_747_bot_Oextremum,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A ) ).
% bot.extremum
thf(fact_748_bot_Oextremum__unique,axiom,
! [A: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ bot_bo3957492148770167129t_unit )
= ( A = bot_bo3957492148770167129t_unit ) ) ).
% bot.extremum_unique
thf(fact_749_bot_Oextremum__unique,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% bot.extremum_unique
thf(fact_750_bot_Oextremum__unique,axiom,
! [A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ bot_bot_set_set_v )
= ( A = bot_bot_set_set_v ) ) ).
% bot.extremum_unique
thf(fact_751_bot_Oextremum__unique,axiom,
! [A: set_v] :
( ( ord_less_eq_set_v @ A @ bot_bot_set_v )
= ( A = bot_bot_set_v ) ) ).
% bot.extremum_unique
thf(fact_752_bot_Oextremum__unique,axiom,
! [A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ bot_bo723834152578015283od_v_v )
= ( A = bot_bo723834152578015283od_v_v ) ) ).
% bot.extremum_unique
thf(fact_753_bot_Oextremum__uniqueI,axiom,
! [A: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ bot_bo3957492148770167129t_unit )
=> ( A = bot_bo3957492148770167129t_unit ) ) ).
% bot.extremum_uniqueI
thf(fact_754_bot_Oextremum__uniqueI,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
=> ( A = bot_bot_set_o ) ) ).
% bot.extremum_uniqueI
thf(fact_755_bot_Oextremum__uniqueI,axiom,
! [A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ bot_bot_set_set_v )
=> ( A = bot_bot_set_set_v ) ) ).
% bot.extremum_uniqueI
thf(fact_756_bot_Oextremum__uniqueI,axiom,
! [A: set_v] :
( ( ord_less_eq_set_v @ A @ bot_bot_set_v )
=> ( A = bot_bot_set_v ) ) ).
% bot.extremum_uniqueI
thf(fact_757_bot_Oextremum__uniqueI,axiom,
! [A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ bot_bo723834152578015283od_v_v )
=> ( A = bot_bo723834152578015283od_v_v ) ) ).
% bot.extremum_uniqueI
thf(fact_758_Un__UNIV__right,axiom,
! [A4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ top_to5429829297380968215od_v_v )
= top_to5429829297380968215od_v_v ) ).
% Un_UNIV_right
thf(fact_759_Un__UNIV__right,axiom,
! [A4: set_v] :
( ( sup_sup_set_v @ A4 @ top_top_set_v )
= top_top_set_v ) ).
% Un_UNIV_right
thf(fact_760_Un__UNIV__right,axiom,
! [A4: set_set_v] :
( ( sup_sup_set_set_v @ A4 @ top_top_set_set_v )
= top_top_set_set_v ) ).
% Un_UNIV_right
thf(fact_761_Un__UNIV__right,axiom,
! [A4: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ A4 @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% Un_UNIV_right
thf(fact_762_Un__UNIV__right,axiom,
! [A4: set_o] :
( ( sup_sup_set_o @ A4 @ top_top_set_o )
= top_top_set_o ) ).
% Un_UNIV_right
thf(fact_763_Un__UNIV__right,axiom,
! [A4: set_nat] :
( ( sup_sup_set_nat @ A4 @ top_top_set_nat )
= top_top_set_nat ) ).
% Un_UNIV_right
thf(fact_764_Un__UNIV__left,axiom,
! [B4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ top_to5429829297380968215od_v_v @ B4 )
= top_to5429829297380968215od_v_v ) ).
% Un_UNIV_left
thf(fact_765_Un__UNIV__left,axiom,
! [B4: set_v] :
( ( sup_sup_set_v @ top_top_set_v @ B4 )
= top_top_set_v ) ).
% Un_UNIV_left
thf(fact_766_Un__UNIV__left,axiom,
! [B4: set_set_v] :
( ( sup_sup_set_set_v @ top_top_set_set_v @ B4 )
= top_top_set_set_v ) ).
% Un_UNIV_left
thf(fact_767_Un__UNIV__left,axiom,
! [B4: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ top_to1996260823553986621t_unit @ B4 )
= top_to1996260823553986621t_unit ) ).
% Un_UNIV_left
thf(fact_768_Un__UNIV__left,axiom,
! [B4: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ B4 )
= top_top_set_o ) ).
% Un_UNIV_left
thf(fact_769_Un__UNIV__left,axiom,
! [B4: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ B4 )
= top_top_set_nat ) ).
% Un_UNIV_left
thf(fact_770_insert__UNIV,axiom,
! [X: v] :
( ( insert_v @ X @ top_top_set_v )
= top_top_set_v ) ).
% insert_UNIV
thf(fact_771_insert__UNIV,axiom,
! [X: product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ top_to5429829297380968215od_v_v )
= top_to5429829297380968215od_v_v ) ).
% insert_UNIV
thf(fact_772_insert__UNIV,axiom,
! [X: set_v] :
( ( insert_set_v @ X @ top_top_set_set_v )
= top_top_set_set_v ) ).
% insert_UNIV
thf(fact_773_insert__UNIV,axiom,
! [X: product_unit] :
( ( insert_Product_unit @ X @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% insert_UNIV
thf(fact_774_insert__UNIV,axiom,
! [X: $o] :
( ( insert_o @ X @ top_top_set_o )
= top_top_set_o ) ).
% insert_UNIV
thf(fact_775_insert__UNIV,axiom,
! [X: nat] :
( ( insert_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% insert_UNIV
thf(fact_776_Un__mono,axiom,
! [A4: set_set_v,C2: set_set_v,B4: set_set_v,D: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A4 @ C2 )
=> ( ( ord_le5216385588623774835_set_v @ B4 @ D )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A4 @ B4 ) @ ( sup_sup_set_set_v @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_777_Un__mono,axiom,
! [A4: set_v,C2: set_v,B4: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A4 @ C2 )
=> ( ( ord_less_eq_set_v @ B4 @ D )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A4 @ B4 ) @ ( sup_sup_set_v @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_778_Un__mono,axiom,
! [A4: set_Product_prod_v_v,C2: set_Product_prod_v_v,B4: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B4 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B4 ) @ ( sup_su414716646722978715od_v_v @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_779_Un__least,axiom,
! [A4: set_set_v,C2: set_set_v,B4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A4 @ C2 )
=> ( ( ord_le5216385588623774835_set_v @ B4 @ C2 )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A4 @ B4 ) @ C2 ) ) ) ).
% Un_least
thf(fact_780_Un__least,axiom,
! [A4: set_v,C2: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ A4 @ C2 )
=> ( ( ord_less_eq_set_v @ B4 @ C2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A4 @ B4 ) @ C2 ) ) ) ).
% Un_least
thf(fact_781_Un__least,axiom,
! [A4: set_Product_prod_v_v,C2: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B4 @ C2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B4 ) @ C2 ) ) ) ).
% Un_least
thf(fact_782_Un__upper1,axiom,
! [A4: set_set_v,B4: set_set_v] : ( ord_le5216385588623774835_set_v @ A4 @ ( sup_sup_set_set_v @ A4 @ B4 ) ) ).
% Un_upper1
thf(fact_783_Un__upper1,axiom,
! [A4: set_v,B4: set_v] : ( ord_less_eq_set_v @ A4 @ ( sup_sup_set_v @ A4 @ B4 ) ) ).
% Un_upper1
thf(fact_784_Un__upper1,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ A4 @ B4 ) ) ).
% Un_upper1
thf(fact_785_Un__upper2,axiom,
! [B4: set_set_v,A4: set_set_v] : ( ord_le5216385588623774835_set_v @ B4 @ ( sup_sup_set_set_v @ A4 @ B4 ) ) ).
% Un_upper2
thf(fact_786_Un__upper2,axiom,
! [B4: set_v,A4: set_v] : ( ord_less_eq_set_v @ B4 @ ( sup_sup_set_v @ A4 @ B4 ) ) ).
% Un_upper2
thf(fact_787_Un__upper2,axiom,
! [B4: set_Product_prod_v_v,A4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B4 @ ( sup_su414716646722978715od_v_v @ A4 @ B4 ) ) ).
% Un_upper2
thf(fact_788_Un__absorb1,axiom,
! [A4: set_set_v,B4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A4 @ B4 )
=> ( ( sup_sup_set_set_v @ A4 @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_789_Un__absorb1,axiom,
! [A4: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ A4 @ B4 )
=> ( ( sup_sup_set_v @ A4 @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_790_Un__absorb1,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B4 )
=> ( ( sup_su414716646722978715od_v_v @ A4 @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_791_Un__absorb2,axiom,
! [B4: set_set_v,A4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B4 @ A4 )
=> ( ( sup_sup_set_set_v @ A4 @ B4 )
= A4 ) ) ).
% Un_absorb2
thf(fact_792_Un__absorb2,axiom,
! [B4: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ B4 @ A4 )
=> ( ( sup_sup_set_v @ A4 @ B4 )
= A4 ) ) ).
% Un_absorb2
thf(fact_793_Un__absorb2,axiom,
! [B4: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B4 @ A4 )
=> ( ( sup_su414716646722978715od_v_v @ A4 @ B4 )
= A4 ) ) ).
% Un_absorb2
thf(fact_794_subset__UnE,axiom,
! [C2: set_set_v,A4: set_set_v,B4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C2 @ ( sup_sup_set_set_v @ A4 @ B4 ) )
=> ~ ! [A7: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A7 @ A4 )
=> ! [B8: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B8 @ B4 )
=> ( C2
!= ( sup_sup_set_set_v @ A7 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_795_subset__UnE,axiom,
! [C2: set_v,A4: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ C2 @ ( sup_sup_set_v @ A4 @ B4 ) )
=> ~ ! [A7: set_v] :
( ( ord_less_eq_set_v @ A7 @ A4 )
=> ! [B8: set_v] :
( ( ord_less_eq_set_v @ B8 @ B4 )
=> ( C2
!= ( sup_sup_set_v @ A7 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_796_subset__UnE,axiom,
! [C2: set_Product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A4 @ B4 ) )
=> ~ ! [A7: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A7 @ A4 )
=> ! [B8: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B8 @ B4 )
=> ( C2
!= ( sup_su414716646722978715od_v_v @ A7 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_797_subset__Un__eq,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [A5: set_set_v,B6: set_set_v] :
( ( sup_sup_set_set_v @ A5 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_798_subset__Un__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B6: set_v] :
( ( sup_sup_set_v @ A5 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_799_subset__Un__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A5 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_800_insert__subsetI,axiom,
! [X: set_v,A4: set_set_v,X6: set_set_v] :
( ( member_set_v @ X @ A4 )
=> ( ( ord_le5216385588623774835_set_v @ X6 @ A4 )
=> ( ord_le5216385588623774835_set_v @ ( insert_set_v @ X @ X6 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_801_insert__subsetI,axiom,
! [X: product_unit,A4: set_Product_unit,X6: set_Product_unit] :
( ( member_Product_unit @ X @ A4 )
=> ( ( ord_le3507040750410214029t_unit @ X6 @ A4 )
=> ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ X @ X6 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_802_insert__subsetI,axiom,
! [X: $o,A4: set_o,X6: set_o] :
( ( member_o @ X @ A4 )
=> ( ( ord_less_eq_set_o @ X6 @ A4 )
=> ( ord_less_eq_set_o @ ( insert_o @ X @ X6 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_803_insert__subsetI,axiom,
! [X: v,A4: set_v,X6: set_v] :
( ( member_v2 @ X @ A4 )
=> ( ( ord_less_eq_set_v @ X6 @ A4 )
=> ( ord_less_eq_set_v @ ( insert_v @ X @ X6 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_804_insert__subsetI,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v,X6: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A4 )
=> ( ( ord_le7336532860387713383od_v_v @ X6 @ A4 )
=> ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X @ X6 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_805_insert__mono,axiom,
! [C2: set_set_v,D: set_set_v,A: set_v] :
( ( ord_le5216385588623774835_set_v @ C2 @ D )
=> ( ord_le5216385588623774835_set_v @ ( insert_set_v @ A @ C2 ) @ ( insert_set_v @ A @ D ) ) ) ).
% insert_mono
thf(fact_806_insert__mono,axiom,
! [C2: set_Product_unit,D: set_Product_unit,A: product_unit] :
( ( ord_le3507040750410214029t_unit @ C2 @ D )
=> ( ord_le3507040750410214029t_unit @ ( insert_Product_unit @ A @ C2 ) @ ( insert_Product_unit @ A @ D ) ) ) ).
% insert_mono
thf(fact_807_insert__mono,axiom,
! [C2: set_o,D: set_o,A: $o] :
( ( ord_less_eq_set_o @ C2 @ D )
=> ( ord_less_eq_set_o @ ( insert_o @ A @ C2 ) @ ( insert_o @ A @ D ) ) ) ).
% insert_mono
thf(fact_808_insert__mono,axiom,
! [C2: set_v,D: set_v,A: v] :
( ( ord_less_eq_set_v @ C2 @ D )
=> ( ord_less_eq_set_v @ ( insert_v @ A @ C2 ) @ ( insert_v @ A @ D ) ) ) ).
% insert_mono
thf(fact_809_insert__mono,axiom,
! [C2: set_Product_prod_v_v,D: set_Product_prod_v_v,A: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ A @ C2 ) @ ( insert1338601472111419319od_v_v @ A @ D ) ) ) ).
% insert_mono
thf(fact_810_subset__insert,axiom,
! [X: set_v,A4: set_set_v,B4: set_set_v] :
( ~ ( member_set_v @ X @ A4 )
=> ( ( ord_le5216385588623774835_set_v @ A4 @ ( insert_set_v @ X @ B4 ) )
= ( ord_le5216385588623774835_set_v @ A4 @ B4 ) ) ) ).
% subset_insert
thf(fact_811_subset__insert,axiom,
! [X: product_unit,A4: set_Product_unit,B4: set_Product_unit] :
( ~ ( member_Product_unit @ X @ A4 )
=> ( ( ord_le3507040750410214029t_unit @ A4 @ ( insert_Product_unit @ X @ B4 ) )
= ( ord_le3507040750410214029t_unit @ A4 @ B4 ) ) ) ).
% subset_insert
thf(fact_812_subset__insert,axiom,
! [X: $o,A4: set_o,B4: set_o] :
( ~ ( member_o @ X @ A4 )
=> ( ( ord_less_eq_set_o @ A4 @ ( insert_o @ X @ B4 ) )
= ( ord_less_eq_set_o @ A4 @ B4 ) ) ) ).
% subset_insert
thf(fact_813_subset__insert,axiom,
! [X: v,A4: set_v,B4: set_v] :
( ~ ( member_v2 @ X @ A4 )
=> ( ( ord_less_eq_set_v @ A4 @ ( insert_v @ X @ B4 ) )
= ( ord_less_eq_set_v @ A4 @ B4 ) ) ) ).
% subset_insert
thf(fact_814_subset__insert,axiom,
! [X: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A4 )
=> ( ( ord_le7336532860387713383od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ X @ B4 ) )
= ( ord_le7336532860387713383od_v_v @ A4 @ B4 ) ) ) ).
% subset_insert
thf(fact_815_subset__insertI,axiom,
! [B4: set_set_v,A: set_v] : ( ord_le5216385588623774835_set_v @ B4 @ ( insert_set_v @ A @ B4 ) ) ).
% subset_insertI
thf(fact_816_subset__insertI,axiom,
! [B4: set_Product_unit,A: product_unit] : ( ord_le3507040750410214029t_unit @ B4 @ ( insert_Product_unit @ A @ B4 ) ) ).
% subset_insertI
thf(fact_817_subset__insertI,axiom,
! [B4: set_o,A: $o] : ( ord_less_eq_set_o @ B4 @ ( insert_o @ A @ B4 ) ) ).
% subset_insertI
thf(fact_818_subset__insertI,axiom,
! [B4: set_v,A: v] : ( ord_less_eq_set_v @ B4 @ ( insert_v @ A @ B4 ) ) ).
% subset_insertI
thf(fact_819_subset__insertI,axiom,
! [B4: set_Product_prod_v_v,A: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B4 @ ( insert1338601472111419319od_v_v @ A @ B4 ) ) ).
% subset_insertI
thf(fact_820_subset__insertI2,axiom,
! [A4: set_set_v,B4: set_set_v,B: set_v] :
( ( ord_le5216385588623774835_set_v @ A4 @ B4 )
=> ( ord_le5216385588623774835_set_v @ A4 @ ( insert_set_v @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_821_subset__insertI2,axiom,
! [A4: set_Product_unit,B4: set_Product_unit,B: product_unit] :
( ( ord_le3507040750410214029t_unit @ A4 @ B4 )
=> ( ord_le3507040750410214029t_unit @ A4 @ ( insert_Product_unit @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_822_subset__insertI2,axiom,
! [A4: set_o,B4: set_o,B: $o] :
( ( ord_less_eq_set_o @ A4 @ B4 )
=> ( ord_less_eq_set_o @ A4 @ ( insert_o @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_823_subset__insertI2,axiom,
! [A4: set_v,B4: set_v,B: v] :
( ( ord_less_eq_set_v @ A4 @ B4 )
=> ( ord_less_eq_set_v @ A4 @ ( insert_v @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_824_subset__insertI2,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v,B: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B4 )
=> ( ord_le7336532860387713383od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_825_splice_Ocases,axiom,
! [X: produc1391462591744249447list_v] :
( ! [Ys3: list_v] :
( X
!= ( produc6795410681906604247list_v @ nil_v @ Ys3 ) )
=> ~ ! [X3: v,Xs3: list_v,Ys3: list_v] :
( X
!= ( produc6795410681906604247list_v @ ( cons_v @ X3 @ Xs3 ) @ Ys3 ) ) ) ).
% splice.cases
thf(fact_826_shuffles_Ocases,axiom,
! [X: produc1391462591744249447list_v] :
( ! [Ys3: list_v] :
( X
!= ( produc6795410681906604247list_v @ nil_v @ Ys3 ) )
=> ( ! [Xs3: list_v] :
( X
!= ( produc6795410681906604247list_v @ Xs3 @ nil_v ) )
=> ~ ! [X3: v,Xs3: list_v,Y3: v,Ys3: list_v] :
( X
!= ( produc6795410681906604247list_v @ ( cons_v @ X3 @ Xs3 ) @ ( cons_v @ Y3 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_827_sorted__wrt_Ocases,axiom,
! [X: produc8237170675765753490list_v] :
( ! [P4: v > v > $o] :
( X
!= ( produc601102195597853570list_v @ P4 @ nil_v ) )
=> ~ ! [P4: v > v > $o,X3: v,Ys3: list_v] :
( X
!= ( produc601102195597853570list_v @ P4 @ ( cons_v @ X3 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_828_successively_Ocases,axiom,
! [X: produc8237170675765753490list_v] :
( ! [P4: v > v > $o] :
( X
!= ( produc601102195597853570list_v @ P4 @ nil_v ) )
=> ( ! [P4: v > v > $o,X3: v] :
( X
!= ( produc601102195597853570list_v @ P4 @ ( cons_v @ X3 @ nil_v ) ) )
=> ~ ! [P4: v > v > $o,X3: v,Y3: v,Xs3: list_v] :
( X
!= ( produc601102195597853570list_v @ P4 @ ( cons_v @ X3 @ ( cons_v @ Y3 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_829_list__nonempty__induct,axiom,
! [Xs: list_v,P2: list_v > $o] :
( ( Xs != nil_v )
=> ( ! [X3: v] : ( P2 @ ( cons_v @ X3 @ nil_v ) )
=> ( ! [X3: v,Xs3: list_v] :
( ( Xs3 != nil_v )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( cons_v @ X3 @ Xs3 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_830_list__induct2_H,axiom,
! [P2: list_v > list_v > $o,Xs: list_v,Ys: list_v] :
( ( P2 @ nil_v @ nil_v )
=> ( ! [X3: v,Xs3: list_v] : ( P2 @ ( cons_v @ X3 @ Xs3 ) @ nil_v )
=> ( ! [Y3: v,Ys3: list_v] : ( P2 @ nil_v @ ( cons_v @ Y3 @ Ys3 ) )
=> ( ! [X3: v,Xs3: list_v,Y3: v,Ys3: list_v] :
( ( P2 @ Xs3 @ Ys3 )
=> ( P2 @ ( cons_v @ X3 @ Xs3 ) @ ( cons_v @ Y3 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_831_neq__Nil__conv,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
= ( ? [Y4: v,Ys4: list_v] :
( Xs
= ( cons_v @ Y4 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_832_remdups__adj_Ocases,axiom,
! [X: list_v] :
( ( X != nil_v )
=> ( ! [X3: v] :
( X
!= ( cons_v @ X3 @ nil_v ) )
=> ~ ! [X3: v,Y3: v,Xs3: list_v] :
( X
!= ( cons_v @ X3 @ ( cons_v @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_833_list_Oexhaust,axiom,
! [Y: list_v] :
( ( Y != nil_v )
=> ~ ! [X212: v,X222: list_v] :
( Y
!= ( cons_v @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_834_list_OdiscI,axiom,
! [List: list_v,X21: v,X22: list_v] :
( ( List
= ( cons_v @ X21 @ X22 ) )
=> ( List != nil_v ) ) ).
% list.discI
thf(fact_835_list_Odistinct_I1_J,axiom,
! [X21: v,X22: list_v] :
( nil_v
!= ( cons_v @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_836_list_Oset__intros_I2_J,axiom,
! [Y: product_prod_v_v,X22: list_P7986770385144383213od_v_v,X21: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ X22 ) )
=> ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_837_list_Oset__intros_I2_J,axiom,
! [Y: v,X22: list_v,X21: v] :
( ( member_v2 @ Y @ ( set_v2 @ X22 ) )
=> ( member_v2 @ Y @ ( set_v2 @ ( cons_v @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_838_list_Oset__intros_I1_J,axiom,
! [X21: product_prod_v_v,X22: list_P7986770385144383213od_v_v] : ( member7453568604450474000od_v_v @ X21 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_839_list_Oset__intros_I1_J,axiom,
! [X21: v,X22: list_v] : ( member_v2 @ X21 @ ( set_v2 @ ( cons_v @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_840_list_Oset__cases,axiom,
! [E: product_prod_v_v,A: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ E @ ( set_Product_prod_v_v2 @ A ) )
=> ( ! [Z22: list_P7986770385144383213od_v_v] :
( A
!= ( cons_P4120604216776828829od_v_v @ E @ Z22 ) )
=> ~ ! [Z1: product_prod_v_v,Z22: list_P7986770385144383213od_v_v] :
( ( A
= ( cons_P4120604216776828829od_v_v @ Z1 @ Z22 ) )
=> ~ ( member7453568604450474000od_v_v @ E @ ( set_Product_prod_v_v2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_841_list_Oset__cases,axiom,
! [E: v,A: list_v] :
( ( member_v2 @ E @ ( set_v2 @ A ) )
=> ( ! [Z22: list_v] :
( A
!= ( cons_v @ E @ Z22 ) )
=> ~ ! [Z1: v,Z22: list_v] :
( ( A
= ( cons_v @ Z1 @ Z22 ) )
=> ~ ( member_v2 @ E @ ( set_v2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_842_set__ConsD,axiom,
! [Y: product_prod_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_843_set__ConsD,axiom,
! [Y: v,X: v,Xs: list_v] :
( ( member_v2 @ Y @ ( set_v2 @ ( cons_v @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_v2 @ Y @ ( set_v2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_844_Un__Diff,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B4 ) @ C2 )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A4 @ C2 ) @ ( minus_4183494784930505774od_v_v @ B4 @ C2 ) ) ) ).
% Un_Diff
thf(fact_845_Un__Diff,axiom,
! [A4: set_set_v,B4: set_set_v,C2: set_set_v] :
( ( minus_7228012346218142266_set_v @ ( sup_sup_set_set_v @ A4 @ B4 ) @ C2 )
= ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ A4 @ C2 ) @ ( minus_7228012346218142266_set_v @ B4 @ C2 ) ) ) ).
% Un_Diff
thf(fact_846_Un__Diff,axiom,
! [A4: set_v,B4: set_v,C2: set_v] :
( ( minus_minus_set_v @ ( sup_sup_set_v @ A4 @ B4 ) @ C2 )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A4 @ C2 ) @ ( minus_minus_set_v @ B4 @ C2 ) ) ) ).
% Un_Diff
thf(fact_847_insert__Diff__if,axiom,
! [X: set_v,B4: set_set_v,A4: set_set_v] :
( ( ( member_set_v @ X @ B4 )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X @ A4 ) @ B4 )
= ( minus_7228012346218142266_set_v @ A4 @ B4 ) ) )
& ( ~ ( member_set_v @ X @ B4 )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X @ A4 ) @ B4 )
= ( insert_set_v @ X @ ( minus_7228012346218142266_set_v @ A4 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_848_insert__Diff__if,axiom,
! [X: product_unit,B4: set_Product_unit,A4: set_Product_unit] :
( ( ( member_Product_unit @ X @ B4 )
=> ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit @ X @ A4 ) @ B4 )
= ( minus_6452836326544984404t_unit @ A4 @ B4 ) ) )
& ( ~ ( member_Product_unit @ X @ B4 )
=> ( ( minus_6452836326544984404t_unit @ ( insert_Product_unit @ X @ A4 ) @ B4 )
= ( insert_Product_unit @ X @ ( minus_6452836326544984404t_unit @ A4 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_849_insert__Diff__if,axiom,
! [X: $o,B4: set_o,A4: set_o] :
( ( ( member_o @ X @ B4 )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A4 ) @ B4 )
= ( minus_minus_set_o @ A4 @ B4 ) ) )
& ( ~ ( member_o @ X @ B4 )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A4 ) @ B4 )
= ( insert_o @ X @ ( minus_minus_set_o @ A4 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_850_insert__Diff__if,axiom,
! [X: product_prod_v_v,B4: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ X @ B4 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A4 ) @ B4 )
= ( minus_4183494784930505774od_v_v @ A4 @ B4 ) ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ B4 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A4 ) @ B4 )
= ( insert1338601472111419319od_v_v @ X @ ( minus_4183494784930505774od_v_v @ A4 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_851_insert__Diff__if,axiom,
! [X: v,B4: set_v,A4: set_v] :
( ( ( member_v2 @ X @ B4 )
=> ( ( minus_minus_set_v @ ( insert_v @ X @ A4 ) @ B4 )
= ( minus_minus_set_v @ A4 @ B4 ) ) )
& ( ~ ( member_v2 @ X @ B4 )
=> ( ( minus_minus_set_v @ ( insert_v @ X @ A4 ) @ B4 )
= ( insert_v @ X @ ( minus_minus_set_v @ A4 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_852_subrelI,axiom,
! [R: set_Pr6425124735969554649t_unit,S: set_Pr6425124735969554649t_unit] :
( ! [X3: v,Y3: sCC_Bl1394983891496994913t_unit] :
( ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X3 @ Y3 ) @ R )
=> ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X3 @ Y3 ) @ S ) )
=> ( ord_le7290744839000465721t_unit @ R @ S ) ) ).
% subrelI
thf(fact_853_subrelI,axiom,
! [R: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ! [X3: v,Y3: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y3 ) @ R )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X3 @ Y3 ) @ S ) )
=> ( ord_le7336532860387713383od_v_v @ R @ S ) ) ).
% subrelI
thf(fact_854_append__Cons,axiom,
! [X: v,Xs: list_v,Ys: list_v] :
( ( append_v @ ( cons_v @ X @ Xs ) @ Ys )
= ( cons_v @ X @ ( append_v @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_855_Cons__eq__appendI,axiom,
! [X: v,Xs1: list_v,Ys: list_v,Xs: list_v,Zs: list_v] :
( ( ( cons_v @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_v @ Xs1 @ Zs ) )
=> ( ( cons_v @ X @ Xs )
= ( append_v @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_856_empty__not__UNIV,axiom,
bot_bo723834152578015283od_v_v != top_to5429829297380968215od_v_v ).
% empty_not_UNIV
thf(fact_857_empty__not__UNIV,axiom,
bot_bot_set_set_v != top_top_set_set_v ).
% empty_not_UNIV
thf(fact_858_empty__not__UNIV,axiom,
bot_bot_set_v != top_top_set_v ).
% empty_not_UNIV
thf(fact_859_empty__not__UNIV,axiom,
bot_bo3957492148770167129t_unit != top_to1996260823553986621t_unit ).
% empty_not_UNIV
thf(fact_860_empty__not__UNIV,axiom,
bot_bot_set_o != top_top_set_o ).
% empty_not_UNIV
thf(fact_861_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_862_subset__emptyI,axiom,
! [A4: set_Product_unit] :
( ! [X3: product_unit] :
~ ( member_Product_unit @ X3 @ A4 )
=> ( ord_le3507040750410214029t_unit @ A4 @ bot_bo3957492148770167129t_unit ) ) ).
% subset_emptyI
thf(fact_863_subset__emptyI,axiom,
! [A4: set_o] :
( ! [X3: $o] :
~ ( member_o @ X3 @ A4 )
=> ( ord_less_eq_set_o @ A4 @ bot_bot_set_o ) ) ).
% subset_emptyI
thf(fact_864_subset__emptyI,axiom,
! [A4: set_set_v] :
( ! [X3: set_v] :
~ ( member_set_v @ X3 @ A4 )
=> ( ord_le5216385588623774835_set_v @ A4 @ bot_bot_set_set_v ) ) ).
% subset_emptyI
thf(fact_865_subset__emptyI,axiom,
! [A4: set_v] :
( ! [X3: v] :
~ ( member_v2 @ X3 @ A4 )
=> ( ord_less_eq_set_v @ A4 @ bot_bot_set_v ) ) ).
% subset_emptyI
thf(fact_866_subset__emptyI,axiom,
! [A4: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ X3 @ A4 )
=> ( ord_le7336532860387713383od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) ).
% subset_emptyI
thf(fact_867_list_Osel_I1_J,axiom,
! [X21: v,X22: list_v] :
( ( hd_v @ ( cons_v @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_868_list_Osel_I3_J,axiom,
! [X21: v,X22: list_v] :
( ( tl_v @ ( cons_v @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_869_precedes__in__tail,axiom,
! [X: v,Z: v,Y: v,Zs: list_v] :
( ( X != Z )
=> ( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( cons_v @ Z @ Zs ) )
= ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Zs ) ) ) ).
% precedes_in_tail
thf(fact_870_member__rec_I1_J,axiom,
! [X: v,Xs: list_v,Y: v] :
( ( member_v @ ( cons_v @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member_v @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_871_Diff__partition,axiom,
! [A4: set_set_v,B4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A4 @ B4 )
=> ( ( sup_sup_set_set_v @ A4 @ ( minus_7228012346218142266_set_v @ B4 @ A4 ) )
= B4 ) ) ).
% Diff_partition
thf(fact_872_Diff__partition,axiom,
! [A4: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ A4 @ B4 )
=> ( ( sup_sup_set_v @ A4 @ ( minus_minus_set_v @ B4 @ A4 ) )
= B4 ) ) ).
% Diff_partition
thf(fact_873_Diff__partition,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B4 )
=> ( ( sup_su414716646722978715od_v_v @ A4 @ ( minus_4183494784930505774od_v_v @ B4 @ A4 ) )
= B4 ) ) ).
% Diff_partition
thf(fact_874_Diff__subset__conv,axiom,
! [A4: set_set_v,B4: set_set_v,C2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A4 @ B4 ) @ C2 )
= ( ord_le5216385588623774835_set_v @ A4 @ ( sup_sup_set_set_v @ B4 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_875_Diff__subset__conv,axiom,
! [A4: set_v,B4: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A4 @ B4 ) @ C2 )
= ( ord_less_eq_set_v @ A4 @ ( sup_sup_set_v @ B4 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_876_Diff__subset__conv,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A4 @ B4 ) @ C2 )
= ( ord_le7336532860387713383od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ B4 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_877_set__subset__Cons,axiom,
! [Xs: list_v,X: v] : ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ ( cons_v @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_878_set__subset__Cons,axiom,
! [Xs: list_P7986770385144383213od_v_v,X: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_879_subset__Diff__insert,axiom,
! [A4: set_set_v,B4: set_set_v,X: set_v,C2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A4 @ ( minus_7228012346218142266_set_v @ B4 @ ( insert_set_v @ X @ C2 ) ) )
= ( ( ord_le5216385588623774835_set_v @ A4 @ ( minus_7228012346218142266_set_v @ B4 @ C2 ) )
& ~ ( member_set_v @ X @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_880_subset__Diff__insert,axiom,
! [A4: set_Product_unit,B4: set_Product_unit,X: product_unit,C2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A4 @ ( minus_6452836326544984404t_unit @ B4 @ ( insert_Product_unit @ X @ C2 ) ) )
= ( ( ord_le3507040750410214029t_unit @ A4 @ ( minus_6452836326544984404t_unit @ B4 @ C2 ) )
& ~ ( member_Product_unit @ X @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_881_subset__Diff__insert,axiom,
! [A4: set_o,B4: set_o,X: $o,C2: set_o] :
( ( ord_less_eq_set_o @ A4 @ ( minus_minus_set_o @ B4 @ ( insert_o @ X @ C2 ) ) )
= ( ( ord_less_eq_set_o @ A4 @ ( minus_minus_set_o @ B4 @ C2 ) )
& ~ ( member_o @ X @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_882_subset__Diff__insert,axiom,
! [A4: set_v,B4: set_v,X: v,C2: set_v] :
( ( ord_less_eq_set_v @ A4 @ ( minus_minus_set_v @ B4 @ ( insert_v @ X @ C2 ) ) )
= ( ( ord_less_eq_set_v @ A4 @ ( minus_minus_set_v @ B4 @ C2 ) )
& ~ ( member_v2 @ X @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_883_subset__Diff__insert,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v,X: product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ ( minus_4183494784930505774od_v_v @ B4 @ ( insert1338601472111419319od_v_v @ X @ C2 ) ) )
= ( ( ord_le7336532860387713383od_v_v @ A4 @ ( minus_4183494784930505774od_v_v @ B4 @ C2 ) )
& ~ ( member7453568604450474000od_v_v @ X @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_884_rev__nonempty__induct,axiom,
! [Xs: list_v,P2: list_v > $o] :
( ( Xs != nil_v )
=> ( ! [X3: v] : ( P2 @ ( cons_v @ X3 @ nil_v ) )
=> ( ! [X3: v,Xs3: list_v] :
( ( Xs3 != nil_v )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( append_v @ Xs3 @ ( cons_v @ X3 @ nil_v ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_885_append__eq__Cons__conv,axiom,
! [Ys: list_v,Zs: list_v,X: v,Xs: list_v] :
( ( ( append_v @ Ys @ Zs )
= ( cons_v @ X @ Xs ) )
= ( ( ( Ys = nil_v )
& ( Zs
= ( cons_v @ X @ Xs ) ) )
| ? [Ys5: list_v] :
( ( Ys
= ( cons_v @ X @ Ys5 ) )
& ( ( append_v @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_886_Cons__eq__append__conv,axiom,
! [X: v,Xs: list_v,Ys: list_v,Zs: list_v] :
( ( ( cons_v @ X @ Xs )
= ( append_v @ Ys @ Zs ) )
= ( ( ( Ys = nil_v )
& ( ( cons_v @ X @ Xs )
= Zs ) )
| ? [Ys5: list_v] :
( ( ( cons_v @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_v @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_887_rev__exhaust,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ~ ! [Ys3: list_v,Y3: v] :
( Xs
!= ( append_v @ Ys3 @ ( cons_v @ Y3 @ nil_v ) ) ) ) ).
% rev_exhaust
thf(fact_888_rev__induct,axiom,
! [P2: list_v > $o,Xs: list_v] :
( ( P2 @ nil_v )
=> ( ! [X3: v,Xs3: list_v] :
( ( P2 @ Xs3 )
=> ( P2 @ ( append_v @ Xs3 @ ( cons_v @ X3 @ nil_v ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_889_split__list,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ? [Ys3: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( Xs
= ( append2138873909117096322od_v_v @ Ys3 @ ( cons_P4120604216776828829od_v_v @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_890_split__list,axiom,
! [X: v,Xs: list_v] :
( ( member_v2 @ X @ ( set_v2 @ Xs ) )
=> ? [Ys3: list_v,Zs2: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_891_split__list__last,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ? [Ys3: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys3 @ ( cons_P4120604216776828829od_v_v @ X @ Zs2 ) ) )
& ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_892_split__list__last,axiom,
! [X: v,Xs: list_v] :
( ( member_v2 @ X @ ( set_v2 @ Xs ) )
=> ? [Ys3: list_v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys3 @ ( cons_v @ X @ Zs2 ) ) )
& ~ ( member_v2 @ X @ ( set_v2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_893_split__list__prop,axiom,
! [Xs: list_v,P2: v > $o] :
( ? [X5: v] :
( ( member_v2 @ X5 @ ( set_v2 @ Xs ) )
& ( P2 @ X5 ) )
=> ? [Ys3: list_v,X3: v] :
( ? [Zs2: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs2 ) ) )
& ( P2 @ X3 ) ) ) ).
% split_list_prop
thf(fact_894_split__list__first,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ? [Ys3: list_P7986770385144383213od_v_v,Zs2: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys3 @ ( cons_P4120604216776828829od_v_v @ X @ Zs2 ) ) )
& ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_895_split__list__first,axiom,
! [X: v,Xs: list_v] :
( ( member_v2 @ X @ ( set_v2 @ Xs ) )
=> ? [Ys3: list_v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys3 @ ( cons_v @ X @ Zs2 ) ) )
& ~ ( member_v2 @ X @ ( set_v2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_896_split__list__propE,axiom,
! [Xs: list_v,P2: v > $o] :
( ? [X5: v] :
( ( member_v2 @ X5 @ ( set_v2 @ Xs ) )
& ( P2 @ X5 ) )
=> ~ ! [Ys3: list_v,X3: v] :
( ? [Zs2: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs2 ) ) )
=> ~ ( P2 @ X3 ) ) ) ).
% split_list_propE
thf(fact_897_append__Cons__eq__iff,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v,Xs5: list_P7986770385144383213od_v_v,Ys6: list_P7986770385144383213od_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys ) )
=> ( ( ( append2138873909117096322od_v_v @ Xs @ ( cons_P4120604216776828829od_v_v @ X @ Ys ) )
= ( append2138873909117096322od_v_v @ Xs5 @ ( cons_P4120604216776828829od_v_v @ X @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_898_append__Cons__eq__iff,axiom,
! [X: v,Xs: list_v,Ys: list_v,Xs5: list_v,Ys6: list_v] :
( ~ ( member_v2 @ X @ ( set_v2 @ Xs ) )
=> ( ~ ( member_v2 @ X @ ( set_v2 @ Ys ) )
=> ( ( ( append_v @ Xs @ ( cons_v @ X @ Ys ) )
= ( append_v @ Xs5 @ ( cons_v @ X @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_899_in__set__conv__decomp,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
= ( ? [Ys4: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( Xs
= ( append2138873909117096322od_v_v @ Ys4 @ ( cons_P4120604216776828829od_v_v @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_900_in__set__conv__decomp,axiom,
! [X: v,Xs: list_v] :
( ( member_v2 @ X @ ( set_v2 @ Xs ) )
= ( ? [Ys4: list_v,Zs3: list_v] :
( Xs
= ( append_v @ Ys4 @ ( cons_v @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_901_split__list__last__prop,axiom,
! [Xs: list_v,P2: v > $o] :
( ? [X5: v] :
( ( member_v2 @ X5 @ ( set_v2 @ Xs ) )
& ( P2 @ X5 ) )
=> ? [Ys3: list_v,X3: v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs2 ) ) )
& ( P2 @ X3 )
& ! [Xa2: v] :
( ( member_v2 @ Xa2 @ ( set_v2 @ Zs2 ) )
=> ~ ( P2 @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_902_split__list__first__prop,axiom,
! [Xs: list_v,P2: v > $o] :
( ? [X5: v] :
( ( member_v2 @ X5 @ ( set_v2 @ Xs ) )
& ( P2 @ X5 ) )
=> ? [Ys3: list_v,X3: v] :
( ? [Zs2: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs2 ) ) )
& ( P2 @ X3 )
& ! [Xa2: v] :
( ( member_v2 @ Xa2 @ ( set_v2 @ Ys3 ) )
=> ~ ( P2 @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_903_split__list__last__propE,axiom,
! [Xs: list_v,P2: v > $o] :
( ? [X5: v] :
( ( member_v2 @ X5 @ ( set_v2 @ Xs ) )
& ( P2 @ X5 ) )
=> ~ ! [Ys3: list_v,X3: v,Zs2: list_v] :
( ( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs2 ) ) )
=> ( ( P2 @ X3 )
=> ~ ! [Xa2: v] :
( ( member_v2 @ Xa2 @ ( set_v2 @ Zs2 ) )
=> ~ ( P2 @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_904_split__list__first__propE,axiom,
! [Xs: list_v,P2: v > $o] :
( ? [X5: v] :
( ( member_v2 @ X5 @ ( set_v2 @ Xs ) )
& ( P2 @ X5 ) )
=> ~ ! [Ys3: list_v,X3: v] :
( ? [Zs2: list_v] :
( Xs
= ( append_v @ Ys3 @ ( cons_v @ X3 @ Zs2 ) ) )
=> ( ( P2 @ X3 )
=> ~ ! [Xa2: v] :
( ( member_v2 @ Xa2 @ ( set_v2 @ Ys3 ) )
=> ~ ( P2 @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_905_in__set__conv__decomp__last,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
= ( ? [Ys4: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys4 @ ( cons_P4120604216776828829od_v_v @ X @ Zs3 ) ) )
& ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_906_in__set__conv__decomp__last,axiom,
! [X: v,Xs: list_v] :
( ( member_v2 @ X @ ( set_v2 @ Xs ) )
= ( ? [Ys4: list_v,Zs3: list_v] :
( ( Xs
= ( append_v @ Ys4 @ ( cons_v @ X @ Zs3 ) ) )
& ~ ( member_v2 @ X @ ( set_v2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_907_in__set__conv__decomp__first,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
= ( ? [Ys4: list_P7986770385144383213od_v_v,Zs3: list_P7986770385144383213od_v_v] :
( ( Xs
= ( append2138873909117096322od_v_v @ Ys4 @ ( cons_P4120604216776828829od_v_v @ X @ Zs3 ) ) )
& ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_908_in__set__conv__decomp__first,axiom,
! [X: v,Xs: list_v] :
( ( member_v2 @ X @ ( set_v2 @ Xs ) )
= ( ? [Ys4: list_v,Zs3: list_v] :
( ( Xs
= ( append_v @ Ys4 @ ( cons_v @ X @ Zs3 ) ) )
& ~ ( member_v2 @ X @ ( set_v2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_909_split__list__last__prop__iff,axiom,
! [Xs: list_v,P2: v > $o] :
( ( ? [X4: v] :
( ( member_v2 @ X4 @ ( set_v2 @ Xs ) )
& ( P2 @ X4 ) ) )
= ( ? [Ys4: list_v,X4: v,Zs3: list_v] :
( ( Xs
= ( append_v @ Ys4 @ ( cons_v @ X4 @ Zs3 ) ) )
& ( P2 @ X4 )
& ! [Y4: v] :
( ( member_v2 @ Y4 @ ( set_v2 @ Zs3 ) )
=> ~ ( P2 @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_910_split__list__first__prop__iff,axiom,
! [Xs: list_v,P2: v > $o] :
( ( ? [X4: v] :
( ( member_v2 @ X4 @ ( set_v2 @ Xs ) )
& ( P2 @ X4 ) ) )
= ( ? [Ys4: list_v,X4: v] :
( ? [Zs3: list_v] :
( Xs
= ( append_v @ Ys4 @ ( cons_v @ X4 @ Zs3 ) ) )
& ( P2 @ X4 )
& ! [Y4: v] :
( ( member_v2 @ Y4 @ ( set_v2 @ Ys4 ) )
=> ~ ( P2 @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_911_tl__Nil,axiom,
! [Xs: list_v] :
( ( ( tl_v @ Xs )
= nil_v )
= ( ( Xs = nil_v )
| ? [X4: v] :
( Xs
= ( cons_v @ X4 @ nil_v ) ) ) ) ).
% tl_Nil
thf(fact_912_Nil__tl,axiom,
! [Xs: list_v] :
( ( nil_v
= ( tl_v @ Xs ) )
= ( ( Xs = nil_v )
| ? [X4: v] :
( Xs
= ( cons_v @ X4 @ nil_v ) ) ) ) ).
% Nil_tl
thf(fact_913_empty__set,axiom,
( bot_bo3957492148770167129t_unit
= ( set_Product_unit2 @ nil_Product_unit ) ) ).
% empty_set
thf(fact_914_empty__set,axiom,
( bot_bo723834152578015283od_v_v
= ( set_Product_prod_v_v2 @ nil_Product_prod_v_v ) ) ).
% empty_set
thf(fact_915_empty__set,axiom,
( bot_bot_set_o
= ( set_o2 @ nil_o ) ) ).
% empty_set
thf(fact_916_empty__set,axiom,
( bot_bot_set_set_v
= ( set_set_v2 @ nil_set_v ) ) ).
% empty_set
thf(fact_917_empty__set,axiom,
( bot_bot_set_v
= ( set_v2 @ nil_v ) ) ).
% empty_set
thf(fact_918_tail__not__precedes,axiom,
! [Y: product_prod_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ Y @ X @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) )
=> ( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( X = Y ) ) ) ).
% tail_not_precedes
thf(fact_919_tail__not__precedes,axiom,
! [Y: v,X: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ Y @ X @ ( cons_v @ X @ Xs ) )
=> ( ~ ( member_v2 @ X @ ( set_v2 @ Xs ) )
=> ( X = Y ) ) ) ).
% tail_not_precedes
thf(fact_920_head__precedes,axiom,
! [Y: product_prod_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) )
=> ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ).
% head_precedes
thf(fact_921_head__precedes,axiom,
! [Y: v,X: v,Xs: list_v] :
( ( member_v2 @ Y @ ( set_v2 @ ( cons_v @ X @ Xs ) ) )
=> ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( cons_v @ X @ Xs ) ) ) ).
% head_precedes
thf(fact_922_list_Oexhaust__sel,axiom,
! [List: list_v] :
( ( List != nil_v )
=> ( List
= ( cons_v @ ( hd_v @ List ) @ ( tl_v @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_923_precedes__def,axiom,
( sCC_Bl2026170059108282219od_v_v
= ( ^ [X4: product_prod_v_v,Y4: product_prod_v_v,Xs2: list_P7986770385144383213od_v_v] :
? [L: list_P7986770385144383213od_v_v,R4: list_P7986770385144383213od_v_v] :
( ( Xs2
= ( append2138873909117096322od_v_v @ L @ ( cons_P4120604216776828829od_v_v @ X4 @ R4 ) ) )
& ( member7453568604450474000od_v_v @ Y4 @ ( set_Product_prod_v_v2 @ ( cons_P4120604216776828829od_v_v @ X4 @ R4 ) ) ) ) ) ) ).
% precedes_def
thf(fact_924_precedes__def,axiom,
( sCC_Bl4022239298816431255edes_v
= ( ^ [X4: v,Y4: v,Xs2: list_v] :
? [L: list_v,R4: list_v] :
( ( Xs2
= ( append_v @ L @ ( cons_v @ X4 @ R4 ) ) )
& ( member_v2 @ Y4 @ ( set_v2 @ ( cons_v @ X4 @ R4 ) ) ) ) ) ) ).
% precedes_def
thf(fact_925_graph_Osubscc__add,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,S3: set_set_v,X: set_v,Y: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl7907073126578335045_set_v @ Successors @ S3 )
=> ( ( member_set_v @ X @ S3 )
=> ( ( sCC_Bl7354734129683093653_set_v @ Successors @ X @ Y )
=> ( ( sCC_Bl7354734129683093653_set_v @ Successors @ Y @ X )
=> ( sCC_Bl7907073126578335045_set_v @ Successors @ ( insert_set_v @ Y @ S3 ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_926_graph_Osubscc__add,axiom,
! [Vertices: set_Product_unit,Successors: product_unit > set_Product_unit,S3: set_Product_unit,X: product_unit,Y: product_unit] :
( ( sCC_Bl1875605551932356204t_unit @ Vertices @ Successors )
=> ( ( sCC_Bl5104783218331194207t_unit @ Successors @ S3 )
=> ( ( member_Product_unit @ X @ S3 )
=> ( ( sCC_Bl6050654579099102895t_unit @ Successors @ X @ Y )
=> ( ( sCC_Bl6050654579099102895t_unit @ Successors @ Y @ X )
=> ( sCC_Bl5104783218331194207t_unit @ Successors @ ( insert_Product_unit @ Y @ S3 ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_927_graph_Osubscc__add,axiom,
! [Vertices: set_o,Successors: $o > set_o,S3: set_o,X: $o,Y: $o] :
( ( sCC_Bloemen_graph_o @ Vertices @ Successors )
=> ( ( sCC_Bl6048899608199235562bscc_o @ Successors @ S3 )
=> ( ( member_o @ X @ S3 )
=> ( ( sCC_Bl1688025099652275002able_o @ Successors @ X @ Y )
=> ( ( sCC_Bl1688025099652275002able_o @ Successors @ Y @ X )
=> ( sCC_Bl6048899608199235562bscc_o @ Successors @ ( insert_o @ Y @ S3 ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_928_graph_Osubscc__add,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S3: set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl2301996248249672505od_v_v @ Successors @ S3 )
=> ( ( member7453568604450474000od_v_v @ X @ S3 )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ X )
=> ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( insert1338601472111419319od_v_v @ Y @ S3 ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_929_graph_Osubscc__add,axiom,
! [Vertices: set_v,Successors: v > set_v,S3: set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S3 )
=> ( ( member_v2 @ X @ S3 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ X )
=> ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( insert_v @ Y @ S3 ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_930_split__list__precedes,axiom,
! [Y: product_prod_v_v,Ys: list_P7986770385144383213od_v_v,X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( member7453568604450474000od_v_v @ Y @ ( set_Product_prod_v_v2 @ ( append2138873909117096322od_v_v @ Ys @ ( cons_P4120604216776828829od_v_v @ X @ nil_Product_prod_v_v ) ) ) )
=> ( sCC_Bl2026170059108282219od_v_v @ Y @ X @ ( append2138873909117096322od_v_v @ Ys @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) ) ) ) ).
% split_list_precedes
thf(fact_931_split__list__precedes,axiom,
! [Y: v,Ys: list_v,X: v,Xs: list_v] :
( ( member_v2 @ Y @ ( set_v2 @ ( append_v @ Ys @ ( cons_v @ X @ nil_v ) ) ) )
=> ( sCC_Bl4022239298816431255edes_v @ Y @ X @ ( append_v @ Ys @ ( cons_v @ X @ Xs ) ) ) ) ).
% split_list_precedes
thf(fact_932_graph_Ois__scc__def,axiom,
! [Vertices: set_Product_unit,Successors: product_unit > set_Product_unit,S3: set_Product_unit] :
( ( sCC_Bl1875605551932356204t_unit @ Vertices @ Successors )
=> ( ( sCC_Bl4845159262847726003t_unit @ Successors @ S3 )
= ( ( S3 != bot_bo3957492148770167129t_unit )
& ( sCC_Bl5104783218331194207t_unit @ Successors @ S3 )
& ! [S4: set_Product_unit] :
( ( ( ord_le3507040750410214029t_unit @ S3 @ S4 )
& ( sCC_Bl5104783218331194207t_unit @ Successors @ S4 ) )
=> ( S4 = S3 ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_933_graph_Ois__scc__def,axiom,
! [Vertices: set_o,Successors: $o > set_o,S3: set_o] :
( ( sCC_Bloemen_graph_o @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_o @ Successors @ S3 )
= ( ( S3 != bot_bot_set_o )
& ( sCC_Bl6048899608199235562bscc_o @ Successors @ S3 )
& ! [S4: set_o] :
( ( ( ord_less_eq_set_o @ S3 @ S4 )
& ( sCC_Bl6048899608199235562bscc_o @ Successors @ S4 ) )
=> ( S4 = S3 ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_934_graph_Ois__scc__def,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,S3: set_set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl1515522642333523865_set_v @ Successors @ S3 )
= ( ( S3 != bot_bot_set_set_v )
& ( sCC_Bl7907073126578335045_set_v @ Successors @ S3 )
& ! [S4: set_set_v] :
( ( ( ord_le5216385588623774835_set_v @ S3 @ S4 )
& ( sCC_Bl7907073126578335045_set_v @ Successors @ S4 ) )
=> ( S4 = S3 ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_935_graph_Ois__scc__def,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S3: set_Product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S3 )
= ( ( S3 != bot_bo723834152578015283od_v_v )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S3 )
& ! [S4: set_Product_prod_v_v] :
( ( ( ord_le7336532860387713383od_v_v @ S3 @ S4 )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S4 ) )
=> ( S4 = S3 ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_936_graph_Ois__scc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S3: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S3 )
= ( ( S3 != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S3 )
& ! [S4: set_v] :
( ( ( ord_less_eq_set_v @ S3 @ S4 )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S4 ) )
=> ( S4 = S3 ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_937_graph_Opre__dfs__def,axiom,
! [Vertices: set_v,Successors: v > set_v,V2: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl36166008131615352t_unit @ Successors @ V2 @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
& ~ ( member_v2 @ V2 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ ( sCC_Bl1090238580953940555t_unit @ E ) @ V2 )
& ( ( sCC_Bl3795065053823578884t_unit @ E @ V2 )
= bot_bot_set_v )
& ! [X4: v] :
( ( member_v2 @ X4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X4 @ V2 ) ) ) ) ) ).
% graph.pre_dfs_def
thf(fact_938_graph_Opre__post_I2_J,axiom,
! [Vertices: set_v,Successors: v > set_v,V2: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ Successors ) @ ( sum_In5289330923152326972t_unit @ ( produc3862955338007567901t_unit @ V2 @ E ) ) )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V2 @ E )
=> ( sCC_Bl6082031138996704384t_unit @ Successors @ V2 @ E @ ( sCC_Bloemen_dfss_v @ Successors @ V2 @ E ) ) ) ) ) ).
% graph.pre_post(2)
thf(fact_939_infinite__UNIV__char__0,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_char_0
thf(fact_940_ex__new__if__finite,axiom,
! [A4: set_Product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ top_to5429829297380968215od_v_v )
=> ( ( finite3348123685078250256od_v_v @ A4 )
=> ? [A3: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ A3 @ A4 ) ) ) ).
% ex_new_if_finite
thf(fact_941_ex__new__if__finite,axiom,
! [A4: set_v] :
( ~ ( finite_finite_v @ top_top_set_v )
=> ( ( finite_finite_v @ A4 )
=> ? [A3: v] :
~ ( member_v2 @ A3 @ A4 ) ) ) ).
% ex_new_if_finite
thf(fact_942_ex__new__if__finite,axiom,
! [A4: set_Product_unit] :
( ~ ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite4290736615968046902t_unit @ A4 )
=> ? [A3: product_unit] :
~ ( member_Product_unit @ A3 @ A4 ) ) ) ).
% ex_new_if_finite
thf(fact_943_ex__new__if__finite,axiom,
! [A4: set_o] :
( ~ ( finite_finite_o @ top_top_set_o )
=> ( ( finite_finite_o @ A4 )
=> ? [A3: $o] :
~ ( member_o @ A3 @ A4 ) ) ) ).
% ex_new_if_finite
thf(fact_944_ex__new__if__finite,axiom,
! [A4: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A4 )
=> ? [A3: nat] :
~ ( member_nat @ A3 @ A4 ) ) ) ).
% ex_new_if_finite
thf(fact_945_finite__UNIV,axiom,
finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ).
% finite_UNIV
thf(fact_946_finite__UNIV,axiom,
finite_finite_o @ top_top_set_o ).
% finite_UNIV
thf(fact_947_Finite__Set_Ofinite__set,axiom,
( ( finite_finite_set_v @ top_top_set_set_v )
= ( finite_finite_v @ top_top_set_v ) ) ).
% Finite_Set.finite_set
thf(fact_948_Finite__Set_Ofinite__set,axiom,
( ( finite1772178364199683094t_unit @ top_to1767297665138865437t_unit )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% Finite_Set.finite_set
thf(fact_949_Finite__Set_Ofinite__set,axiom,
( ( finite_finite_set_o @ top_top_set_set_o )
= ( finite_finite_o @ top_top_set_o ) ) ).
% Finite_Set.finite_set
thf(fact_950_Finite__Set_Ofinite__set,axiom,
( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% Finite_Set.finite_set
thf(fact_951_graph_Opre__dfss__def,axiom,
! [Vertices: set_v,Successors: v > set_v,V2: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V2 @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
& ( member_v2 @ V2 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
& ( member_v2 @ V2 @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
& ! [X4: v] :
( ( member_v2 @ X4 @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) )
=> ( member_v2 @ X4 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
& ! [X4: v] :
( ( member_v2 @ X4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X4 @ V2 ) )
& ? [Ns: list_v] :
( ( sCC_Bl9201514103433284750t_unit @ E )
= ( cons_v @ V2 @ Ns ) ) ) ) ) ).
% graph.pre_dfss_def
thf(fact_952_boolean__algebra_Odisj__one__right,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ top_to5429829297380968215od_v_v )
= top_to5429829297380968215od_v_v ) ).
% boolean_algebra.disj_one_right
thf(fact_953_boolean__algebra_Odisj__one__right,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ top_top_set_v )
= top_top_set_v ) ).
% boolean_algebra.disj_one_right
thf(fact_954_boolean__algebra_Odisj__one__right,axiom,
! [X: set_set_v] :
( ( sup_sup_set_set_v @ X @ top_top_set_set_v )
= top_top_set_set_v ) ).
% boolean_algebra.disj_one_right
thf(fact_955_boolean__algebra_Odisj__one__right,axiom,
! [X: product_unit] :
( ( sup_sup_Product_unit @ X @ top_top_Product_unit )
= top_top_Product_unit ) ).
% boolean_algebra.disj_one_right
thf(fact_956_boolean__algebra_Odisj__one__right,axiom,
! [X: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ X @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% boolean_algebra.disj_one_right
thf(fact_957_boolean__algebra_Odisj__one__right,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ X @ top_top_set_o )
= top_top_set_o ) ).
% boolean_algebra.disj_one_right
thf(fact_958_boolean__algebra_Odisj__one__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_right
thf(fact_959_boolean__algebra_Odisj__one__left,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ top_to5429829297380968215od_v_v @ X )
= top_to5429829297380968215od_v_v ) ).
% boolean_algebra.disj_one_left
thf(fact_960_boolean__algebra_Odisj__one__left,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ top_top_set_v @ X )
= top_top_set_v ) ).
% boolean_algebra.disj_one_left
thf(fact_961_boolean__algebra_Odisj__one__left,axiom,
! [X: set_set_v] :
( ( sup_sup_set_set_v @ top_top_set_set_v @ X )
= top_top_set_set_v ) ).
% boolean_algebra.disj_one_left
thf(fact_962_boolean__algebra_Odisj__one__left,axiom,
! [X: product_unit] :
( ( sup_sup_Product_unit @ top_top_Product_unit @ X )
= top_top_Product_unit ) ).
% boolean_algebra.disj_one_left
thf(fact_963_boolean__algebra_Odisj__one__left,axiom,
! [X: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ top_to1996260823553986621t_unit @ X )
= top_to1996260823553986621t_unit ) ).
% boolean_algebra.disj_one_left
thf(fact_964_boolean__algebra_Odisj__one__left,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ X )
= top_top_set_o ) ).
% boolean_algebra.disj_one_left
thf(fact_965_boolean__algebra_Odisj__one__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_left
thf(fact_966_sup__top__right,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ top_to5429829297380968215od_v_v )
= top_to5429829297380968215od_v_v ) ).
% sup_top_right
thf(fact_967_sup__top__right,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ top_top_set_v )
= top_top_set_v ) ).
% sup_top_right
thf(fact_968_sup__top__right,axiom,
! [X: set_set_v] :
( ( sup_sup_set_set_v @ X @ top_top_set_set_v )
= top_top_set_set_v ) ).
% sup_top_right
thf(fact_969_sup__top__right,axiom,
! [X: product_unit] :
( ( sup_sup_Product_unit @ X @ top_top_Product_unit )
= top_top_Product_unit ) ).
% sup_top_right
thf(fact_970_sup__top__right,axiom,
! [X: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ X @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% sup_top_right
thf(fact_971_sup__top__right,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ X @ top_top_set_o )
= top_top_set_o ) ).
% sup_top_right
thf(fact_972_sup__top__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% sup_top_right
thf(fact_973_sup__top__left,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ top_to5429829297380968215od_v_v @ X )
= top_to5429829297380968215od_v_v ) ).
% sup_top_left
thf(fact_974_sup__top__left,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ top_top_set_v @ X )
= top_top_set_v ) ).
% sup_top_left
thf(fact_975_sup__top__left,axiom,
! [X: set_set_v] :
( ( sup_sup_set_set_v @ top_top_set_set_v @ X )
= top_top_set_set_v ) ).
% sup_top_left
thf(fact_976_sup__top__left,axiom,
! [X: product_unit] :
( ( sup_sup_Product_unit @ top_top_Product_unit @ X )
= top_top_Product_unit ) ).
% sup_top_left
thf(fact_977_sup__top__left,axiom,
! [X: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ top_to1996260823553986621t_unit @ X )
= top_to1996260823553986621t_unit ) ).
% sup_top_left
thf(fact_978_sup__top__left,axiom,
! [X: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ X )
= top_top_set_o ) ).
% sup_top_left
thf(fact_979_sup__top__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X )
= top_top_set_nat ) ).
% sup_top_left
thf(fact_980_set__union,axiom,
! [Xs: list_P7986770385144383213od_v_v,Ys: list_P7986770385144383213od_v_v] :
( ( set_Product_prod_v_v2 @ ( union_4602324378607836129od_v_v @ Xs @ Ys ) )
= ( sup_su414716646722978715od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ ( set_Product_prod_v_v2 @ Ys ) ) ) ).
% set_union
thf(fact_981_set__union,axiom,
! [Xs: list_v,Ys: list_v] :
( ( set_v2 @ ( union_v @ Xs @ Ys ) )
= ( sup_sup_set_v @ ( set_v2 @ Xs ) @ ( set_v2 @ Ys ) ) ) ).
% set_union
thf(fact_982_set__union,axiom,
! [Xs: list_set_v,Ys: list_set_v] :
( ( set_set_v2 @ ( union_set_v @ Xs @ Ys ) )
= ( sup_sup_set_set_v @ ( set_set_v2 @ Xs ) @ ( set_set_v2 @ Ys ) ) ) ).
% set_union
thf(fact_983_equality,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R3: sCC_Bl1394983891496994913t_unit] :
( ( ( sCC_Bl1090238580953940555t_unit @ R )
= ( sCC_Bl1090238580953940555t_unit @ R3 ) )
=> ( ( ( sCC_Bl1280885523602775798t_unit @ R )
= ( sCC_Bl1280885523602775798t_unit @ R3 ) )
=> ( ( ( sCC_Bl157864678168468314t_unit @ R )
= ( sCC_Bl157864678168468314t_unit @ R3 ) )
=> ( ( ( sCC_Bl4645233313691564917t_unit @ R )
= ( sCC_Bl4645233313691564917t_unit @ R3 ) )
=> ( ( ( sCC_Bl3795065053823578884t_unit @ R )
= ( sCC_Bl3795065053823578884t_unit @ R3 ) )
=> ( ( ( sCC_Bl2536197123907397897t_unit @ R )
= ( sCC_Bl2536197123907397897t_unit @ R3 ) )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ R )
= ( sCC_Bl8828226123343373779t_unit @ R3 ) )
=> ( ( ( sCC_Bl9201514103433284750t_unit @ R )
= ( sCC_Bl9201514103433284750t_unit @ R3 ) )
=> ( ( ( sCC_Bl3567736435408124606t_unit @ R )
= ( sCC_Bl3567736435408124606t_unit @ R3 ) )
=> ( R = R3 ) ) ) ) ) ) ) ) ) ) ).
% equality
thf(fact_984_bot__set__def,axiom,
( bot_bo3957492148770167129t_unit
= ( collect_Product_unit @ bot_bo4642748612307482820unit_o ) ) ).
% bot_set_def
thf(fact_985_bot__set__def,axiom,
( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ bot_bo8461541820394803818_v_v_o ) ) ).
% bot_set_def
thf(fact_986_bot__set__def,axiom,
( bot_bot_set_o
= ( collect_o @ bot_bot_o_o ) ) ).
% bot_set_def
thf(fact_987_bot__set__def,axiom,
( bot_bot_set_set_v
= ( collect_set_v @ bot_bot_set_v_o ) ) ).
% bot_set_def
thf(fact_988_bot__set__def,axiom,
( bot_bot_set_v
= ( collect_v @ bot_bot_v_o ) ) ).
% bot_set_def
thf(fact_989_measures__lesseq,axiom,
! [F: v > nat,X: v,Y: v,Fs: list_v_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ ( measures_v @ Fs ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ ( measures_v @ ( cons_v_nat @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_990_surjective,axiom,
! [R: sCC_Bl1394983891496994913t_unit] :
( R
= ( sCC_Bl8064756265740546429t_unit @ ( sCC_Bl1090238580953940555t_unit @ R ) @ ( sCC_Bl1280885523602775798t_unit @ R ) @ ( sCC_Bl157864678168468314t_unit @ R ) @ ( sCC_Bl4645233313691564917t_unit @ R ) @ ( sCC_Bl3795065053823578884t_unit @ R ) @ ( sCC_Bl2536197123907397897t_unit @ R ) @ ( sCC_Bl8828226123343373779t_unit @ R ) @ ( sCC_Bl9201514103433284750t_unit @ R ) @ ( sCC_Bl3567736435408124606t_unit @ R ) ) ) ).
% surjective
thf(fact_991_avoiding__explored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,X: v,Y: v,E5: set_Product_prod_v_v,W: v,V2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E5 )
=> ( ~ ( member_v2 @ Y @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v2 @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ ( sup_su414716646722978715od_v_v @ E5 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V2 @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).
% avoiding_explored
thf(fact_992_scc__partition,axiom,
! [S3: set_v,S5: set_v,X: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S3 )
=> ( ( sCC_Bloemen_is_scc_v @ successors @ S5 )
=> ( ( member_v2 @ X @ ( inf_inf_set_v @ S3 @ S5 ) )
=> ( S3 = S5 ) ) ) ) ).
% scc_partition
thf(fact_993_the__elem__eq,axiom,
! [X: product_unit] :
( ( the_el608902216710161154t_unit @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= X ) ).
% the_elem_eq
thf(fact_994_the__elem__eq,axiom,
! [X: product_prod_v_v] :
( ( the_el5392834299063928540od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= X ) ).
% the_elem_eq
thf(fact_995_the__elem__eq,axiom,
! [X: $o] :
( ( the_elem_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% the_elem_eq
thf(fact_996_the__elem__eq,axiom,
! [X: set_v] :
( ( the_elem_set_v @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= X ) ).
% the_elem_eq
thf(fact_997_the__elem__eq,axiom,
! [X: v] :
( ( the_elem_v @ ( insert_v @ X @ bot_bot_set_v ) )
= X ) ).
% the_elem_eq
thf(fact_998_ra__mono,axiom,
! [X: v,Y: v,E5: set_Product_prod_v_v,E6: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E5 )
=> ( ( ord_le7336532860387713383od_v_v @ E6 @ E5 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E6 ) ) ) ).
% ra_mono
thf(fact_999_ra__trans,axiom,
! [X: v,Y: v,E5: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E5 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ Y @ Z @ E5 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z @ E5 ) ) ) ).
% ra_trans
thf(fact_1000_ra__refl,axiom,
! [X: v,E5: set_Product_prod_v_v] : ( sCC_Bl4291963740693775144ding_v @ successors @ X @ X @ E5 ) ).
% ra_refl
thf(fact_1001_ra__reachable,axiom,
! [X: v,Y: v,E5: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E5 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% ra_reachable
thf(fact_1002_ra__cases,axiom,
! [X: v,Y: v,E5: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E5 )
=> ( ( X = Y )
| ? [Z3: v] :
( ( member_v2 @ Z3 @ ( successors @ X ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Z3 ) @ E5 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ Z3 @ Y @ E5 ) ) ) ) ).
% ra_cases
thf(fact_1003_edge__ra,axiom,
! [Y: v,X: v,E5: set_Product_prod_v_v] :
( ( member_v2 @ Y @ ( successors @ X ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ E5 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E5 ) ) ) ).
% edge_ra
thf(fact_1004_reachable__avoiding_Osimps,axiom,
! [A1: v,A22: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A22 @ A32 )
= ( ? [X4: v,E7: set_Product_prod_v_v] :
( ( A1 = X4 )
& ( A22 = X4 )
& ( A32 = E7 ) )
| ? [X4: v,Y4: v,E7: set_Product_prod_v_v,Z2: v] :
( ( A1 = X4 )
& ( A22 = Z2 )
& ( A32 = E7 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ X4 @ Y4 @ E7 )
& ( member_v2 @ Z2 @ ( successors @ Y4 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y4 @ Z2 ) @ E7 ) ) ) ) ).
% reachable_avoiding.simps
thf(fact_1005_ra__succ,axiom,
! [X: v,Y: v,E5: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E5 )
=> ( ( member_v2 @ Z @ ( successors @ Y ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Z ) @ E5 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z @ E5 ) ) ) ) ).
% ra_succ
thf(fact_1006_reachable__avoiding_Ocases,axiom,
! [A1: v,A22: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A22 @ A32 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ Y3 @ A32 )
=> ( ( member_v2 @ A22 @ ( successors @ Y3 ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ A22 ) @ A32 ) ) ) ) ) ).
% reachable_avoiding.cases
thf(fact_1007_IntI,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A4 )
=> ( ( member7453568604450474000od_v_v @ C @ B4 )
=> ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) ) ) ) ).
% IntI
thf(fact_1008_IntI,axiom,
! [C: v,A4: set_v,B4: set_v] :
( ( member_v2 @ C @ A4 )
=> ( ( member_v2 @ C @ B4 )
=> ( member_v2 @ C @ ( inf_inf_set_v @ A4 @ B4 ) ) ) ) ).
% IntI
thf(fact_1009_Int__iff,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) )
= ( ( member7453568604450474000od_v_v @ C @ A4 )
& ( member7453568604450474000od_v_v @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_1010_Int__iff,axiom,
! [C: v,A4: set_v,B4: set_v] :
( ( member_v2 @ C @ ( inf_inf_set_v @ A4 @ B4 ) )
= ( ( member_v2 @ C @ A4 )
& ( member_v2 @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_1011_ra__empty,axiom,
! [X: v,Y: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% ra_empty
thf(fact_1012_inf__top_Oright__neutral,axiom,
! [A: set_v] :
( ( inf_inf_set_v @ A @ top_top_set_v )
= A ) ).
% inf_top.right_neutral
thf(fact_1013_inf__top_Oright__neutral,axiom,
! [A: product_unit] :
( ( inf_inf_Product_unit @ A @ top_top_Product_unit )
= A ) ).
% inf_top.right_neutral
thf(fact_1014_inf__top_Oright__neutral,axiom,
! [A: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ A @ top_to1996260823553986621t_unit )
= A ) ).
% inf_top.right_neutral
thf(fact_1015_inf__top_Oright__neutral,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ A @ top_top_set_o )
= A ) ).
% inf_top.right_neutral
thf(fact_1016_inf__top_Oright__neutral,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ top_top_set_nat )
= A ) ).
% inf_top.right_neutral
thf(fact_1017_inf__top_Oneutr__eq__iff,axiom,
! [A: set_v,B: set_v] :
( ( top_top_set_v
= ( inf_inf_set_v @ A @ B ) )
= ( ( A = top_top_set_v )
& ( B = top_top_set_v ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1018_inf__top_Oneutr__eq__iff,axiom,
! [A: product_unit,B: product_unit] :
( ( top_top_Product_unit
= ( inf_inf_Product_unit @ A @ B ) )
= ( ( A = top_top_Product_unit )
& ( B = top_top_Product_unit ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1019_inf__top_Oneutr__eq__iff,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( top_to1996260823553986621t_unit
= ( inf_in4660618365625256667t_unit @ A @ B ) )
= ( ( A = top_to1996260823553986621t_unit )
& ( B = top_to1996260823553986621t_unit ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1020_inf__top_Oneutr__eq__iff,axiom,
! [A: set_o,B: set_o] :
( ( top_top_set_o
= ( inf_inf_set_o @ A @ B ) )
= ( ( A = top_top_set_o )
& ( B = top_top_set_o ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1021_inf__top_Oneutr__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ A @ B ) )
= ( ( A = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_1022_inf__top_Oleft__neutral,axiom,
! [A: set_v] :
( ( inf_inf_set_v @ top_top_set_v @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_1023_inf__top_Oleft__neutral,axiom,
! [A: product_unit] :
( ( inf_inf_Product_unit @ top_top_Product_unit @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_1024_inf__top_Oleft__neutral,axiom,
! [A: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_1025_inf__top_Oleft__neutral,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ top_top_set_o @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_1026_inf__top_Oleft__neutral,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ A )
= A ) ).
% inf_top.left_neutral
thf(fact_1027_inf__top_Oeq__neutr__iff,axiom,
! [A: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A @ B )
= top_top_set_v )
= ( ( A = top_top_set_v )
& ( B = top_top_set_v ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1028_inf__top_Oeq__neutr__iff,axiom,
! [A: product_unit,B: product_unit] :
( ( ( inf_inf_Product_unit @ A @ B )
= top_top_Product_unit )
= ( ( A = top_top_Product_unit )
& ( B = top_top_Product_unit ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1029_inf__top_Oeq__neutr__iff,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A @ B )
= top_to1996260823553986621t_unit )
= ( ( A = top_to1996260823553986621t_unit )
& ( B = top_to1996260823553986621t_unit ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1030_inf__top_Oeq__neutr__iff,axiom,
! [A: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A @ B )
= top_top_set_o )
= ( ( A = top_top_set_o )
& ( B = top_top_set_o ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1031_inf__top_Oeq__neutr__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= top_top_set_nat )
= ( ( A = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_1032_top__eq__inf__iff,axiom,
! [X: set_v,Y: set_v] :
( ( top_top_set_v
= ( inf_inf_set_v @ X @ Y ) )
= ( ( X = top_top_set_v )
& ( Y = top_top_set_v ) ) ) ).
% top_eq_inf_iff
thf(fact_1033_top__eq__inf__iff,axiom,
! [X: product_unit,Y: product_unit] :
( ( top_top_Product_unit
= ( inf_inf_Product_unit @ X @ Y ) )
= ( ( X = top_top_Product_unit )
& ( Y = top_top_Product_unit ) ) ) ).
% top_eq_inf_iff
thf(fact_1034_top__eq__inf__iff,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( top_to1996260823553986621t_unit
= ( inf_in4660618365625256667t_unit @ X @ Y ) )
= ( ( X = top_to1996260823553986621t_unit )
& ( Y = top_to1996260823553986621t_unit ) ) ) ).
% top_eq_inf_iff
thf(fact_1035_top__eq__inf__iff,axiom,
! [X: set_o,Y: set_o] :
( ( top_top_set_o
= ( inf_inf_set_o @ X @ Y ) )
= ( ( X = top_top_set_o )
& ( Y = top_top_set_o ) ) ) ).
% top_eq_inf_iff
thf(fact_1036_top__eq__inf__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ X @ Y ) )
= ( ( X = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_1037_inf__eq__top__iff,axiom,
! [X: set_v,Y: set_v] :
( ( ( inf_inf_set_v @ X @ Y )
= top_top_set_v )
= ( ( X = top_top_set_v )
& ( Y = top_top_set_v ) ) ) ).
% inf_eq_top_iff
thf(fact_1038_inf__eq__top__iff,axiom,
! [X: product_unit,Y: product_unit] :
( ( ( inf_inf_Product_unit @ X @ Y )
= top_top_Product_unit )
= ( ( X = top_top_Product_unit )
& ( Y = top_top_Product_unit ) ) ) ).
% inf_eq_top_iff
thf(fact_1039_inf__eq__top__iff,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ X @ Y )
= top_to1996260823553986621t_unit )
= ( ( X = top_to1996260823553986621t_unit )
& ( Y = top_to1996260823553986621t_unit ) ) ) ).
% inf_eq_top_iff
thf(fact_1040_inf__eq__top__iff,axiom,
! [X: set_o,Y: set_o] :
( ( ( inf_inf_set_o @ X @ Y )
= top_top_set_o )
= ( ( X = top_top_set_o )
& ( Y = top_top_set_o ) ) ) ).
% inf_eq_top_iff
thf(fact_1041_inf__eq__top__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X @ Y )
= top_top_set_nat )
= ( ( X = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_1042_inf__top__right,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ top_top_set_v )
= X ) ).
% inf_top_right
thf(fact_1043_inf__top__right,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ X @ top_top_Product_unit )
= X ) ).
% inf_top_right
thf(fact_1044_inf__top__right,axiom,
! [X: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ X @ top_to1996260823553986621t_unit )
= X ) ).
% inf_top_right
thf(fact_1045_inf__top__right,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ X @ top_top_set_o )
= X ) ).
% inf_top_right
thf(fact_1046_inf__top__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ top_top_set_nat )
= X ) ).
% inf_top_right
thf(fact_1047_inf__top__left,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ top_top_set_v @ X )
= X ) ).
% inf_top_left
thf(fact_1048_inf__top__left,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ top_top_Product_unit @ X )
= X ) ).
% inf_top_left
thf(fact_1049_inf__top__left,axiom,
! [X: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ X )
= X ) ).
% inf_top_left
thf(fact_1050_inf__top__left,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ top_top_set_o @ X )
= X ) ).
% inf_top_left
thf(fact_1051_inf__top__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ X )
= X ) ).
% inf_top_left
thf(fact_1052_Int__UNIV,axiom,
! [A4: set_v,B4: set_v] :
( ( ( inf_inf_set_v @ A4 @ B4 )
= top_top_set_v )
= ( ( A4 = top_top_set_v )
& ( B4 = top_top_set_v ) ) ) ).
% Int_UNIV
thf(fact_1053_Int__UNIV,axiom,
! [A4: set_Product_unit,B4: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A4 @ B4 )
= top_to1996260823553986621t_unit )
= ( ( A4 = top_to1996260823553986621t_unit )
& ( B4 = top_to1996260823553986621t_unit ) ) ) ).
% Int_UNIV
thf(fact_1054_Int__UNIV,axiom,
! [A4: set_o,B4: set_o] :
( ( ( inf_inf_set_o @ A4 @ B4 )
= top_top_set_o )
= ( ( A4 = top_top_set_o )
& ( B4 = top_top_set_o ) ) ) ).
% Int_UNIV
thf(fact_1055_Int__UNIV,axiom,
! [A4: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ A4 @ B4 )
= top_top_set_nat )
= ( ( A4 = top_top_set_nat )
& ( B4 = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_1056_Int__subset__iff,axiom,
! [C2: set_v,A4: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A4 @ B4 ) )
= ( ( ord_less_eq_set_v @ C2 @ A4 )
& ( ord_less_eq_set_v @ C2 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_1057_Int__subset__iff,axiom,
! [C2: set_Product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) )
= ( ( ord_le7336532860387713383od_v_v @ C2 @ A4 )
& ( ord_le7336532860387713383od_v_v @ C2 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_1058_Int__insert__left__if0,axiom,
! [A: set_v,C2: set_set_v,B4: set_set_v] :
( ~ ( member_set_v @ A @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B4 ) @ C2 )
= ( inf_inf_set_set_v @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1059_Int__insert__left__if0,axiom,
! [A: product_unit,C2: set_Product_unit,B4: set_Product_unit] :
( ~ ( member_Product_unit @ A @ C2 )
=> ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A @ B4 ) @ C2 )
= ( inf_in4660618365625256667t_unit @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1060_Int__insert__left__if0,axiom,
! [A: $o,C2: set_o,B4: set_o] :
( ~ ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B4 ) @ C2 )
= ( inf_inf_set_o @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1061_Int__insert__left__if0,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B4 ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1062_Int__insert__left__if0,axiom,
! [A: v,C2: set_v,B4: set_v] :
( ~ ( member_v2 @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B4 ) @ C2 )
= ( inf_inf_set_v @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1063_Int__insert__left__if1,axiom,
! [A: set_v,C2: set_set_v,B4: set_set_v] :
( ( member_set_v @ A @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B4 ) @ C2 )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1064_Int__insert__left__if1,axiom,
! [A: product_unit,C2: set_Product_unit,B4: set_Product_unit] :
( ( member_Product_unit @ A @ C2 )
=> ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A @ B4 ) @ C2 )
= ( insert_Product_unit @ A @ ( inf_in4660618365625256667t_unit @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1065_Int__insert__left__if1,axiom,
! [A: $o,C2: set_o,B4: set_o] :
( ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B4 ) @ C2 )
= ( insert_o @ A @ ( inf_inf_set_o @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1066_Int__insert__left__if1,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B4 ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1067_Int__insert__left__if1,axiom,
! [A: v,C2: set_v,B4: set_v] :
( ( member_v2 @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B4 ) @ C2 )
= ( insert_v @ A @ ( inf_inf_set_v @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1068_insert__inter__insert,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A4 ) @ ( insert1338601472111419319od_v_v @ A @ B4 ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_1069_insert__inter__insert,axiom,
! [A: set_v,A4: set_set_v,B4: set_set_v] :
( ( inf_inf_set_set_v @ ( insert_set_v @ A @ A4 ) @ ( insert_set_v @ A @ B4 ) )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ A4 @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_1070_insert__inter__insert,axiom,
! [A: product_unit,A4: set_Product_unit,B4: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A @ A4 ) @ ( insert_Product_unit @ A @ B4 ) )
= ( insert_Product_unit @ A @ ( inf_in4660618365625256667t_unit @ A4 @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_1071_insert__inter__insert,axiom,
! [A: $o,A4: set_o,B4: set_o] :
( ( inf_inf_set_o @ ( insert_o @ A @ A4 ) @ ( insert_o @ A @ B4 ) )
= ( insert_o @ A @ ( inf_inf_set_o @ A4 @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_1072_insert__inter__insert,axiom,
! [A: v,A4: set_v,B4: set_v] :
( ( inf_inf_set_v @ ( insert_v @ A @ A4 ) @ ( insert_v @ A @ B4 ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A4 @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_1073_Int__insert__right__if0,axiom,
! [A: set_v,A4: set_set_v,B4: set_set_v] :
( ~ ( member_set_v @ A @ A4 )
=> ( ( inf_inf_set_set_v @ A4 @ ( insert_set_v @ A @ B4 ) )
= ( inf_inf_set_set_v @ A4 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_1074_Int__insert__right__if0,axiom,
! [A: product_unit,A4: set_Product_unit,B4: set_Product_unit] :
( ~ ( member_Product_unit @ A @ A4 )
=> ( ( inf_in4660618365625256667t_unit @ A4 @ ( insert_Product_unit @ A @ B4 ) )
= ( inf_in4660618365625256667t_unit @ A4 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_1075_Int__insert__right__if0,axiom,
! [A: $o,A4: set_o,B4: set_o] :
( ~ ( member_o @ A @ A4 )
=> ( ( inf_inf_set_o @ A4 @ ( insert_o @ A @ B4 ) )
= ( inf_inf_set_o @ A4 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_1076_Int__insert__right__if0,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A4 )
=> ( ( inf_in6271465464967711157od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A @ B4 ) )
= ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_1077_Int__insert__right__if0,axiom,
! [A: v,A4: set_v,B4: set_v] :
( ~ ( member_v2 @ A @ A4 )
=> ( ( inf_inf_set_v @ A4 @ ( insert_v @ A @ B4 ) )
= ( inf_inf_set_v @ A4 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_1078_Int__insert__right__if1,axiom,
! [A: set_v,A4: set_set_v,B4: set_set_v] :
( ( member_set_v @ A @ A4 )
=> ( ( inf_inf_set_set_v @ A4 @ ( insert_set_v @ A @ B4 ) )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ A4 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1079_Int__insert__right__if1,axiom,
! [A: product_unit,A4: set_Product_unit,B4: set_Product_unit] :
( ( member_Product_unit @ A @ A4 )
=> ( ( inf_in4660618365625256667t_unit @ A4 @ ( insert_Product_unit @ A @ B4 ) )
= ( insert_Product_unit @ A @ ( inf_in4660618365625256667t_unit @ A4 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1080_Int__insert__right__if1,axiom,
! [A: $o,A4: set_o,B4: set_o] :
( ( member_o @ A @ A4 )
=> ( ( inf_inf_set_o @ A4 @ ( insert_o @ A @ B4 ) )
= ( insert_o @ A @ ( inf_inf_set_o @ A4 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1081_Int__insert__right__if1,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A4 )
=> ( ( inf_in6271465464967711157od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A @ B4 ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1082_Int__insert__right__if1,axiom,
! [A: v,A4: set_v,B4: set_v] :
( ( member_v2 @ A @ A4 )
=> ( ( inf_inf_set_v @ A4 @ ( insert_v @ A @ B4 ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A4 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1083_Int__Un__eq_I4_J,axiom,
! [T2: set_Product_prod_v_v,S3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ T2 @ ( inf_in6271465464967711157od_v_v @ S3 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_1084_Int__Un__eq_I4_J,axiom,
! [T2: set_v,S3: set_v] :
( ( sup_sup_set_v @ T2 @ ( inf_inf_set_v @ S3 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_1085_Int__Un__eq_I4_J,axiom,
! [T2: set_set_v,S3: set_set_v] :
( ( sup_sup_set_set_v @ T2 @ ( inf_inf_set_set_v @ S3 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_1086_Int__Un__eq_I3_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ S3 @ ( inf_in6271465464967711157od_v_v @ S3 @ T2 ) )
= S3 ) ).
% Int_Un_eq(3)
thf(fact_1087_Int__Un__eq_I3_J,axiom,
! [S3: set_v,T2: set_v] :
( ( sup_sup_set_v @ S3 @ ( inf_inf_set_v @ S3 @ T2 ) )
= S3 ) ).
% Int_Un_eq(3)
thf(fact_1088_Int__Un__eq_I3_J,axiom,
! [S3: set_set_v,T2: set_set_v] :
( ( sup_sup_set_set_v @ S3 @ ( inf_inf_set_set_v @ S3 @ T2 ) )
= S3 ) ).
% Int_Un_eq(3)
thf(fact_1089_Int__Un__eq_I2_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_1090_Int__Un__eq_I2_J,axiom,
! [S3: set_v,T2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_1091_Int__Un__eq_I2_J,axiom,
! [S3: set_set_v,T2: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_1092_Int__Un__eq_I1_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Int_Un_eq(1)
thf(fact_1093_Int__Un__eq_I1_J,axiom,
! [S3: set_v,T2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Int_Un_eq(1)
thf(fact_1094_Int__Un__eq_I1_J,axiom,
! [S3: set_set_v,T2: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Int_Un_eq(1)
thf(fact_1095_Un__Int__eq_I4_J,axiom,
! [T2: set_Product_prod_v_v,S3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ T2 @ ( sup_su414716646722978715od_v_v @ S3 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_1096_Un__Int__eq_I4_J,axiom,
! [T2: set_v,S3: set_v] :
( ( inf_inf_set_v @ T2 @ ( sup_sup_set_v @ S3 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_1097_Un__Int__eq_I4_J,axiom,
! [T2: set_set_v,S3: set_set_v] :
( ( inf_inf_set_set_v @ T2 @ ( sup_sup_set_set_v @ S3 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_1098_Un__Int__eq_I3_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ S3 @ ( sup_su414716646722978715od_v_v @ S3 @ T2 ) )
= S3 ) ).
% Un_Int_eq(3)
thf(fact_1099_Un__Int__eq_I3_J,axiom,
! [S3: set_v,T2: set_v] :
( ( inf_inf_set_v @ S3 @ ( sup_sup_set_v @ S3 @ T2 ) )
= S3 ) ).
% Un_Int_eq(3)
thf(fact_1100_Un__Int__eq_I3_J,axiom,
! [S3: set_set_v,T2: set_set_v] :
( ( inf_inf_set_set_v @ S3 @ ( sup_sup_set_set_v @ S3 @ T2 ) )
= S3 ) ).
% Un_Int_eq(3)
thf(fact_1101_Un__Int__eq_I2_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_1102_Un__Int__eq_I2_J,axiom,
! [S3: set_v,T2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_1103_Un__Int__eq_I2_J,axiom,
! [S3: set_set_v,T2: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ S3 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_1104_Un__Int__eq_I1_J,axiom,
! [S3: set_Product_prod_v_v,T2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Un_Int_eq(1)
thf(fact_1105_Un__Int__eq_I1_J,axiom,
! [S3: set_v,T2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Un_Int_eq(1)
thf(fact_1106_Un__Int__eq_I1_J,axiom,
! [S3: set_set_v,T2: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ S3 @ T2 ) @ S3 )
= S3 ) ).
% Un_Int_eq(1)
thf(fact_1107_ra__add__edge,axiom,
! [X: v,Y: v,E5: set_Product_prod_v_v,V2: v,W: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E5 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ ( sup_su414716646722978715od_v_v @ E5 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V2 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ V2 @ ( sup_su414716646722978715od_v_v @ E5 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V2 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ successors @ W @ Y @ ( sup_su414716646722978715od_v_v @ E5 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V2 @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ).
% ra_add_edge
thf(fact_1108_disjoint__insert_I2_J,axiom,
! [A4: set_Product_unit,B: product_unit,B4: set_Product_unit] :
( ( bot_bo3957492148770167129t_unit
= ( inf_in4660618365625256667t_unit @ A4 @ ( insert_Product_unit @ B @ B4 ) ) )
= ( ~ ( member_Product_unit @ B @ A4 )
& ( bot_bo3957492148770167129t_unit
= ( inf_in4660618365625256667t_unit @ A4 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1109_disjoint__insert_I2_J,axiom,
! [A4: set_Product_prod_v_v,B: product_prod_v_v,B4: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ B @ B4 ) ) )
= ( ~ ( member7453568604450474000od_v_v @ B @ A4 )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1110_disjoint__insert_I2_J,axiom,
! [A4: set_o,B: $o,B4: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ A4 @ ( insert_o @ B @ B4 ) ) )
= ( ~ ( member_o @ B @ A4 )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A4 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1111_disjoint__insert_I2_J,axiom,
! [A4: set_set_v,B: set_v,B4: set_set_v] :
( ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A4 @ ( insert_set_v @ B @ B4 ) ) )
= ( ~ ( member_set_v @ B @ A4 )
& ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A4 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1112_disjoint__insert_I2_J,axiom,
! [A4: set_v,B: v,B4: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ A4 @ ( insert_v @ B @ B4 ) ) )
= ( ~ ( member_v2 @ B @ A4 )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A4 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1113_disjoint__insert_I1_J,axiom,
! [B4: set_Product_unit,A: product_unit,A4: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ B4 @ ( insert_Product_unit @ A @ A4 ) )
= bot_bo3957492148770167129t_unit )
= ( ~ ( member_Product_unit @ A @ B4 )
& ( ( inf_in4660618365625256667t_unit @ B4 @ A4 )
= bot_bo3957492148770167129t_unit ) ) ) ).
% disjoint_insert(1)
thf(fact_1114_disjoint__insert_I1_J,axiom,
! [B4: set_Product_prod_v_v,A: product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ B4 @ ( insert1338601472111419319od_v_v @ A @ A4 ) )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B4 )
& ( ( inf_in6271465464967711157od_v_v @ B4 @ A4 )
= bot_bo723834152578015283od_v_v ) ) ) ).
% disjoint_insert(1)
thf(fact_1115_disjoint__insert_I1_J,axiom,
! [B4: set_o,A: $o,A4: set_o] :
( ( ( inf_inf_set_o @ B4 @ ( insert_o @ A @ A4 ) )
= bot_bot_set_o )
= ( ~ ( member_o @ A @ B4 )
& ( ( inf_inf_set_o @ B4 @ A4 )
= bot_bot_set_o ) ) ) ).
% disjoint_insert(1)
thf(fact_1116_disjoint__insert_I1_J,axiom,
! [B4: set_set_v,A: set_v,A4: set_set_v] :
( ( ( inf_inf_set_set_v @ B4 @ ( insert_set_v @ A @ A4 ) )
= bot_bot_set_set_v )
= ( ~ ( member_set_v @ A @ B4 )
& ( ( inf_inf_set_set_v @ B4 @ A4 )
= bot_bot_set_set_v ) ) ) ).
% disjoint_insert(1)
thf(fact_1117_disjoint__insert_I1_J,axiom,
! [B4: set_v,A: v,A4: set_v] :
( ( ( inf_inf_set_v @ B4 @ ( insert_v @ A @ A4 ) )
= bot_bot_set_v )
= ( ~ ( member_v2 @ A @ B4 )
& ( ( inf_inf_set_v @ B4 @ A4 )
= bot_bot_set_v ) ) ) ).
% disjoint_insert(1)
thf(fact_1118_insert__disjoint_I2_J,axiom,
! [A: product_unit,A4: set_Product_unit,B4: set_Product_unit] :
( ( bot_bo3957492148770167129t_unit
= ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A @ A4 ) @ B4 ) )
= ( ~ ( member_Product_unit @ A @ B4 )
& ( bot_bo3957492148770167129t_unit
= ( inf_in4660618365625256667t_unit @ A4 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1119_insert__disjoint_I2_J,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A4 ) @ B4 ) )
= ( ~ ( member7453568604450474000od_v_v @ A @ B4 )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1120_insert__disjoint_I2_J,axiom,
! [A: $o,A4: set_o,B4: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ ( insert_o @ A @ A4 ) @ B4 ) )
= ( ~ ( member_o @ A @ B4 )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A4 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1121_insert__disjoint_I2_J,axiom,
! [A: set_v,A4: set_set_v,B4: set_set_v] :
( ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ ( insert_set_v @ A @ A4 ) @ B4 ) )
= ( ~ ( member_set_v @ A @ B4 )
& ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A4 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1122_insert__disjoint_I2_J,axiom,
! [A: v,A4: set_v,B4: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ ( insert_v @ A @ A4 ) @ B4 ) )
= ( ~ ( member_v2 @ A @ B4 )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A4 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1123_insert__disjoint_I1_J,axiom,
! [A: product_unit,A4: set_Product_unit,B4: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A @ A4 ) @ B4 )
= bot_bo3957492148770167129t_unit )
= ( ~ ( member_Product_unit @ A @ B4 )
& ( ( inf_in4660618365625256667t_unit @ A4 @ B4 )
= bot_bo3957492148770167129t_unit ) ) ) ).
% insert_disjoint(1)
thf(fact_1124_insert__disjoint_I1_J,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A4 ) @ B4 )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B4 )
& ( ( inf_in6271465464967711157od_v_v @ A4 @ B4 )
= bot_bo723834152578015283od_v_v ) ) ) ).
% insert_disjoint(1)
thf(fact_1125_insert__disjoint_I1_J,axiom,
! [A: $o,A4: set_o,B4: set_o] :
( ( ( inf_inf_set_o @ ( insert_o @ A @ A4 ) @ B4 )
= bot_bot_set_o )
= ( ~ ( member_o @ A @ B4 )
& ( ( inf_inf_set_o @ A4 @ B4 )
= bot_bot_set_o ) ) ) ).
% insert_disjoint(1)
thf(fact_1126_insert__disjoint_I1_J,axiom,
! [A: set_v,A4: set_set_v,B4: set_set_v] :
( ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ A4 ) @ B4 )
= bot_bot_set_set_v )
= ( ~ ( member_set_v @ A @ B4 )
& ( ( inf_inf_set_set_v @ A4 @ B4 )
= bot_bot_set_set_v ) ) ) ).
% insert_disjoint(1)
thf(fact_1127_insert__disjoint_I1_J,axiom,
! [A: v,A4: set_v,B4: set_v] :
( ( ( inf_inf_set_v @ ( insert_v @ A @ A4 ) @ B4 )
= bot_bot_set_v )
= ( ~ ( member_v2 @ A @ B4 )
& ( ( inf_inf_set_v @ A4 @ B4 )
= bot_bot_set_v ) ) ) ).
% insert_disjoint(1)
thf(fact_1128_Diff__disjoint,axiom,
! [A4: set_Product_unit,B4: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ A4 @ ( minus_6452836326544984404t_unit @ B4 @ A4 ) )
= bot_bo3957492148770167129t_unit ) ).
% Diff_disjoint
thf(fact_1129_Diff__disjoint,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A4 @ ( minus_4183494784930505774od_v_v @ B4 @ A4 ) )
= bot_bo723834152578015283od_v_v ) ).
% Diff_disjoint
thf(fact_1130_Diff__disjoint,axiom,
! [A4: set_o,B4: set_o] :
( ( inf_inf_set_o @ A4 @ ( minus_minus_set_o @ B4 @ A4 ) )
= bot_bot_set_o ) ).
% Diff_disjoint
thf(fact_1131_Diff__disjoint,axiom,
! [A4: set_set_v,B4: set_set_v] :
( ( inf_inf_set_set_v @ A4 @ ( minus_7228012346218142266_set_v @ B4 @ A4 ) )
= bot_bot_set_set_v ) ).
% Diff_disjoint
thf(fact_1132_Diff__disjoint,axiom,
! [A4: set_v,B4: set_v] :
( ( inf_inf_set_v @ A4 @ ( minus_minus_set_v @ B4 @ A4 ) )
= bot_bot_set_v ) ).
% Diff_disjoint
thf(fact_1133_boolean__algebra_Oconj__one__right,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ top_top_set_v )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_1134_boolean__algebra_Oconj__one__right,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ X @ top_top_Product_unit )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_1135_boolean__algebra_Oconj__one__right,axiom,
! [X: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ X @ top_to1996260823553986621t_unit )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_1136_boolean__algebra_Oconj__one__right,axiom,
! [X: set_o] :
( ( inf_inf_set_o @ X @ top_top_set_o )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_1137_boolean__algebra_Oconj__one__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ top_top_set_nat )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_1138_disjoint__iff__not__equal,axiom,
! [A4: set_Product_unit,B4: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A4 @ B4 )
= bot_bo3957492148770167129t_unit )
= ( ! [X4: product_unit] :
( ( member_Product_unit @ X4 @ A4 )
=> ! [Y4: product_unit] :
( ( member_Product_unit @ Y4 @ B4 )
=> ( X4 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1139_disjoint__iff__not__equal,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A4 @ B4 )
= bot_bo723834152578015283od_v_v )
= ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A4 )
=> ! [Y4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y4 @ B4 )
=> ( X4 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1140_disjoint__iff__not__equal,axiom,
! [A4: set_o,B4: set_o] :
( ( ( inf_inf_set_o @ A4 @ B4 )
= bot_bot_set_o )
= ( ! [X4: $o] :
( ( member_o @ X4 @ A4 )
=> ! [Y4: $o] :
( ( member_o @ Y4 @ B4 )
=> ( X4 = (~ Y4) ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1141_disjoint__iff__not__equal,axiom,
! [A4: set_set_v,B4: set_set_v] :
( ( ( inf_inf_set_set_v @ A4 @ B4 )
= bot_bot_set_set_v )
= ( ! [X4: set_v] :
( ( member_set_v @ X4 @ A4 )
=> ! [Y4: set_v] :
( ( member_set_v @ Y4 @ B4 )
=> ( X4 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1142_disjoint__iff__not__equal,axiom,
! [A4: set_v,B4: set_v] :
( ( ( inf_inf_set_v @ A4 @ B4 )
= bot_bot_set_v )
= ( ! [X4: v] :
( ( member_v2 @ X4 @ A4 )
=> ! [Y4: v] :
( ( member_v2 @ Y4 @ B4 )
=> ( X4 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1143_Int__empty__right,axiom,
! [A4: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ A4 @ bot_bo3957492148770167129t_unit )
= bot_bo3957492148770167129t_unit ) ).
% Int_empty_right
thf(fact_1144_Int__empty__right,axiom,
! [A4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A4 @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_right
thf(fact_1145_Int__empty__right,axiom,
! [A4: set_o] :
( ( inf_inf_set_o @ A4 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% Int_empty_right
thf(fact_1146_Int__empty__right,axiom,
! [A4: set_set_v] :
( ( inf_inf_set_set_v @ A4 @ bot_bot_set_set_v )
= bot_bot_set_set_v ) ).
% Int_empty_right
thf(fact_1147_Int__empty__right,axiom,
! [A4: set_v] :
( ( inf_inf_set_v @ A4 @ bot_bot_set_v )
= bot_bot_set_v ) ).
% Int_empty_right
thf(fact_1148_Int__empty__left,axiom,
! [B4: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ bot_bo3957492148770167129t_unit @ B4 )
= bot_bo3957492148770167129t_unit ) ).
% Int_empty_left
thf(fact_1149_Int__empty__left,axiom,
! [B4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ B4 )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_left
thf(fact_1150_Int__empty__left,axiom,
! [B4: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ B4 )
= bot_bot_set_o ) ).
% Int_empty_left
thf(fact_1151_Int__empty__left,axiom,
! [B4: set_set_v] :
( ( inf_inf_set_set_v @ bot_bot_set_set_v @ B4 )
= bot_bot_set_set_v ) ).
% Int_empty_left
thf(fact_1152_Int__empty__left,axiom,
! [B4: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ B4 )
= bot_bot_set_v ) ).
% Int_empty_left
thf(fact_1153_disjoint__iff,axiom,
! [A4: set_Product_unit,B4: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A4 @ B4 )
= bot_bo3957492148770167129t_unit )
= ( ! [X4: product_unit] :
( ( member_Product_unit @ X4 @ A4 )
=> ~ ( member_Product_unit @ X4 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_1154_disjoint__iff,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A4 @ B4 )
= bot_bo723834152578015283od_v_v )
= ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A4 )
=> ~ ( member7453568604450474000od_v_v @ X4 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_1155_disjoint__iff,axiom,
! [A4: set_o,B4: set_o] :
( ( ( inf_inf_set_o @ A4 @ B4 )
= bot_bot_set_o )
= ( ! [X4: $o] :
( ( member_o @ X4 @ A4 )
=> ~ ( member_o @ X4 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_1156_disjoint__iff,axiom,
! [A4: set_set_v,B4: set_set_v] :
( ( ( inf_inf_set_set_v @ A4 @ B4 )
= bot_bot_set_set_v )
= ( ! [X4: set_v] :
( ( member_set_v @ X4 @ A4 )
=> ~ ( member_set_v @ X4 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_1157_disjoint__iff,axiom,
! [A4: set_v,B4: set_v] :
( ( ( inf_inf_set_v @ A4 @ B4 )
= bot_bot_set_v )
= ( ! [X4: v] :
( ( member_v2 @ X4 @ A4 )
=> ~ ( member_v2 @ X4 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_1158_Int__emptyI,axiom,
! [A4: set_Product_unit,B4: set_Product_unit] :
( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A4 )
=> ~ ( member_Product_unit @ X3 @ B4 ) )
=> ( ( inf_in4660618365625256667t_unit @ A4 @ B4 )
= bot_bo3957492148770167129t_unit ) ) ).
% Int_emptyI
thf(fact_1159_Int__emptyI,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A4 )
=> ~ ( member7453568604450474000od_v_v @ X3 @ B4 ) )
=> ( ( inf_in6271465464967711157od_v_v @ A4 @ B4 )
= bot_bo723834152578015283od_v_v ) ) ).
% Int_emptyI
thf(fact_1160_Int__emptyI,axiom,
! [A4: set_o,B4: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A4 )
=> ~ ( member_o @ X3 @ B4 ) )
=> ( ( inf_inf_set_o @ A4 @ B4 )
= bot_bot_set_o ) ) ).
% Int_emptyI
thf(fact_1161_Int__emptyI,axiom,
! [A4: set_set_v,B4: set_set_v] :
( ! [X3: set_v] :
( ( member_set_v @ X3 @ A4 )
=> ~ ( member_set_v @ X3 @ B4 ) )
=> ( ( inf_inf_set_set_v @ A4 @ B4 )
= bot_bot_set_set_v ) ) ).
% Int_emptyI
thf(fact_1162_Int__emptyI,axiom,
! [A4: set_v,B4: set_v] :
( ! [X3: v] :
( ( member_v2 @ X3 @ A4 )
=> ~ ( member_v2 @ X3 @ B4 ) )
=> ( ( inf_inf_set_v @ A4 @ B4 )
= bot_bot_set_v ) ) ).
% Int_emptyI
thf(fact_1163_Int__UNIV__left,axiom,
! [B4: set_v] :
( ( inf_inf_set_v @ top_top_set_v @ B4 )
= B4 ) ).
% Int_UNIV_left
thf(fact_1164_Int__UNIV__left,axiom,
! [B4: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ B4 )
= B4 ) ).
% Int_UNIV_left
thf(fact_1165_Int__UNIV__left,axiom,
! [B4: set_o] :
( ( inf_inf_set_o @ top_top_set_o @ B4 )
= B4 ) ).
% Int_UNIV_left
thf(fact_1166_Int__UNIV__left,axiom,
! [B4: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B4 )
= B4 ) ).
% Int_UNIV_left
thf(fact_1167_Int__UNIV__right,axiom,
! [A4: set_v] :
( ( inf_inf_set_v @ A4 @ top_top_set_v )
= A4 ) ).
% Int_UNIV_right
thf(fact_1168_Int__UNIV__right,axiom,
! [A4: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ A4 @ top_to1996260823553986621t_unit )
= A4 ) ).
% Int_UNIV_right
thf(fact_1169_Int__UNIV__right,axiom,
! [A4: set_o] :
( ( inf_inf_set_o @ A4 @ top_top_set_o )
= A4 ) ).
% Int_UNIV_right
thf(fact_1170_Int__UNIV__right,axiom,
! [A4: set_nat] :
( ( inf_inf_set_nat @ A4 @ top_top_set_nat )
= A4 ) ).
% Int_UNIV_right
thf(fact_1171_Int__Collect__mono,axiom,
! [A4: set_set_v,B4: set_set_v,P2: set_v > $o,Q: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ A4 @ B4 )
=> ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A4 )
=> ( ( P2 @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le5216385588623774835_set_v @ ( inf_inf_set_set_v @ A4 @ ( collect_set_v @ P2 ) ) @ ( inf_inf_set_set_v @ B4 @ ( collect_set_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1172_Int__Collect__mono,axiom,
! [A4: set_v,B4: set_v,P2: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ A4 @ B4 )
=> ( ! [X3: v] :
( ( member_v2 @ X3 @ A4 )
=> ( ( P2 @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A4 @ ( collect_v @ P2 ) ) @ ( inf_inf_set_v @ B4 @ ( collect_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1173_Int__Collect__mono,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v,P2: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B4 )
=> ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A4 )
=> ( ( P2 @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ ( collec140062887454715474od_v_v @ P2 ) ) @ ( inf_in6271465464967711157od_v_v @ B4 @ ( collec140062887454715474od_v_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1174_Int__greatest,axiom,
! [C2: set_v,A4: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ C2 @ A4 )
=> ( ( ord_less_eq_set_v @ C2 @ B4 )
=> ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A4 @ B4 ) ) ) ) ).
% Int_greatest
thf(fact_1175_Int__greatest,axiom,
! [C2: set_Product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ A4 )
=> ( ( ord_le7336532860387713383od_v_v @ C2 @ B4 )
=> ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) ) ) ) ).
% Int_greatest
thf(fact_1176_Int__absorb2,axiom,
! [A4: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ A4 @ B4 )
=> ( ( inf_inf_set_v @ A4 @ B4 )
= A4 ) ) ).
% Int_absorb2
thf(fact_1177_Int__absorb2,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ B4 )
=> ( ( inf_in6271465464967711157od_v_v @ A4 @ B4 )
= A4 ) ) ).
% Int_absorb2
thf(fact_1178_Int__absorb1,axiom,
! [B4: set_v,A4: set_v] :
( ( ord_less_eq_set_v @ B4 @ A4 )
=> ( ( inf_inf_set_v @ A4 @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_1179_Int__absorb1,axiom,
! [B4: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B4 @ A4 )
=> ( ( inf_in6271465464967711157od_v_v @ A4 @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_1180_Int__lower2,axiom,
! [A4: set_v,B4: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A4 @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_1181_Int__lower2,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_1182_Int__lower1,axiom,
! [A4: set_v,B4: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A4 @ B4 ) @ A4 ) ).
% Int_lower1
thf(fact_1183_Int__lower1,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) @ A4 ) ).
% Int_lower1
thf(fact_1184_Int__mono,axiom,
! [A4: set_v,C2: set_v,B4: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A4 @ C2 )
=> ( ( ord_less_eq_set_v @ B4 @ D )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A4 @ B4 ) @ ( inf_inf_set_v @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_1185_Int__mono,axiom,
! [A4: set_Product_prod_v_v,C2: set_Product_prod_v_v,B4: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A4 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B4 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) @ ( inf_in6271465464967711157od_v_v @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_1186_Int__insert__right,axiom,
! [A: set_v,A4: set_set_v,B4: set_set_v] :
( ( ( member_set_v @ A @ A4 )
=> ( ( inf_inf_set_set_v @ A4 @ ( insert_set_v @ A @ B4 ) )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ A4 @ B4 ) ) ) )
& ( ~ ( member_set_v @ A @ A4 )
=> ( ( inf_inf_set_set_v @ A4 @ ( insert_set_v @ A @ B4 ) )
= ( inf_inf_set_set_v @ A4 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_1187_Int__insert__right,axiom,
! [A: product_unit,A4: set_Product_unit,B4: set_Product_unit] :
( ( ( member_Product_unit @ A @ A4 )
=> ( ( inf_in4660618365625256667t_unit @ A4 @ ( insert_Product_unit @ A @ B4 ) )
= ( insert_Product_unit @ A @ ( inf_in4660618365625256667t_unit @ A4 @ B4 ) ) ) )
& ( ~ ( member_Product_unit @ A @ A4 )
=> ( ( inf_in4660618365625256667t_unit @ A4 @ ( insert_Product_unit @ A @ B4 ) )
= ( inf_in4660618365625256667t_unit @ A4 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_1188_Int__insert__right,axiom,
! [A: $o,A4: set_o,B4: set_o] :
( ( ( member_o @ A @ A4 )
=> ( ( inf_inf_set_o @ A4 @ ( insert_o @ A @ B4 ) )
= ( insert_o @ A @ ( inf_inf_set_o @ A4 @ B4 ) ) ) )
& ( ~ ( member_o @ A @ A4 )
=> ( ( inf_inf_set_o @ A4 @ ( insert_o @ A @ B4 ) )
= ( inf_inf_set_o @ A4 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_1189_Int__insert__right,axiom,
! [A: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ A4 )
=> ( ( inf_in6271465464967711157od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A @ B4 ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ A4 )
=> ( ( inf_in6271465464967711157od_v_v @ A4 @ ( insert1338601472111419319od_v_v @ A @ B4 ) )
= ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_1190_Int__insert__right,axiom,
! [A: v,A4: set_v,B4: set_v] :
( ( ( member_v2 @ A @ A4 )
=> ( ( inf_inf_set_v @ A4 @ ( insert_v @ A @ B4 ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A4 @ B4 ) ) ) )
& ( ~ ( member_v2 @ A @ A4 )
=> ( ( inf_inf_set_v @ A4 @ ( insert_v @ A @ B4 ) )
= ( inf_inf_set_v @ A4 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_1191_Int__insert__left,axiom,
! [A: set_v,C2: set_set_v,B4: set_set_v] :
( ( ( member_set_v @ A @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B4 ) @ C2 )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ B4 @ C2 ) ) ) )
& ( ~ ( member_set_v @ A @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B4 ) @ C2 )
= ( inf_inf_set_set_v @ B4 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1192_Int__insert__left,axiom,
! [A: product_unit,C2: set_Product_unit,B4: set_Product_unit] :
( ( ( member_Product_unit @ A @ C2 )
=> ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A @ B4 ) @ C2 )
= ( insert_Product_unit @ A @ ( inf_in4660618365625256667t_unit @ B4 @ C2 ) ) ) )
& ( ~ ( member_Product_unit @ A @ C2 )
=> ( ( inf_in4660618365625256667t_unit @ ( insert_Product_unit @ A @ B4 ) @ C2 )
= ( inf_in4660618365625256667t_unit @ B4 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1193_Int__insert__left,axiom,
! [A: $o,C2: set_o,B4: set_o] :
( ( ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B4 ) @ C2 )
= ( insert_o @ A @ ( inf_inf_set_o @ B4 @ C2 ) ) ) )
& ( ~ ( member_o @ A @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A @ B4 ) @ C2 )
= ( inf_inf_set_o @ B4 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1194_Int__insert__left,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B4 ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B4 @ C2 ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B4 ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ B4 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1195_Int__insert__left,axiom,
! [A: v,C2: set_v,B4: set_v] :
( ( ( member_v2 @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B4 ) @ C2 )
= ( insert_v @ A @ ( inf_inf_set_v @ B4 @ C2 ) ) ) )
& ( ~ ( member_v2 @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B4 ) @ C2 )
= ( inf_inf_set_v @ B4 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1196_Un__Int__distrib2,axiom,
! [B4: set_Product_prod_v_v,C2: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B4 @ C2 ) @ A4 )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B4 @ A4 ) @ ( sup_su414716646722978715od_v_v @ C2 @ A4 ) ) ) ).
% Un_Int_distrib2
thf(fact_1197_Un__Int__distrib2,axiom,
! [B4: set_v,C2: set_v,A4: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ B4 @ C2 ) @ A4 )
= ( inf_inf_set_v @ ( sup_sup_set_v @ B4 @ A4 ) @ ( sup_sup_set_v @ C2 @ A4 ) ) ) ).
% Un_Int_distrib2
thf(fact_1198_Un__Int__distrib2,axiom,
! [B4: set_set_v,C2: set_set_v,A4: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ B4 @ C2 ) @ A4 )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ B4 @ A4 ) @ ( sup_sup_set_set_v @ C2 @ A4 ) ) ) ).
% Un_Int_distrib2
thf(fact_1199_Int__Un__distrib2,axiom,
! [B4: set_Product_prod_v_v,C2: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B4 @ C2 ) @ A4 )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B4 @ A4 ) @ ( inf_in6271465464967711157od_v_v @ C2 @ A4 ) ) ) ).
% Int_Un_distrib2
thf(fact_1200_Int__Un__distrib2,axiom,
! [B4: set_v,C2: set_v,A4: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ B4 @ C2 ) @ A4 )
= ( sup_sup_set_v @ ( inf_inf_set_v @ B4 @ A4 ) @ ( inf_inf_set_v @ C2 @ A4 ) ) ) ).
% Int_Un_distrib2
thf(fact_1201_Int__Un__distrib2,axiom,
! [B4: set_set_v,C2: set_set_v,A4: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ B4 @ C2 ) @ A4 )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ B4 @ A4 ) @ ( inf_inf_set_set_v @ C2 @ A4 ) ) ) ).
% Int_Un_distrib2
thf(fact_1202_Un__Int__distrib,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ ( inf_in6271465464967711157od_v_v @ B4 @ C2 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B4 ) @ ( sup_su414716646722978715od_v_v @ A4 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1203_Un__Int__distrib,axiom,
! [A4: set_v,B4: set_v,C2: set_v] :
( ( sup_sup_set_v @ A4 @ ( inf_inf_set_v @ B4 @ C2 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ A4 @ B4 ) @ ( sup_sup_set_v @ A4 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1204_Un__Int__distrib,axiom,
! [A4: set_set_v,B4: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ A4 @ ( inf_inf_set_set_v @ B4 @ C2 ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ A4 @ B4 ) @ ( sup_sup_set_set_v @ A4 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1205_Int__Un__distrib,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ B4 @ C2 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) @ ( inf_in6271465464967711157od_v_v @ A4 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1206_Int__Un__distrib,axiom,
! [A4: set_v,B4: set_v,C2: set_v] :
( ( inf_inf_set_v @ A4 @ ( sup_sup_set_v @ B4 @ C2 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ A4 @ B4 ) @ ( inf_inf_set_v @ A4 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1207_Int__Un__distrib,axiom,
! [A4: set_set_v,B4: set_set_v,C2: set_set_v] :
( ( inf_inf_set_set_v @ A4 @ ( sup_sup_set_set_v @ B4 @ C2 ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A4 @ B4 ) @ ( inf_inf_set_set_v @ A4 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1208_Un__Int__crazy,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) @ ( inf_in6271465464967711157od_v_v @ B4 @ C2 ) ) @ ( inf_in6271465464967711157od_v_v @ C2 @ A4 ) )
= ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A4 @ B4 ) @ ( sup_su414716646722978715od_v_v @ B4 @ C2 ) ) @ ( sup_su414716646722978715od_v_v @ C2 @ A4 ) ) ) ).
% Un_Int_crazy
thf(fact_1209_Un__Int__crazy,axiom,
! [A4: set_v,B4: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ A4 @ B4 ) @ ( inf_inf_set_v @ B4 @ C2 ) ) @ ( inf_inf_set_v @ C2 @ A4 ) )
= ( inf_inf_set_v @ ( inf_inf_set_v @ ( sup_sup_set_v @ A4 @ B4 ) @ ( sup_sup_set_v @ B4 @ C2 ) ) @ ( sup_sup_set_v @ C2 @ A4 ) ) ) ).
% Un_Int_crazy
thf(fact_1210_Un__Int__crazy,axiom,
! [A4: set_set_v,B4: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A4 @ B4 ) @ ( inf_inf_set_set_v @ B4 @ C2 ) ) @ ( inf_inf_set_set_v @ C2 @ A4 ) )
= ( inf_inf_set_set_v @ ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ A4 @ B4 ) @ ( sup_sup_set_set_v @ B4 @ C2 ) ) @ ( sup_sup_set_set_v @ C2 @ A4 ) ) ) ).
% Un_Int_crazy
thf(fact_1211_Diff__Int__distrib2,axiom,
! [A4: set_v,B4: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( minus_minus_set_v @ A4 @ B4 ) @ C2 )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A4 @ C2 ) @ ( inf_inf_set_v @ B4 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1212_Diff__Int__distrib,axiom,
! [C2: set_v,A4: set_v,B4: set_v] :
( ( inf_inf_set_v @ C2 @ ( minus_minus_set_v @ A4 @ B4 ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ C2 @ A4 ) @ ( inf_inf_set_v @ C2 @ B4 ) ) ) ).
% Diff_Int_distrib
thf(fact_1213_Diff__Diff__Int,axiom,
! [A4: set_v,B4: set_v] :
( ( minus_minus_set_v @ A4 @ ( minus_minus_set_v @ A4 @ B4 ) )
= ( inf_inf_set_v @ A4 @ B4 ) ) ).
% Diff_Diff_Int
thf(fact_1214_Diff__Int2,axiom,
! [A4: set_v,C2: set_v,B4: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A4 @ C2 ) @ ( inf_inf_set_v @ B4 @ C2 ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A4 @ C2 ) @ B4 ) ) ).
% Diff_Int2
thf(fact_1215_Int__Diff,axiom,
! [A4: set_v,B4: set_v,C2: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A4 @ B4 ) @ C2 )
= ( inf_inf_set_v @ A4 @ ( minus_minus_set_v @ B4 @ C2 ) ) ) ).
% Int_Diff
thf(fact_1216_graph_Oreachable__avoiding_Ocong,axiom,
sCC_Bl4291963740693775144ding_v = sCC_Bl4291963740693775144ding_v ).
% graph.reachable_avoiding.cong
thf(fact_1217_IntE,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A4 )
=> ~ ( member7453568604450474000od_v_v @ C @ B4 ) ) ) ).
% IntE
thf(fact_1218_IntE,axiom,
! [C: v,A4: set_v,B4: set_v] :
( ( member_v2 @ C @ ( inf_inf_set_v @ A4 @ B4 ) )
=> ~ ( ( member_v2 @ C @ A4 )
=> ~ ( member_v2 @ C @ B4 ) ) ) ).
% IntE
thf(fact_1219_IntD1,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) )
=> ( member7453568604450474000od_v_v @ C @ A4 ) ) ).
% IntD1
thf(fact_1220_IntD1,axiom,
! [C: v,A4: set_v,B4: set_v] :
( ( member_v2 @ C @ ( inf_inf_set_v @ A4 @ B4 ) )
=> ( member_v2 @ C @ A4 ) ) ).
% IntD1
thf(fact_1221_IntD2,axiom,
! [C: product_prod_v_v,A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) )
=> ( member7453568604450474000od_v_v @ C @ B4 ) ) ).
% IntD2
thf(fact_1222_IntD2,axiom,
! [C: v,A4: set_v,B4: set_v] :
( ( member_v2 @ C @ ( inf_inf_set_v @ A4 @ B4 ) )
=> ( member_v2 @ C @ B4 ) ) ).
% IntD2
thf(fact_1223_Int__assoc,axiom,
! [A4: set_v,B4: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A4 @ B4 ) @ C2 )
= ( inf_inf_set_v @ A4 @ ( inf_inf_set_v @ B4 @ C2 ) ) ) ).
% Int_assoc
thf(fact_1224_Int__absorb,axiom,
! [A4: set_v] :
( ( inf_inf_set_v @ A4 @ A4 )
= A4 ) ).
% Int_absorb
thf(fact_1225_Int__commute,axiom,
( inf_inf_set_v
= ( ^ [A5: set_v,B6: set_v] : ( inf_inf_set_v @ B6 @ A5 ) ) ) ).
% Int_commute
thf(fact_1226_Int__left__absorb,axiom,
! [A4: set_v,B4: set_v] :
( ( inf_inf_set_v @ A4 @ ( inf_inf_set_v @ A4 @ B4 ) )
= ( inf_inf_set_v @ A4 @ B4 ) ) ).
% Int_left_absorb
thf(fact_1227_Int__left__commute,axiom,
! [A4: set_v,B4: set_v,C2: set_v] :
( ( inf_inf_set_v @ A4 @ ( inf_inf_set_v @ B4 @ C2 ) )
= ( inf_inf_set_v @ B4 @ ( inf_inf_set_v @ A4 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_1228_select__convs_I7_J,axiom,
! [Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Stack ) ).
% select_convs(7)
thf(fact_1229_select__convs_I2_J,axiom,
! [Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= S6 ) ).
% select_convs(2)
thf(fact_1230_select__convs_I4_J,axiom,
! [Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl4645233313691564917t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Visited ) ).
% select_convs(4)
thf(fact_1231_select__convs_I5_J,axiom,
! [Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Vsuccs ) ).
% select_convs(5)
thf(fact_1232_select__convs_I3_J,axiom,
! [Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl157864678168468314t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Explored ) ).
% select_convs(3)
thf(fact_1233_select__convs_I8_J,axiom,
! [Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl9201514103433284750t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Cstack ) ).
% select_convs(8)
thf(fact_1234_graph_Ora__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,E5: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ X @ E5 ) ) ).
% graph.ra_refl
thf(fact_1235_graph_Ora__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E5: set_Product_prod_v_v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E5 )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ Y @ Z @ E5 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Z @ E5 ) ) ) ) ).
% graph.ra_trans
thf(fact_1236_update__convs_I8_J,axiom,
! [Cstack2: list_v > list_v,Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl7876664385711583351t_unit @ Cstack2 @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ ( Cstack2 @ Cstack ) @ More ) ) ).
% update_convs(8)
thf(fact_1237_select__convs_I6_J,axiom,
! [Root: v,S6: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl2536197123907397897t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S6 @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Sccs ) ).
% select_convs(6)
thf(fact_1238_Int__Diff__disjoint,axiom,
! [A4: set_Product_unit,B4: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ ( inf_in4660618365625256667t_unit @ A4 @ B4 ) @ ( minus_6452836326544984404t_unit @ A4 @ B4 ) )
= bot_bo3957492148770167129t_unit ) ).
% Int_Diff_disjoint
thf(fact_1239_Int__Diff__disjoint,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) @ ( minus_4183494784930505774od_v_v @ A4 @ B4 ) )
= bot_bo723834152578015283od_v_v ) ).
% Int_Diff_disjoint
thf(fact_1240_Int__Diff__disjoint,axiom,
! [A4: set_o,B4: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ A4 @ B4 ) @ ( minus_minus_set_o @ A4 @ B4 ) )
= bot_bot_set_o ) ).
% Int_Diff_disjoint
thf(fact_1241_Int__Diff__disjoint,axiom,
! [A4: set_set_v,B4: set_set_v] :
( ( inf_inf_set_set_v @ ( inf_inf_set_set_v @ A4 @ B4 ) @ ( minus_7228012346218142266_set_v @ A4 @ B4 ) )
= bot_bot_set_set_v ) ).
% Int_Diff_disjoint
thf(fact_1242_Int__Diff__disjoint,axiom,
! [A4: set_v,B4: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A4 @ B4 ) @ ( minus_minus_set_v @ A4 @ B4 ) )
= bot_bot_set_v ) ).
% Int_Diff_disjoint
thf(fact_1243_Diff__triv,axiom,
! [A4: set_Product_unit,B4: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A4 @ B4 )
= bot_bo3957492148770167129t_unit )
=> ( ( minus_6452836326544984404t_unit @ A4 @ B4 )
= A4 ) ) ).
% Diff_triv
thf(fact_1244_Diff__triv,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A4 @ B4 )
= bot_bo723834152578015283od_v_v )
=> ( ( minus_4183494784930505774od_v_v @ A4 @ B4 )
= A4 ) ) ).
% Diff_triv
thf(fact_1245_Diff__triv,axiom,
! [A4: set_o,B4: set_o] :
( ( ( inf_inf_set_o @ A4 @ B4 )
= bot_bot_set_o )
=> ( ( minus_minus_set_o @ A4 @ B4 )
= A4 ) ) ).
% Diff_triv
thf(fact_1246_Diff__triv,axiom,
! [A4: set_set_v,B4: set_set_v] :
( ( ( inf_inf_set_set_v @ A4 @ B4 )
= bot_bot_set_set_v )
=> ( ( minus_7228012346218142266_set_v @ A4 @ B4 )
= A4 ) ) ).
% Diff_triv
thf(fact_1247_Diff__triv,axiom,
! [A4: set_v,B4: set_v] :
( ( ( inf_inf_set_v @ A4 @ B4 )
= bot_bot_set_v )
=> ( ( minus_minus_set_v @ A4 @ B4 )
= A4 ) ) ).
% Diff_triv
thf(fact_1248_Un__Int__assoc__eq,axiom,
! [A4: set_set_v,B4: set_set_v,C2: set_set_v] :
( ( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A4 @ B4 ) @ C2 )
= ( inf_inf_set_set_v @ A4 @ ( sup_sup_set_set_v @ B4 @ C2 ) ) )
= ( ord_le5216385588623774835_set_v @ C2 @ A4 ) ) ).
% Un_Int_assoc_eq
thf(fact_1249_Un__Int__assoc__eq,axiom,
! [A4: set_v,B4: set_v,C2: set_v] :
( ( ( sup_sup_set_v @ ( inf_inf_set_v @ A4 @ B4 ) @ C2 )
= ( inf_inf_set_v @ A4 @ ( sup_sup_set_v @ B4 @ C2 ) ) )
= ( ord_less_eq_set_v @ C2 @ A4 ) ) ).
% Un_Int_assoc_eq
thf(fact_1250_Un__Int__assoc__eq,axiom,
! [A4: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A4 @ B4 ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ A4 @ ( sup_su414716646722978715od_v_v @ B4 @ C2 ) ) )
= ( ord_le7336532860387713383od_v_v @ C2 @ A4 ) ) ).
% Un_Int_assoc_eq
thf(fact_1251_Un__Diff__Int,axiom,
! [A4: set_v,B4: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ A4 @ B4 ) @ ( inf_inf_set_v @ A4 @ B4 ) )
= A4 ) ).
% Un_Diff_Int
thf(fact_1252_pre__dfs__implies__post__dfs,axiom,
! [V2: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl36166008131615352t_unit @ successors @ V2 @ E )
=> ( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V2 @ E ) ) )
=> ( ( sCC_Bl6082031138996704384t_unit @ successors @ V2
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V2 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V2 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V2 @ bot_bot_set_v ) )
@ E ) ) )
@ ( sCC_Bloemen_dfss_v @ successors @ V2
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V2 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V2 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V2 @ bot_bot_set_v ) )
@ E ) ) ) ) )
=> ( sCC_Bl8953792750115413617t_unit @ successors @ V2 @ E @ ( sCC_Bloemen_dfs_v @ successors @ V2 @ E ) ) ) ) ) ).
% pre_dfs_implies_post_dfs
thf(fact_1253_dfs__dfss_Odomintros_I1_J,axiom,
! [V2: v,E: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors )
@ ( sum_In5289330923152326972t_unit
@ ( produc3862955338007567901t_unit @ V2
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V2 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V2 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( insert_v @ V2 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ E ) ) ) ) ) )
=> ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V2 @ E ) ) ) ) ).
% dfs_dfss.domintros(1)
thf(fact_1254_pre__dfss__post__dfs__pre__dfss,axiom,
! [V2: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V2 @ E )
=> ( ( member_v2 @ W @ ( successors @ V2 ) )
=> ( ~ ( member_v2 @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ W @ E @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) )
=> ( sCC_Bl1748261141445803503t_unit @ successors @ V2
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X4: v] : ( if_set_v @ ( X4 = V2 ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) @ V2 ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) @ X4 ) )
@ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) ) ) ) ) ) ) ).
% pre_dfss_post_dfs_pre_dfss
thf(fact_1255_pre__dfss__unite__pre__dfss,axiom,
! [V2: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V2 @ E )
=> ( ( member_v2 @ W @ ( successors @ V2 ) )
=> ( ~ ( member_v2 @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) )
=> ( ( member_v2 @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v2 @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl1748261141445803503t_unit @ successors @ V2
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X4: v] : ( if_set_v @ ( X4 = V2 ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_unite_v @ V2 @ W @ E ) @ V2 ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_unite_v @ V2 @ W @ E ) @ X4 ) )
@ ( sCC_Bloemen_unite_v @ V2 @ W @ E ) ) ) ) ) ) ) ) ).
% pre_dfss_unite_pre_dfss
thf(fact_1256_dfs_Opsimps,axiom,
! [V2: v,E: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V2 @ E ) ) )
=> ( ( sCC_Bloemen_dfs_v @ successors @ V2 @ E )
= ( if_SCC4926449794303880475t_unit
@ ( V2
= ( hd_v
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V2
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V2 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V2 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V2 @ bot_bot_set_v ) )
@ E ) ) ) ) ) ) )
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V2
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V2 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V2 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V2 @ bot_bot_set_v ) )
@ E ) ) ) ) ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V2
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V2 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V2 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V2 @ bot_bot_set_v ) )
@ E ) ) ) ) ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] :
( sup_sup_set_v
@ ( sCC_Bl157864678168468314t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V2
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V2 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V2 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V2 @ bot_bot_set_v ) )
@ E ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V2
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V2 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V2 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V2 @ bot_bot_set_v ) )
@ E ) ) ) )
@ V2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] :
( sup_sup_set_set_v
@ ( sCC_Bl2536197123907397897t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V2
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V2 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V2 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V2 @ bot_bot_set_v ) )
@ E ) ) ) ) )
@ ( insert_set_v
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V2
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V2 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V2 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V2 @ bot_bot_set_v ) )
@ E ) ) ) )
@ V2 )
@ bot_bot_set_set_v ) )
@ ( sCC_Bloemen_dfss_v @ successors @ V2
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V2 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V2 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V2 @ bot_bot_set_v ) )
@ E ) ) ) ) ) ) ) )
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V2
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V2 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V2 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V2 @ bot_bot_set_v ) )
@ E ) ) ) ) ) )
@ ( sCC_Bloemen_dfss_v @ successors @ V2
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V2 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V2 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V2 @ bot_bot_set_v ) )
@ E ) ) ) ) ) ) ) ) ).
% dfs.psimps
thf(fact_1257_dfss_Opsimps,axiom,
! [V2: v,E: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In5289330923152326972t_unit @ ( produc3862955338007567901t_unit @ V2 @ E ) ) )
=> ( ( sCC_Bloemen_dfss_v @ successors @ V2 @ E )
= ( if_SCC4926449794303880475t_unit
@ ( ( minus_minus_set_v @ ( successors @ V2 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) )
= bot_bot_set_v )
@ E
@ ( sCC_Bloemen_dfss_v @ successors @ V2
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X4: v] :
( if_set_v @ ( X4 = V2 )
@ ( sup_sup_set_v
@ ( sCC_Bl3795065053823578884t_unit
@ ( if_SCC4926449794303880475t_unit
@ ( member_v2
@ ( fChoice_v
@ ^ [Y4: v] : ( member_v2 @ Y4 @ ( minus_minus_set_v @ ( successors @ V2 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E ) )
@ E
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v2
@ ( fChoice_v
@ ^ [Y4: v] : ( member_v2 @ Y4 @ ( minus_minus_set_v @ ( successors @ V2 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [Y4: v] : ( member_v2 @ Y4 @ ( minus_minus_set_v @ ( successors @ V2 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) ) ) )
@ E )
@ ( sCC_Bloemen_unite_v @ V2
@ ( fChoice_v
@ ^ [Y4: v] : ( member_v2 @ Y4 @ ( minus_minus_set_v @ ( successors @ V2 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) ) ) )
@ E ) ) )
@ V2 )
@ ( insert_v
@ ( fChoice_v
@ ^ [Y4: v] : ( member_v2 @ Y4 @ ( minus_minus_set_v @ ( successors @ V2 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) ) ) )
@ bot_bot_set_v ) )
@ ( sCC_Bl3795065053823578884t_unit
@ ( if_SCC4926449794303880475t_unit
@ ( member_v2
@ ( fChoice_v
@ ^ [Y4: v] : ( member_v2 @ Y4 @ ( minus_minus_set_v @ ( successors @ V2 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E ) )
@ E
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v2
@ ( fChoice_v
@ ^ [Y4: v] : ( member_v2 @ Y4 @ ( minus_minus_set_v @ ( successors @ V2 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [Y4: v] : ( member_v2 @ Y4 @ ( minus_minus_set_v @ ( successors @ V2 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) ) ) )
@ E )
@ ( sCC_Bloemen_unite_v @ V2
@ ( fChoice_v
@ ^ [Y4: v] : ( member_v2 @ Y4 @ ( minus_minus_set_v @ ( successors @ V2 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) ) ) )
@ E ) ) )
@ X4 ) )
@ ( if_SCC4926449794303880475t_unit
@ ( member_v2
@ ( fChoice_v
@ ^ [X4: v] : ( member_v2 @ X4 @ ( minus_minus_set_v @ ( successors @ V2 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E ) )
@ E
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v2
@ ( fChoice_v
@ ^ [X4: v] : ( member_v2 @ X4 @ ( minus_minus_set_v @ ( successors @ V2 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [X4: v] : ( member_v2 @ X4 @ ( minus_minus_set_v @ ( successors @ V2 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) ) ) )
@ E )
@ ( sCC_Bloemen_unite_v @ V2
@ ( fChoice_v
@ ^ [X4: v] : ( member_v2 @ X4 @ ( minus_minus_set_v @ ( successors @ V2 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) ) ) )
@ E ) ) ) ) ) ) ) ) ).
% dfss.psimps
thf(fact_1258_unite__S__equal,axiom,
! [E: sCC_Bl1394983891496994913t_unit,W: v,V2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v2 @ W @ ( successors @ V2 ) )
=> ( ~ ( member_v2 @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V2 ) )
=> ( ( member_v2 @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v2 @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N3: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V2 @ W @ E ) @ N3 ) )
& ( member_v2 @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V2 @ W @ E ) ) ) ) ) ) )
= ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N3: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E @ N3 ) )
& ( member_v2 @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) ) ) ) ) ) ) ) ).
% unite_S_equal
thf(fact_1259_unite__def,axiom,
( sCC_Bloemen_unite_v
= ( ^ [V4: v,W2: v,E8: sCC_Bl1394983891496994913t_unit] :
( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] :
( dropWhile_v
@ ^ [X4: v] :
~ ( member_v2 @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ X4 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) )
@ ( sCC_Bl3155122997657187039t_unit
@ ^ [Uu: v > set_v,X4: v] :
( if_set_v
@ ( member_v2 @ X4
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uv: set_v] :
? [Y4: v] :
( ( Uv
= ( sCC_Bl1280885523602775798t_unit @ E8 @ Y4 ) )
& ( member_v2 @ Y4
@ ( sup_sup_set_v
@ ( set_v2
@ ( takeWhile_v
@ ^ [Z2: v] :
~ ( member_v2 @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
@ ( insert_v
@ ( hd_v
@ ( dropWhile_v
@ ^ [Z2: v] :
~ ( member_v2 @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
@ bot_bot_set_v ) ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uv: set_v] :
? [Y4: v] :
( ( Uv
= ( sCC_Bl1280885523602775798t_unit @ E8 @ Y4 ) )
& ( member_v2 @ Y4
@ ( sup_sup_set_v
@ ( set_v2
@ ( takeWhile_v
@ ^ [Z2: v] :
~ ( member_v2 @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
@ ( insert_v
@ ( hd_v
@ ( dropWhile_v
@ ^ [Z2: v] :
~ ( member_v2 @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
@ bot_bot_set_v ) ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit @ E8 @ X4 ) )
@ E8 ) ) ) ) ).
% unite_def
thf(fact_1260_inf__unit__def,axiom,
( inf_inf_Product_unit
= ( ^ [Uu2: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% inf_unit_def
thf(fact_1261_top__unit__def,axiom,
top_top_Product_unit = product_Unity ).
% top_unit_def
thf(fact_1262_old_Ounit_Oexhaust,axiom,
! [Y: product_unit] : ( Y = product_Unity ) ).
% old.unit.exhaust
thf(fact_1263_eq__sym__Unity__conv,axiom,
! [X: $o] :
( ( X
= ( product_Unity = product_Unity ) )
= X ) ).
% eq_sym_Unity_conv
thf(fact_1264_unit__all__impI,axiom,
! [P2: product_unit > $o,Q: product_unit > $o] :
( ( ( P2 @ product_Unity )
=> ( Q @ product_Unity ) )
=> ! [X5: product_unit] :
( ( P2 @ X5 )
=> ( Q @ X5 ) ) ) ).
% unit_all_impI
thf(fact_1265_sup__unit__def,axiom,
( sup_sup_Product_unit
= ( ^ [Uu2: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% sup_unit_def
thf(fact_1266_bot__unit__def,axiom,
bot_bot_Product_unit = product_Unity ).
% bot_unit_def
thf(fact_1267_UNIV__unit,axiom,
( top_to1996260823553986621t_unit
= ( insert_Product_unit @ product_Unity @ bot_bo3957492148770167129t_unit ) ) ).
% UNIV_unit
thf(fact_1268_Sup__unit__def,axiom,
( comple4687483117567863418t_unit
= ( ^ [Uu2: set_Product_unit] : product_Unity ) ) ).
% Sup_unit_def
thf(fact_1269_default__unit__def,axiom,
defaul566961228789861419t_unit = product_Unity ).
% default_unit_def
thf(fact_1270_UNIV__bool,axiom,
( top_top_set_o
= ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% UNIV_bool
thf(fact_1271_Inf__unit__def,axiom,
( comple2584293577114468500t_unit
= ( ^ [Uu2: set_Product_unit] : product_Unity ) ) ).
% Inf_unit_def
thf(fact_1272_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
% Helper facts (6)
thf(help_fChoice_1_1_fChoice_001tf__v_T,axiom,
! [P2: v > $o] :
( ( P2 @ ( fChoice_v @ P2 ) )
= ( ? [X7: v] : ( P2 @ X7 ) ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X: set_v,Y: set_v] :
( ( if_set_v @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_Itf__v_J_T,axiom,
! [X: set_v,Y: set_v] :
( ( if_set_v @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_T,axiom,
! [X: sCC_Bl1394983891496994913t_unit,Y: sCC_Bl1394983891496994913t_unit] :
( ( if_SCC4926449794303880475t_unit @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__SCC____Bloemen____Sequential__Oenv__Oenv____ext_Itf__v_Mt__Product____Type__Ounit_J_T,axiom,
! [X: sCC_Bl1394983891496994913t_unit,Y: sCC_Bl1394983891496994913t_unit] :
( ( if_SCC4926449794303880475t_unit @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
$false ).
%------------------------------------------------------------------------------