TPTP Problem File: SLH0854^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : SCC_Bloemen_Sequential/0000_SCC_Bloemen_Sequential/prob_02470_084957__6357718_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1309 ( 681 unt; 163 typ; 0 def)
% Number of atoms : 2956 (1301 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 11218 ( 308 ~; 51 |; 247 &;9606 @)
% ( 0 <=>;1006 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 6 avg)
% Number of types : 17 ( 16 usr)
% Number of type conns : 512 ( 512 >; 0 *; 0 +; 0 <<)
% Number of symbols : 150 ( 147 usr; 17 con; 0-4 aty)
% Number of variables : 3323 ( 506 ^;2748 !; 69 ?;3323 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:54:08.846
%------------------------------------------------------------------------------
% Could-be-implicit typings (16)
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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_001t__Set__Oset_Itf__v_J_001t__Product____Type__Ounit,type,
sCC_Bl1947314421851746340t_unit: sCC_Bl337355980704484737t_unit > set_v > set_set_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ovsuccs_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl3795065053823578884t_unit: sCC_Bl1394983891496994913t_unit > v > set_v ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ovsuccs__update_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit,type,
sCC_Bl2958793191457503513t_unit: ( ( product_prod_v_v > set_Product_prod_v_v ) > product_prod_v_v > set_Product_prod_v_v ) > sCC_Bl7326425374436813197t_unit > sCC_Bl7326425374436813197t_unit ).
thf(sy_c_SCC__Bloemen__Sequential_Oenv_Ovsuccs__update_001t__Set__Oset_Itf__v_J_001t__Product____Type__Ounit,type,
sCC_Bl8982985682160534541t_unit: ( ( set_v > set_set_v ) > set_v > set_set_v ) > sCC_Bl337355980704484737t_unit > sCC_Bl337355980704484737t_unit ).
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_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__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_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
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__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_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__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 ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Opost__dfs_001tf__v_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
sCC_Bl8953792750115413617t_unit: ( v > set_v ) > v > sCC_Bl1394983891496994913t_unit > sCC_Bl1394983891496994913t_unit > $o ).
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,
sCC_Bl3607325323686918683t_unit: ( product_prod_v_v > set_Product_prod_v_v ) > product_prod_v_v > sCC_Bl7326425374436813197t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Opre__dfss_001t__Set__Oset_Itf__v_J_001t__Product____Type__Ounit,type,
sCC_Bl7011918439528173327t_unit: ( set_v > set_set_v ) > set_v > sCC_Bl337355980704484737t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Opre__dfss_001tf__v_001t__Product____Type__Ounit,type,
sCC_Bl1748261141445803503t_unit: ( v > set_v ) > v > sCC_Bl1394983891496994913t_unit > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Oreachable_001t__Product____Type__Oprod_Itf__v_Mtf__v_J,type,
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__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_001tf__v,type,
sCC_Bl770211535891879572_end_v: ( v > set_v ) > v > v > $o ).
thf(sy_c_SCC__Bloemen__Sequential_Ograph_Osub__env_001t__Product____Type__Oprod_Itf__v_Mtf__v_J_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
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_001t__Set__Oset_Itf__v_J,type,
sCC_Bl2303001396873455969_set_v: set_v > set_v > sCC_Bl337355980704484737t_unit > sCC_Bl337355980704484737t_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_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__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
collec8263177866097347122od_v_v: ( set_Product_prod_v_v > $o ) > set_se8455005133513928103od_v_v ).
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_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__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
insert7504383016908236695od_v_v: set_Product_prod_v_v > set_se8455005133513928103od_v_v > set_se8455005133513928103od_v_v ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_Itf__v_J_J,type,
insert_set_set_v: set_set_v > set_set_set_v > set_set_set_v ).
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_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_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__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J,type,
member8406446414694345712od_v_v: set_Product_prod_v_v > set_se8455005133513928103od_v_v > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__v_J_J,type,
member_set_set_v: set_set_v > set_set_set_v > $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_v: v > set_v > $o ).
thf(sy_v_e,type,
e: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_e1,type,
e1: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_e_H,type,
e2: sCC_Bl1394983891496994913t_unit ).
thf(sy_v_e_H_H,type,
e3: 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 (1135)
thf(fact_0_dfs__dfss__rel_Ocong,axiom,
sCC_Bl907557413677168252_rel_v = sCC_Bl907557413677168252_rel_v ).
% dfs_dfss_rel.cong
thf(fact_1_e_H_H__def,axiom,
( e3
= ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ e2 ) ) ).
% e''_def
thf(fact_2_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_3_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_4_precedes__refl,axiom,
! [X: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ X @ Xs )
= ( member_v @ X @ ( set_v2 @ Xs ) ) ) ).
% precedes_refl
thf(fact_5_calculation_I5_J,axiom,
! [N: v,M: v] :
( ( member_v @ M
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ N ) )
= ( ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ N )
= ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ M ) ) ) ).
% calculation(5)
thf(fact_6_calculation_I11_J,axiom,
! [N: v,M: v] :
( ( sCC_Bl4022239298816431255edes_v @ N @ M
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) )
=> ( sCC_Bl4022239298816431255edes_v @ N @ M
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ) ).
% calculation(11)
thf(fact_7_True,axiom,
( v2
= ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) ) ) ).
% True
thf(fact_8_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_9_precedes__mem_I2_J,axiom,
! [X: v,Y: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( member_v @ Y @ ( set_v2 @ Xs ) ) ) ).
% precedes_mem(2)
thf(fact_10_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_11_precedes__mem_I1_J,axiom,
! [X: v,Y: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( member_v @ X @ ( set_v2 @ Xs ) ) ) ).
% precedes_mem(1)
thf(fact_12_fold__congs_I8_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: list_v,F: list_v > list_v,F2: list_v > list_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl9201514103433284750t_unit @ R2 )
= V )
=> ( ! [V2: list_v] :
( ( V = V2 )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl7876664385711583351t_unit @ F @ R )
= ( sCC_Bl7876664385711583351t_unit @ F2 @ R2 ) ) ) ) ) ).
% fold_congs(8)
thf(fact_13_fold__congs_I7_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: list_v,F: list_v > list_v,F2: list_v > list_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ R2 )
= V )
=> ( ! [V2: list_v] :
( ( V = V2 )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl349061681862590396t_unit @ F @ R )
= ( sCC_Bl349061681862590396t_unit @ F2 @ R2 ) ) ) ) ) ).
% fold_congs(7)
thf(fact_14_fold__congs_I6_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: set_set_v,F: set_set_v > set_set_v,F2: set_set_v > set_set_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl2536197123907397897t_unit @ R2 )
= V )
=> ( ! [V2: set_set_v] :
( ( V = V2 )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl6816368539212994290t_unit @ F @ R )
= ( sCC_Bl6816368539212994290t_unit @ F2 @ R2 ) ) ) ) ) ).
% fold_congs(6)
thf(fact_15_fold__congs_I3_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: set_v,F: set_v > set_v,F2: set_v > set_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl157864678168468314t_unit @ R2 )
= V )
=> ( ! [V2: set_v] :
( ( V = V2 )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl2708505634401380163t_unit @ F @ R )
= ( sCC_Bl2708505634401380163t_unit @ F2 @ R2 ) ) ) ) ) ).
% fold_congs(3)
thf(fact_16_unfold__congs_I8_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: list_v,F: list_v > list_v,F2: list_v > list_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl9201514103433284750t_unit @ R2 )
= V )
=> ( ! [V2: list_v] :
( ( V2 = V )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl7876664385711583351t_unit @ F @ R )
= ( sCC_Bl7876664385711583351t_unit @ F2 @ R2 ) ) ) ) ) ).
% unfold_congs(8)
thf(fact_17_unfold__congs_I7_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: list_v,F: list_v > list_v,F2: list_v > list_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl8828226123343373779t_unit @ R2 )
= V )
=> ( ! [V2: list_v] :
( ( V2 = V )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl349061681862590396t_unit @ F @ R )
= ( sCC_Bl349061681862590396t_unit @ F2 @ R2 ) ) ) ) ) ).
% unfold_congs(7)
thf(fact_18_unfold__congs_I6_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: set_set_v,F: set_set_v > set_set_v,F2: set_set_v > set_set_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl2536197123907397897t_unit @ R2 )
= V )
=> ( ! [V2: set_set_v] :
( ( V2 = V )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl6816368539212994290t_unit @ F @ R )
= ( sCC_Bl6816368539212994290t_unit @ F2 @ R2 ) ) ) ) ) ).
% unfold_congs(6)
thf(fact_19_unfold__congs_I3_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: set_v,F: set_v > set_v,F2: set_v > set_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl157864678168468314t_unit @ R2 )
= V )
=> ( ! [V2: set_v] :
( ( V2 = V )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl2708505634401380163t_unit @ F @ R )
= ( sCC_Bl2708505634401380163t_unit @ F2 @ R2 ) ) ) ) ) ).
% unfold_congs(3)
thf(fact_20_stack2,axiom,
( ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) )
= ( sCC_Bl8828226123343373779t_unit @ e ) ) ).
% stack2
thf(fact_21__092_060open_062v_A_092_060in_062_Avisited_A_Ie_H_092_060lparr_062sccs_A_058_061_Asccs_Ae_H_A_092_060union_062_A_123_092_060S_062_Ae_H_Av_125_M_Aexplored_A_058_061_Aexplored_Ae_H_A_092_060union_062_A_092_060S_062_Ae_H_Av_M_Astack_A_058_061_Atl_A_Istack_Ae_H_J_M_Acstack_A_058_061_Atl_A_Icstack_Ae_H_J_092_060rparr_062_J_092_060close_062,axiom,
( member_v @ v2
@ ( sCC_Bl4645233313691564917t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ).
% \<open>v \<in> visited (e'\<lparr>sccs := sccs e' \<union> {\<S> e' v}, explored := explored e' \<union> \<S> e' v, stack := tl (stack e'), cstack := tl (cstack e')\<rparr>)\<close>
thf(fact_22_calculation_I8_J,axiom,
( ( comple2307003700295860064_set_v
@ ( sCC_Bl2536197123907397897t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) )
= ( sCC_Bl157864678168468314t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ).
% calculation(8)
thf(fact_23_calculation_I9_J,axiom,
( distinct_v
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ).
% calculation(9)
thf(fact_24_calculation_I2_J,axiom,
( distinct_v
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ).
% calculation(2)
thf(fact_25_singleton__conv,axiom,
! [A: set_v] :
( ( collect_set_v
@ ^ [X2: set_v] : ( X2 = A ) )
= ( insert_set_v @ A @ bot_bot_set_set_v ) ) ).
% singleton_conv
thf(fact_26_singleton__conv,axiom,
! [A: v] :
( ( collect_v
@ ^ [X2: v] : ( X2 = A ) )
= ( insert_v @ A @ bot_bot_set_v ) ) ).
% singleton_conv
thf(fact_27_singleton__conv,axiom,
! [A: product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] : ( X2 = A ) )
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% singleton_conv
thf(fact_28_singleton__conv2,axiom,
! [A: set_v] :
( ( collect_set_v
@ ( ^ [Y2: set_v,Z: set_v] : ( Y2 = Z )
@ A ) )
= ( insert_set_v @ A @ bot_bot_set_set_v ) ) ).
% singleton_conv2
thf(fact_29_singleton__conv2,axiom,
! [A: v] :
( ( collect_v
@ ( ^ [Y2: v,Z: v] : ( Y2 = Z )
@ A ) )
= ( insert_v @ A @ bot_bot_set_v ) ) ).
% singleton_conv2
thf(fact_30_singleton__conv2,axiom,
! [A: product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ( ^ [Y2: product_prod_v_v,Z: product_prod_v_v] : ( Y2 = Z )
@ A ) )
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% singleton_conv2
thf(fact_31_subenv,axiom,
( sCC_Bl5768913643336123637t_unit @ e
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ).
% subenv
thf(fact_32_Un__insert__left,axiom,
! [A: v,B: set_v,C: set_v] :
( ( sup_sup_set_v @ ( insert_v @ A @ B ) @ C )
= ( insert_v @ A @ ( sup_sup_set_v @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_33_Un__insert__left,axiom,
! [A: set_v,B: set_set_v,C: set_set_v] :
( ( sup_sup_set_set_v @ ( insert_set_v @ A @ B ) @ C )
= ( insert_set_v @ A @ ( sup_sup_set_set_v @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_34_Un__insert__left,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_35_Un__insert__right,axiom,
! [A2: set_v,A: v,B: set_v] :
( ( sup_sup_set_v @ A2 @ ( insert_v @ A @ B ) )
= ( insert_v @ A @ ( sup_sup_set_v @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_36_Un__insert__right,axiom,
! [A2: set_set_v,A: set_v,B: set_set_v] :
( ( sup_sup_set_set_v @ A2 @ ( insert_set_v @ A @ B ) )
= ( insert_set_v @ A @ ( sup_sup_set_set_v @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_37_Un__insert__right,axiom,
! [A2: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_38_Un__empty,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( ( sup_sup_set_set_v @ A2 @ B )
= bot_bot_set_set_v )
= ( ( A2 = bot_bot_set_set_v )
& ( B = bot_bot_set_set_v ) ) ) ).
% Un_empty
thf(fact_39_Un__empty,axiom,
! [A2: set_v,B: set_v] :
( ( ( sup_sup_set_v @ A2 @ B )
= bot_bot_set_v )
= ( ( A2 = bot_bot_set_v )
& ( B = bot_bot_set_v ) ) ) ).
% Un_empty
thf(fact_40_Un__empty,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A2 @ B )
= bot_bo723834152578015283od_v_v )
= ( ( A2 = bot_bo723834152578015283od_v_v )
& ( B = bot_bo723834152578015283od_v_v ) ) ) ).
% Un_empty
thf(fact_41_singletonI,axiom,
! [A: set_v] : ( member_set_v @ A @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) ).
% singletonI
thf(fact_42_singletonI,axiom,
! [A: v] : ( member_v @ A @ ( insert_v @ A @ bot_bot_set_v ) ) ).
% singletonI
thf(fact_43_singletonI,axiom,
! [A: product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% singletonI
thf(fact_44_empty__Collect__eq,axiom,
! [P: set_v > $o] :
( ( bot_bot_set_set_v
= ( collect_set_v @ P ) )
= ( ! [X2: set_v] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_45_empty__Collect__eq,axiom,
! [P: v > $o] :
( ( bot_bot_set_v
= ( collect_v @ P ) )
= ( ! [X2: v] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_46_empty__Collect__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ P ) )
= ( ! [X2: product_prod_v_v] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_47_Collect__empty__eq,axiom,
! [P: set_v > $o] :
( ( ( collect_set_v @ P )
= bot_bot_set_set_v )
= ( ! [X2: set_v] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_48_Collect__empty__eq,axiom,
! [P: v > $o] :
( ( ( collect_v @ P )
= bot_bot_set_v )
= ( ! [X2: v] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_49_Collect__empty__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( ( collec140062887454715474od_v_v @ P )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_50_all__not__in__conv,axiom,
! [A2: set_set_v] :
( ( ! [X2: set_v] :
~ ( member_set_v @ X2 @ A2 ) )
= ( A2 = bot_bot_set_set_v ) ) ).
% all_not_in_conv
thf(fact_51_all__not__in__conv,axiom,
! [A2: set_v] :
( ( ! [X2: v] :
~ ( member_v @ X2 @ A2 ) )
= ( A2 = bot_bot_set_v ) ) ).
% all_not_in_conv
thf(fact_52_all__not__in__conv,axiom,
! [A2: set_Product_prod_v_v] :
( ( ! [X2: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ X2 @ A2 ) )
= ( A2 = bot_bo723834152578015283od_v_v ) ) ).
% all_not_in_conv
thf(fact_53_empty__iff,axiom,
! [C2: set_v] :
~ ( member_set_v @ C2 @ bot_bot_set_set_v ) ).
% empty_iff
thf(fact_54_empty__iff,axiom,
! [C2: v] :
~ ( member_v @ C2 @ bot_bot_set_v ) ).
% empty_iff
thf(fact_55_empty__iff,axiom,
! [C2: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ C2 @ bot_bo723834152578015283od_v_v ) ).
% empty_iff
thf(fact_56_insert__absorb2,axiom,
! [X: set_v,A2: set_set_v] :
( ( insert_set_v @ X @ ( insert_set_v @ X @ A2 ) )
= ( insert_set_v @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_57_insert__absorb2,axiom,
! [X: v,A2: set_v] :
( ( insert_v @ X @ ( insert_v @ X @ A2 ) )
= ( insert_v @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_58_insert__absorb2,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( insert1338601472111419319od_v_v @ X @ A2 ) )
= ( insert1338601472111419319od_v_v @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_59_insert__iff,axiom,
! [A: set_v,B2: set_v,A2: set_set_v] :
( ( member_set_v @ A @ ( insert_set_v @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_set_v @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_60_insert__iff,axiom,
! [A: v,B2: v,A2: set_v] :
( ( member_v @ A @ ( insert_v @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_v @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_61_insert__iff,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member7453568604450474000od_v_v @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_62_insertCI,axiom,
! [A: set_v,B: set_set_v,B2: set_v] :
( ( ~ ( member_set_v @ A @ B )
=> ( A = B2 ) )
=> ( member_set_v @ A @ ( insert_set_v @ B2 @ B ) ) ) ).
% insertCI
thf(fact_63_insertCI,axiom,
! [A: v,B: set_v,B2: v] :
( ( ~ ( member_v @ A @ B )
=> ( A = B2 ) )
=> ( member_v @ A @ ( insert_v @ B2 @ B ) ) ) ).
% insertCI
thf(fact_64_insertCI,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ A @ B )
=> ( A = B2 ) )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).
% insertCI
thf(fact_65_Un__iff,axiom,
! [C2: v,A2: set_v,B: set_v] :
( ( member_v @ C2 @ ( sup_sup_set_v @ A2 @ B ) )
= ( ( member_v @ C2 @ A2 )
| ( member_v @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_66_Un__iff,axiom,
! [C2: set_v,A2: set_set_v,B: set_set_v] :
( ( member_set_v @ C2 @ ( sup_sup_set_set_v @ A2 @ B ) )
= ( ( member_set_v @ C2 @ A2 )
| ( member_set_v @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_67_Un__iff,axiom,
! [C2: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A2 @ B ) )
= ( ( member7453568604450474000od_v_v @ C2 @ A2 )
| ( member7453568604450474000od_v_v @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_68_UnCI,axiom,
! [C2: v,B: set_v,A2: set_v] :
( ( ~ ( member_v @ C2 @ B )
=> ( member_v @ C2 @ A2 ) )
=> ( member_v @ C2 @ ( sup_sup_set_v @ A2 @ B ) ) ) ).
% UnCI
thf(fact_69_UnCI,axiom,
! [C2: set_v,B: set_set_v,A2: set_set_v] :
( ( ~ ( member_set_v @ C2 @ B )
=> ( member_set_v @ C2 @ A2 ) )
=> ( member_set_v @ C2 @ ( sup_sup_set_set_v @ A2 @ B ) ) ) ).
% UnCI
thf(fact_70_UnCI,axiom,
! [C2: product_prod_v_v,B: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ C2 @ B )
=> ( member7453568604450474000od_v_v @ C2 @ A2 ) )
=> ( member7453568604450474000od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A2 @ B ) ) ) ).
% UnCI
thf(fact_71_v,axiom,
~ ( member_v @ v2 @ ( sCC_Bl4645233313691564917t_unit @ e ) ) ).
% v
thf(fact_72__092_060open_062sub__env_Ae_Ae_H_092_060close_062,axiom,
sCC_Bl5768913643336123637t_unit @ e @ e2 ).
% \<open>sub_env e e'\<close>
thf(fact_73_calculation_I6_J,axiom,
! [N: v] :
( ~ ( member_v @ N
@ ( sCC_Bl4645233313691564917t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ N )
= ( insert_v @ N @ bot_bot_set_v ) ) ) ).
% calculation(6)
thf(fact_74_sub,axiom,
sCC_Bl5768913643336123637t_unit @ e @ e1 ).
% sub
thf(fact_75_precedes__antisym,axiom,
! [X: v,Y: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( ( sCC_Bl4022239298816431255edes_v @ Y @ X @ Xs )
=> ( ( distinct_v @ Xs )
=> ( X = Y ) ) ) ) ).
% precedes_antisym
thf(fact_76_precedes__antisym,axiom,
! [X: product_prod_v_v,Y: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ Xs )
=> ( ( sCC_Bl2026170059108282219od_v_v @ Y @ X @ Xs )
=> ( ( distin6159370996967099744od_v_v @ Xs )
=> ( X = Y ) ) ) ) ).
% precedes_antisym
thf(fact_77_precedes__trans,axiom,
! [X: v,Y: v,Xs: list_v,Z2: v] :
( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Xs )
=> ( ( sCC_Bl4022239298816431255edes_v @ Y @ Z2 @ Xs )
=> ( ( distinct_v @ Xs )
=> ( sCC_Bl4022239298816431255edes_v @ X @ Z2 @ Xs ) ) ) ) ).
% precedes_trans
thf(fact_78_precedes__trans,axiom,
! [X: product_prod_v_v,Y: product_prod_v_v,Xs: list_P7986770385144383213od_v_v,Z2: product_prod_v_v] :
( ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ Xs )
=> ( ( sCC_Bl2026170059108282219od_v_v @ Y @ Z2 @ Xs )
=> ( ( distin6159370996967099744od_v_v @ Xs )
=> ( sCC_Bl2026170059108282219od_v_v @ X @ Z2 @ Xs ) ) ) ) ).
% precedes_trans
thf(fact_79_mem__Collect__eq,axiom,
! [A: v,P: v > $o] :
( ( member_v @ A @ ( collect_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_80_mem__Collect__eq,axiom,
! [A: product_prod_v_v,P: product_prod_v_v > $o] :
( ( member7453568604450474000od_v_v @ A @ ( collec140062887454715474od_v_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_81_mem__Collect__eq,axiom,
! [A: set_v,P: set_v > $o] :
( ( member_set_v @ A @ ( collect_set_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_82_Collect__mem__eq,axiom,
! [A2: set_v] :
( ( collect_v
@ ^ [X2: v] : ( member_v @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_83_Collect__mem__eq,axiom,
! [A2: set_Product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_84_Collect__mem__eq,axiom,
! [A2: set_set_v] :
( ( collect_set_v
@ ^ [X2: set_v] : ( member_set_v @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_85_Collect__cong,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ! [X3: set_v] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_set_v @ P )
= ( collect_set_v @ Q ) ) ) ).
% Collect_cong
thf(fact_86_ex__in__conv,axiom,
! [A2: set_set_v] :
( ( ? [X2: set_v] : ( member_set_v @ X2 @ A2 ) )
= ( A2 != bot_bot_set_set_v ) ) ).
% ex_in_conv
thf(fact_87_ex__in__conv,axiom,
! [A2: set_v] :
( ( ? [X2: v] : ( member_v @ X2 @ A2 ) )
= ( A2 != bot_bot_set_v ) ) ).
% ex_in_conv
thf(fact_88_ex__in__conv,axiom,
! [A2: set_Product_prod_v_v] :
( ( ? [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A2 ) )
= ( A2 != bot_bo723834152578015283od_v_v ) ) ).
% ex_in_conv
thf(fact_89_equals0I,axiom,
! [A2: set_set_v] :
( ! [Y3: set_v] :
~ ( member_set_v @ Y3 @ A2 )
=> ( A2 = bot_bot_set_set_v ) ) ).
% equals0I
thf(fact_90_equals0I,axiom,
! [A2: set_v] :
( ! [Y3: v] :
~ ( member_v @ Y3 @ A2 )
=> ( A2 = bot_bot_set_v ) ) ).
% equals0I
thf(fact_91_equals0I,axiom,
! [A2: set_Product_prod_v_v] :
( ! [Y3: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ Y3 @ A2 )
=> ( A2 = bot_bo723834152578015283od_v_v ) ) ).
% equals0I
thf(fact_92_equals0D,axiom,
! [A2: set_set_v,A: set_v] :
( ( A2 = bot_bot_set_set_v )
=> ~ ( member_set_v @ A @ A2 ) ) ).
% equals0D
thf(fact_93_equals0D,axiom,
! [A2: set_v,A: v] :
( ( A2 = bot_bot_set_v )
=> ~ ( member_v @ A @ A2 ) ) ).
% equals0D
thf(fact_94_equals0D,axiom,
! [A2: set_Product_prod_v_v,A: product_prod_v_v] :
( ( A2 = bot_bo723834152578015283od_v_v )
=> ~ ( member7453568604450474000od_v_v @ A @ A2 ) ) ).
% equals0D
thf(fact_95_emptyE,axiom,
! [A: set_v] :
~ ( member_set_v @ A @ bot_bot_set_set_v ) ).
% emptyE
thf(fact_96_emptyE,axiom,
! [A: v] :
~ ( member_v @ A @ bot_bot_set_v ) ).
% emptyE
thf(fact_97_emptyE,axiom,
! [A: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ A @ bot_bo723834152578015283od_v_v ) ).
% emptyE
thf(fact_98_mk__disjoint__insert,axiom,
! [A: set_v,A2: set_set_v] :
( ( member_set_v @ A @ A2 )
=> ? [B3: set_set_v] :
( ( A2
= ( insert_set_v @ A @ B3 ) )
& ~ ( member_set_v @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_99_mk__disjoint__insert,axiom,
! [A: v,A2: set_v] :
( ( member_v @ A @ A2 )
=> ? [B3: set_v] :
( ( A2
= ( insert_v @ A @ B3 ) )
& ~ ( member_v @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_100_mk__disjoint__insert,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A2 )
=> ? [B3: set_Product_prod_v_v] :
( ( A2
= ( insert1338601472111419319od_v_v @ A @ B3 ) )
& ~ ( member7453568604450474000od_v_v @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_101_insert__commute,axiom,
! [X: set_v,Y: set_v,A2: set_set_v] :
( ( insert_set_v @ X @ ( insert_set_v @ Y @ A2 ) )
= ( insert_set_v @ Y @ ( insert_set_v @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_102_insert__commute,axiom,
! [X: v,Y: v,A2: set_v] :
( ( insert_v @ X @ ( insert_v @ Y @ A2 ) )
= ( insert_v @ Y @ ( insert_v @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_103_insert__commute,axiom,
! [X: product_prod_v_v,Y: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( insert1338601472111419319od_v_v @ Y @ A2 ) )
= ( insert1338601472111419319od_v_v @ Y @ ( insert1338601472111419319od_v_v @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_104_insert__eq__iff,axiom,
! [A: set_v,A2: set_set_v,B2: set_v,B: set_set_v] :
( ~ ( member_set_v @ A @ A2 )
=> ( ~ ( member_set_v @ B2 @ B )
=> ( ( ( insert_set_v @ A @ A2 )
= ( insert_set_v @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_set_v] :
( ( A2
= ( insert_set_v @ B2 @ C3 ) )
& ~ ( member_set_v @ B2 @ C3 )
& ( B
= ( insert_set_v @ A @ C3 ) )
& ~ ( member_set_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_105_insert__eq__iff,axiom,
! [A: v,A2: set_v,B2: v,B: set_v] :
( ~ ( member_v @ A @ A2 )
=> ( ~ ( member_v @ B2 @ B )
=> ( ( ( insert_v @ A @ A2 )
= ( insert_v @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_v] :
( ( A2
= ( insert_v @ B2 @ C3 ) )
& ~ ( member_v @ B2 @ C3 )
& ( B
= ( insert_v @ A @ C3 ) )
& ~ ( member_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_106_insert__eq__iff,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B2: product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ~ ( member7453568604450474000od_v_v @ B2 @ B )
=> ( ( ( insert1338601472111419319od_v_v @ A @ A2 )
= ( insert1338601472111419319od_v_v @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_Product_prod_v_v] :
( ( A2
= ( insert1338601472111419319od_v_v @ B2 @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ B2 @ C3 )
& ( B
= ( insert1338601472111419319od_v_v @ A @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_107_insert__absorb,axiom,
! [A: set_v,A2: set_set_v] :
( ( member_set_v @ A @ A2 )
=> ( ( insert_set_v @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_108_insert__absorb,axiom,
! [A: v,A2: set_v] :
( ( member_v @ A @ A2 )
=> ( ( insert_v @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_109_insert__absorb,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( insert1338601472111419319od_v_v @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_110_insert__ident,axiom,
! [X: set_v,A2: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ X @ A2 )
=> ( ~ ( member_set_v @ X @ B )
=> ( ( ( insert_set_v @ X @ A2 )
= ( insert_set_v @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_111_insert__ident,axiom,
! [X: v,A2: set_v,B: set_v] :
( ~ ( member_v @ X @ A2 )
=> ( ~ ( member_v @ X @ B )
=> ( ( ( insert_v @ X @ A2 )
= ( insert_v @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_112_insert__ident,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A2 )
=> ( ~ ( member7453568604450474000od_v_v @ X @ B )
=> ( ( ( insert1338601472111419319od_v_v @ X @ A2 )
= ( insert1338601472111419319od_v_v @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_113_Set_Oset__insert,axiom,
! [X: set_v,A2: set_set_v] :
( ( member_set_v @ X @ A2 )
=> ~ ! [B3: set_set_v] :
( ( A2
= ( insert_set_v @ X @ B3 ) )
=> ( member_set_v @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_114_Set_Oset__insert,axiom,
! [X: v,A2: set_v] :
( ( member_v @ X @ A2 )
=> ~ ! [B3: set_v] :
( ( A2
= ( insert_v @ X @ B3 ) )
=> ( member_v @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_115_Set_Oset__insert,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A2 )
=> ~ ! [B3: set_Product_prod_v_v] :
( ( A2
= ( insert1338601472111419319od_v_v @ X @ B3 ) )
=> ( member7453568604450474000od_v_v @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_116_insertI2,axiom,
! [A: set_v,B: set_set_v,B2: set_v] :
( ( member_set_v @ A @ B )
=> ( member_set_v @ A @ ( insert_set_v @ B2 @ B ) ) ) ).
% insertI2
thf(fact_117_insertI2,axiom,
! [A: v,B: set_v,B2: v] :
( ( member_v @ A @ B )
=> ( member_v @ A @ ( insert_v @ B2 @ B ) ) ) ).
% insertI2
thf(fact_118_insertI2,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ B )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).
% insertI2
thf(fact_119_insertI1,axiom,
! [A: set_v,B: set_set_v] : ( member_set_v @ A @ ( insert_set_v @ A @ B ) ) ).
% insertI1
thf(fact_120_insertI1,axiom,
! [A: v,B: set_v] : ( member_v @ A @ ( insert_v @ A @ B ) ) ).
% insertI1
thf(fact_121_insertI1,axiom,
! [A: product_prod_v_v,B: set_Product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ B ) ) ).
% insertI1
thf(fact_122_insertE,axiom,
! [A: set_v,B2: set_v,A2: set_set_v] :
( ( member_set_v @ A @ ( insert_set_v @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_set_v @ A @ A2 ) ) ) ).
% insertE
thf(fact_123_insertE,axiom,
! [A: v,B2: v,A2: set_v] :
( ( member_v @ A @ ( insert_v @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_v @ A @ A2 ) ) ) ).
% insertE
thf(fact_124_insertE,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member7453568604450474000od_v_v @ A @ A2 ) ) ) ).
% insertE
thf(fact_125_Un__left__commute,axiom,
! [A2: set_v,B: set_v,C: set_v] :
( ( sup_sup_set_v @ A2 @ ( sup_sup_set_v @ B @ C ) )
= ( sup_sup_set_v @ B @ ( sup_sup_set_v @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_126_Un__left__commute,axiom,
! [A2: set_set_v,B: set_set_v,C: set_set_v] :
( ( sup_sup_set_set_v @ A2 @ ( sup_sup_set_set_v @ B @ C ) )
= ( sup_sup_set_set_v @ B @ ( sup_sup_set_set_v @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_127_Un__left__commute,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B @ C ) )
= ( sup_su414716646722978715od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_128_Un__left__absorb,axiom,
! [A2: set_v,B: set_v] :
( ( sup_sup_set_v @ A2 @ ( sup_sup_set_v @ A2 @ B ) )
= ( sup_sup_set_v @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_129_Un__left__absorb,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( sup_sup_set_set_v @ A2 @ ( sup_sup_set_set_v @ A2 @ B ) )
= ( sup_sup_set_set_v @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_130_Un__left__absorb,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ A2 @ B ) )
= ( sup_su414716646722978715od_v_v @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_131_Un__commute,axiom,
( sup_sup_set_v
= ( ^ [A3: set_v,B4: set_v] : ( sup_sup_set_v @ B4 @ A3 ) ) ) ).
% Un_commute
thf(fact_132_Un__commute,axiom,
( sup_sup_set_set_v
= ( ^ [A3: set_set_v,B4: set_set_v] : ( sup_sup_set_set_v @ B4 @ A3 ) ) ) ).
% Un_commute
thf(fact_133_Un__commute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B4 @ A3 ) ) ) ).
% Un_commute
thf(fact_134_Un__absorb,axiom,
! [A2: set_v] :
( ( sup_sup_set_v @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_135_Un__absorb,axiom,
! [A2: set_set_v] :
( ( sup_sup_set_set_v @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_136_Un__absorb,axiom,
! [A2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_137_Un__assoc,axiom,
! [A2: set_v,B: set_v,C: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A2 @ B ) @ C )
= ( sup_sup_set_v @ A2 @ ( sup_sup_set_v @ B @ C ) ) ) ).
% Un_assoc
thf(fact_138_Un__assoc,axiom,
! [A2: set_set_v,B: set_set_v,C: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ A2 @ B ) @ C )
= ( sup_sup_set_set_v @ A2 @ ( sup_sup_set_set_v @ B @ C ) ) ) ).
% Un_assoc
thf(fact_139_Un__assoc,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B ) @ C )
= ( sup_su414716646722978715od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B @ C ) ) ) ).
% Un_assoc
thf(fact_140_ball__Un,axiom,
! [A2: set_v,B: set_v,P: v > $o] :
( ( ! [X2: v] :
( ( member_v @ X2 @ ( sup_sup_set_v @ A2 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: v] :
( ( member_v @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: v] :
( ( member_v @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_141_ball__Un,axiom,
! [A2: set_set_v,B: set_set_v,P: set_v > $o] :
( ( ! [X2: set_v] :
( ( member_set_v @ X2 @ ( sup_sup_set_set_v @ A2 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: set_v] :
( ( member_set_v @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_142_ball__Un,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ A2 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_143_bex__Un,axiom,
! [A2: set_v,B: set_v,P: v > $o] :
( ( ? [X2: v] :
( ( member_v @ X2 @ ( sup_sup_set_v @ A2 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: v] :
( ( member_v @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: v] :
( ( member_v @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_144_bex__Un,axiom,
! [A2: set_set_v,B: set_set_v,P: set_v > $o] :
( ( ? [X2: set_v] :
( ( member_set_v @ X2 @ ( sup_sup_set_set_v @ A2 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: set_v] :
( ( member_set_v @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_145_bex__Un,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( sup_su414716646722978715od_v_v @ A2 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_146_UnI2,axiom,
! [C2: v,B: set_v,A2: set_v] :
( ( member_v @ C2 @ B )
=> ( member_v @ C2 @ ( sup_sup_set_v @ A2 @ B ) ) ) ).
% UnI2
thf(fact_147_UnI2,axiom,
! [C2: set_v,B: set_set_v,A2: set_set_v] :
( ( member_set_v @ C2 @ B )
=> ( member_set_v @ C2 @ ( sup_sup_set_set_v @ A2 @ B ) ) ) ).
% UnI2
thf(fact_148_UnI2,axiom,
! [C2: product_prod_v_v,B: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ B )
=> ( member7453568604450474000od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A2 @ B ) ) ) ).
% UnI2
thf(fact_149_UnI1,axiom,
! [C2: v,A2: set_v,B: set_v] :
( ( member_v @ C2 @ A2 )
=> ( member_v @ C2 @ ( sup_sup_set_v @ A2 @ B ) ) ) ).
% UnI1
thf(fact_150_UnI1,axiom,
! [C2: set_v,A2: set_set_v,B: set_set_v] :
( ( member_set_v @ C2 @ A2 )
=> ( member_set_v @ C2 @ ( sup_sup_set_set_v @ A2 @ B ) ) ) ).
% UnI1
thf(fact_151_UnI1,axiom,
! [C2: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ A2 )
=> ( member7453568604450474000od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A2 @ B ) ) ) ).
% UnI1
thf(fact_152_UnE,axiom,
! [C2: v,A2: set_v,B: set_v] :
( ( member_v @ C2 @ ( sup_sup_set_v @ A2 @ B ) )
=> ( ~ ( member_v @ C2 @ A2 )
=> ( member_v @ C2 @ B ) ) ) ).
% UnE
thf(fact_153_UnE,axiom,
! [C2: set_v,A2: set_set_v,B: set_set_v] :
( ( member_set_v @ C2 @ ( sup_sup_set_set_v @ A2 @ B ) )
=> ( ~ ( member_set_v @ C2 @ A2 )
=> ( member_set_v @ C2 @ B ) ) ) ).
% UnE
thf(fact_154_UnE,axiom,
! [C2: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A2 @ B ) )
=> ( ~ ( member7453568604450474000od_v_v @ C2 @ A2 )
=> ( member7453568604450474000od_v_v @ C2 @ B ) ) ) ).
% UnE
thf(fact_155_empty__def,axiom,
( bot_bot_set_set_v
= ( collect_set_v
@ ^ [X2: set_v] : $false ) ) ).
% empty_def
thf(fact_156_empty__def,axiom,
( bot_bot_set_v
= ( collect_v
@ ^ [X2: v] : $false ) ) ).
% empty_def
thf(fact_157_empty__def,axiom,
( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] : $false ) ) ).
% empty_def
thf(fact_158_insert__Collect,axiom,
! [A: v,P: v > $o] :
( ( insert_v @ A @ ( collect_v @ P ) )
= ( collect_v
@ ^ [U: v] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_159_insert__Collect,axiom,
! [A: product_prod_v_v,P: product_prod_v_v > $o] :
( ( insert1338601472111419319od_v_v @ A @ ( collec140062887454715474od_v_v @ P ) )
= ( collec140062887454715474od_v_v
@ ^ [U: product_prod_v_v] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_160_insert__Collect,axiom,
! [A: set_v,P: set_v > $o] :
( ( insert_set_v @ A @ ( collect_set_v @ P ) )
= ( collect_set_v
@ ^ [U: set_v] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_161_insert__compr,axiom,
( insert_v
= ( ^ [A4: v,B4: set_v] :
( collect_v
@ ^ [X2: v] :
( ( X2 = A4 )
| ( member_v @ X2 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_162_insert__compr,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A4: product_prod_v_v,B4: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( X2 = A4 )
| ( member7453568604450474000od_v_v @ X2 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_163_insert__compr,axiom,
( insert_set_v
= ( ^ [A4: set_v,B4: set_set_v] :
( collect_set_v
@ ^ [X2: set_v] :
( ( X2 = A4 )
| ( member_set_v @ X2 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_164_Collect__disj__eq,axiom,
! [P: v > $o,Q: v > $o] :
( ( collect_v
@ ^ [X2: v] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_165_Collect__disj__eq,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ( collect_set_v
@ ^ [X2: set_v] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_166_Collect__disj__eq,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_su414716646722978715od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_167_Un__def,axiom,
( sup_sup_set_v
= ( ^ [A3: set_v,B4: set_v] :
( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ A3 )
| ( member_v @ X2 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_168_Un__def,axiom,
( sup_sup_set_set_v
= ( ^ [A3: set_set_v,B4: set_set_v] :
( collect_set_v
@ ^ [X2: set_v] :
( ( member_set_v @ X2 @ A3 )
| ( member_set_v @ X2 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_169_Un__def,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A3 )
| ( member7453568604450474000od_v_v @ X2 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_170_singleton__inject,axiom,
! [A: set_v,B2: set_v] :
( ( ( insert_set_v @ A @ bot_bot_set_set_v )
= ( insert_set_v @ B2 @ bot_bot_set_set_v ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_171_singleton__inject,axiom,
! [A: v,B2: v] :
( ( ( insert_v @ A @ bot_bot_set_v )
= ( insert_v @ B2 @ bot_bot_set_v ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_172_singleton__inject,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_173_insert__not__empty,axiom,
! [A: set_v,A2: set_set_v] :
( ( insert_set_v @ A @ A2 )
!= bot_bot_set_set_v ) ).
% insert_not_empty
thf(fact_174_insert__not__empty,axiom,
! [A: v,A2: set_v] :
( ( insert_v @ A @ A2 )
!= bot_bot_set_v ) ).
% insert_not_empty
thf(fact_175_insert__not__empty,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ A2 )
!= bot_bo723834152578015283od_v_v ) ).
% insert_not_empty
thf(fact_176_doubleton__eq__iff,axiom,
! [A: set_v,B2: set_v,C2: set_v,D: set_v] :
( ( ( insert_set_v @ A @ ( insert_set_v @ B2 @ bot_bot_set_set_v ) )
= ( insert_set_v @ C2 @ ( insert_set_v @ D @ bot_bot_set_set_v ) ) )
= ( ( ( A = C2 )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_177_doubleton__eq__iff,axiom,
! [A: v,B2: v,C2: v,D: v] :
( ( ( insert_v @ A @ ( insert_v @ B2 @ bot_bot_set_v ) )
= ( insert_v @ C2 @ ( insert_v @ D @ bot_bot_set_v ) ) )
= ( ( ( A = C2 )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_178_doubleton__eq__iff,axiom,
! [A: product_prod_v_v,B2: product_prod_v_v,C2: product_prod_v_v,D: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
= ( insert1338601472111419319od_v_v @ C2 @ ( insert1338601472111419319od_v_v @ D @ bot_bo723834152578015283od_v_v ) ) )
= ( ( ( A = C2 )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_179_singleton__iff,axiom,
! [B2: set_v,A: set_v] :
( ( member_set_v @ B2 @ ( insert_set_v @ A @ bot_bot_set_set_v ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_180_singleton__iff,axiom,
! [B2: v,A: v] :
( ( member_v @ B2 @ ( insert_v @ A @ bot_bot_set_v ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_181_singleton__iff,axiom,
! [B2: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_182_singletonD,axiom,
! [B2: set_v,A: set_v] :
( ( member_set_v @ B2 @ ( insert_set_v @ A @ bot_bot_set_set_v ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_183_singletonD,axiom,
! [B2: v,A: v] :
( ( member_v @ B2 @ ( insert_v @ A @ bot_bot_set_v ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_184_singletonD,axiom,
! [B2: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_185_Un__empty__right,axiom,
! [A2: set_set_v] :
( ( sup_sup_set_set_v @ A2 @ bot_bot_set_set_v )
= A2 ) ).
% Un_empty_right
thf(fact_186_Un__empty__right,axiom,
! [A2: set_v] :
( ( sup_sup_set_v @ A2 @ bot_bot_set_v )
= A2 ) ).
% Un_empty_right
thf(fact_187_Un__empty__right,axiom,
! [A2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ bot_bo723834152578015283od_v_v )
= A2 ) ).
% Un_empty_right
thf(fact_188_Un__empty__left,axiom,
! [B: set_set_v] :
( ( sup_sup_set_set_v @ bot_bot_set_set_v @ B )
= B ) ).
% Un_empty_left
thf(fact_189_Un__empty__left,axiom,
! [B: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ B )
= B ) ).
% Un_empty_left
thf(fact_190_Un__empty__left,axiom,
! [B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ B )
= B ) ).
% Un_empty_left
thf(fact_191_Collect__conv__if2,axiom,
! [P: set_v > $o,A: set_v] :
( ( ( P @ A )
=> ( ( collect_set_v
@ ^ [X2: set_v] :
( ( A = X2 )
& ( P @ X2 ) ) )
= ( insert_set_v @ A @ bot_bot_set_set_v ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_v
@ ^ [X2: set_v] :
( ( A = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_set_v ) ) ) ).
% Collect_conv_if2
thf(fact_192_Collect__conv__if2,axiom,
! [P: v > $o,A: v] :
( ( ( P @ A )
=> ( ( collect_v
@ ^ [X2: v] :
( ( A = X2 )
& ( P @ X2 ) ) )
= ( insert_v @ A @ bot_bot_set_v ) ) )
& ( ~ ( P @ A )
=> ( ( collect_v
@ ^ [X2: v] :
( ( A = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_v ) ) ) ).
% Collect_conv_if2
thf(fact_193_Collect__conv__if2,axiom,
! [P: product_prod_v_v > $o,A: product_prod_v_v] :
( ( ( P @ A )
=> ( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( A = X2 )
& ( P @ X2 ) ) )
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
& ( ~ ( P @ A )
=> ( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( A = X2 )
& ( P @ X2 ) ) )
= bot_bo723834152578015283od_v_v ) ) ) ).
% Collect_conv_if2
thf(fact_194_Collect__conv__if,axiom,
! [P: set_v > $o,A: set_v] :
( ( ( P @ A )
=> ( ( collect_set_v
@ ^ [X2: set_v] :
( ( X2 = A )
& ( P @ X2 ) ) )
= ( insert_set_v @ A @ bot_bot_set_set_v ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_v
@ ^ [X2: set_v] :
( ( X2 = A )
& ( P @ X2 ) ) )
= bot_bot_set_set_v ) ) ) ).
% Collect_conv_if
thf(fact_195_Collect__conv__if,axiom,
! [P: v > $o,A: v] :
( ( ( P @ A )
=> ( ( collect_v
@ ^ [X2: v] :
( ( X2 = A )
& ( P @ X2 ) ) )
= ( insert_v @ A @ bot_bot_set_v ) ) )
& ( ~ ( P @ A )
=> ( ( collect_v
@ ^ [X2: v] :
( ( X2 = A )
& ( P @ X2 ) ) )
= bot_bot_set_v ) ) ) ).
% Collect_conv_if
thf(fact_196_Collect__conv__if,axiom,
! [P: product_prod_v_v > $o,A: product_prod_v_v] :
( ( ( P @ A )
=> ( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( X2 = A )
& ( P @ X2 ) ) )
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
& ( ~ ( P @ A )
=> ( ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( X2 = A )
& ( P @ X2 ) ) )
= bot_bo723834152578015283od_v_v ) ) ) ).
% Collect_conv_if
thf(fact_197_insert__def,axiom,
( insert_v
= ( ^ [A4: v] :
( sup_sup_set_v
@ ( collect_v
@ ^ [X2: v] : ( X2 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_198_insert__def,axiom,
( insert_set_v
= ( ^ [A4: set_v] :
( sup_sup_set_set_v
@ ( collect_set_v
@ ^ [X2: set_v] : ( X2 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_199_insert__def,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A4: product_prod_v_v] :
( sup_su414716646722978715od_v_v
@ ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] : ( X2 = A4 ) ) ) ) ) ).
% insert_def
thf(fact_200_singleton__Un__iff,axiom,
! [X: set_v,A2: set_set_v,B: set_set_v] :
( ( ( insert_set_v @ X @ bot_bot_set_set_v )
= ( sup_sup_set_set_v @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_set_v )
& ( B
= ( insert_set_v @ X @ bot_bot_set_set_v ) ) )
| ( ( A2
= ( insert_set_v @ X @ bot_bot_set_set_v ) )
& ( B = bot_bot_set_set_v ) )
| ( ( A2
= ( insert_set_v @ X @ bot_bot_set_set_v ) )
& ( B
= ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_201_singleton__Un__iff,axiom,
! [X: v,A2: set_v,B: set_v] :
( ( ( insert_v @ X @ bot_bot_set_v )
= ( sup_sup_set_v @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_v )
& ( B
= ( insert_v @ X @ bot_bot_set_v ) ) )
| ( ( A2
= ( insert_v @ X @ bot_bot_set_v ) )
& ( B = bot_bot_set_v ) )
| ( ( A2
= ( insert_v @ X @ bot_bot_set_v ) )
& ( B
= ( insert_v @ X @ bot_bot_set_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_202_singleton__Un__iff,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v )
= ( sup_su414716646722978715od_v_v @ A2 @ B ) )
= ( ( ( A2 = bot_bo723834152578015283od_v_v )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B = bot_bo723834152578015283od_v_v ) )
| ( ( A2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_203_Un__singleton__iff,axiom,
! [A2: set_set_v,B: set_set_v,X: set_v] :
( ( ( sup_sup_set_set_v @ A2 @ B )
= ( insert_set_v @ X @ bot_bot_set_set_v ) )
= ( ( ( A2 = bot_bot_set_set_v )
& ( B
= ( insert_set_v @ X @ bot_bot_set_set_v ) ) )
| ( ( A2
= ( insert_set_v @ X @ bot_bot_set_set_v ) )
& ( B = bot_bot_set_set_v ) )
| ( ( A2
= ( insert_set_v @ X @ bot_bot_set_set_v ) )
& ( B
= ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_204_Un__singleton__iff,axiom,
! [A2: set_v,B: set_v,X: v] :
( ( ( sup_sup_set_v @ A2 @ B )
= ( insert_v @ X @ bot_bot_set_v ) )
= ( ( ( A2 = bot_bot_set_v )
& ( B
= ( insert_v @ X @ bot_bot_set_v ) ) )
| ( ( A2
= ( insert_v @ X @ bot_bot_set_v ) )
& ( B = bot_bot_set_v ) )
| ( ( A2
= ( insert_v @ X @ bot_bot_set_v ) )
& ( B
= ( insert_v @ X @ bot_bot_set_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_205_Un__singleton__iff,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A2 @ B )
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= ( ( ( A2 = bot_bo723834152578015283od_v_v )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B = bot_bo723834152578015283od_v_v ) )
| ( ( A2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_206_insert__is__Un,axiom,
( insert_set_v
= ( ^ [A4: set_v] : ( sup_sup_set_set_v @ ( insert_set_v @ A4 @ bot_bot_set_set_v ) ) ) ) ).
% insert_is_Un
thf(fact_207_insert__is__Un,axiom,
( insert_v
= ( ^ [A4: v] : ( sup_sup_set_v @ ( insert_v @ A4 @ bot_bot_set_v ) ) ) ) ).
% insert_is_Un
thf(fact_208_insert__is__Un,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A4: product_prod_v_v] : ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% insert_is_Un
thf(fact_209_calculation_I13_J,axiom,
( ord_less_eq_set_v
@ ( set_v2
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ).
% calculation(13)
thf(fact_210_calculation_I12_J,axiom,
( ord_less_eq_set_v
@ ( sCC_Bl157864678168468314t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ).
% calculation(12)
thf(fact_211_calculation_I4_J,axiom,
! [N: v] :
( ~ ( member_v @ N
@ ( sCC_Bl4645233313691564917t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) )
=> ( ( sCC_Bl3795065053823578884t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ N )
= bot_bot_set_v ) ) ).
% calculation(4)
thf(fact_212_calculation_I17_J,axiom,
! [N: v,M: v] :
( ( ( member_v @ N
@ ( set_v2
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) )
& ( member_v @ M
@ ( set_v2
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) )
& ( N != M ) )
=> ( ( inf_inf_set_v
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ N )
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ M ) )
= bot_bot_set_v ) ) ).
% calculation(17)
thf(fact_213_Union__Un__distrib,axiom,
! [A2: set_set_set_v,B: set_set_set_v] :
( ( comple5450237519782573632_set_v @ ( sup_su4471370308589719943_set_v @ A2 @ B ) )
= ( sup_sup_set_set_v @ ( comple5450237519782573632_set_v @ A2 ) @ ( comple5450237519782573632_set_v @ B ) ) ) ).
% Union_Un_distrib
thf(fact_214_Union__Un__distrib,axiom,
! [A2: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( comple5788137035815166516od_v_v @ ( sup_su335656005089752955od_v_v @ A2 @ B ) )
= ( sup_su414716646722978715od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Union_Un_distrib
thf(fact_215_Union__Un__distrib,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( comple2307003700295860064_set_v @ ( sup_sup_set_set_v @ A2 @ B ) )
= ( sup_sup_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Union_Un_distrib
thf(fact_216_Sup__insert,axiom,
! [A: set_set_v,A2: set_set_set_v] :
( ( comple5450237519782573632_set_v @ ( insert_set_set_v @ A @ A2 ) )
= ( sup_sup_set_set_v @ A @ ( comple5450237519782573632_set_v @ A2 ) ) ) ).
% Sup_insert
thf(fact_217_Sup__insert,axiom,
! [A: set_Product_prod_v_v,A2: set_se8455005133513928103od_v_v] :
( ( comple5788137035815166516od_v_v @ ( insert7504383016908236695od_v_v @ A @ A2 ) )
= ( sup_su414716646722978715od_v_v @ A @ ( comple5788137035815166516od_v_v @ A2 ) ) ) ).
% Sup_insert
thf(fact_218_Sup__insert,axiom,
! [A: set_v,A2: set_set_v] :
( ( comple2307003700295860064_set_v @ ( insert_set_v @ A @ A2 ) )
= ( sup_sup_set_v @ A @ ( comple2307003700295860064_set_v @ A2 ) ) ) ).
% Sup_insert
thf(fact_219_cSup__singleton,axiom,
! [X: set_v] :
( ( comple2307003700295860064_set_v @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= X ) ).
% cSup_singleton
thf(fact_220_ccpo__Sup__singleton,axiom,
! [X: set_v] :
( ( comple2307003700295860064_set_v @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= X ) ).
% ccpo_Sup_singleton
thf(fact_221_Sup__empty,axiom,
( ( comple5450237519782573632_set_v @ bot_bo5775917114081396255_set_v )
= bot_bot_set_set_v ) ).
% Sup_empty
thf(fact_222_Sup__empty,axiom,
( ( comple5788137035815166516od_v_v @ bot_bo3497076220358800403od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Sup_empty
thf(fact_223_Sup__empty,axiom,
( ( comple2307003700295860064_set_v @ bot_bot_set_set_v )
= bot_bot_set_v ) ).
% Sup_empty
thf(fact_224_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_set_v] :
( ( ( comple5450237519782573632_set_v @ A2 )
= bot_bot_set_set_v )
= ( ! [X2: set_set_v] :
( ( member_set_set_v @ X2 @ A2 )
=> ( X2 = bot_bot_set_set_v ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_225_Sup__bot__conv_I1_J,axiom,
! [A2: set_se8455005133513928103od_v_v] :
( ( ( comple5788137035815166516od_v_v @ A2 )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A2 )
=> ( X2 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_226_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_v] :
( ( ( comple2307003700295860064_set_v @ A2 )
= bot_bot_set_v )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
=> ( X2 = bot_bot_set_v ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_227_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_set_v] :
( ( bot_bot_set_set_v
= ( comple5450237519782573632_set_v @ A2 ) )
= ( ! [X2: set_set_v] :
( ( member_set_set_v @ X2 @ A2 )
=> ( X2 = bot_bot_set_set_v ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_228_Sup__bot__conv_I2_J,axiom,
! [A2: set_se8455005133513928103od_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( comple5788137035815166516od_v_v @ A2 ) )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A2 )
=> ( X2 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_229_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_v] :
( ( bot_bot_set_v
= ( comple2307003700295860064_set_v @ A2 ) )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
=> ( X2 = bot_bot_set_v ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_230_subsetI,axiom,
! [A2: set_v,B: set_v] :
( ! [X3: v] :
( ( member_v @ X3 @ A2 )
=> ( member_v @ X3 @ B ) )
=> ( ord_less_eq_set_v @ A2 @ B ) ) ).
% subsetI
thf(fact_231_subsetI,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A2 )
=> ( member7453568604450474000od_v_v @ X3 @ B ) )
=> ( ord_le7336532860387713383od_v_v @ A2 @ B ) ) ).
% subsetI
thf(fact_232_subset__antisym,axiom,
! [A2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A2 @ B )
=> ( ( ord_less_eq_set_v @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_233_subset__antisym,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_234_IntI,axiom,
! [C2: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ A2 )
=> ( ( member7453568604450474000od_v_v @ C2 @ B )
=> ( member7453568604450474000od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_235_IntI,axiom,
! [C2: v,A2: set_v,B: set_v] :
( ( member_v @ C2 @ A2 )
=> ( ( member_v @ C2 @ B )
=> ( member_v @ C2 @ ( inf_inf_set_v @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_236_Int__iff,axiom,
! [C2: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) )
= ( ( member7453568604450474000od_v_v @ C2 @ A2 )
& ( member7453568604450474000od_v_v @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_237_Int__iff,axiom,
! [C2: v,A2: set_v,B: set_v] :
( ( member_v @ C2 @ ( inf_inf_set_v @ A2 @ B ) )
= ( ( member_v @ C2 @ A2 )
& ( member_v @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_238_Union__iff,axiom,
! [A2: product_prod_v_v,C: set_se8455005133513928103od_v_v] :
( ( member7453568604450474000od_v_v @ A2 @ ( comple5788137035815166516od_v_v @ C ) )
= ( ? [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ C )
& ( member7453568604450474000od_v_v @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_239_Union__iff,axiom,
! [A2: v,C: set_set_v] :
( ( member_v @ A2 @ ( comple2307003700295860064_set_v @ C ) )
= ( ? [X2: set_v] :
( ( member_set_v @ X2 @ C )
& ( member_v @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_240_UnionI,axiom,
! [X4: set_Product_prod_v_v,C: set_se8455005133513928103od_v_v,A2: product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ C )
=> ( ( member7453568604450474000od_v_v @ A2 @ X4 )
=> ( member7453568604450474000od_v_v @ A2 @ ( comple5788137035815166516od_v_v @ C ) ) ) ) ).
% UnionI
thf(fact_241_UnionI,axiom,
! [X4: set_v,C: set_set_v,A2: v] :
( ( member_set_v @ X4 @ C )
=> ( ( member_v @ A2 @ X4 )
=> ( member_v @ A2 @ ( comple2307003700295860064_set_v @ C ) ) ) ) ).
% UnionI
thf(fact_242_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_v,P: v > $o] :
( ( ! [X2: v] :
( ( member_v @ X2 @ ( comple2307003700295860064_set_v @ A2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
=> ! [Y4: v] :
( ( member_v @ Y4 @ X2 )
=> ( P @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_243_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_v,P: v > $o] :
( ( ? [X2: v] :
( ( member_v @ X2 @ ( comple2307003700295860064_set_v @ A2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
& ? [Y4: v] :
( ( member_v @ Y4 @ X2 )
& ( P @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_244_subset__empty,axiom,
! [A2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ bot_bot_set_set_v )
= ( A2 = bot_bot_set_set_v ) ) ).
% subset_empty
thf(fact_245_subset__empty,axiom,
! [A2: set_v] :
( ( ord_less_eq_set_v @ A2 @ bot_bot_set_v )
= ( A2 = bot_bot_set_v ) ) ).
% subset_empty
thf(fact_246_subset__empty,axiom,
! [A2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ bot_bo723834152578015283od_v_v )
= ( A2 = bot_bo723834152578015283od_v_v ) ) ).
% subset_empty
thf(fact_247_empty__subsetI,axiom,
! [A2: set_set_v] : ( ord_le5216385588623774835_set_v @ bot_bot_set_set_v @ A2 ) ).
% empty_subsetI
thf(fact_248_empty__subsetI,axiom,
! [A2: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A2 ) ).
% empty_subsetI
thf(fact_249_empty__subsetI,axiom,
! [A2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A2 ) ).
% empty_subsetI
thf(fact_250_insert__subset,axiom,
! [X: set_v,A2: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( insert_set_v @ X @ A2 ) @ B )
= ( ( member_set_v @ X @ B )
& ( ord_le5216385588623774835_set_v @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_251_insert__subset,axiom,
! [X: v,A2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ ( insert_v @ X @ A2 ) @ B )
= ( ( member_v @ X @ B )
& ( ord_less_eq_set_v @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_252_insert__subset,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X @ A2 ) @ B )
= ( ( member7453568604450474000od_v_v @ X @ B )
& ( ord_le7336532860387713383od_v_v @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_253_Un__subset__iff,axiom,
! [A2: set_set_v,B: set_set_v,C: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A2 @ B ) @ C )
= ( ( ord_le5216385588623774835_set_v @ A2 @ C )
& ( ord_le5216385588623774835_set_v @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_254_Un__subset__iff,axiom,
! [A2: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A2 @ B ) @ C )
= ( ( ord_less_eq_set_v @ A2 @ C )
& ( ord_less_eq_set_v @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_255_Un__subset__iff,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B ) @ C )
= ( ( ord_le7336532860387713383od_v_v @ A2 @ C )
& ( ord_le7336532860387713383od_v_v @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_256_Int__subset__iff,axiom,
! [C: set_v,A2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C @ ( inf_inf_set_v @ A2 @ B ) )
= ( ( ord_less_eq_set_v @ C @ A2 )
& ( ord_less_eq_set_v @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_257_Int__subset__iff,axiom,
! [C: set_Product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) )
= ( ( ord_le7336532860387713383od_v_v @ C @ A2 )
& ( ord_le7336532860387713383od_v_v @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_258_Int__insert__left__if0,axiom,
! [A: set_v,C: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ A @ C )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B ) @ C )
= ( inf_inf_set_set_v @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_259_Int__insert__left__if0,axiom,
! [A: product_prod_v_v,C: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ C )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C )
= ( inf_in6271465464967711157od_v_v @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_260_Int__insert__left__if0,axiom,
! [A: v,C: set_v,B: set_v] :
( ~ ( member_v @ A @ C )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B ) @ C )
= ( inf_inf_set_v @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_261_Int__insert__left__if1,axiom,
! [A: set_v,C: set_set_v,B: set_set_v] :
( ( member_set_v @ A @ C )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B ) @ C )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_262_Int__insert__left__if1,axiom,
! [A: product_prod_v_v,C: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ C )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_263_Int__insert__left__if1,axiom,
! [A: v,C: set_v,B: set_v] :
( ( member_v @ A @ C )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B ) @ C )
= ( insert_v @ A @ ( inf_inf_set_v @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_264_insert__inter__insert,axiom,
! [A: set_v,A2: set_set_v,B: set_set_v] :
( ( inf_inf_set_set_v @ ( insert_set_v @ A @ A2 ) @ ( insert_set_v @ A @ B ) )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_265_insert__inter__insert,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A2 ) @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_266_insert__inter__insert,axiom,
! [A: v,A2: set_v,B: set_v] :
( ( inf_inf_set_v @ ( insert_v @ A @ A2 ) @ ( insert_v @ A @ B ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_267_Int__insert__right__if0,axiom,
! [A: set_v,A2: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ A @ A2 )
=> ( ( inf_inf_set_set_v @ A2 @ ( insert_set_v @ A @ B ) )
= ( inf_inf_set_set_v @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_268_Int__insert__right__if0,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( inf_in6271465464967711157od_v_v @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_269_Int__insert__right__if0,axiom,
! [A: v,A2: set_v,B: set_v] :
( ~ ( member_v @ A @ A2 )
=> ( ( inf_inf_set_v @ A2 @ ( insert_v @ A @ B ) )
= ( inf_inf_set_v @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_270_Int__insert__right__if1,axiom,
! [A: set_v,A2: set_set_v,B: set_set_v] :
( ( member_set_v @ A @ A2 )
=> ( ( inf_inf_set_set_v @ A2 @ ( insert_set_v @ A @ B ) )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_271_Int__insert__right__if1,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_272_Int__insert__right__if1,axiom,
! [A: v,A2: set_v,B: set_v] :
( ( member_v @ A @ A2 )
=> ( ( inf_inf_set_v @ A2 @ ( insert_v @ A @ B ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_273_Int__Un__eq_I4_J,axiom,
! [T: set_v,S: set_v] :
( ( sup_sup_set_v @ T @ ( inf_inf_set_v @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_274_Int__Un__eq_I4_J,axiom,
! [T: set_set_v,S: set_set_v] :
( ( sup_sup_set_set_v @ T @ ( inf_inf_set_set_v @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_275_Int__Un__eq_I4_J,axiom,
! [T: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ T @ ( inf_in6271465464967711157od_v_v @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_276_Int__Un__eq_I3_J,axiom,
! [S: set_v,T: set_v] :
( ( sup_sup_set_v @ S @ ( inf_inf_set_v @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_277_Int__Un__eq_I3_J,axiom,
! [S: set_set_v,T: set_set_v] :
( ( sup_sup_set_set_v @ S @ ( inf_inf_set_set_v @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_278_Int__Un__eq_I3_J,axiom,
! [S: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ S @ ( inf_in6271465464967711157od_v_v @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_279_Int__Un__eq_I2_J,axiom,
! [S: set_v,T: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_280_Int__Un__eq_I2_J,axiom,
! [S: set_set_v,T: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_281_Int__Un__eq_I2_J,axiom,
! [S: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_282_Int__Un__eq_I1_J,axiom,
! [S: set_v,T: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_283_Int__Un__eq_I1_J,axiom,
! [S: set_set_v,T: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_284_Int__Un__eq_I1_J,axiom,
! [S: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_285_Un__Int__eq_I4_J,axiom,
! [T: set_v,S: set_v] :
( ( inf_inf_set_v @ T @ ( sup_sup_set_v @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_286_Un__Int__eq_I4_J,axiom,
! [T: set_set_v,S: set_set_v] :
( ( inf_inf_set_set_v @ T @ ( sup_sup_set_set_v @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_287_Un__Int__eq_I4_J,axiom,
! [T: set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ T @ ( sup_su414716646722978715od_v_v @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_288_Un__Int__eq_I3_J,axiom,
! [S: set_v,T: set_v] :
( ( inf_inf_set_v @ S @ ( sup_sup_set_v @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_289_Un__Int__eq_I3_J,axiom,
! [S: set_set_v,T: set_set_v] :
( ( inf_inf_set_set_v @ S @ ( sup_sup_set_set_v @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_290_Un__Int__eq_I3_J,axiom,
! [S: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ S @ ( sup_su414716646722978715od_v_v @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_291_Un__Int__eq_I2_J,axiom,
! [S: set_v,T: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_292_Un__Int__eq_I2_J,axiom,
! [S: set_set_v,T: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_293_Un__Int__eq_I2_J,axiom,
! [S: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_294_Un__Int__eq_I1_J,axiom,
! [S: set_v,T: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_295_Un__Int__eq_I1_J,axiom,
! [S: set_set_v,T: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_296_Un__Int__eq_I1_J,axiom,
! [S: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_297_singleton__insert__inj__eq,axiom,
! [B2: set_v,A: set_v,A2: set_set_v] :
( ( ( insert_set_v @ B2 @ bot_bot_set_set_v )
= ( insert_set_v @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le5216385588623774835_set_v @ A2 @ ( insert_set_v @ B2 @ bot_bot_set_set_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_298_singleton__insert__inj__eq,axiom,
! [B2: v,A: v,A2: set_v] :
( ( ( insert_v @ B2 @ bot_bot_set_v )
= ( insert_v @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_v @ A2 @ ( insert_v @ B2 @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_299_singleton__insert__inj__eq,axiom,
! [B2: product_prod_v_v,A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_300_singleton__insert__inj__eq_H,axiom,
! [A: set_v,A2: set_set_v,B2: set_v] :
( ( ( insert_set_v @ A @ A2 )
= ( insert_set_v @ B2 @ bot_bot_set_set_v ) )
= ( ( A = B2 )
& ( ord_le5216385588623774835_set_v @ A2 @ ( insert_set_v @ B2 @ bot_bot_set_set_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_301_singleton__insert__inj__eq_H,axiom,
! [A: v,A2: set_v,B2: v] :
( ( ( insert_v @ A @ A2 )
= ( insert_v @ B2 @ bot_bot_set_v ) )
= ( ( A = B2 )
& ( ord_less_eq_set_v @ A2 @ ( insert_v @ B2 @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_302_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ A2 )
= ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) )
= ( ( A = B2 )
& ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ B2 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_303_disjoint__insert_I2_J,axiom,
! [A2: set_set_v,B2: set_v,B: set_set_v] :
( ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A2 @ ( insert_set_v @ B2 @ B ) ) )
= ( ~ ( member_set_v @ B2 @ A2 )
& ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_304_disjoint__insert_I2_J,axiom,
! [A2: set_v,B2: v,B: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ A2 @ ( insert_v @ B2 @ B ) ) )
= ( ~ ( member_v @ B2 @ A2 )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_305_disjoint__insert_I2_J,axiom,
! [A2: set_Product_prod_v_v,B2: product_prod_v_v,B: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) )
= ( ~ ( member7453568604450474000od_v_v @ B2 @ A2 )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_306_disjoint__insert_I1_J,axiom,
! [B: set_set_v,A: set_v,A2: set_set_v] :
( ( ( inf_inf_set_set_v @ B @ ( insert_set_v @ A @ A2 ) )
= bot_bot_set_set_v )
= ( ~ ( member_set_v @ A @ B )
& ( ( inf_inf_set_set_v @ B @ A2 )
= bot_bot_set_set_v ) ) ) ).
% disjoint_insert(1)
thf(fact_307_disjoint__insert_I1_J,axiom,
! [B: set_v,A: v,A2: set_v] :
( ( ( inf_inf_set_v @ B @ ( insert_v @ A @ A2 ) )
= bot_bot_set_v )
= ( ~ ( member_v @ A @ B )
& ( ( inf_inf_set_v @ B @ A2 )
= bot_bot_set_v ) ) ) ).
% disjoint_insert(1)
thf(fact_308_disjoint__insert_I1_J,axiom,
! [B: set_Product_prod_v_v,A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ A2 ) )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B )
& ( ( inf_in6271465464967711157od_v_v @ B @ A2 )
= bot_bo723834152578015283od_v_v ) ) ) ).
% disjoint_insert(1)
thf(fact_309_insert__disjoint_I2_J,axiom,
! [A: set_v,A2: set_set_v,B: set_set_v] :
( ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ ( insert_set_v @ A @ A2 ) @ B ) )
= ( ~ ( member_set_v @ A @ B )
& ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_310_insert__disjoint_I2_J,axiom,
! [A: v,A2: set_v,B: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ ( insert_v @ A @ A2 ) @ B ) )
= ( ~ ( member_v @ A @ B )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_311_insert__disjoint_I2_J,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A2 ) @ B ) )
= ( ~ ( member7453568604450474000od_v_v @ A @ B )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_312_insert__disjoint_I1_J,axiom,
! [A: set_v,A2: set_set_v,B: set_set_v] :
( ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ A2 ) @ B )
= bot_bot_set_set_v )
= ( ~ ( member_set_v @ A @ B )
& ( ( inf_inf_set_set_v @ A2 @ B )
= bot_bot_set_set_v ) ) ) ).
% insert_disjoint(1)
thf(fact_313_insert__disjoint_I1_J,axiom,
! [A: v,A2: set_v,B: set_v] :
( ( ( inf_inf_set_v @ ( insert_v @ A @ A2 ) @ B )
= bot_bot_set_v )
= ( ~ ( member_v @ A @ B )
& ( ( inf_inf_set_v @ A2 @ B )
= bot_bot_set_v ) ) ) ).
% insert_disjoint(1)
thf(fact_314_insert__disjoint_I1_J,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A2 ) @ B )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B )
& ( ( inf_in6271465464967711157od_v_v @ A2 @ B )
= bot_bo723834152578015283od_v_v ) ) ) ).
% insert_disjoint(1)
thf(fact_315_Sup__subset__mono,axiom,
! [A2: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A2 @ B )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Sup_subset_mono
thf(fact_316_Sup__subset__mono,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ B )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Sup_subset_mono
thf(fact_317_Sup__inter__less__eq,axiom,
! [A2: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] : ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ ( inf_in6058586722421118357od_v_v @ A2 @ B ) ) @ ( inf_in6271465464967711157od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_318_Sup__inter__less__eq,axiom,
! [A2: set_set_v,B: set_set_v] : ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( inf_inf_set_set_v @ A2 @ B ) ) @ ( inf_inf_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_319_IntE,axiom,
! [C2: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) )
=> ~ ( ( member7453568604450474000od_v_v @ C2 @ A2 )
=> ~ ( member7453568604450474000od_v_v @ C2 @ B ) ) ) ).
% IntE
thf(fact_320_IntE,axiom,
! [C2: v,A2: set_v,B: set_v] :
( ( member_v @ C2 @ ( inf_inf_set_v @ A2 @ B ) )
=> ~ ( ( member_v @ C2 @ A2 )
=> ~ ( member_v @ C2 @ B ) ) ) ).
% IntE
thf(fact_321_IntD1,axiom,
! [C2: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) )
=> ( member7453568604450474000od_v_v @ C2 @ A2 ) ) ).
% IntD1
thf(fact_322_IntD1,axiom,
! [C2: v,A2: set_v,B: set_v] :
( ( member_v @ C2 @ ( inf_inf_set_v @ A2 @ B ) )
=> ( member_v @ C2 @ A2 ) ) ).
% IntD1
thf(fact_323_IntD2,axiom,
! [C2: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) )
=> ( member7453568604450474000od_v_v @ C2 @ B ) ) ).
% IntD2
thf(fact_324_IntD2,axiom,
! [C2: v,A2: set_v,B: set_v] :
( ( member_v @ C2 @ ( inf_inf_set_v @ A2 @ B ) )
=> ( member_v @ C2 @ B ) ) ).
% IntD2
thf(fact_325_in__mono,axiom,
! [A2: set_v,B: set_v,X: v] :
( ( ord_less_eq_set_v @ A2 @ B )
=> ( ( member_v @ X @ A2 )
=> ( member_v @ X @ B ) ) ) ).
% in_mono
thf(fact_326_in__mono,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B )
=> ( ( member7453568604450474000od_v_v @ X @ A2 )
=> ( member7453568604450474000od_v_v @ X @ B ) ) ) ).
% in_mono
thf(fact_327_subsetD,axiom,
! [A2: set_v,B: set_v,C2: v] :
( ( ord_less_eq_set_v @ A2 @ B )
=> ( ( member_v @ C2 @ A2 )
=> ( member_v @ C2 @ B ) ) ) ).
% subsetD
thf(fact_328_subsetD,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C2: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B )
=> ( ( member7453568604450474000od_v_v @ C2 @ A2 )
=> ( member7453568604450474000od_v_v @ C2 @ B ) ) ) ).
% subsetD
thf(fact_329_Int__mono,axiom,
! [A2: set_v,C: set_v,B: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A2 @ C )
=> ( ( ord_less_eq_set_v @ B @ D2 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A2 @ B ) @ ( inf_inf_set_v @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_330_Int__mono,axiom,
! [A2: set_Product_prod_v_v,C: set_Product_prod_v_v,B: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) @ ( inf_in6271465464967711157od_v_v @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_331_Int__assoc,axiom,
! [A2: set_v,B: set_v,C: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A2 @ B ) @ C )
= ( inf_inf_set_v @ A2 @ ( inf_inf_set_v @ B @ C ) ) ) ).
% Int_assoc
thf(fact_332_equalityE,axiom,
! [A2: set_v,B: set_v] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_v @ A2 @ B )
=> ~ ( ord_less_eq_set_v @ B @ A2 ) ) ) ).
% equalityE
thf(fact_333_equalityE,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A2 = B )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A2 @ B )
=> ~ ( ord_le7336532860387713383od_v_v @ B @ A2 ) ) ) ).
% equalityE
thf(fact_334_subset__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A3: set_v,B4: set_v] :
! [X2: v] :
( ( member_v @ X2 @ A3 )
=> ( member_v @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_335_subset__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A3 )
=> ( member7453568604450474000od_v_v @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_336_Int__absorb,axiom,
! [A2: set_v] :
( ( inf_inf_set_v @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_337_Int__lower1,axiom,
! [A2: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_338_Int__lower1,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_339_Int__lower2,axiom,
! [A2: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_340_Int__lower2,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_341_equalityD1,axiom,
! [A2: set_v,B: set_v] :
( ( A2 = B )
=> ( ord_less_eq_set_v @ A2 @ B ) ) ).
% equalityD1
thf(fact_342_equalityD1,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A2 = B )
=> ( ord_le7336532860387713383od_v_v @ A2 @ B ) ) ).
% equalityD1
thf(fact_343_equalityD2,axiom,
! [A2: set_v,B: set_v] :
( ( A2 = B )
=> ( ord_less_eq_set_v @ B @ A2 ) ) ).
% equalityD2
thf(fact_344_equalityD2,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A2 = B )
=> ( ord_le7336532860387713383od_v_v @ B @ A2 ) ) ).
% equalityD2
thf(fact_345_subset__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A3: set_v,B4: set_v] :
! [T2: v] :
( ( member_v @ T2 @ A3 )
=> ( member_v @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_346_subset__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
! [T2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ T2 @ A3 )
=> ( member7453568604450474000od_v_v @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_347_Int__absorb1,axiom,
! [B: set_v,A2: set_v] :
( ( ord_less_eq_set_v @ B @ A2 )
=> ( ( inf_inf_set_v @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_348_Int__absorb1,axiom,
! [B: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_349_Int__absorb2,axiom,
! [A2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A2 @ B )
=> ( ( inf_inf_set_v @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_350_Int__absorb2,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_351_Int__commute,axiom,
( inf_inf_set_v
= ( ^ [A3: set_v,B4: set_v] : ( inf_inf_set_v @ B4 @ A3 ) ) ) ).
% Int_commute
thf(fact_352_subset__refl,axiom,
! [A2: set_v] : ( ord_less_eq_set_v @ A2 @ A2 ) ).
% subset_refl
thf(fact_353_subset__refl,axiom,
! [A2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A2 @ A2 ) ).
% subset_refl
thf(fact_354_Collect__mono,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ! [X3: set_v] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) ) ) ).
% Collect_mono
thf(fact_355_Collect__mono,axiom,
! [P: v > $o,Q: v > $o] :
( ! [X3: v] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) ) ) ).
% Collect_mono
thf(fact_356_Collect__mono,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ! [X3: product_prod_v_v] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) ) ) ).
% Collect_mono
thf(fact_357_Int__greatest,axiom,
! [C: set_v,A2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C @ A2 )
=> ( ( ord_less_eq_set_v @ C @ B )
=> ( ord_less_eq_set_v @ C @ ( inf_inf_set_v @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_358_Int__greatest,axiom,
! [C: set_Product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A2 )
=> ( ( ord_le7336532860387713383od_v_v @ C @ B )
=> ( ord_le7336532860387713383od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_359_subset__trans,axiom,
! [A2: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A2 @ B )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ord_less_eq_set_v @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_360_subset__trans,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ord_le7336532860387713383od_v_v @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_361_set__eq__subset,axiom,
( ( ^ [Y2: set_v,Z: set_v] : ( Y2 = Z ) )
= ( ^ [A3: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ A3 @ B4 )
& ( ord_less_eq_set_v @ B4 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_362_set__eq__subset,axiom,
( ( ^ [Y2: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( Y2 = Z ) )
= ( ^ [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B4 )
& ( ord_le7336532860387713383od_v_v @ B4 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_363_Int__left__absorb,axiom,
! [A2: set_v,B: set_v] :
( ( inf_inf_set_v @ A2 @ ( inf_inf_set_v @ A2 @ B ) )
= ( inf_inf_set_v @ A2 @ B ) ) ).
% Int_left_absorb
thf(fact_364_Collect__mono__iff,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) )
= ( ! [X2: set_v] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_365_Collect__mono__iff,axiom,
! [P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) )
= ( ! [X2: v] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_366_Collect__mono__iff,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) )
= ( ! [X2: product_prod_v_v] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_367_Int__Collect__mono,axiom,
! [A2: set_set_v,B: set_set_v,P: set_v > $o,Q: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ A2 @ B )
=> ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le5216385588623774835_set_v @ ( inf_inf_set_set_v @ A2 @ ( collect_set_v @ P ) ) @ ( inf_inf_set_set_v @ B @ ( collect_set_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_368_Int__Collect__mono,axiom,
! [A2: set_v,B: set_v,P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ A2 @ B )
=> ( ! [X3: v] :
( ( member_v @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A2 @ ( collect_v @ P ) ) @ ( inf_inf_set_v @ B @ ( collect_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_369_Int__Collect__mono,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B )
=> ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ ( collec140062887454715474od_v_v @ P ) ) @ ( inf_in6271465464967711157od_v_v @ B @ ( collec140062887454715474od_v_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_370_Int__left__commute,axiom,
! [A2: set_v,B: set_v,C: set_v] :
( ( inf_inf_set_v @ A2 @ ( inf_inf_set_v @ B @ C ) )
= ( inf_inf_set_v @ B @ ( inf_inf_set_v @ A2 @ C ) ) ) ).
% Int_left_commute
thf(fact_371_Union__mono,axiom,
! [A2: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A2 @ B )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Union_mono
thf(fact_372_Union__mono,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ B )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Union_mono
thf(fact_373_Union__Int__subset,axiom,
! [A2: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] : ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ ( inf_in6058586722421118357od_v_v @ A2 @ B ) ) @ ( inf_in6271465464967711157od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Union_Int_subset
thf(fact_374_Union__Int__subset,axiom,
! [A2: set_set_v,B: set_set_v] : ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ ( inf_inf_set_set_v @ A2 @ B ) ) @ ( inf_inf_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Union_Int_subset
thf(fact_375_Un__Int__assoc__eq,axiom,
! [A2: set_set_v,B: set_set_v,C: set_set_v] :
( ( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A2 @ B ) @ C )
= ( inf_inf_set_set_v @ A2 @ ( sup_sup_set_set_v @ B @ C ) ) )
= ( ord_le5216385588623774835_set_v @ C @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_376_Un__Int__assoc__eq,axiom,
! [A2: set_v,B: set_v,C: set_v] :
( ( ( sup_sup_set_v @ ( inf_inf_set_v @ A2 @ B ) @ C )
= ( inf_inf_set_v @ A2 @ ( sup_sup_set_v @ B @ C ) ) )
= ( ord_less_eq_set_v @ C @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_377_Un__Int__assoc__eq,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) @ C )
= ( inf_in6271465464967711157od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B @ C ) ) )
= ( ord_le7336532860387713383od_v_v @ C @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_378_Collect__conj__eq,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ( collect_set_v
@ ^ [X2: set_v] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_379_Collect__conj__eq,axiom,
! [P: v > $o,Q: v > $o] :
( ( collect_v
@ ^ [X2: v] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_380_Int__Collect,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( member7453568604450474000od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A2 @ ( collec140062887454715474od_v_v @ P ) ) )
= ( ( member7453568604450474000od_v_v @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_381_Int__Collect,axiom,
! [X: set_v,A2: set_set_v,P: set_v > $o] :
( ( member_set_v @ X @ ( inf_inf_set_set_v @ A2 @ ( collect_set_v @ P ) ) )
= ( ( member_set_v @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_382_Int__Collect,axiom,
! [X: v,A2: set_v,P: v > $o] :
( ( member_v @ X @ ( inf_inf_set_v @ A2 @ ( collect_v @ P ) ) )
= ( ( member_v @ X @ A2 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_383_Int__def,axiom,
( inf_in6271465464967711157od_v_v
= ( ^ [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A3 )
& ( member7453568604450474000od_v_v @ X2 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_384_Int__def,axiom,
( inf_inf_set_set_v
= ( ^ [A3: set_set_v,B4: set_set_v] :
( collect_set_v
@ ^ [X2: set_v] :
( ( member_set_v @ X2 @ A3 )
& ( member_set_v @ X2 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_385_Int__def,axiom,
( inf_inf_set_v
= ( ^ [A3: set_v,B4: set_v] :
( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ A3 )
& ( member_v @ X2 @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_386_Collect__subset,axiom,
! [A2: set_set_v,P: set_v > $o] :
( ord_le5216385588623774835_set_v
@ ( collect_set_v
@ ^ [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_387_Collect__subset,axiom,
! [A2: set_v,P: v > $o] :
( ord_less_eq_set_v
@ ( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_388_Collect__subset,axiom,
! [A2: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ord_le7336532860387713383od_v_v
@ ( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_389_cSup__eq__maximum,axiom,
! [Z2: set_Product_prod_v_v,X4: set_se8455005133513928103od_v_v] :
( ( member8406446414694345712od_v_v @ Z2 @ X4 )
=> ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ X4 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ Z2 ) )
=> ( ( comple5788137035815166516od_v_v @ X4 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_390_cSup__eq__maximum,axiom,
! [Z2: set_v,X4: set_set_v] :
( ( member_set_v @ Z2 @ X4 )
=> ( ! [X3: set_v] :
( ( member_set_v @ X3 @ X4 )
=> ( ord_less_eq_set_v @ X3 @ Z2 ) )
=> ( ( comple2307003700295860064_set_v @ X4 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_391_Sup__upper2,axiom,
! [U2: set_Product_prod_v_v,A2: set_se8455005133513928103od_v_v,V3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ U2 @ A2 )
=> ( ( ord_le7336532860387713383od_v_v @ V3 @ U2 )
=> ( ord_le7336532860387713383od_v_v @ V3 @ ( comple5788137035815166516od_v_v @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_392_Sup__upper2,axiom,
! [U2: set_v,A2: set_set_v,V3: set_v] :
( ( member_set_v @ U2 @ A2 )
=> ( ( ord_less_eq_set_v @ V3 @ U2 )
=> ( ord_less_eq_set_v @ V3 @ ( comple2307003700295860064_set_v @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_393_Sup__le__iff,axiom,
! [A2: set_se8455005133513928103od_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ B2 )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ X2 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_394_Sup__le__iff,axiom,
! [A2: set_set_v,B2: set_v] :
( ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ B2 )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
=> ( ord_less_eq_set_v @ X2 @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_395_Sup__upper,axiom,
! [X: set_Product_prod_v_v,A2: set_se8455005133513928103od_v_v] :
( ( member8406446414694345712od_v_v @ X @ A2 )
=> ( ord_le7336532860387713383od_v_v @ X @ ( comple5788137035815166516od_v_v @ A2 ) ) ) ).
% Sup_upper
thf(fact_396_Sup__upper,axiom,
! [X: set_v,A2: set_set_v] :
( ( member_set_v @ X @ A2 )
=> ( ord_less_eq_set_v @ X @ ( comple2307003700295860064_set_v @ A2 ) ) ) ).
% Sup_upper
thf(fact_397_Sup__least,axiom,
! [A2: set_se8455005133513928103od_v_v,Z2: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ Z2 ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ Z2 ) ) ).
% Sup_least
thf(fact_398_Sup__least,axiom,
! [A2: set_set_v,Z2: set_v] :
( ! [X3: set_v] :
( ( member_set_v @ X3 @ A2 )
=> ( ord_less_eq_set_v @ X3 @ Z2 ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ Z2 ) ) ).
% Sup_least
thf(fact_399_Sup__mono,axiom,
! [A2: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ! [A5: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A5 @ A2 )
=> ? [X5: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X5 @ B )
& ( ord_le7336532860387713383od_v_v @ A5 @ X5 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Sup_mono
thf(fact_400_Sup__mono,axiom,
! [A2: set_set_v,B: set_set_v] :
( ! [A5: set_v] :
( ( member_set_v @ A5 @ A2 )
=> ? [X5: set_v] :
( ( member_set_v @ X5 @ B )
& ( ord_less_eq_set_v @ A5 @ X5 ) ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Sup_mono
thf(fact_401_Sup__eqI,axiom,
! [A2: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ! [Y3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Y3 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ Y3 @ X ) )
=> ( ! [Y3: set_Product_prod_v_v] :
( ! [Z3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Z3 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ Z3 @ Y3 ) )
=> ( ord_le7336532860387713383od_v_v @ X @ Y3 ) )
=> ( ( comple5788137035815166516od_v_v @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_402_Sup__eqI,axiom,
! [A2: set_set_v,X: set_v] :
( ! [Y3: set_v] :
( ( member_set_v @ Y3 @ A2 )
=> ( ord_less_eq_set_v @ Y3 @ X ) )
=> ( ! [Y3: set_v] :
( ! [Z3: set_v] :
( ( member_set_v @ Z3 @ A2 )
=> ( ord_less_eq_set_v @ Z3 @ Y3 ) )
=> ( ord_less_eq_set_v @ X @ Y3 ) )
=> ( ( comple2307003700295860064_set_v @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_403_Union__subsetI,axiom,
! [A2: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A2 )
=> ? [Y5: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Y5 @ B )
& ( ord_le7336532860387713383od_v_v @ X3 @ Y5 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Union_subsetI
thf(fact_404_Union__subsetI,axiom,
! [A2: set_set_v,B: set_set_v] :
( ! [X3: set_v] :
( ( member_set_v @ X3 @ A2 )
=> ? [Y5: set_v] :
( ( member_set_v @ Y5 @ B )
& ( ord_less_eq_set_v @ X3 @ Y5 ) ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Union_subsetI
thf(fact_405_Union__upper,axiom,
! [B: set_Product_prod_v_v,A2: set_se8455005133513928103od_v_v] :
( ( member8406446414694345712od_v_v @ B @ A2 )
=> ( ord_le7336532860387713383od_v_v @ B @ ( comple5788137035815166516od_v_v @ A2 ) ) ) ).
% Union_upper
thf(fact_406_Union__upper,axiom,
! [B: set_v,A2: set_set_v] :
( ( member_set_v @ B @ A2 )
=> ( ord_less_eq_set_v @ B @ ( comple2307003700295860064_set_v @ A2 ) ) ) ).
% Union_upper
thf(fact_407_Union__least,axiom,
! [A2: set_se8455005133513928103od_v_v,C: set_Product_prod_v_v] :
( ! [X6: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X6 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ X6 @ C ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ C ) ) ).
% Union_least
thf(fact_408_Union__least,axiom,
! [A2: set_set_v,C: set_v] :
( ! [X6: set_v] :
( ( member_set_v @ X6 @ A2 )
=> ( ord_less_eq_set_v @ X6 @ C ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ C ) ) ).
% Union_least
thf(fact_409_bot__set__def,axiom,
( bot_bot_set_set_v
= ( collect_set_v @ bot_bot_set_v_o ) ) ).
% bot_set_def
thf(fact_410_bot__set__def,axiom,
( bot_bot_set_v
= ( collect_v @ bot_bot_v_o ) ) ).
% bot_set_def
thf(fact_411_bot__set__def,axiom,
( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ bot_bo8461541820394803818_v_v_o ) ) ).
% bot_set_def
thf(fact_412_Int__emptyI,axiom,
! [A2: set_set_v,B: set_set_v] :
( ! [X3: set_v] :
( ( member_set_v @ X3 @ A2 )
=> ~ ( member_set_v @ X3 @ B ) )
=> ( ( inf_inf_set_set_v @ A2 @ B )
= bot_bot_set_set_v ) ) ).
% Int_emptyI
thf(fact_413_Int__emptyI,axiom,
! [A2: set_v,B: set_v] :
( ! [X3: v] :
( ( member_v @ X3 @ A2 )
=> ~ ( member_v @ X3 @ B ) )
=> ( ( inf_inf_set_v @ A2 @ B )
= bot_bot_set_v ) ) ).
% Int_emptyI
thf(fact_414_Int__emptyI,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A2 )
=> ~ ( member7453568604450474000od_v_v @ X3 @ B ) )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ B )
= bot_bo723834152578015283od_v_v ) ) ).
% Int_emptyI
thf(fact_415_disjoint__iff,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( ( inf_inf_set_set_v @ A2 @ B )
= bot_bot_set_set_v )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
=> ~ ( member_set_v @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_416_disjoint__iff,axiom,
! [A2: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A2 @ B )
= bot_bot_set_v )
= ( ! [X2: v] :
( ( member_v @ X2 @ A2 )
=> ~ ( member_v @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_417_disjoint__iff,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A2 @ B )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A2 )
=> ~ ( member7453568604450474000od_v_v @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_418_Int__empty__left,axiom,
! [B: set_set_v] :
( ( inf_inf_set_set_v @ bot_bot_set_set_v @ B )
= bot_bot_set_set_v ) ).
% Int_empty_left
thf(fact_419_Int__empty__left,axiom,
! [B: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ B )
= bot_bot_set_v ) ).
% Int_empty_left
thf(fact_420_Int__empty__left,axiom,
! [B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ B )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_left
thf(fact_421_Int__empty__right,axiom,
! [A2: set_set_v] :
( ( inf_inf_set_set_v @ A2 @ bot_bot_set_set_v )
= bot_bot_set_set_v ) ).
% Int_empty_right
thf(fact_422_Int__empty__right,axiom,
! [A2: set_v] :
( ( inf_inf_set_v @ A2 @ bot_bot_set_v )
= bot_bot_set_v ) ).
% Int_empty_right
thf(fact_423_Int__empty__right,axiom,
! [A2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A2 @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_right
thf(fact_424_disjoint__iff__not__equal,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( ( inf_inf_set_set_v @ A2 @ B )
= bot_bot_set_set_v )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
=> ! [Y4: set_v] :
( ( member_set_v @ Y4 @ B )
=> ( X2 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_425_disjoint__iff__not__equal,axiom,
! [A2: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A2 @ B )
= bot_bot_set_v )
= ( ! [X2: v] :
( ( member_v @ X2 @ A2 )
=> ! [Y4: v] :
( ( member_v @ Y4 @ B )
=> ( X2 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_426_disjoint__iff__not__equal,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A2 @ B )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A2 )
=> ! [Y4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y4 @ B )
=> ( X2 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_427_Int__insert__left,axiom,
! [A: set_v,C: set_set_v,B: set_set_v] :
( ( ( member_set_v @ A @ C )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B ) @ C )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ B @ C ) ) ) )
& ( ~ ( member_set_v @ A @ C )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B ) @ C )
= ( inf_inf_set_set_v @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_428_Int__insert__left,axiom,
! [A: product_prod_v_v,C: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ C )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ C )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B ) @ C )
= ( inf_in6271465464967711157od_v_v @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_429_Int__insert__left,axiom,
! [A: v,C: set_v,B: set_v] :
( ( ( member_v @ A @ C )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B ) @ C )
= ( insert_v @ A @ ( inf_inf_set_v @ B @ C ) ) ) )
& ( ~ ( member_v @ A @ C )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B ) @ C )
= ( inf_inf_set_v @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_430_Int__insert__right,axiom,
! [A: set_v,A2: set_set_v,B: set_set_v] :
( ( ( member_set_v @ A @ A2 )
=> ( ( inf_inf_set_set_v @ A2 @ ( insert_set_v @ A @ B ) )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ A2 @ B ) ) ) )
& ( ~ ( member_set_v @ A @ A2 )
=> ( ( inf_inf_set_set_v @ A2 @ ( insert_set_v @ A @ B ) )
= ( inf_inf_set_set_v @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_431_Int__insert__right,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( inf_in6271465464967711157od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( inf_in6271465464967711157od_v_v @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_432_Int__insert__right,axiom,
! [A: v,A2: set_v,B: set_v] :
( ( ( member_v @ A @ A2 )
=> ( ( inf_inf_set_v @ A2 @ ( insert_v @ A @ B ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A2 @ B ) ) ) )
& ( ~ ( member_v @ A @ A2 )
=> ( ( inf_inf_set_v @ A2 @ ( insert_v @ A @ B ) )
= ( inf_inf_set_v @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_433_Un__Int__crazy,axiom,
! [A2: set_v,B: set_v,C: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ A2 @ B ) @ ( inf_inf_set_v @ B @ C ) ) @ ( inf_inf_set_v @ C @ A2 ) )
= ( inf_inf_set_v @ ( inf_inf_set_v @ ( sup_sup_set_v @ A2 @ B ) @ ( sup_sup_set_v @ B @ C ) ) @ ( sup_sup_set_v @ C @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_434_Un__Int__crazy,axiom,
! [A2: set_set_v,B: set_set_v,C: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A2 @ B ) @ ( inf_inf_set_set_v @ B @ C ) ) @ ( inf_inf_set_set_v @ C @ A2 ) )
= ( inf_inf_set_set_v @ ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ A2 @ B ) @ ( sup_sup_set_set_v @ B @ C ) ) @ ( sup_sup_set_set_v @ C @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_435_Un__Int__crazy,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) @ ( inf_in6271465464967711157od_v_v @ B @ C ) ) @ ( inf_in6271465464967711157od_v_v @ C @ A2 ) )
= ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B ) @ ( sup_su414716646722978715od_v_v @ B @ C ) ) @ ( sup_su414716646722978715od_v_v @ C @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_436_Int__Un__distrib,axiom,
! [A2: set_v,B: set_v,C: set_v] :
( ( inf_inf_set_v @ A2 @ ( sup_sup_set_v @ B @ C ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ A2 @ B ) @ ( inf_inf_set_v @ A2 @ C ) ) ) ).
% Int_Un_distrib
thf(fact_437_Int__Un__distrib,axiom,
! [A2: set_set_v,B: set_set_v,C: set_set_v] :
( ( inf_inf_set_set_v @ A2 @ ( sup_sup_set_set_v @ B @ C ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A2 @ B ) @ ( inf_inf_set_set_v @ A2 @ C ) ) ) ).
% Int_Un_distrib
thf(fact_438_Int__Un__distrib,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B @ C ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) @ ( inf_in6271465464967711157od_v_v @ A2 @ C ) ) ) ).
% Int_Un_distrib
thf(fact_439_Un__Int__distrib,axiom,
! [A2: set_v,B: set_v,C: set_v] :
( ( sup_sup_set_v @ A2 @ ( inf_inf_set_v @ B @ C ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ A2 @ B ) @ ( sup_sup_set_v @ A2 @ C ) ) ) ).
% Un_Int_distrib
thf(fact_440_Un__Int__distrib,axiom,
! [A2: set_set_v,B: set_set_v,C: set_set_v] :
( ( sup_sup_set_set_v @ A2 @ ( inf_inf_set_set_v @ B @ C ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ A2 @ B ) @ ( sup_sup_set_set_v @ A2 @ C ) ) ) ).
% Un_Int_distrib
thf(fact_441_Un__Int__distrib,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( inf_in6271465464967711157od_v_v @ B @ C ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B ) @ ( sup_su414716646722978715od_v_v @ A2 @ C ) ) ) ).
% Un_Int_distrib
thf(fact_442_Int__Un__distrib2,axiom,
! [B: set_v,C: set_v,A2: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ B @ C ) @ A2 )
= ( sup_sup_set_v @ ( inf_inf_set_v @ B @ A2 ) @ ( inf_inf_set_v @ C @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_443_Int__Un__distrib2,axiom,
! [B: set_set_v,C: set_set_v,A2: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ B @ C ) @ A2 )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ B @ A2 ) @ ( inf_inf_set_set_v @ C @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_444_Int__Un__distrib2,axiom,
! [B: set_Product_prod_v_v,C: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C ) @ A2 )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B @ A2 ) @ ( inf_in6271465464967711157od_v_v @ C @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_445_Un__Int__distrib2,axiom,
! [B: set_v,C: set_v,A2: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ B @ C ) @ A2 )
= ( inf_inf_set_v @ ( sup_sup_set_v @ B @ A2 ) @ ( sup_sup_set_v @ C @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_446_Un__Int__distrib2,axiom,
! [B: set_set_v,C: set_set_v,A2: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ B @ C ) @ A2 )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ B @ A2 ) @ ( sup_sup_set_set_v @ C @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_447_Un__Int__distrib2,axiom,
! [B: set_Product_prod_v_v,C: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B @ C ) @ A2 )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B @ A2 ) @ ( sup_su414716646722978715od_v_v @ C @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_448_insert__mono,axiom,
! [C: set_set_v,D2: set_set_v,A: set_v] :
( ( ord_le5216385588623774835_set_v @ C @ D2 )
=> ( ord_le5216385588623774835_set_v @ ( insert_set_v @ A @ C ) @ ( insert_set_v @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_449_insert__mono,axiom,
! [C: set_v,D2: set_v,A: v] :
( ( ord_less_eq_set_v @ C @ D2 )
=> ( ord_less_eq_set_v @ ( insert_v @ A @ C ) @ ( insert_v @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_450_insert__mono,axiom,
! [C: set_Product_prod_v_v,D2: set_Product_prod_v_v,A: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ A @ C ) @ ( insert1338601472111419319od_v_v @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_451_subset__insert,axiom,
! [X: set_v,A2: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ X @ A2 )
=> ( ( ord_le5216385588623774835_set_v @ A2 @ ( insert_set_v @ X @ B ) )
= ( ord_le5216385588623774835_set_v @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_452_subset__insert,axiom,
! [X: v,A2: set_v,B: set_v] :
( ~ ( member_v @ X @ A2 )
=> ( ( ord_less_eq_set_v @ A2 @ ( insert_v @ X @ B ) )
= ( ord_less_eq_set_v @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_453_subset__insert,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A2 )
=> ( ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X @ B ) )
= ( ord_le7336532860387713383od_v_v @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_454_subset__insertI,axiom,
! [B: set_set_v,A: set_v] : ( ord_le5216385588623774835_set_v @ B @ ( insert_set_v @ A @ B ) ) ).
% subset_insertI
thf(fact_455_subset__insertI,axiom,
! [B: set_v,A: v] : ( ord_less_eq_set_v @ B @ ( insert_v @ A @ B ) ) ).
% subset_insertI
thf(fact_456_subset__insertI,axiom,
! [B: set_Product_prod_v_v,A: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ B ) ) ).
% subset_insertI
thf(fact_457_subset__insertI2,axiom,
! [A2: set_set_v,B: set_set_v,B2: set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ B )
=> ( ord_le5216385588623774835_set_v @ A2 @ ( insert_set_v @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_458_subset__insertI2,axiom,
! [A2: set_v,B: set_v,B2: v] :
( ( ord_less_eq_set_v @ A2 @ B )
=> ( ord_less_eq_set_v @ A2 @ ( insert_v @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_459_subset__insertI2,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,B2: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B )
=> ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_460_Un__mono,axiom,
! [A2: set_set_v,C: set_set_v,B: set_set_v,D2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ C )
=> ( ( ord_le5216385588623774835_set_v @ B @ D2 )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A2 @ B ) @ ( sup_sup_set_set_v @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_461_Un__mono,axiom,
! [A2: set_v,C: set_v,B: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A2 @ C )
=> ( ( ord_less_eq_set_v @ B @ D2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A2 @ B ) @ ( sup_sup_set_v @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_462_Un__mono,axiom,
! [A2: set_Product_prod_v_v,C: set_Product_prod_v_v,B: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B ) @ ( sup_su414716646722978715od_v_v @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_463_Un__least,axiom,
! [A2: set_set_v,C: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ C )
=> ( ( ord_le5216385588623774835_set_v @ B @ C )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A2 @ B ) @ C ) ) ) ).
% Un_least
thf(fact_464_Un__least,axiom,
! [A2: set_v,C: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A2 @ C )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A2 @ B ) @ C ) ) ) ).
% Un_least
thf(fact_465_Un__least,axiom,
! [A2: set_Product_prod_v_v,C: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B ) @ C ) ) ) ).
% Un_least
thf(fact_466_Un__upper1,axiom,
! [A2: set_set_v,B: set_set_v] : ( ord_le5216385588623774835_set_v @ A2 @ ( sup_sup_set_set_v @ A2 @ B ) ) ).
% Un_upper1
thf(fact_467_Un__upper1,axiom,
! [A2: set_v,B: set_v] : ( ord_less_eq_set_v @ A2 @ ( sup_sup_set_v @ A2 @ B ) ) ).
% Un_upper1
thf(fact_468_Un__upper1,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ A2 @ B ) ) ).
% Un_upper1
thf(fact_469_Un__upper2,axiom,
! [B: set_set_v,A2: set_set_v] : ( ord_le5216385588623774835_set_v @ B @ ( sup_sup_set_set_v @ A2 @ B ) ) ).
% Un_upper2
thf(fact_470_Un__upper2,axiom,
! [B: set_v,A2: set_v] : ( ord_less_eq_set_v @ B @ ( sup_sup_set_v @ A2 @ B ) ) ).
% Un_upper2
thf(fact_471_Un__upper2,axiom,
! [B: set_Product_prod_v_v,A2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A2 @ B ) ) ).
% Un_upper2
thf(fact_472_Un__absorb1,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ B )
=> ( ( sup_sup_set_set_v @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_473_Un__absorb1,axiom,
! [A2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A2 @ B )
=> ( ( sup_sup_set_v @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_474_Un__absorb1,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B )
=> ( ( sup_su414716646722978715od_v_v @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_475_Un__absorb2,axiom,
! [B: set_set_v,A2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B @ A2 )
=> ( ( sup_sup_set_set_v @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_476_Un__absorb2,axiom,
! [B: set_v,A2: set_v] :
( ( ord_less_eq_set_v @ B @ A2 )
=> ( ( sup_sup_set_v @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_477_Un__absorb2,axiom,
! [B: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A2 )
=> ( ( sup_su414716646722978715od_v_v @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_478_subset__UnE,axiom,
! [C: set_set_v,A2: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C @ ( sup_sup_set_set_v @ A2 @ B ) )
=> ~ ! [A6: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A6 @ A2 )
=> ! [B5: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B5 @ B )
=> ( C
!= ( sup_sup_set_set_v @ A6 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_479_subset__UnE,axiom,
! [C: set_v,A2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A2 @ B ) )
=> ~ ! [A6: set_v] :
( ( ord_less_eq_set_v @ A6 @ A2 )
=> ! [B5: set_v] :
( ( ord_less_eq_set_v @ B5 @ B )
=> ( C
!= ( sup_sup_set_v @ A6 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_480_subset__UnE,axiom,
! [C: set_Product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A2 @ B ) )
=> ~ ! [A6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A6 @ A2 )
=> ! [B5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B5 @ B )
=> ( C
!= ( sup_su414716646722978715od_v_v @ A6 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_481_subset__Un__eq,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [A3: set_set_v,B4: set_set_v] :
( ( sup_sup_set_set_v @ A3 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_482_subset__Un__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A3: set_v,B4: set_v] :
( ( sup_sup_set_v @ A3 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_483_subset__Un__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_484_Union__disjoint,axiom,
! [C: set_set_set_v,A2: set_set_v] :
( ( ( inf_inf_set_set_v @ ( comple5450237519782573632_set_v @ C ) @ A2 )
= bot_bot_set_set_v )
= ( ! [X2: set_set_v] :
( ( member_set_set_v @ X2 @ C )
=> ( ( inf_inf_set_set_v @ X2 @ A2 )
= bot_bot_set_set_v ) ) ) ) ).
% Union_disjoint
thf(fact_485_Union__disjoint,axiom,
! [C: set_se8455005133513928103od_v_v,A2: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ ( comple5788137035815166516od_v_v @ C ) @ A2 )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ C )
=> ( ( inf_in6271465464967711157od_v_v @ X2 @ A2 )
= bot_bo723834152578015283od_v_v ) ) ) ) ).
% Union_disjoint
thf(fact_486_Union__disjoint,axiom,
! [C: set_set_v,A2: set_v] :
( ( ( inf_inf_set_v @ ( comple2307003700295860064_set_v @ C ) @ A2 )
= bot_bot_set_v )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ C )
=> ( ( inf_inf_set_v @ X2 @ A2 )
= bot_bot_set_v ) ) ) ) ).
% Union_disjoint
thf(fact_487_cSup__eq__non__empty,axiom,
! [X4: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( X4 != bot_bo3497076220358800403od_v_v )
=> ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ X4 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ A ) )
=> ( ! [Y3: set_Product_prod_v_v] :
( ! [X5: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X5 @ X4 )
=> ( ord_le7336532860387713383od_v_v @ X5 @ Y3 ) )
=> ( ord_le7336532860387713383od_v_v @ A @ Y3 ) )
=> ( ( comple5788137035815166516od_v_v @ X4 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_488_cSup__eq__non__empty,axiom,
! [X4: set_set_v,A: set_v] :
( ( X4 != bot_bot_set_set_v )
=> ( ! [X3: set_v] :
( ( member_set_v @ X3 @ X4 )
=> ( ord_less_eq_set_v @ X3 @ A ) )
=> ( ! [Y3: set_v] :
( ! [X5: set_v] :
( ( member_set_v @ X5 @ X4 )
=> ( ord_less_eq_set_v @ X5 @ Y3 ) )
=> ( ord_less_eq_set_v @ A @ Y3 ) )
=> ( ( comple2307003700295860064_set_v @ X4 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_489_cSup__least,axiom,
! [X4: set_se8455005133513928103od_v_v,Z2: set_Product_prod_v_v] :
( ( X4 != bot_bo3497076220358800403od_v_v )
=> ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ X4 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ Z2 ) )
=> ( ord_le7336532860387713383od_v_v @ ( comple5788137035815166516od_v_v @ X4 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_490_cSup__least,axiom,
! [X4: set_set_v,Z2: set_v] :
( ( X4 != bot_bot_set_set_v )
=> ( ! [X3: set_v] :
( ( member_set_v @ X3 @ X4 )
=> ( ord_less_eq_set_v @ X3 @ Z2 ) )
=> ( ord_less_eq_set_v @ ( comple2307003700295860064_set_v @ X4 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_491_less__eq__Sup,axiom,
! [A2: set_se8455005133513928103od_v_v,U2: set_Product_prod_v_v] :
( ! [V2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ V2 @ A2 )
=> ( ord_le7336532860387713383od_v_v @ U2 @ V2 ) )
=> ( ( A2 != bot_bo3497076220358800403od_v_v )
=> ( ord_le7336532860387713383od_v_v @ U2 @ ( comple5788137035815166516od_v_v @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_492_less__eq__Sup,axiom,
! [A2: set_set_v,U2: set_v] :
( ! [V2: set_v] :
( ( member_set_v @ V2 @ A2 )
=> ( ord_less_eq_set_v @ U2 @ V2 ) )
=> ( ( A2 != bot_bot_set_set_v )
=> ( ord_less_eq_set_v @ U2 @ ( comple2307003700295860064_set_v @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_493_sup__set__def,axiom,
( sup_sup_set_v
= ( ^ [A3: set_v,B4: set_v] :
( collect_v
@ ( sup_sup_v_o
@ ^ [X2: v] : ( member_v @ X2 @ A3 )
@ ^ [X2: v] : ( member_v @ X2 @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_494_sup__set__def,axiom,
( sup_sup_set_set_v
= ( ^ [A3: set_set_v,B4: set_set_v] :
( collect_set_v
@ ( sup_sup_set_v_o
@ ^ [X2: set_v] : ( member_set_v @ X2 @ A3 )
@ ^ [X2: set_v] : ( member_set_v @ X2 @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_495_sup__set__def,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ( sup_su5941406310530359554_v_v_o
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A3 )
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_496_subset__singletonD,axiom,
! [A2: set_set_v,X: set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
=> ( ( A2 = bot_bot_set_set_v )
| ( A2
= ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ).
% subset_singletonD
thf(fact_497_subset__singletonD,axiom,
! [A2: set_v,X: v] :
( ( ord_less_eq_set_v @ A2 @ ( insert_v @ X @ bot_bot_set_v ) )
=> ( ( A2 = bot_bot_set_v )
| ( A2
= ( insert_v @ X @ bot_bot_set_v ) ) ) ) ).
% subset_singletonD
thf(fact_498_subset__singletonD,axiom,
! [A2: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
=> ( ( A2 = bot_bo723834152578015283od_v_v )
| ( A2
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singletonD
thf(fact_499_subset__singleton__iff,axiom,
! [X4: set_set_v,A: set_v] :
( ( ord_le5216385588623774835_set_v @ X4 @ ( insert_set_v @ A @ bot_bot_set_set_v ) )
= ( ( X4 = bot_bot_set_set_v )
| ( X4
= ( insert_set_v @ A @ bot_bot_set_set_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_500_subset__singleton__iff,axiom,
! [X4: set_v,A: v] :
( ( ord_less_eq_set_v @ X4 @ ( insert_v @ A @ bot_bot_set_v ) )
= ( ( X4 = bot_bot_set_v )
| ( X4
= ( insert_v @ A @ bot_bot_set_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_501_subset__singleton__iff,axiom,
! [X4: set_Product_prod_v_v,A: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
= ( ( X4 = bot_bo723834152578015283od_v_v )
| ( X4
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_502_UnionE,axiom,
! [A2: product_prod_v_v,C: set_se8455005133513928103od_v_v] :
( ( member7453568604450474000od_v_v @ A2 @ ( comple5788137035815166516od_v_v @ C ) )
=> ~ ! [X6: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A2 @ X6 )
=> ~ ( member8406446414694345712od_v_v @ X6 @ C ) ) ) ).
% UnionE
thf(fact_503_UnionE,axiom,
! [A2: v,C: set_set_v] :
( ( member_v @ A2 @ ( comple2307003700295860064_set_v @ C ) )
=> ~ ! [X6: set_v] :
( ( member_v @ A2 @ X6 )
=> ~ ( member_set_v @ X6 @ C ) ) ) ).
% UnionE
thf(fact_504_Union__empty__conv,axiom,
! [A2: set_set_set_v] :
( ( ( comple5450237519782573632_set_v @ A2 )
= bot_bot_set_set_v )
= ( ! [X2: set_set_v] :
( ( member_set_set_v @ X2 @ A2 )
=> ( X2 = bot_bot_set_set_v ) ) ) ) ).
% Union_empty_conv
thf(fact_505_Union__empty__conv,axiom,
! [A2: set_se8455005133513928103od_v_v] :
( ( ( comple5788137035815166516od_v_v @ A2 )
= bot_bo723834152578015283od_v_v )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A2 )
=> ( X2 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% Union_empty_conv
thf(fact_506_Union__empty__conv,axiom,
! [A2: set_set_v] :
( ( ( comple2307003700295860064_set_v @ A2 )
= bot_bot_set_v )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
=> ( X2 = bot_bot_set_v ) ) ) ) ).
% Union_empty_conv
thf(fact_507_empty__Union__conv,axiom,
! [A2: set_set_set_v] :
( ( bot_bot_set_set_v
= ( comple5450237519782573632_set_v @ A2 ) )
= ( ! [X2: set_set_v] :
( ( member_set_set_v @ X2 @ A2 )
=> ( X2 = bot_bot_set_set_v ) ) ) ) ).
% empty_Union_conv
thf(fact_508_empty__Union__conv,axiom,
! [A2: set_se8455005133513928103od_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( comple5788137035815166516od_v_v @ A2 ) )
= ( ! [X2: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X2 @ A2 )
=> ( X2 = bot_bo723834152578015283od_v_v ) ) ) ) ).
% empty_Union_conv
thf(fact_509_empty__Union__conv,axiom,
! [A2: set_set_v] :
( ( bot_bot_set_v
= ( comple2307003700295860064_set_v @ A2 ) )
= ( ! [X2: set_v] :
( ( member_set_v @ X2 @ A2 )
=> ( X2 = bot_bot_set_v ) ) ) ) ).
% empty_Union_conv
thf(fact_510_Sup__union__distrib,axiom,
! [A2: set_set_set_v,B: set_set_set_v] :
( ( comple5450237519782573632_set_v @ ( sup_su4471370308589719943_set_v @ A2 @ B ) )
= ( sup_sup_set_set_v @ ( comple5450237519782573632_set_v @ A2 ) @ ( comple5450237519782573632_set_v @ B ) ) ) ).
% Sup_union_distrib
thf(fact_511_Sup__union__distrib,axiom,
! [A2: set_se8455005133513928103od_v_v,B: set_se8455005133513928103od_v_v] :
( ( comple5788137035815166516od_v_v @ ( sup_su335656005089752955od_v_v @ A2 @ B ) )
= ( sup_su414716646722978715od_v_v @ ( comple5788137035815166516od_v_v @ A2 ) @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Sup_union_distrib
thf(fact_512_Sup__union__distrib,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( comple2307003700295860064_set_v @ ( sup_sup_set_set_v @ A2 @ B ) )
= ( sup_sup_set_v @ ( comple2307003700295860064_set_v @ A2 ) @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Sup_union_distrib
thf(fact_513_Union__empty,axiom,
( ( comple5450237519782573632_set_v @ bot_bo5775917114081396255_set_v )
= bot_bot_set_set_v ) ).
% Union_empty
thf(fact_514_Union__empty,axiom,
( ( comple5788137035815166516od_v_v @ bot_bo3497076220358800403od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Union_empty
thf(fact_515_Union__empty,axiom,
( ( comple2307003700295860064_set_v @ bot_bot_set_set_v )
= bot_bot_set_v ) ).
% Union_empty
thf(fact_516_Union__insert,axiom,
! [A: set_set_v,B: set_set_set_v] :
( ( comple5450237519782573632_set_v @ ( insert_set_set_v @ A @ B ) )
= ( sup_sup_set_set_v @ A @ ( comple5450237519782573632_set_v @ B ) ) ) ).
% Union_insert
thf(fact_517_Union__insert,axiom,
! [A: set_Product_prod_v_v,B: set_se8455005133513928103od_v_v] :
( ( comple5788137035815166516od_v_v @ ( insert7504383016908236695od_v_v @ A @ B ) )
= ( sup_su414716646722978715od_v_v @ A @ ( comple5788137035815166516od_v_v @ B ) ) ) ).
% Union_insert
thf(fact_518_Union__insert,axiom,
! [A: set_v,B: set_set_v] :
( ( comple2307003700295860064_set_v @ ( insert_set_v @ A @ B ) )
= ( sup_sup_set_v @ A @ ( comple2307003700295860064_set_v @ B ) ) ) ).
% Union_insert
thf(fact_519_e1__def,axiom,
( e1
= ( 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 ) ) ) ) ).
% e1_def
thf(fact_520_calculation_I3_J,axiom,
! [N: v] :
( ord_less_eq_set_v
@ ( sCC_Bl3795065053823578884t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ N )
@ ( inf_inf_set_v @ ( successors @ N )
@ ( sCC_Bl4645233313691564917t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ) ).
% calculation(3)
thf(fact_521_explored__vsuccs,axiom,
! [U2: v] :
( ( member_v @ U2
@ ( sCC_Bl157864678168468314t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) )
=> ( ( sCC_Bl3795065053823578884t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ U2 )
= ( successors @ U2 ) ) ) ).
% explored_vsuccs
thf(fact_522_inf__sup__absorb,axiom,
! [X: set_v,Y: set_v] :
( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_523_inf__sup__absorb,axiom,
! [X: set_set_v,Y: set_set_v] :
( ( inf_inf_set_set_v @ X @ ( sup_sup_set_set_v @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_524_inf__sup__absorb,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_525_sup__inf__absorb,axiom,
! [X: set_v,Y: set_v] :
( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_526_sup__inf__absorb,axiom,
! [X: set_set_v,Y: set_set_v] :
( ( sup_sup_set_set_v @ X @ ( inf_inf_set_set_v @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_527_sup__inf__absorb,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_528_inf__bot__left,axiom,
! [X: set_set_v] :
( ( inf_inf_set_set_v @ bot_bot_set_set_v @ X )
= bot_bot_set_set_v ) ).
% inf_bot_left
thf(fact_529_inf__bot__left,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ X )
= bot_bot_set_v ) ).
% inf_bot_left
thf(fact_530_inf__bot__left,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X )
= bot_bo723834152578015283od_v_v ) ).
% inf_bot_left
thf(fact_531_inf__bot__right,axiom,
! [X: set_set_v] :
( ( inf_inf_set_set_v @ X @ bot_bot_set_set_v )
= bot_bot_set_set_v ) ).
% inf_bot_right
thf(fact_532_inf__bot__right,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ bot_bot_set_v )
= bot_bot_set_v ) ).
% inf_bot_right
thf(fact_533_inf__bot__right,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% inf_bot_right
thf(fact_534_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_set_v] :
( ( inf_inf_set_set_v @ bot_bot_set_set_v @ X )
= bot_bot_set_set_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_535_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ X )
= bot_bot_set_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_536_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ X )
= bot_bo723834152578015283od_v_v ) ).
% boolean_algebra.conj_zero_left
thf(fact_537_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_set_v] :
( ( inf_inf_set_set_v @ X @ bot_bot_set_set_v )
= bot_bot_set_set_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_538_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ bot_bot_set_v )
= bot_bot_set_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_539_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% boolean_algebra.conj_zero_right
thf(fact_540_sup__bot__left,axiom,
! [X: set_set_v] :
( ( sup_sup_set_set_v @ bot_bot_set_set_v @ X )
= X ) ).
% sup_bot_left
thf(fact_541_sup__bot__left,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ X )
= X ) ).
% sup_bot_left
thf(fact_542_sup__bot__left,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ X )
= X ) ).
% sup_bot_left
thf(fact_543_sup_Oright__idem,axiom,
! [A: set_v,B2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A @ B2 ) @ B2 )
= ( sup_sup_set_v @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_544_sup_Oright__idem,axiom,
! [A: set_set_v,B2: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ A @ B2 ) @ B2 )
= ( sup_sup_set_set_v @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_545_sup_Oright__idem,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ B2 )
= ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_546_sup__left__idem,axiom,
! [X: set_v,Y: set_v] :
( ( sup_sup_set_v @ X @ ( sup_sup_set_v @ X @ Y ) )
= ( sup_sup_set_v @ X @ Y ) ) ).
% sup_left_idem
thf(fact_547_sup__left__idem,axiom,
! [X: set_set_v,Y: set_set_v] :
( ( sup_sup_set_set_v @ X @ ( sup_sup_set_set_v @ X @ Y ) )
= ( sup_sup_set_set_v @ X @ Y ) ) ).
% sup_left_idem
thf(fact_548_sup__left__idem,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y ) )
= ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% sup_left_idem
thf(fact_549_sup_Oleft__idem,axiom,
! [A: set_v,B2: set_v] :
( ( sup_sup_set_v @ A @ ( sup_sup_set_v @ A @ B2 ) )
= ( sup_sup_set_v @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_550_sup_Oleft__idem,axiom,
! [A: set_set_v,B2: set_set_v] :
( ( sup_sup_set_set_v @ A @ ( sup_sup_set_set_v @ A @ B2 ) )
= ( sup_sup_set_set_v @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_551_sup_Oleft__idem,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B2 ) )
= ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_552_sup__idem,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ X )
= X ) ).
% sup_idem
thf(fact_553_sup__idem,axiom,
! [X: set_set_v] :
( ( sup_sup_set_set_v @ X @ X )
= X ) ).
% sup_idem
thf(fact_554_sup__idem,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ X )
= X ) ).
% sup_idem
thf(fact_555_sup_Oidem,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ A @ A )
= A ) ).
% sup.idem
thf(fact_556_sup_Oidem,axiom,
! [A: set_set_v] :
( ( sup_sup_set_set_v @ A @ A )
= A ) ).
% sup.idem
thf(fact_557_sup_Oidem,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ A )
= A ) ).
% sup.idem
thf(fact_558_inf_Oidem,axiom,
! [A: set_v] :
( ( inf_inf_set_v @ A @ A )
= A ) ).
% inf.idem
thf(fact_559_inf__idem,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ X )
= X ) ).
% inf_idem
thf(fact_560_inf_Oleft__idem,axiom,
! [A: set_v,B2: set_v] :
( ( inf_inf_set_v @ A @ ( inf_inf_set_v @ A @ B2 ) )
= ( inf_inf_set_v @ A @ B2 ) ) ).
% inf.left_idem
thf(fact_561_inf__left__idem,axiom,
! [X: set_v,Y: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ X @ Y ) )
= ( inf_inf_set_v @ X @ Y ) ) ).
% inf_left_idem
thf(fact_562_inf_Oright__idem,axiom,
! [A: set_v,B2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B2 ) @ B2 )
= ( inf_inf_set_v @ A @ B2 ) ) ).
% inf.right_idem
thf(fact_563_inf__right__idem,axiom,
! [X: set_v,Y: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y ) @ Y )
= ( inf_inf_set_v @ X @ Y ) ) ).
% inf_right_idem
thf(fact_564__C1_C,axiom,
sCC_Bl36166008131615352t_unit @ successors @ v2 @ e ).
% "1"
thf(fact_565__C3_C,axiom,
sCC_Bl6082031138996704384t_unit @ successors @ v2 @ e1 @ e2 ).
% "3"
thf(fact_566_notempty,axiom,
( ( sCC_Bl8828226123343373779t_unit @ e2 )
= ( cons_v @ v2 @ ( sCC_Bl8828226123343373779t_unit @ e ) ) ) ).
% notempty
thf(fact_567_cst_H,axiom,
( ( sCC_Bl9201514103433284750t_unit @ e2 )
= ( cons_v @ v2 @ ( sCC_Bl9201514103433284750t_unit @ e ) ) ) ).
% cst'
thf(fact_568_e_H__def,axiom,
( e2
= ( sCC_Bloemen_dfss_v @ successors @ v2 @ e1 ) ) ).
% e'_def
thf(fact_569_sup_Obounded__iff,axiom,
! [B2: set_set_v,C2: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ B2 @ C2 ) @ A )
= ( ( ord_le5216385588623774835_set_v @ B2 @ A )
& ( ord_le5216385588623774835_set_v @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_570_sup_Obounded__iff,axiom,
! [B2: set_v,C2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C2 ) @ A )
= ( ( ord_less_eq_set_v @ B2 @ A )
& ( ord_less_eq_set_v @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_571_sup_Obounded__iff,axiom,
! [B2: set_Product_prod_v_v,C2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) @ A )
= ( ( ord_le7336532860387713383od_v_v @ B2 @ A )
& ( ord_le7336532860387713383od_v_v @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_572_le__sup__iff,axiom,
! [X: set_set_v,Y: set_set_v,Z2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ X @ Y ) @ Z2 )
= ( ( ord_le5216385588623774835_set_v @ X @ Z2 )
& ( ord_le5216385588623774835_set_v @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_573_le__sup__iff,axiom,
! [X: set_v,Y: set_v,Z2: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_set_v @ X @ Z2 )
& ( ord_less_eq_set_v @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_574_le__sup__iff,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ Z2 )
= ( ( ord_le7336532860387713383od_v_v @ X @ Z2 )
& ( ord_le7336532860387713383od_v_v @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_575_le__inf__iff,axiom,
! [X: set_v,Y: set_v,Z2: set_v] :
( ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ Y @ Z2 ) )
= ( ( ord_less_eq_set_v @ X @ Y )
& ( ord_less_eq_set_v @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_576_le__inf__iff,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z2 ) )
= ( ( ord_le7336532860387713383od_v_v @ X @ Y )
& ( ord_le7336532860387713383od_v_v @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_577_inf_Obounded__iff,axiom,
! [A: set_v,B2: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B2 @ C2 ) )
= ( ( ord_less_eq_set_v @ A @ B2 )
& ( ord_less_eq_set_v @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_578_inf_Obounded__iff,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) )
= ( ( ord_le7336532860387713383od_v_v @ A @ B2 )
& ( ord_le7336532860387713383od_v_v @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_579_sup__bot_Oright__neutral,axiom,
! [A: set_set_v] :
( ( sup_sup_set_set_v @ A @ bot_bot_set_set_v )
= A ) ).
% sup_bot.right_neutral
thf(fact_580_sup__bot_Oright__neutral,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ A @ bot_bot_set_v )
= A ) ).
% sup_bot.right_neutral
thf(fact_581_sup__bot_Oright__neutral,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A @ bot_bo723834152578015283od_v_v )
= A ) ).
% sup_bot.right_neutral
thf(fact_582_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_set_v,B2: set_set_v] :
( ( bot_bot_set_set_v
= ( sup_sup_set_set_v @ A @ B2 ) )
= ( ( A = bot_bot_set_set_v )
& ( B2 = bot_bot_set_set_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_583_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_v,B2: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ A @ B2 ) )
= ( ( A = bot_bot_set_v )
& ( B2 = bot_bot_set_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_584_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ A @ B2 ) )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B2 = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_585_sup__bot_Oleft__neutral,axiom,
! [A: set_set_v] :
( ( sup_sup_set_set_v @ bot_bot_set_set_v @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_586_sup__bot_Oleft__neutral,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_587_sup__bot_Oleft__neutral,axiom,
! [A: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_588_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_set_v,B2: set_set_v] :
( ( ( sup_sup_set_set_v @ A @ B2 )
= bot_bot_set_set_v )
= ( ( A = bot_bot_set_set_v )
& ( B2 = bot_bot_set_set_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_589_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_v,B2: set_v] :
( ( ( sup_sup_set_v @ A @ B2 )
= bot_bot_set_v )
= ( ( A = bot_bot_set_v )
& ( B2 = bot_bot_set_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_590_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A @ B2 )
= bot_bo723834152578015283od_v_v )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B2 = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_591_sup__eq__bot__iff,axiom,
! [X: set_set_v,Y: set_set_v] :
( ( ( sup_sup_set_set_v @ X @ Y )
= bot_bot_set_set_v )
= ( ( X = bot_bot_set_set_v )
& ( Y = bot_bot_set_set_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_592_sup__eq__bot__iff,axiom,
! [X: set_v,Y: set_v] :
( ( ( sup_sup_set_v @ X @ Y )
= bot_bot_set_v )
= ( ( X = bot_bot_set_v )
& ( Y = bot_bot_set_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_593_sup__eq__bot__iff,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ X @ Y )
= bot_bo723834152578015283od_v_v )
= ( ( X = bot_bo723834152578015283od_v_v )
& ( Y = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_eq_bot_iff
thf(fact_594_bot__eq__sup__iff,axiom,
! [X: set_set_v,Y: set_set_v] :
( ( bot_bot_set_set_v
= ( sup_sup_set_set_v @ X @ Y ) )
= ( ( X = bot_bot_set_set_v )
& ( Y = bot_bot_set_set_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_595_bot__eq__sup__iff,axiom,
! [X: set_v,Y: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ X @ Y ) )
= ( ( X = bot_bot_set_v )
& ( Y = bot_bot_set_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_596_bot__eq__sup__iff,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ X @ Y ) )
= ( ( X = bot_bo723834152578015283od_v_v )
& ( Y = bot_bo723834152578015283od_v_v ) ) ) ).
% bot_eq_sup_iff
thf(fact_597_sup__bot__right,axiom,
! [X: set_set_v] :
( ( sup_sup_set_set_v @ X @ bot_bot_set_set_v )
= X ) ).
% sup_bot_right
thf(fact_598_sup__bot__right,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ bot_bot_set_v )
= X ) ).
% sup_bot_right
thf(fact_599_sup__bot__right,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ bot_bo723834152578015283od_v_v )
= X ) ).
% sup_bot_right
thf(fact_600_sclosed,axiom,
! [X5: v] :
( ( member_v @ X5 @ vertices )
=> ( ord_less_eq_set_v @ ( successors @ X5 ) @ vertices ) ) ).
% sclosed
thf(fact_601_scc__partition,axiom,
! [S: set_v,S2: set_v,X: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
=> ( ( sCC_Bloemen_is_scc_v @ successors @ S2 )
=> ( ( member_v @ X @ ( inf_inf_set_v @ S @ S2 ) )
=> ( S = S2 ) ) ) ) ).
% scc_partition
thf(fact_602_precedes__in__tail,axiom,
! [X: v,Z2: v,Y: v,Zs: list_v] :
( ( X != Z2 )
=> ( ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( cons_v @ Z2 @ Zs ) )
= ( sCC_Bl4022239298816431255edes_v @ X @ Y @ Zs ) ) ) ).
% precedes_in_tail
thf(fact_603_precedes__in__tail,axiom,
! [X: product_prod_v_v,Z2: product_prod_v_v,Y: product_prod_v_v,Zs: list_P7986770385144383213od_v_v] :
( ( X != Z2 )
=> ( ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ ( cons_P4120604216776828829od_v_v @ Z2 @ Zs ) )
= ( sCC_Bl2026170059108282219od_v_v @ X @ Y @ Zs ) ) ) ).
% precedes_in_tail
thf(fact_604_inf__set__def,axiom,
( inf_in6271465464967711157od_v_v
= ( ^ [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ( inf_in6860806757119575912_v_v_o
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A3 )
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_605_inf__set__def,axiom,
( inf_inf_set_set_v
= ( ^ [A3: set_set_v,B4: set_set_v] :
( collect_set_v
@ ( inf_inf_set_v_o
@ ^ [X2: set_v] : ( member_set_v @ X2 @ A3 )
@ ^ [X2: set_v] : ( member_set_v @ X2 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_606_inf__set__def,axiom,
( inf_inf_set_v
= ( ^ [A3: set_v,B4: set_v] :
( collect_v
@ ( inf_inf_v_o
@ ^ [X2: v] : ( member_v @ X2 @ A3 )
@ ^ [X2: v] : ( member_v @ X2 @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_607_less__eq__set__def,axiom,
( ord_less_eq_set_v
= ( ^ [A3: set_v,B4: set_v] :
( ord_less_eq_v_o
@ ^ [X2: v] : ( member_v @ X2 @ A3 )
@ ^ [X2: v] : ( member_v @ X2 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_608_less__eq__set__def,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ord_le5892402249245633078_v_v_o
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A3 )
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_609_tail__not__precedes,axiom,
! [Y: v,X: v,Xs: list_v] :
( ( sCC_Bl4022239298816431255edes_v @ Y @ X @ ( cons_v @ X @ Xs ) )
=> ( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
=> ( X = Y ) ) ) ).
% tail_not_precedes
thf(fact_610_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_611_head__precedes,axiom,
! [Y: v,X: v,Xs: list_v] :
( ( member_v @ Y @ ( set_v2 @ ( cons_v @ X @ Xs ) ) )
=> ( sCC_Bl4022239298816431255edes_v @ X @ Y @ ( cons_v @ X @ Xs ) ) ) ).
% head_precedes
thf(fact_612_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_613_fold__congs_I4_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: set_v,F: set_v > set_v,F2: set_v > set_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl4645233313691564917t_unit @ R2 )
= V )
=> ( ! [V2: set_v] :
( ( V = V2 )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl7870604408699998558t_unit @ F @ R )
= ( sCC_Bl7870604408699998558t_unit @ F2 @ R2 ) ) ) ) ) ).
% fold_congs(4)
thf(fact_614_unfold__congs_I4_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: set_v,F: set_v > set_v,F2: set_v > set_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl4645233313691564917t_unit @ R2 )
= V )
=> ( ! [V2: set_v] :
( ( V2 = V )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl7870604408699998558t_unit @ F @ R )
= ( sCC_Bl7870604408699998558t_unit @ F2 @ R2 ) ) ) ) ) ).
% unfold_congs(4)
thf(fact_615_sup__left__commute,axiom,
! [X: set_v,Y: set_v,Z2: set_v] :
( ( sup_sup_set_v @ X @ ( sup_sup_set_v @ Y @ Z2 ) )
= ( sup_sup_set_v @ Y @ ( sup_sup_set_v @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_616_sup__left__commute,axiom,
! [X: set_set_v,Y: set_set_v,Z2: set_set_v] :
( ( sup_sup_set_set_v @ X @ ( sup_sup_set_set_v @ Y @ Z2 ) )
= ( sup_sup_set_set_v @ Y @ ( sup_sup_set_set_v @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_617_sup__left__commute,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z2 ) )
= ( sup_su414716646722978715od_v_v @ Y @ ( sup_su414716646722978715od_v_v @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_618_sup_Oleft__commute,axiom,
! [B2: set_v,A: set_v,C2: set_v] :
( ( sup_sup_set_v @ B2 @ ( sup_sup_set_v @ A @ C2 ) )
= ( sup_sup_set_v @ A @ ( sup_sup_set_v @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_619_sup_Oleft__commute,axiom,
! [B2: set_set_v,A: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ B2 @ ( sup_sup_set_set_v @ A @ C2 ) )
= ( sup_sup_set_set_v @ A @ ( sup_sup_set_set_v @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_620_sup_Oleft__commute,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A @ C2 ) )
= ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_621_boolean__algebra__cancel_Osup2,axiom,
! [B: set_v,K: set_v,B2: set_v,A: set_v] :
( ( B
= ( sup_sup_set_v @ K @ B2 ) )
=> ( ( sup_sup_set_v @ A @ B )
= ( sup_sup_set_v @ K @ ( sup_sup_set_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_622_boolean__algebra__cancel_Osup2,axiom,
! [B: set_set_v,K: set_set_v,B2: set_set_v,A: set_set_v] :
( ( B
= ( sup_sup_set_set_v @ K @ B2 ) )
=> ( ( sup_sup_set_set_v @ A @ B )
= ( sup_sup_set_set_v @ K @ ( sup_sup_set_set_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_623_boolean__algebra__cancel_Osup2,axiom,
! [B: set_Product_prod_v_v,K: set_Product_prod_v_v,B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( B
= ( sup_su414716646722978715od_v_v @ K @ B2 ) )
=> ( ( sup_su414716646722978715od_v_v @ A @ B )
= ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_624_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_v,K: set_v,A: set_v,B2: set_v] :
( ( A2
= ( sup_sup_set_v @ K @ A ) )
=> ( ( sup_sup_set_v @ A2 @ B2 )
= ( sup_sup_set_v @ K @ ( sup_sup_set_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_625_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_set_v,K: set_set_v,A: set_set_v,B2: set_set_v] :
( ( A2
= ( sup_sup_set_set_v @ K @ A ) )
=> ( ( sup_sup_set_set_v @ A2 @ B2 )
= ( sup_sup_set_set_v @ K @ ( sup_sup_set_set_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_626_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_Product_prod_v_v,K: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A2
= ( sup_su414716646722978715od_v_v @ K @ A ) )
=> ( ( sup_su414716646722978715od_v_v @ A2 @ B2 )
= ( sup_su414716646722978715od_v_v @ K @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_627_sup__commute,axiom,
( sup_sup_set_v
= ( ^ [X2: set_v,Y4: set_v] : ( sup_sup_set_v @ Y4 @ X2 ) ) ) ).
% sup_commute
thf(fact_628_sup__commute,axiom,
( sup_sup_set_set_v
= ( ^ [X2: set_set_v,Y4: set_set_v] : ( sup_sup_set_set_v @ Y4 @ X2 ) ) ) ).
% sup_commute
thf(fact_629_sup__commute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y4: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ Y4 @ X2 ) ) ) ).
% sup_commute
thf(fact_630_sup_Ocommute,axiom,
( sup_sup_set_v
= ( ^ [A4: set_v,B6: set_v] : ( sup_sup_set_v @ B6 @ A4 ) ) ) ).
% sup.commute
thf(fact_631_sup_Ocommute,axiom,
( sup_sup_set_set_v
= ( ^ [A4: set_set_v,B6: set_set_v] : ( sup_sup_set_set_v @ B6 @ A4 ) ) ) ).
% sup.commute
thf(fact_632_sup_Ocommute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A4: set_Product_prod_v_v,B6: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B6 @ A4 ) ) ) ).
% sup.commute
thf(fact_633_sup__assoc,axiom,
! [X: set_v,Y: set_v,Z2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ X @ Y ) @ Z2 )
= ( sup_sup_set_v @ X @ ( sup_sup_set_v @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_634_sup__assoc,axiom,
! [X: set_set_v,Y: set_set_v,Z2: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ X @ Y ) @ Z2 )
= ( sup_sup_set_set_v @ X @ ( sup_sup_set_set_v @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_635_sup__assoc,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ Z2 )
= ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_636_sup_Oassoc,axiom,
! [A: set_v,B2: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A @ B2 ) @ C2 )
= ( sup_sup_set_v @ A @ ( sup_sup_set_v @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_637_sup_Oassoc,axiom,
! [A: set_set_v,B2: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ A @ B2 ) @ C2 )
= ( sup_sup_set_set_v @ A @ ( sup_sup_set_set_v @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_638_sup_Oassoc,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ C2 )
= ( sup_su414716646722978715od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_639_inf__sup__aci_I5_J,axiom,
( sup_sup_set_v
= ( ^ [X2: set_v,Y4: set_v] : ( sup_sup_set_v @ Y4 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_640_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_v
= ( ^ [X2: set_set_v,Y4: set_set_v] : ( sup_sup_set_set_v @ Y4 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_641_inf__sup__aci_I5_J,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y4: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ Y4 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_642_inf__sup__aci_I6_J,axiom,
! [X: set_v,Y: set_v,Z2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ X @ Y ) @ Z2 )
= ( sup_sup_set_v @ X @ ( sup_sup_set_v @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_643_inf__sup__aci_I6_J,axiom,
! [X: set_set_v,Y: set_set_v,Z2: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ X @ Y ) @ Z2 )
= ( sup_sup_set_set_v @ X @ ( sup_sup_set_set_v @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_644_inf__sup__aci_I6_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ Z2 )
= ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_645_inf__sup__aci_I7_J,axiom,
! [X: set_v,Y: set_v,Z2: set_v] :
( ( sup_sup_set_v @ X @ ( sup_sup_set_v @ Y @ Z2 ) )
= ( sup_sup_set_v @ Y @ ( sup_sup_set_v @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_646_inf__sup__aci_I7_J,axiom,
! [X: set_set_v,Y: set_set_v,Z2: set_set_v] :
( ( sup_sup_set_set_v @ X @ ( sup_sup_set_set_v @ Y @ Z2 ) )
= ( sup_sup_set_set_v @ Y @ ( sup_sup_set_set_v @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_647_inf__sup__aci_I7_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z2 ) )
= ( sup_su414716646722978715od_v_v @ Y @ ( sup_su414716646722978715od_v_v @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_648_inf__sup__aci_I8_J,axiom,
! [X: set_v,Y: set_v] :
( ( sup_sup_set_v @ X @ ( sup_sup_set_v @ X @ Y ) )
= ( sup_sup_set_v @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_649_inf__sup__aci_I8_J,axiom,
! [X: set_set_v,Y: set_set_v] :
( ( sup_sup_set_set_v @ X @ ( sup_sup_set_set_v @ X @ Y ) )
= ( sup_sup_set_set_v @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_650_inf__sup__aci_I8_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y ) )
= ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_651_inf__sup__aci_I4_J,axiom,
! [X: set_v,Y: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ X @ Y ) )
= ( inf_inf_set_v @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_652_inf__sup__aci_I3_J,axiom,
! [X: set_v,Y: set_v,Z2: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y @ Z2 ) )
= ( inf_inf_set_v @ Y @ ( inf_inf_set_v @ X @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_653_inf__sup__aci_I2_J,axiom,
! [X: set_v,Y: set_v,Z2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y ) @ Z2 )
= ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_654_inf__sup__aci_I1_J,axiom,
( inf_inf_set_v
= ( ^ [X2: set_v,Y4: set_v] : ( inf_inf_set_v @ Y4 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_655_inf_Oassoc,axiom,
! [A: set_v,B2: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B2 ) @ C2 )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B2 @ C2 ) ) ) ).
% inf.assoc
thf(fact_656_inf__assoc,axiom,
! [X: set_v,Y: set_v,Z2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y ) @ Z2 )
= ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y @ Z2 ) ) ) ).
% inf_assoc
thf(fact_657_inf_Ocommute,axiom,
( inf_inf_set_v
= ( ^ [A4: set_v,B6: set_v] : ( inf_inf_set_v @ B6 @ A4 ) ) ) ).
% inf.commute
thf(fact_658_inf__commute,axiom,
( inf_inf_set_v
= ( ^ [X2: set_v,Y4: set_v] : ( inf_inf_set_v @ Y4 @ X2 ) ) ) ).
% inf_commute
thf(fact_659_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_v,K: set_v,A: set_v,B2: set_v] :
( ( A2
= ( inf_inf_set_v @ K @ A ) )
=> ( ( inf_inf_set_v @ A2 @ B2 )
= ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_660_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_v,K: set_v,B2: set_v,A: set_v] :
( ( B
= ( inf_inf_set_v @ K @ B2 ) )
=> ( ( inf_inf_set_v @ A @ B )
= ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_661_inf_Oleft__commute,axiom,
! [B2: set_v,A: set_v,C2: set_v] :
( ( inf_inf_set_v @ B2 @ ( inf_inf_set_v @ A @ C2 ) )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B2 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_662_inf__left__commute,axiom,
! [X: set_v,Y: set_v,Z2: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y @ Z2 ) )
= ( inf_inf_set_v @ Y @ ( inf_inf_set_v @ X @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_663_sup_OcoboundedI2,axiom,
! [C2: set_set_v,B2: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C2 @ B2 )
=> ( ord_le5216385588623774835_set_v @ C2 @ ( sup_sup_set_set_v @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_664_sup_OcoboundedI2,axiom,
! [C2: set_v,B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ C2 @ B2 )
=> ( ord_less_eq_set_v @ C2 @ ( sup_sup_set_v @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_665_sup_OcoboundedI2,axiom,
! [C2: set_Product_prod_v_v,B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ B2 )
=> ( ord_le7336532860387713383od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_666_sup_OcoboundedI1,axiom,
! [C2: set_set_v,A: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C2 @ A )
=> ( ord_le5216385588623774835_set_v @ C2 @ ( sup_sup_set_set_v @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_667_sup_OcoboundedI1,axiom,
! [C2: set_v,A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C2 @ A )
=> ( ord_less_eq_set_v @ C2 @ ( sup_sup_set_v @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_668_sup_OcoboundedI1,axiom,
! [C2: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ A )
=> ( ord_le7336532860387713383od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_669_sup_Oabsorb__iff2,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [A4: set_set_v,B6: set_set_v] :
( ( sup_sup_set_set_v @ A4 @ B6 )
= B6 ) ) ) ).
% sup.absorb_iff2
thf(fact_670_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B6: set_v] :
( ( sup_sup_set_v @ A4 @ B6 )
= B6 ) ) ) ).
% sup.absorb_iff2
thf(fact_671_sup_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ B6 )
= B6 ) ) ) ).
% sup.absorb_iff2
thf(fact_672_sup_Oabsorb__iff1,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [B6: set_set_v,A4: set_set_v] :
( ( sup_sup_set_set_v @ A4 @ B6 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_673_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [B6: set_v,A4: set_v] :
( ( sup_sup_set_v @ A4 @ B6 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_674_sup_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B6: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A4 @ B6 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_675_sup_Ocobounded2,axiom,
! [B2: set_set_v,A: set_set_v] : ( ord_le5216385588623774835_set_v @ B2 @ ( sup_sup_set_set_v @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_676_sup_Ocobounded2,axiom,
! [B2: set_v,A: set_v] : ( ord_less_eq_set_v @ B2 @ ( sup_sup_set_v @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_677_sup_Ocobounded2,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B2 @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_678_sup_Ocobounded1,axiom,
! [A: set_set_v,B2: set_set_v] : ( ord_le5216385588623774835_set_v @ A @ ( sup_sup_set_set_v @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_679_sup_Ocobounded1,axiom,
! [A: set_v,B2: set_v] : ( ord_less_eq_set_v @ A @ ( sup_sup_set_v @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_680_sup_Ocobounded1,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_681_sup_Oorder__iff,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [B6: set_set_v,A4: set_set_v] :
( A4
= ( sup_sup_set_set_v @ A4 @ B6 ) ) ) ) ).
% sup.order_iff
thf(fact_682_sup_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [B6: set_v,A4: set_v] :
( A4
= ( sup_sup_set_v @ A4 @ B6 ) ) ) ) ).
% sup.order_iff
thf(fact_683_sup_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B6: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( A4
= ( sup_su414716646722978715od_v_v @ A4 @ B6 ) ) ) ) ).
% sup.order_iff
thf(fact_684_sup_OboundedI,axiom,
! [B2: set_set_v,A: set_set_v,C2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B2 @ A )
=> ( ( ord_le5216385588623774835_set_v @ C2 @ A )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ B2 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_685_sup_OboundedI,axiom,
! [B2: set_v,A: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( ( ord_less_eq_set_v @ C2 @ A )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_686_sup_OboundedI,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C2 @ A )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_687_sup_OboundedE,axiom,
! [B2: set_set_v,C2: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ B2 @ C2 ) @ A )
=> ~ ( ( ord_le5216385588623774835_set_v @ B2 @ A )
=> ~ ( ord_le5216385588623774835_set_v @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_688_sup_OboundedE,axiom,
! [B2: set_v,C2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B2 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_set_v @ B2 @ A )
=> ~ ( ord_less_eq_set_v @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_689_sup_OboundedE,axiom,
! [B2: set_Product_prod_v_v,C2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B2 @ C2 ) @ A )
=> ~ ( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ~ ( ord_le7336532860387713383od_v_v @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_690_sup__absorb2,axiom,
! [X: set_set_v,Y: set_set_v] :
( ( ord_le5216385588623774835_set_v @ X @ Y )
=> ( ( sup_sup_set_set_v @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_691_sup__absorb2,axiom,
! [X: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( sup_sup_set_v @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_692_sup__absorb2,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_693_sup__absorb1,axiom,
! [Y: set_set_v,X: set_set_v] :
( ( ord_le5216385588623774835_set_v @ Y @ X )
=> ( ( sup_sup_set_set_v @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_694_sup__absorb1,axiom,
! [Y: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y @ X )
=> ( ( sup_sup_set_v @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_695_sup__absorb1,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_696_sup_Oabsorb2,axiom,
! [A: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ B2 )
=> ( ( sup_sup_set_set_v @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_697_sup_Oabsorb2,axiom,
! [A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( sup_sup_set_v @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_698_sup_Oabsorb2,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( sup_su414716646722978715od_v_v @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_699_sup_Oabsorb1,axiom,
! [B2: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B2 @ A )
=> ( ( sup_sup_set_set_v @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_700_sup_Oabsorb1,axiom,
! [B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( ( sup_sup_set_v @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_701_sup_Oabsorb1,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( ( sup_su414716646722978715od_v_v @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_702_sup__unique,axiom,
! [F: set_set_v > set_set_v > set_set_v,X: set_set_v,Y: set_set_v] :
( ! [X3: set_set_v,Y3: set_set_v] : ( ord_le5216385588623774835_set_v @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_set_v,Y3: set_set_v] : ( ord_le5216385588623774835_set_v @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_set_v,Y3: set_set_v,Z4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ Y3 @ X3 )
=> ( ( ord_le5216385588623774835_set_v @ Z4 @ X3 )
=> ( ord_le5216385588623774835_set_v @ ( F @ Y3 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup_set_set_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_703_sup__unique,axiom,
! [F: set_v > set_v > set_v,X: set_v,Y: set_v] :
( ! [X3: set_v,Y3: set_v] : ( ord_less_eq_set_v @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_v,Y3: set_v] : ( ord_less_eq_set_v @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_v,Y3: set_v,Z4: set_v] :
( ( ord_less_eq_set_v @ Y3 @ X3 )
=> ( ( ord_less_eq_set_v @ Z4 @ X3 )
=> ( ord_less_eq_set_v @ ( F @ Y3 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup_set_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_704_sup__unique,axiom,
! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y3 @ ( F @ X3 @ Y3 ) )
=> ( ! [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v,Z4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y3 @ X3 )
=> ( ( ord_le7336532860387713383od_v_v @ Z4 @ X3 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ Y3 @ Z4 ) @ X3 ) ) )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_705_sup_OorderI,axiom,
! [A: set_set_v,B2: set_set_v] :
( ( A
= ( sup_sup_set_set_v @ A @ B2 ) )
=> ( ord_le5216385588623774835_set_v @ B2 @ A ) ) ).
% sup.orderI
thf(fact_706_sup_OorderI,axiom,
! [A: set_v,B2: set_v] :
( ( A
= ( sup_sup_set_v @ A @ B2 ) )
=> ( ord_less_eq_set_v @ B2 @ A ) ) ).
% sup.orderI
thf(fact_707_sup_OorderI,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A
= ( sup_su414716646722978715od_v_v @ A @ B2 ) )
=> ( ord_le7336532860387713383od_v_v @ B2 @ A ) ) ).
% sup.orderI
thf(fact_708_sup_OorderE,axiom,
! [B2: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B2 @ A )
=> ( A
= ( sup_sup_set_set_v @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_709_sup_OorderE,axiom,
! [B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( A
= ( sup_sup_set_v @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_710_sup_OorderE,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( A
= ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_711_le__iff__sup,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [X2: set_set_v,Y4: set_set_v] :
( ( sup_sup_set_set_v @ X2 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_712_le__iff__sup,axiom,
( ord_less_eq_set_v
= ( ^ [X2: set_v,Y4: set_v] :
( ( sup_sup_set_v @ X2 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_713_le__iff__sup,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X2 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_714_sup__least,axiom,
! [Y: set_set_v,X: set_set_v,Z2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ Y @ X )
=> ( ( ord_le5216385588623774835_set_v @ Z2 @ X )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_715_sup__least,axiom,
! [Y: set_v,X: set_v,Z2: set_v] :
( ( ord_less_eq_set_v @ Y @ X )
=> ( ( ord_less_eq_set_v @ Z2 @ X )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_716_sup__least,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( ( ord_le7336532860387713383od_v_v @ Z2 @ X )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_717_sup__mono,axiom,
! [A: set_set_v,C2: set_set_v,B2: set_set_v,D: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ C2 )
=> ( ( ord_le5216385588623774835_set_v @ B2 @ D )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A @ B2 ) @ ( sup_sup_set_set_v @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_718_sup__mono,axiom,
! [A: set_v,C2: set_v,B2: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A @ C2 )
=> ( ( ord_less_eq_set_v @ B2 @ D )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B2 ) @ ( sup_sup_set_v @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_719_sup__mono,axiom,
! [A: set_Product_prod_v_v,C2: set_Product_prod_v_v,B2: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ ( sup_su414716646722978715od_v_v @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_720_sup_Omono,axiom,
! [C2: set_set_v,A: set_set_v,D: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C2 @ A )
=> ( ( ord_le5216385588623774835_set_v @ D @ B2 )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ C2 @ D ) @ ( sup_sup_set_set_v @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_721_sup_Omono,axiom,
! [C2: set_v,A: set_v,D: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ C2 @ A )
=> ( ( ord_less_eq_set_v @ D @ B2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ C2 @ D ) @ ( sup_sup_set_v @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_722_sup_Omono,axiom,
! [C2: set_Product_prod_v_v,A: set_Product_prod_v_v,D: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ A )
=> ( ( ord_le7336532860387713383od_v_v @ D @ B2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ C2 @ D ) @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_723_le__supI2,axiom,
! [X: set_set_v,B2: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ X @ B2 )
=> ( ord_le5216385588623774835_set_v @ X @ ( sup_sup_set_set_v @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_724_le__supI2,axiom,
! [X: set_v,B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ X @ B2 )
=> ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_725_le__supI2,axiom,
! [X: set_Product_prod_v_v,B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ B2 )
=> ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_726_le__supI1,axiom,
! [X: set_set_v,A: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ X @ A )
=> ( ord_le5216385588623774835_set_v @ X @ ( sup_sup_set_set_v @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_727_le__supI1,axiom,
! [X: set_v,A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ X @ A )
=> ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_728_le__supI1,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_729_sup__ge2,axiom,
! [Y: set_set_v,X: set_set_v] : ( ord_le5216385588623774835_set_v @ Y @ ( sup_sup_set_set_v @ X @ Y ) ) ).
% sup_ge2
thf(fact_730_sup__ge2,axiom,
! [Y: set_v,X: set_v] : ( ord_less_eq_set_v @ Y @ ( sup_sup_set_v @ X @ Y ) ) ).
% sup_ge2
thf(fact_731_sup__ge2,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y @ ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% sup_ge2
thf(fact_732_sup__ge1,axiom,
! [X: set_set_v,Y: set_set_v] : ( ord_le5216385588623774835_set_v @ X @ ( sup_sup_set_set_v @ X @ Y ) ) ).
% sup_ge1
thf(fact_733_sup__ge1,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ X @ Y ) ) ).
% sup_ge1
thf(fact_734_sup__ge1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% sup_ge1
thf(fact_735_le__supI,axiom,
! [A: set_set_v,X: set_set_v,B2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ X )
=> ( ( ord_le5216385588623774835_set_v @ B2 @ X )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_736_le__supI,axiom,
! [A: set_v,X: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ X )
=> ( ( ord_less_eq_set_v @ B2 @ X )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_737_le__supI,axiom,
! [A: set_Product_prod_v_v,X: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ X )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_738_le__supE,axiom,
! [A: set_set_v,B2: set_set_v,X: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A @ B2 ) @ X )
=> ~ ( ( ord_le5216385588623774835_set_v @ A @ X )
=> ~ ( ord_le5216385588623774835_set_v @ B2 @ X ) ) ) ).
% le_supE
thf(fact_739_le__supE,axiom,
! [A: set_v,B2: set_v,X: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_v @ A @ X )
=> ~ ( ord_less_eq_set_v @ B2 @ X ) ) ) ).
% le_supE
thf(fact_740_le__supE,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B2 ) @ X )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ~ ( ord_le7336532860387713383od_v_v @ B2 @ X ) ) ) ).
% le_supE
thf(fact_741_inf__sup__ord_I3_J,axiom,
! [X: set_set_v,Y: set_set_v] : ( ord_le5216385588623774835_set_v @ X @ ( sup_sup_set_set_v @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_742_inf__sup__ord_I3_J,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_743_inf__sup__ord_I3_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_744_inf__sup__ord_I4_J,axiom,
! [Y: set_set_v,X: set_set_v] : ( ord_le5216385588623774835_set_v @ Y @ ( sup_sup_set_set_v @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_745_inf__sup__ord_I4_J,axiom,
! [Y: set_v,X: set_v] : ( ord_less_eq_set_v @ Y @ ( sup_sup_set_v @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_746_inf__sup__ord_I4_J,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y @ ( sup_su414716646722978715od_v_v @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_747_inf__sup__ord_I2_J,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_748_inf__sup__ord_I2_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_749_inf__sup__ord_I1_J,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_750_inf__sup__ord_I1_J,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_751_inf__le1,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_752_inf__le1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_753_inf__le2,axiom,
! [X: set_v,Y: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_754_inf__le2,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_755_le__infE,axiom,
! [X: set_v,A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ A @ B2 ) )
=> ~ ( ( ord_less_eq_set_v @ X @ A )
=> ~ ( ord_less_eq_set_v @ X @ B2 ) ) ) ).
% le_infE
thf(fact_756_le__infE,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ~ ( ord_le7336532860387713383od_v_v @ X @ B2 ) ) ) ).
% le_infE
thf(fact_757_le__infI,axiom,
! [X: set_v,A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ X @ A )
=> ( ( ord_less_eq_set_v @ X @ B2 )
=> ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_758_le__infI,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ( ( ord_le7336532860387713383od_v_v @ X @ B2 )
=> ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_759_inf__mono,axiom,
! [A: set_v,C2: set_v,B2: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A @ C2 )
=> ( ( ord_less_eq_set_v @ B2 @ D )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ ( inf_inf_set_v @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_760_inf__mono,axiom,
! [A: set_Product_prod_v_v,C2: set_Product_prod_v_v,B2: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B2 @ D )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ ( inf_in6271465464967711157od_v_v @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_761_le__infI1,axiom,
! [A: set_v,X: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ X )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_762_le__infI1,axiom,
! [A: set_Product_prod_v_v,X: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_763_le__infI2,axiom,
! [B2: set_v,X: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ X )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_764_le__infI2,axiom,
! [B2: set_Product_prod_v_v,X: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ X )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_765_inf_OorderE,axiom,
! [A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( A
= ( inf_inf_set_v @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_766_inf_OorderE,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( A
= ( inf_in6271465464967711157od_v_v @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_767_inf_OorderI,axiom,
! [A: set_v,B2: set_v] :
( ( A
= ( inf_inf_set_v @ A @ B2 ) )
=> ( ord_less_eq_set_v @ A @ B2 ) ) ).
% inf.orderI
thf(fact_768_inf_OorderI,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( A
= ( inf_in6271465464967711157od_v_v @ A @ B2 ) )
=> ( ord_le7336532860387713383od_v_v @ A @ B2 ) ) ).
% inf.orderI
thf(fact_769_inf__unique,axiom,
! [F: set_v > set_v > set_v,X: set_v,Y: set_v] :
( ! [X3: set_v,Y3: set_v] : ( ord_less_eq_set_v @ ( F @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: set_v,Y3: set_v] : ( ord_less_eq_set_v @ ( F @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: set_v,Y3: set_v,Z4: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y3 )
=> ( ( ord_less_eq_set_v @ X3 @ Z4 )
=> ( ord_less_eq_set_v @ X3 @ ( F @ Y3 @ Z4 ) ) ) )
=> ( ( inf_inf_set_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_770_inf__unique,axiom,
! [F: set_Product_prod_v_v > set_Product_prod_v_v > set_Product_prod_v_v,X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v,Z4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y3 )
=> ( ( ord_le7336532860387713383od_v_v @ X3 @ Z4 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ ( F @ Y3 @ Z4 ) ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_771_le__iff__inf,axiom,
( ord_less_eq_set_v
= ( ^ [X2: set_v,Y4: set_v] :
( ( inf_inf_set_v @ X2 @ Y4 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_772_le__iff__inf,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [X2: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X2 @ Y4 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_773_inf_Oabsorb1,axiom,
! [A: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( inf_inf_set_v @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_774_inf_Oabsorb1,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_775_inf_Oabsorb2,axiom,
! [B2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ A )
=> ( ( inf_inf_set_v @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_776_inf_Oabsorb2,axiom,
! [B2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ A )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_777_inf__absorb1,axiom,
! [X: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( inf_inf_set_v @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_778_inf__absorb1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_779_inf__absorb2,axiom,
! [Y: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y @ X )
=> ( ( inf_inf_set_v @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_780_inf__absorb2,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_781_inf_OboundedE,axiom,
! [A: set_v,B2: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_set_v @ A @ B2 )
=> ~ ( ord_less_eq_set_v @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_782_inf_OboundedE,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ~ ( ord_le7336532860387713383od_v_v @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_783_inf_OboundedI,axiom,
! [A: set_v,B2: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A @ B2 )
=> ( ( ord_less_eq_set_v @ A @ C2 )
=> ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_784_inf_OboundedI,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B2 )
=> ( ( ord_le7336532860387713383od_v_v @ A @ C2 )
=> ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_785_inf__greatest,axiom,
! [X: set_v,Y: set_v,Z2: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( ord_less_eq_set_v @ X @ Z2 )
=> ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_786_inf__greatest,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ X @ Z2 )
=> ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_787_inf_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B6: set_v] :
( A4
= ( inf_inf_set_v @ A4 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_788_inf_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( A4
= ( inf_in6271465464967711157od_v_v @ A4 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_789_inf_Ocobounded1,axiom,
! [A: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_790_inf_Ocobounded1,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_791_inf_Ocobounded2,axiom,
! [A: set_v,B2: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_792_inf_Ocobounded2,axiom,
! [A: set_Product_prod_v_v,B2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_793_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [A4: set_v,B6: set_v] :
( ( inf_inf_set_v @ A4 @ B6 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_794_inf_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A4: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A4 @ B6 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_795_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [B6: set_v,A4: set_v] :
( ( inf_inf_set_v @ A4 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_796_inf_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B6: set_Product_prod_v_v,A4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A4 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_797_inf_OcoboundedI1,axiom,
! [A: set_v,C2: set_v,B2: set_v] :
( ( ord_less_eq_set_v @ A @ C2 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_798_inf_OcoboundedI1,axiom,
! [A: set_Product_prod_v_v,C2: set_Product_prod_v_v,B2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C2 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_799_inf_OcoboundedI2,axiom,
! [B2: set_v,C2: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B2 @ C2 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_800_inf_OcoboundedI2,axiom,
! [B2: set_Product_prod_v_v,C2: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B2 @ C2 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_801_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_set_v] :
( ( sup_sup_set_set_v @ X @ bot_bot_set_set_v )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_802_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ bot_bot_set_v )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_803_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ bot_bo723834152578015283od_v_v )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_804_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_v,Z2: set_v,X: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ Z2 ) @ X )
= ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ X ) @ ( sup_sup_set_v @ Z2 @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_805_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_set_v,Z2: set_set_v,X: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y @ Z2 ) @ X )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y @ X ) @ ( sup_sup_set_set_v @ Z2 @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_806_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ Z2 ) @ X )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ X ) @ ( sup_su414716646722978715od_v_v @ Z2 @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_807_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_v,Z2: set_v,X: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ Z2 ) @ X )
= ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ X ) @ ( inf_inf_set_v @ Z2 @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_808_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_set_v,Z2: set_set_v,X: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y @ Z2 ) @ X )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y @ X ) @ ( inf_inf_set_set_v @ Z2 @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_809_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ Z2 ) @ X )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ X ) @ ( inf_in6271465464967711157od_v_v @ Z2 @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_810_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_v,Y: set_v,Z2: set_v] :
( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y @ Z2 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y ) @ ( sup_sup_set_v @ X @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_811_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_set_v,Y: set_set_v,Z2: set_set_v] :
( ( sup_sup_set_set_v @ X @ ( inf_inf_set_set_v @ Y @ Z2 ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X @ Y ) @ ( sup_sup_set_set_v @ X @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_812_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z2 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ ( sup_su414716646722978715od_v_v @ X @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_813_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_v,Y: set_v,Z2: set_v] :
( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y @ Z2 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y ) @ ( inf_inf_set_v @ X @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_814_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_set_v,Y: set_set_v,Z2: set_set_v] :
( ( inf_inf_set_set_v @ X @ ( sup_sup_set_set_v @ Y @ Z2 ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X @ Y ) @ ( inf_inf_set_set_v @ X @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_815_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z2 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ ( inf_in6271465464967711157od_v_v @ X @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_816_sup__inf__distrib2,axiom,
! [Y: set_v,Z2: set_v,X: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ Z2 ) @ X )
= ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ X ) @ ( sup_sup_set_v @ Z2 @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_817_sup__inf__distrib2,axiom,
! [Y: set_set_v,Z2: set_set_v,X: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y @ Z2 ) @ X )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y @ X ) @ ( sup_sup_set_set_v @ Z2 @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_818_sup__inf__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ Z2 ) @ X )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ X ) @ ( sup_su414716646722978715od_v_v @ Z2 @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_819_sup__inf__distrib1,axiom,
! [X: set_v,Y: set_v,Z2: set_v] :
( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y @ Z2 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y ) @ ( sup_sup_set_v @ X @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_820_sup__inf__distrib1,axiom,
! [X: set_set_v,Y: set_set_v,Z2: set_set_v] :
( ( sup_sup_set_set_v @ X @ ( inf_inf_set_set_v @ Y @ Z2 ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X @ Y ) @ ( sup_sup_set_set_v @ X @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_821_sup__inf__distrib1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z2 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ ( sup_su414716646722978715od_v_v @ X @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_822_inf__sup__distrib2,axiom,
! [Y: set_v,Z2: set_v,X: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ Z2 ) @ X )
= ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ X ) @ ( inf_inf_set_v @ Z2 @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_823_inf__sup__distrib2,axiom,
! [Y: set_set_v,Z2: set_set_v,X: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y @ Z2 ) @ X )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y @ X ) @ ( inf_inf_set_set_v @ Z2 @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_824_inf__sup__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z2: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ Z2 ) @ X )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ X ) @ ( inf_in6271465464967711157od_v_v @ Z2 @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_825_inf__sup__distrib1,axiom,
! [X: set_v,Y: set_v,Z2: set_v] :
( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y @ Z2 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y ) @ ( inf_inf_set_v @ X @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_826_inf__sup__distrib1,axiom,
! [X: set_set_v,Y: set_set_v,Z2: set_set_v] :
( ( inf_inf_set_set_v @ X @ ( sup_sup_set_set_v @ Y @ Z2 ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X @ Y ) @ ( inf_inf_set_set_v @ X @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_827_inf__sup__distrib1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z2 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ ( inf_in6271465464967711157od_v_v @ X @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_828_distrib__imp2,axiom,
! [X: set_v,Y: set_v,Z2: set_v] :
( ! [X3: set_v,Y3: set_v,Z4: set_v] :
( ( sup_sup_set_v @ X3 @ ( inf_inf_set_v @ Y3 @ Z4 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X3 @ Y3 ) @ ( sup_sup_set_v @ X3 @ Z4 ) ) )
=> ( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y @ Z2 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y ) @ ( inf_inf_set_v @ X @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_829_distrib__imp2,axiom,
! [X: set_set_v,Y: set_set_v,Z2: set_set_v] :
( ! [X3: set_set_v,Y3: set_set_v,Z4: set_set_v] :
( ( sup_sup_set_set_v @ X3 @ ( inf_inf_set_set_v @ Y3 @ Z4 ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X3 @ Y3 ) @ ( sup_sup_set_set_v @ X3 @ Z4 ) ) )
=> ( ( inf_inf_set_set_v @ X @ ( sup_sup_set_set_v @ Y @ Z2 ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X @ Y ) @ ( inf_inf_set_set_v @ X @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_830_distrib__imp2,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v,Z4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X3 @ ( inf_in6271465464967711157od_v_v @ Y3 @ Z4 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X3 @ Y3 ) @ ( sup_su414716646722978715od_v_v @ X3 @ Z4 ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z2 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ ( inf_in6271465464967711157od_v_v @ X @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_831_distrib__imp1,axiom,
! [X: set_v,Y: set_v,Z2: set_v] :
( ! [X3: set_v,Y3: set_v,Z4: set_v] :
( ( inf_inf_set_v @ X3 @ ( sup_sup_set_v @ Y3 @ Z4 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X3 @ Y3 ) @ ( inf_inf_set_v @ X3 @ Z4 ) ) )
=> ( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y @ Z2 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y ) @ ( sup_sup_set_v @ X @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_832_distrib__imp1,axiom,
! [X: set_set_v,Y: set_set_v,Z2: set_set_v] :
( ! [X3: set_set_v,Y3: set_set_v,Z4: set_set_v] :
( ( inf_inf_set_set_v @ X3 @ ( sup_sup_set_set_v @ Y3 @ Z4 ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X3 @ Y3 ) @ ( inf_inf_set_set_v @ X3 @ Z4 ) ) )
=> ( ( sup_sup_set_set_v @ X @ ( inf_inf_set_set_v @ Y @ Z2 ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X @ Y ) @ ( sup_sup_set_set_v @ X @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_833_distrib__imp1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z2: set_Product_prod_v_v] :
( ! [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v,Z4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ Y3 @ Z4 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X3 @ Y3 ) @ ( inf_in6271465464967711157od_v_v @ X3 @ Z4 ) ) )
=> ( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z2 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ ( sup_su414716646722978715od_v_v @ X @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_834_distrib__sup__le,axiom,
! [X: set_set_v,Y: set_set_v,Z2: set_set_v] : ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ X @ ( inf_inf_set_set_v @ Y @ Z2 ) ) @ ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X @ Y ) @ ( sup_sup_set_set_v @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_835_distrib__sup__le,axiom,
! [X: set_v,Y: set_v,Z2: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y @ Z2 ) ) @ ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y ) @ ( sup_sup_set_v @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_836_distrib__sup__le,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z2 ) ) @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ ( sup_su414716646722978715od_v_v @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_837_distrib__inf__le,axiom,
! [X: set_set_v,Y: set_set_v,Z2: set_set_v] : ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X @ Y ) @ ( inf_inf_set_set_v @ X @ Z2 ) ) @ ( inf_inf_set_set_v @ X @ ( sup_sup_set_set_v @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_838_distrib__inf__le,axiom,
! [X: set_v,Y: set_v,Z2: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y ) @ ( inf_inf_set_v @ X @ Z2 ) ) @ ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_839_distrib__inf__le,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z2: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ ( inf_in6271465464967711157od_v_v @ X @ Z2 ) ) @ ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_840_e2,axiom,
( ( sCC_Bloemen_dfs_v @ successors @ v2 @ e )
= ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ).
% e2
thf(fact_841_calculation_I7_J,axiom,
! [N: v] :
( sCC_Bl5398416737448265317bscc_v @ successors
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ N ) ) ).
% calculation(7)
thf(fact_842_calculation_I16_J,axiom,
! [X5: v] :
( ( member_v @ X5
@ ( minus_minus_set_v
@ ( sCC_Bl4645233313691564917t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) )
@ ( set_v2
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ) )
=> ( ( sCC_Bl3795065053823578884t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ X5 )
= ( successors @ X5 ) ) ) ).
% calculation(16)
thf(fact_843_calculation_I10_J,axiom,
! [N: v,M: v] :
( ( sCC_Bl4022239298816431255edes_v @ N @ M
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ M @ N ) ) ).
% calculation(10)
thf(fact_844_calculation_I15_J,axiom,
! [X5: v] :
( ( member_v @ X5
@ ( sCC_Bl157864678168468314t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) )
=> ! [M: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X5 @ M )
=> ( member_v @ M
@ ( sCC_Bl157864678168468314t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ) ) ).
% calculation(15)
thf(fact_845_visited__unexplored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,M2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ M2 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ M2 @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ~ ! [N2: v] :
( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ M2 @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ) ) ).
% visited_unexplored
thf(fact_846_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_847_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_848_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_849_succ__reachable,axiom,
! [X: v,Y: v,Z2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( member_v @ Z2 @ ( successors @ Y ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z2 ) ) ) ).
% succ_reachable
thf(fact_850_reachable__trans,axiom,
! [X: v,Y: v,Z2: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ Z2 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z2 ) ) ) ).
% reachable_trans
thf(fact_851_reachable__end__induct,axiom,
! [X: v,Y: v,P: v > v > $o] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ! [X3: v] : ( P @ X3 @ X3 )
=> ( ! [X3: v,Y3: v,Z4: v] :
( ( P @ X3 @ Y3 )
=> ( ( member_v @ Z4 @ ( successors @ Y3 ) )
=> ( P @ X3 @ Z4 ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% reachable_end_induct
thf(fact_852_reachable__edge,axiom,
! [Y: v,X: v] :
( ( member_v @ Y @ ( successors @ X ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% reachable_edge
thf(fact_853_reachable_Osimps,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A22 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: v,Y4: v,Z5: v] :
( ( A1 = X2 )
& ( A22 = Z5 )
& ( member_v @ Y4 @ ( successors @ X2 ) )
& ( sCC_Bl649662514949026229able_v @ successors @ Y4 @ Z5 ) ) ) ) ).
% reachable.simps
thf(fact_854_reachable__succ,axiom,
! [Y: v,X: v,Z2: v] :
( ( member_v @ Y @ ( successors @ X ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ Z2 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z2 ) ) ) ).
% reachable_succ
thf(fact_855_reachable__refl,axiom,
! [X: v] : ( sCC_Bl649662514949026229able_v @ successors @ X @ X ) ).
% reachable_refl
thf(fact_856_reachable_Ocases,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y3: v] :
( ( member_v @ Y3 @ ( successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Y3 @ A22 ) ) ) ) ).
% reachable.cases
thf(fact_857_wf_H,axiom,
sCC_Bl9196236973127232072t_unit @ successors @ e2 ).
% wf'
thf(fact_858_local_Owf,axiom,
sCC_Bl9196236973127232072t_unit @ successors @ e ).
% local.wf
thf(fact_859_S__reflexive,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N3: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( member_v @ N3 @ ( sCC_Bl1280885523602775798t_unit @ E @ N3 ) ) ) ).
% S_reflexive
thf(fact_860_is__subscc__def,axiom,
! [S: set_v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
= ( ! [X2: v] :
( ( member_v @ X2 @ S )
=> ! [Y4: v] :
( ( member_v @ Y4 @ S )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ Y4 ) ) ) ) ) ).
% is_subscc_def
thf(fact_861_sccE,axiom,
! [S: set_v,X: v,Y: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ X )
=> ( member_v @ Y @ S ) ) ) ) ) ).
% sccE
thf(fact_862_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_863_DiffI,axiom,
! [C2: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ A2 )
=> ( ~ ( member7453568604450474000od_v_v @ C2 @ B )
=> ( member7453568604450474000od_v_v @ C2 @ ( minus_4183494784930505774od_v_v @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_864_DiffI,axiom,
! [C2: v,A2: set_v,B: set_v] :
( ( member_v @ C2 @ A2 )
=> ( ~ ( member_v @ C2 @ B )
=> ( member_v @ C2 @ ( minus_minus_set_v @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_865_Diff__iff,axiom,
! [C2: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( minus_4183494784930505774od_v_v @ A2 @ B ) )
= ( ( member7453568604450474000od_v_v @ C2 @ A2 )
& ~ ( member7453568604450474000od_v_v @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_866_Diff__iff,axiom,
! [C2: v,A2: set_v,B: set_v] :
( ( member_v @ C2 @ ( minus_minus_set_v @ A2 @ B ) )
= ( ( member_v @ C2 @ A2 )
& ~ ( member_v @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_867_Diff__idemp,axiom,
! [A2: set_v,B: set_v] :
( ( minus_minus_set_v @ ( minus_minus_set_v @ A2 @ B ) @ B )
= ( minus_minus_set_v @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_868_subscc__add,axiom,
! [S: set_v,X: v,Y: v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ X )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( insert_v @ Y @ S ) ) ) ) ) ) ).
% subscc_add
thf(fact_869_stack__unexplored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N3: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ N3 @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ).
% stack_unexplored
thf(fact_870_stack__visited,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N3: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( member_v @ N3 @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ).
% stack_visited
thf(fact_871_is__scc__def,axiom,
! [S: set_v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S )
= ( ( S != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S )
& ! [S3: set_v] :
( ( ( ord_less_eq_set_v @ S @ S3 )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S3 ) )
=> ( S3 = S ) ) ) ) ).
% is_scc_def
thf(fact_872_reachable__visited,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V3: v,W: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V3 @ W )
=> ( ! [X3: v] :
( ( member_v @ X3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa: v] :
( ( member_v @ Xa @ ( minus_minus_set_v @ ( successors @ X3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X3 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V3 @ X3 )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Xa @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ).
% reachable_visited
thf(fact_873_Diff__empty,axiom,
! [A2: set_set_v] :
( ( minus_7228012346218142266_set_v @ A2 @ bot_bot_set_set_v )
= A2 ) ).
% Diff_empty
thf(fact_874_Diff__empty,axiom,
! [A2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ bot_bo723834152578015283od_v_v )
= A2 ) ).
% Diff_empty
thf(fact_875_Diff__empty,axiom,
! [A2: set_v] :
( ( minus_minus_set_v @ A2 @ bot_bot_set_v )
= A2 ) ).
% Diff_empty
thf(fact_876_empty__Diff,axiom,
! [A2: set_set_v] :
( ( minus_7228012346218142266_set_v @ bot_bot_set_set_v @ A2 )
= bot_bot_set_set_v ) ).
% empty_Diff
thf(fact_877_empty__Diff,axiom,
! [A2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ bot_bo723834152578015283od_v_v @ A2 )
= bot_bo723834152578015283od_v_v ) ).
% empty_Diff
thf(fact_878_empty__Diff,axiom,
! [A2: set_v] :
( ( minus_minus_set_v @ bot_bot_set_v @ A2 )
= bot_bot_set_v ) ).
% empty_Diff
thf(fact_879_Diff__cancel,axiom,
! [A2: set_set_v] :
( ( minus_7228012346218142266_set_v @ A2 @ A2 )
= bot_bot_set_set_v ) ).
% Diff_cancel
thf(fact_880_Diff__cancel,axiom,
! [A2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ A2 )
= bot_bo723834152578015283od_v_v ) ).
% Diff_cancel
thf(fact_881_Diff__cancel,axiom,
! [A2: set_v] :
( ( minus_minus_set_v @ A2 @ A2 )
= bot_bot_set_v ) ).
% Diff_cancel
thf(fact_882_Diff__insert0,axiom,
! [X: set_v,A2: set_set_v,B: set_set_v] :
( ~ ( member_set_v @ X @ A2 )
=> ( ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v @ X @ B ) )
= ( minus_7228012346218142266_set_v @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_883_Diff__insert0,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A2 )
=> ( ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X @ B ) )
= ( minus_4183494784930505774od_v_v @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_884_Diff__insert0,axiom,
! [X: v,A2: set_v,B: set_v] :
( ~ ( member_v @ X @ A2 )
=> ( ( minus_minus_set_v @ A2 @ ( insert_v @ X @ B ) )
= ( minus_minus_set_v @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_885_insert__Diff1,axiom,
! [X: set_v,B: set_set_v,A2: set_set_v] :
( ( member_set_v @ X @ B )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X @ A2 ) @ B )
= ( minus_7228012346218142266_set_v @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_886_insert__Diff1,axiom,
! [X: product_prod_v_v,B: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A2 ) @ B )
= ( minus_4183494784930505774od_v_v @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_887_insert__Diff1,axiom,
! [X: v,B: set_v,A2: set_v] :
( ( member_v @ X @ B )
=> ( ( minus_minus_set_v @ ( insert_v @ X @ A2 ) @ B )
= ( minus_minus_set_v @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_888_Un__Diff__cancel,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( sup_sup_set_set_v @ A2 @ ( minus_7228012346218142266_set_v @ B @ A2 ) )
= ( sup_sup_set_set_v @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_889_Un__Diff__cancel,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B @ A2 ) )
= ( sup_su414716646722978715od_v_v @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_890_Un__Diff__cancel,axiom,
! [A2: set_v,B: set_v] :
( ( sup_sup_set_v @ A2 @ ( minus_minus_set_v @ B @ A2 ) )
= ( sup_sup_set_v @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_891_Un__Diff__cancel2,axiom,
! [B: set_set_v,A2: set_set_v] :
( ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ B @ A2 ) @ A2 )
= ( sup_sup_set_set_v @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_892_Un__Diff__cancel2,axiom,
! [B: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ B @ A2 ) @ A2 )
= ( sup_su414716646722978715od_v_v @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_893_Un__Diff__cancel2,axiom,
! [B: set_v,A2: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ B @ A2 ) @ A2 )
= ( sup_sup_set_v @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_894_init__env__pre__dfs,axiom,
! [V3: v] : ( sCC_Bl36166008131615352t_unit @ successors @ V3 @ ( sCC_Bl7693227186847904995_env_v @ V3 ) ) ).
% init_env_pre_dfs
thf(fact_895_stack__class,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N3: v,M2: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( ( member_v @ M2 @ ( sCC_Bl1280885523602775798t_unit @ E @ N3 ) )
=> ( member_v @ M2 @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ).
% stack_class
thf(fact_896_Diff__eq__empty__iff,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( ( minus_7228012346218142266_set_v @ A2 @ B )
= bot_bot_set_set_v )
= ( ord_le5216385588623774835_set_v @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_897_Diff__eq__empty__iff,axiom,
! [A2: set_v,B: set_v] :
( ( ( minus_minus_set_v @ A2 @ B )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_898_Diff__eq__empty__iff,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ A2 @ B )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_899_insert__Diff__single,axiom,
! [A: set_v,A2: set_set_v] :
( ( insert_set_v @ A @ ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) )
= ( insert_set_v @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_900_insert__Diff__single,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= ( insert1338601472111419319od_v_v @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_901_insert__Diff__single,axiom,
! [A: v,A2: set_v] :
( ( insert_v @ A @ ( minus_minus_set_v @ A2 @ ( insert_v @ A @ bot_bot_set_v ) ) )
= ( insert_v @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_902_Diff__disjoint,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( inf_inf_set_set_v @ A2 @ ( minus_7228012346218142266_set_v @ B @ A2 ) )
= bot_bot_set_set_v ) ).
% Diff_disjoint
thf(fact_903_Diff__disjoint,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B @ A2 ) )
= bot_bo723834152578015283od_v_v ) ).
% Diff_disjoint
thf(fact_904_Diff__disjoint,axiom,
! [A2: set_v,B: set_v] :
( ( inf_inf_set_v @ A2 @ ( minus_minus_set_v @ B @ A2 ) )
= bot_bot_set_v ) ).
% Diff_disjoint
thf(fact_905_graph__axioms,axiom,
sCC_Bloemen_graph_v @ vertices @ successors ).
% graph_axioms
thf(fact_906_set__diff__eq,axiom,
( minus_4183494784930505774od_v_v
= ( ^ [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ^ [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ A3 )
& ~ ( member7453568604450474000od_v_v @ X2 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_907_set__diff__eq,axiom,
( minus_7228012346218142266_set_v
= ( ^ [A3: set_set_v,B4: set_set_v] :
( collect_set_v
@ ^ [X2: set_v] :
( ( member_set_v @ X2 @ A3 )
& ~ ( member_set_v @ X2 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_908_set__diff__eq,axiom,
( minus_minus_set_v
= ( ^ [A3: set_v,B4: set_v] :
( collect_v
@ ^ [X2: v] :
( ( member_v @ X2 @ A3 )
& ~ ( member_v @ X2 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_909_graph_Ois__scc_Ocong,axiom,
sCC_Bloemen_is_scc_v = sCC_Bloemen_is_scc_v ).
% graph.is_scc.cong
thf(fact_910_graph_Opre__dfs_Ocong,axiom,
sCC_Bl36166008131615352t_unit = sCC_Bl36166008131615352t_unit ).
% graph.pre_dfs.cong
thf(fact_911_graph_Odfss_Ocong,axiom,
sCC_Bloemen_dfss_v = sCC_Bloemen_dfss_v ).
% graph.dfss.cong
thf(fact_912_DiffE,axiom,
! [C2: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( minus_4183494784930505774od_v_v @ A2 @ B ) )
=> ~ ( ( member7453568604450474000od_v_v @ C2 @ A2 )
=> ( member7453568604450474000od_v_v @ C2 @ B ) ) ) ).
% DiffE
thf(fact_913_DiffE,axiom,
! [C2: v,A2: set_v,B: set_v] :
( ( member_v @ C2 @ ( minus_minus_set_v @ A2 @ B ) )
=> ~ ( ( member_v @ C2 @ A2 )
=> ( member_v @ C2 @ B ) ) ) ).
% DiffE
thf(fact_914_DiffD1,axiom,
! [C2: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( minus_4183494784930505774od_v_v @ A2 @ B ) )
=> ( member7453568604450474000od_v_v @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_915_DiffD1,axiom,
! [C2: v,A2: set_v,B: set_v] :
( ( member_v @ C2 @ ( minus_minus_set_v @ A2 @ B ) )
=> ( member_v @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_916_DiffD2,axiom,
! [C2: product_prod_v_v,A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C2 @ ( minus_4183494784930505774od_v_v @ A2 @ B ) )
=> ~ ( member7453568604450474000od_v_v @ C2 @ B ) ) ).
% DiffD2
thf(fact_917_DiffD2,axiom,
! [C2: v,A2: set_v,B: set_v] :
( ( member_v @ C2 @ ( minus_minus_set_v @ A2 @ B ) )
=> ~ ( member_v @ C2 @ B ) ) ).
% DiffD2
thf(fact_918_graph_Oreachable_Ocong,axiom,
sCC_Bl649662514949026229able_v = sCC_Bl649662514949026229able_v ).
% graph.reachable.cong
thf(fact_919_graph_Owf__env_Ocong,axiom,
sCC_Bl9196236973127232072t_unit = sCC_Bl9196236973127232072t_unit ).
% graph.wf_env.cong
thf(fact_920_graph_Ois__subscc_Ocong,axiom,
sCC_Bl5398416737448265317bscc_v = sCC_Bl5398416737448265317bscc_v ).
% graph.is_subscc.cong
thf(fact_921_graph_Odfs_Ocong,axiom,
sCC_Bloemen_dfs_v = sCC_Bloemen_dfs_v ).
% graph.dfs.cong
thf(fact_922_graph_Opost__dfss_Ocong,axiom,
sCC_Bl6082031138996704384t_unit = sCC_Bl6082031138996704384t_unit ).
% graph.post_dfss.cong
thf(fact_923_Diff__mono,axiom,
! [A2: set_v,C: set_v,D2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A2 @ C )
=> ( ( ord_less_eq_set_v @ D2 @ B )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ B ) @ ( minus_minus_set_v @ C @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_924_Diff__mono,axiom,
! [A2: set_Product_prod_v_v,C: set_Product_prod_v_v,D2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ C )
=> ( ( ord_le7336532860387713383od_v_v @ D2 @ B )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B ) @ ( minus_4183494784930505774od_v_v @ C @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_925_Diff__subset,axiom,
! [A2: set_v,B: set_v] : ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_926_Diff__subset,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_927_double__diff,axiom,
! [A2: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A2 @ B )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ( minus_minus_set_v @ B @ ( minus_minus_set_v @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_928_double__diff,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ( minus_4183494784930505774od_v_v @ B @ ( minus_4183494784930505774od_v_v @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_929_insert__Diff__if,axiom,
! [X: set_v,B: set_set_v,A2: set_set_v] :
( ( ( member_set_v @ X @ B )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X @ A2 ) @ B )
= ( minus_7228012346218142266_set_v @ A2 @ B ) ) )
& ( ~ ( member_set_v @ X @ B )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X @ A2 ) @ B )
= ( insert_set_v @ X @ ( minus_7228012346218142266_set_v @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_930_insert__Diff__if,axiom,
! [X: product_prod_v_v,B: set_Product_prod_v_v,A2: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ X @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A2 ) @ B )
= ( minus_4183494784930505774od_v_v @ A2 @ B ) ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ B )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A2 ) @ B )
= ( insert1338601472111419319od_v_v @ X @ ( minus_4183494784930505774od_v_v @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_931_insert__Diff__if,axiom,
! [X: v,B: set_v,A2: set_v] :
( ( ( member_v @ X @ B )
=> ( ( minus_minus_set_v @ ( insert_v @ X @ A2 ) @ B )
= ( minus_minus_set_v @ A2 @ B ) ) )
& ( ~ ( member_v @ X @ B )
=> ( ( minus_minus_set_v @ ( insert_v @ X @ A2 ) @ B )
= ( insert_v @ X @ ( minus_minus_set_v @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_932_Un__Diff,axiom,
! [A2: set_set_v,B: set_set_v,C: set_set_v] :
( ( minus_7228012346218142266_set_v @ ( sup_sup_set_set_v @ A2 @ B ) @ C )
= ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ A2 @ C ) @ ( minus_7228012346218142266_set_v @ B @ C ) ) ) ).
% Un_Diff
thf(fact_933_Un__Diff,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ ( sup_su414716646722978715od_v_v @ A2 @ B ) @ C )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ C ) @ ( minus_4183494784930505774od_v_v @ B @ C ) ) ) ).
% Un_Diff
thf(fact_934_Un__Diff,axiom,
! [A2: set_v,B: set_v,C: set_v] :
( ( minus_minus_set_v @ ( sup_sup_set_v @ A2 @ B ) @ C )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A2 @ C ) @ ( minus_minus_set_v @ B @ C ) ) ) ).
% Un_Diff
thf(fact_935_Int__Diff,axiom,
! [A2: set_v,B: set_v,C: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A2 @ B ) @ C )
= ( inf_inf_set_v @ A2 @ ( minus_minus_set_v @ B @ C ) ) ) ).
% Int_Diff
thf(fact_936_Diff__Int2,axiom,
! [A2: set_v,C: set_v,B: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A2 @ C ) @ ( inf_inf_set_v @ B @ C ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A2 @ C ) @ B ) ) ).
% Diff_Int2
thf(fact_937_Diff__Diff__Int,axiom,
! [A2: set_v,B: set_v] :
( ( minus_minus_set_v @ A2 @ ( minus_minus_set_v @ A2 @ B ) )
= ( inf_inf_set_v @ A2 @ B ) ) ).
% Diff_Diff_Int
thf(fact_938_Diff__Int__distrib,axiom,
! [C: set_v,A2: set_v,B: set_v] :
( ( inf_inf_set_v @ C @ ( minus_minus_set_v @ A2 @ B ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ C @ A2 ) @ ( inf_inf_set_v @ C @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_939_Diff__Int__distrib2,axiom,
! [A2: set_v,B: set_v,C: set_v] :
( ( inf_inf_set_v @ ( minus_minus_set_v @ A2 @ B ) @ C )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A2 @ C ) @ ( inf_inf_set_v @ B @ C ) ) ) ).
% Diff_Int_distrib2
thf(fact_940_diff__shunt__var,axiom,
! [X: set_set_v,Y: set_set_v] :
( ( ( minus_7228012346218142266_set_v @ X @ Y )
= bot_bot_set_set_v )
= ( ord_le5216385588623774835_set_v @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_941_diff__shunt__var,axiom,
! [X: set_v,Y: set_v] :
( ( ( minus_minus_set_v @ X @ Y )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_942_diff__shunt__var,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ X @ Y )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_943_Diff__insert,axiom,
! [A2: set_set_v,A: set_v,B: set_set_v] :
( ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v @ A @ B ) )
= ( minus_7228012346218142266_set_v @ ( minus_7228012346218142266_set_v @ A2 @ B ) @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) ) ).
% Diff_insert
thf(fact_944_Diff__insert,axiom,
! [A2: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B ) @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ).
% Diff_insert
thf(fact_945_Diff__insert,axiom,
! [A2: set_v,A: v,B: set_v] :
( ( minus_minus_set_v @ A2 @ ( insert_v @ A @ B ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A2 @ B ) @ ( insert_v @ A @ bot_bot_set_v ) ) ) ).
% Diff_insert
thf(fact_946_insert__Diff,axiom,
! [A: set_v,A2: set_set_v] :
( ( member_set_v @ A @ A2 )
=> ( ( insert_set_v @ A @ ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_947_insert__Diff,axiom,
! [A: product_prod_v_v,A2: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A2 )
=> ( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_948_insert__Diff,axiom,
! [A: v,A2: set_v] :
( ( member_v @ A @ A2 )
=> ( ( insert_v @ A @ ( minus_minus_set_v @ A2 @ ( insert_v @ A @ bot_bot_set_v ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_949_Diff__insert2,axiom,
! [A2: set_set_v,A: set_v,B: set_set_v] :
( ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v @ A @ B ) )
= ( minus_7228012346218142266_set_v @ ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) @ B ) ) ).
% Diff_insert2
thf(fact_950_Diff__insert2,axiom,
! [A2: set_Product_prod_v_v,A: product_prod_v_v,B: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ B ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) @ B ) ) ).
% Diff_insert2
thf(fact_951_Diff__insert2,axiom,
! [A2: set_v,A: v,B: set_v] :
( ( minus_minus_set_v @ A2 @ ( insert_v @ A @ B ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A2 @ ( insert_v @ A @ bot_bot_set_v ) ) @ B ) ) ).
% Diff_insert2
thf(fact_952_Diff__insert__absorb,axiom,
! [X: set_v,A2: set_set_v] :
( ~ ( member_set_v @ X @ A2 )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X @ A2 ) @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_953_Diff__insert__absorb,axiom,
! [X: product_prod_v_v,A2: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A2 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A2 ) @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_954_Diff__insert__absorb,axiom,
! [X: v,A2: set_v] :
( ~ ( member_v @ X @ A2 )
=> ( ( minus_minus_set_v @ ( insert_v @ X @ A2 ) @ ( insert_v @ X @ bot_bot_set_v ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_955_subset__Diff__insert,axiom,
! [A2: set_set_v,B: set_set_v,X: set_v,C: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ ( minus_7228012346218142266_set_v @ B @ ( insert_set_v @ X @ C ) ) )
= ( ( ord_le5216385588623774835_set_v @ A2 @ ( minus_7228012346218142266_set_v @ B @ C ) )
& ~ ( member_set_v @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_956_subset__Diff__insert,axiom,
! [A2: set_v,B: set_v,X: v,C: set_v] :
( ( ord_less_eq_set_v @ A2 @ ( minus_minus_set_v @ B @ ( insert_v @ X @ C ) ) )
= ( ( ord_less_eq_set_v @ A2 @ ( minus_minus_set_v @ B @ C ) )
& ~ ( member_v @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_957_subset__Diff__insert,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,X: product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B @ ( insert1338601472111419319od_v_v @ X @ C ) ) )
= ( ( ord_le7336532860387713383od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B @ C ) )
& ~ ( member7453568604450474000od_v_v @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_958_Int__Diff__disjoint,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( inf_inf_set_set_v @ ( inf_inf_set_set_v @ A2 @ B ) @ ( minus_7228012346218142266_set_v @ A2 @ B ) )
= bot_bot_set_set_v ) ).
% Int_Diff_disjoint
thf(fact_959_Int__Diff__disjoint,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) @ ( minus_4183494784930505774od_v_v @ A2 @ B ) )
= bot_bo723834152578015283od_v_v ) ).
% Int_Diff_disjoint
thf(fact_960_Int__Diff__disjoint,axiom,
! [A2: set_v,B: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A2 @ B ) @ ( minus_minus_set_v @ A2 @ B ) )
= bot_bot_set_v ) ).
% Int_Diff_disjoint
thf(fact_961_Diff__triv,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( ( inf_inf_set_set_v @ A2 @ B )
= bot_bot_set_set_v )
=> ( ( minus_7228012346218142266_set_v @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_962_Diff__triv,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A2 @ B )
= bot_bo723834152578015283od_v_v )
=> ( ( minus_4183494784930505774od_v_v @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_963_Diff__triv,axiom,
! [A2: set_v,B: set_v] :
( ( ( inf_inf_set_v @ A2 @ B )
= bot_bot_set_v )
=> ( ( minus_minus_set_v @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_964_Diff__subset__conv,axiom,
! [A2: set_set_v,B: set_set_v,C: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A2 @ B ) @ C )
= ( ord_le5216385588623774835_set_v @ A2 @ ( sup_sup_set_set_v @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_965_Diff__subset__conv,axiom,
! [A2: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ B ) @ C )
= ( ord_less_eq_set_v @ A2 @ ( sup_sup_set_v @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_966_Diff__subset__conv,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B ) @ C )
= ( ord_le7336532860387713383od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_967_Diff__partition,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ B )
=> ( ( sup_sup_set_set_v @ A2 @ ( minus_7228012346218142266_set_v @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_968_Diff__partition,axiom,
! [A2: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A2 @ B )
=> ( ( sup_sup_set_v @ A2 @ ( minus_minus_set_v @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_969_Diff__partition,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ B )
=> ( ( sup_su414716646722978715od_v_v @ A2 @ ( minus_4183494784930505774od_v_v @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_970_Un__Diff__Int,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ A2 @ B ) @ ( inf_inf_set_set_v @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_971_Un__Diff__Int,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B ) @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_972_Un__Diff__Int,axiom,
! [A2: set_v,B: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ A2 @ B ) @ ( inf_inf_set_v @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_973_Int__Diff__Un,axiom,
! [A2: set_set_v,B: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A2 @ B ) @ ( minus_7228012346218142266_set_v @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_974_Int__Diff__Un,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A2 @ B ) @ ( minus_4183494784930505774od_v_v @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_975_Int__Diff__Un,axiom,
! [A2: set_v,B: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ A2 @ B ) @ ( minus_minus_set_v @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_976_Diff__Int,axiom,
! [A2: set_set_v,B: set_set_v,C: set_set_v] :
( ( minus_7228012346218142266_set_v @ A2 @ ( inf_inf_set_set_v @ B @ C ) )
= ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ A2 @ B ) @ ( minus_7228012346218142266_set_v @ A2 @ C ) ) ) ).
% Diff_Int
thf(fact_977_Diff__Int,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ ( inf_in6271465464967711157od_v_v @ B @ C ) )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B ) @ ( minus_4183494784930505774od_v_v @ A2 @ C ) ) ) ).
% Diff_Int
thf(fact_978_Diff__Int,axiom,
! [A2: set_v,B: set_v,C: set_v] :
( ( minus_minus_set_v @ A2 @ ( inf_inf_set_v @ B @ C ) )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A2 @ B ) @ ( minus_minus_set_v @ A2 @ C ) ) ) ).
% Diff_Int
thf(fact_979_Diff__Un,axiom,
! [A2: set_set_v,B: set_set_v,C: set_set_v] :
( ( minus_7228012346218142266_set_v @ A2 @ ( sup_sup_set_set_v @ B @ C ) )
= ( inf_inf_set_set_v @ ( minus_7228012346218142266_set_v @ A2 @ B ) @ ( minus_7228012346218142266_set_v @ A2 @ C ) ) ) ).
% Diff_Un
thf(fact_980_Diff__Un,axiom,
! [A2: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A2 @ ( sup_su414716646722978715od_v_v @ B @ C ) )
= ( inf_in6271465464967711157od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ B ) @ ( minus_4183494784930505774od_v_v @ A2 @ C ) ) ) ).
% Diff_Un
thf(fact_981_Diff__Un,axiom,
! [A2: set_v,B: set_v,C: set_v] :
( ( minus_minus_set_v @ A2 @ ( sup_sup_set_v @ B @ C ) )
= ( inf_inf_set_v @ ( minus_minus_set_v @ A2 @ B ) @ ( minus_minus_set_v @ A2 @ C ) ) ) ).
% Diff_Un
thf(fact_982_subset__insert__iff,axiom,
! [A2: set_set_v,X: set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A2 @ ( insert_set_v @ X @ B ) )
= ( ( ( member_set_v @ X @ A2 )
=> ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) @ B ) )
& ( ~ ( member_set_v @ X @ A2 )
=> ( ord_le5216385588623774835_set_v @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_983_subset__insert__iff,axiom,
! [A2: set_v,X: v,B: set_v] :
( ( ord_less_eq_set_v @ A2 @ ( insert_v @ X @ B ) )
= ( ( ( member_v @ X @ A2 )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ ( insert_v @ X @ bot_bot_set_v ) ) @ B ) )
& ( ~ ( member_v @ X @ A2 )
=> ( ord_less_eq_set_v @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_984_subset__insert__iff,axiom,
! [A2: set_Product_prod_v_v,X: product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X @ B ) )
= ( ( ( member7453568604450474000od_v_v @ X @ A2 )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ A2 )
=> ( ord_le7336532860387713383od_v_v @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_985_Diff__single__insert,axiom,
! [A2: set_set_v,X: set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A2 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) @ B )
=> ( ord_le5216385588623774835_set_v @ A2 @ ( insert_set_v @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_986_Diff__single__insert,axiom,
! [A2: set_v,X: v,B: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A2 @ ( insert_v @ X @ bot_bot_set_v ) ) @ B )
=> ( ord_less_eq_set_v @ A2 @ ( insert_v @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_987_Diff__single__insert,axiom,
! [A2: set_Product_prod_v_v,X: product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B )
=> ( ord_le7336532860387713383od_v_v @ A2 @ ( insert1338601472111419319od_v_v @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_988_not__Cons__self2,axiom,
! [X: v,Xs: list_v] :
( ( cons_v @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_989_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_990_list_Oset__intros_I2_J,axiom,
! [Y: v,X22: list_v,X21: v] :
( ( member_v @ Y @ ( set_v2 @ X22 ) )
=> ( member_v @ Y @ ( set_v2 @ ( cons_v @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_991_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_992_list_Oset__intros_I1_J,axiom,
! [X21: v,X22: list_v] : ( member_v @ X21 @ ( set_v2 @ ( cons_v @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_993_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_994_list_Oset__cases,axiom,
! [E: v,A: list_v] :
( ( member_v @ E @ ( set_v2 @ A ) )
=> ( ! [Z22: list_v] :
( A
!= ( cons_v @ E @ Z22 ) )
=> ~ ! [Z1: v,Z22: list_v] :
( ( A
= ( cons_v @ Z1 @ Z22 ) )
=> ~ ( member_v @ E @ ( set_v2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_995_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_996_set__ConsD,axiom,
! [Y: v,X: v,Xs: list_v] :
( ( member_v @ Y @ ( set_v2 @ ( cons_v @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_v @ Y @ ( set_v2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_997_subset__code_I1_J,axiom,
! [Xs: list_v,B: set_v] :
( ( ord_less_eq_set_v @ ( set_v2 @ Xs ) @ B )
= ( ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ Xs ) )
=> ( member_v @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_998_subset__code_I1_J,axiom,
! [Xs: list_P7986770385144383213od_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( set_Product_prod_v_v2 @ Xs ) @ B )
= ( ! [X2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X2 @ ( set_Product_prod_v_v2 @ Xs ) )
=> ( member7453568604450474000od_v_v @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_999_list_Osel_I3_J,axiom,
! [X21: v,X22: list_v] :
( ( tl_v @ ( cons_v @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_1000_distinct__length__2__or__more,axiom,
! [A: v,B2: v,Xs: list_v] :
( ( distinct_v @ ( cons_v @ A @ ( cons_v @ B2 @ Xs ) ) )
= ( ( A != B2 )
& ( distinct_v @ ( cons_v @ A @ Xs ) )
& ( distinct_v @ ( cons_v @ B2 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_1001_list_Osel_I1_J,axiom,
! [X21: v,X22: list_v] :
( ( hd_v @ ( cons_v @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_1002_distinct__tl,axiom,
! [Xs: list_v] :
( ( distinct_v @ Xs )
=> ( distinct_v @ ( tl_v @ Xs ) ) ) ).
% distinct_tl
thf(fact_1003_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_1004_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_1005_distinct_Osimps_I2_J,axiom,
! [X: product_prod_v_v,Xs: list_P7986770385144383213od_v_v] :
( ( distin6159370996967099744od_v_v @ ( cons_P4120604216776828829od_v_v @ X @ Xs ) )
= ( ~ ( member7453568604450474000od_v_v @ X @ ( set_Product_prod_v_v2 @ Xs ) )
& ( distin6159370996967099744od_v_v @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1006_distinct_Osimps_I2_J,axiom,
! [X: v,Xs: list_v] :
( ( distinct_v @ ( cons_v @ X @ Xs ) )
= ( ~ ( member_v @ X @ ( set_v2 @ Xs ) )
& ( distinct_v @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1007_pre__dfs__def,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl36166008131615352t_unit @ successors @ V3 @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
& ~ ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( sCC_Bl649662514949026229able_v @ successors @ ( sCC_Bl1090238580953940555t_unit @ E ) @ V3 )
& ( ( sCC_Bl3795065053823578884t_unit @ E @ V3 )
= bot_bot_set_v )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V3 ) ) ) ) ).
% pre_dfs_def
thf(fact_1008_calculation_I1_J,axiom,
! [X5: v] :
( ( member_v @ X5
@ ( sCC_Bl4645233313691564917t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors
@ ( sCC_Bl1090238580953940555t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) )
@ X5 ) ) ).
% calculation(1)
thf(fact_1009_pre__dfs__pre__dfss,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl36166008131615352t_unit @ successors @ V3 @ E )
=> ( sCC_Bl1748261141445803503t_unit @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) ) ).
% pre_dfs_pre_dfss
thf(fact_1010_pre__dfss__pre__dfs,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( sCC_Bl36166008131615352t_unit @ successors @ W @ E ) ) ) ) ).
% pre_dfss_pre_dfs
thf(fact_1011_unite__S__tl,axiom,
! [E: sCC_Bl1394983891496994913t_unit,W: v,V3: v,N3: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ N3 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ N3 )
= ( sCC_Bl1280885523602775798t_unit @ E @ N3 ) ) ) ) ) ) ) ) ).
% unite_S_tl
thf(fact_1012_unite__wf__env,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl9196236973127232072t_unit @ successors @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ).
% unite_wf_env
thf(fact_1013_unite__sub__env,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5768913643336123637t_unit @ E @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ).
% unite_sub_env
thf(fact_1014_unite__subscc,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ) ) ).
% unite_subscc
thf(fact_1015_pre__dfss__explored__pre__dfss,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl1748261141445803503t_unit @ successors @ V3
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X2: v] : ( if_set_v @ ( X2 = V3 ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X2 ) )
@ E ) ) ) ) ) ).
% pre_dfss_explored_pre_dfss
thf(fact_1016_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_1017_graph_Osclosed,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ! [X5: v] :
( ( member_v @ X5 @ Vertices )
=> ( ord_less_eq_set_v @ ( Successors @ X5 ) @ Vertices ) ) ) ).
% graph.sclosed
thf(fact_1018_graph_Opre__dfss_Ocong,axiom,
sCC_Bl1748261141445803503t_unit = sCC_Bl1748261141445803503t_unit ).
% graph.pre_dfss.cong
thf(fact_1019_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_1020_graph_Oreachable__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y @ ( Successors @ X ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y ) ) ) ).
% graph.reachable_edge
thf(fact_1021_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,Z2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Z2 ) ) ) ) ).
% graph.succ_reachable
thf(fact_1022_graph_Osucc__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,Z2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( member_v @ Z2 @ ( Successors @ Y ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z2 ) ) ) ) ).
% graph.succ_reachable
thf(fact_1023_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_1024_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_v @ Y3 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Y3 @ A22 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_1025_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 )
= ( ? [X2: product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: product_prod_v_v,Y4: product_prod_v_v,Z5: product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = Z5 )
& ( member7453568604450474000od_v_v @ Y4 @ ( Successors @ X2 ) )
& ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y4 @ Z5 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_1026_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 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: v,Y4: v,Z5: v] :
( ( A1 = X2 )
& ( A22 = Z5 )
& ( member_v @ Y4 @ ( Successors @ X2 ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ Y4 @ Z5 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_1027_graph_Oreachable__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,Z2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ Z2 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z2 ) ) ) ) ).
% graph.reachable_trans
thf(fact_1028_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,P: 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] : ( P @ X3 @ X3 )
=> ( ! [X3: product_prod_v_v,Y3: product_prod_v_v,Z4: product_prod_v_v] :
( ( P @ X3 @ Y3 )
=> ( ( member7453568604450474000od_v_v @ Z4 @ ( Successors @ Y3 ) )
=> ( P @ X3 @ Z4 ) ) )
=> ( P @ X @ Y ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_1029_graph_Oreachable__end__induct,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,P: v > v > $o] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ! [X3: v] : ( P @ X3 @ X3 )
=> ( ! [X3: v,Y3: v,Z4: v] :
( ( P @ X3 @ Y3 )
=> ( ( member_v @ Z4 @ ( Successors @ Y3 ) )
=> ( P @ X3 @ Z4 ) ) )
=> ( P @ X @ Y ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_1030_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_1031_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,Z2: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ Z2 )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Z2 ) ) ) ) ).
% graph.reachable_succ
thf(fact_1032_graph_Oreachable__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v,Z2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ Z2 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z2 ) ) ) ) ).
% graph.reachable_succ
thf(fact_1033_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_1034_minus__set__def,axiom,
( minus_4183494784930505774od_v_v
= ( ^ [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( collec140062887454715474od_v_v
@ ( minus_9095120230875558447_v_v_o
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ A3 )
@ ^ [X2: product_prod_v_v] : ( member7453568604450474000od_v_v @ X2 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_1035_minus__set__def,axiom,
( minus_7228012346218142266_set_v
= ( ^ [A3: set_set_v,B4: set_set_v] :
( collect_set_v
@ ( minus_minus_set_v_o
@ ^ [X2: set_v] : ( member_set_v @ X2 @ A3 )
@ ^ [X2: set_v] : ( member_set_v @ X2 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_1036_minus__set__def,axiom,
( minus_minus_set_v
= ( ^ [A3: set_v,B4: set_v] :
( collect_v
@ ( minus_minus_v_o
@ ^ [X2: v] : ( member_v @ X2 @ A3 )
@ ^ [X2: v] : ( member_v @ X2 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_1037_graph_Oinit__env__pre__dfs,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl36166008131615352t_unit @ Successors @ V3 @ ( sCC_Bl7693227186847904995_env_v @ V3 ) ) ) ).
% graph.init_env_pre_dfs
thf(fact_1038_graph_Opre__dfss__pre__dfs,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V3 @ E )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( member_v @ W @ ( Successors @ V3 ) )
=> ( sCC_Bl36166008131615352t_unit @ Successors @ W @ E ) ) ) ) ) ).
% graph.pre_dfss_pre_dfs
thf(fact_1039_graph_OS__reflexive,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N3: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( member_v @ N3 @ ( sCC_Bl1280885523602775798t_unit @ E @ N3 ) ) ) ) ).
% graph.S_reflexive
thf(fact_1040_graph_Oscc__partition,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v,S2: set_Product_prod_v_v,X: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S2 )
=> ( ( member7453568604450474000od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ S @ S2 ) )
=> ( S = S2 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_1041_graph_Oscc__partition,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,S2: set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S2 )
=> ( ( member_v @ X @ ( inf_inf_set_v @ S @ S2 ) )
=> ( S = S2 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_1042_graph_Ois__subscc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
= ( ! [X2: v] :
( ( member_v @ X2 @ S )
=> ! [Y4: v] :
( ( member_v @ Y4 @ S )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ Y4 ) ) ) ) ) ) ).
% graph.is_subscc_def
thf(fact_1043_graph_OsccE,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: 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 @ S )
=> ( ( member7453568604450474000od_v_v @ X @ S )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ X )
=> ( member7453568604450474000od_v_v @ Y @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_1044_graph_OsccE,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ X )
=> ( member_v @ Y @ S ) ) ) ) ) ) ).
% graph.sccE
thf(fact_1045_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,V3: product_prod_v_v,N3: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl7798947040364291444t_unit @ Successors @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V3 ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( ( member7453568604450474000od_v_v @ N3 @ ( set_Product_prod_v_v2 @ ( tl_Product_prod_v_v @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) ) ) ) )
=> ( ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) @ N3 )
= ( sCC_Bl8440648026628373538t_unit @ E @ N3 ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_1046_graph_Ounite__S__tl,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,W: v,V3: v,N3: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ N3 @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ N3 )
= ( sCC_Bl1280885523602775798t_unit @ E @ N3 ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_tl
thf(fact_1047_graph_Osubscc__add,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,S: set_set_v,X: set_v,Y: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl7907073126578335045_set_v @ Successors @ S )
=> ( ( member_set_v @ X @ S )
=> ( ( sCC_Bl7354734129683093653_set_v @ Successors @ X @ Y )
=> ( ( sCC_Bl7354734129683093653_set_v @ Successors @ Y @ X )
=> ( sCC_Bl7907073126578335045_set_v @ Successors @ ( insert_set_v @ Y @ S ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_1048_graph_Osubscc__add,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: 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 @ S )
=> ( ( member7453568604450474000od_v_v @ X @ S )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ X )
=> ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( insert1338601472111419319od_v_v @ Y @ S ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_1049_graph_Osubscc__add,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
=> ( ( member_v @ X @ S )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ X )
=> ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( insert_v @ Y @ S ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_1050_graph_Ostack__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N3: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ N3 @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ).
% graph.stack_unexplored
thf(fact_1051_graph_Ostack__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N3: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( member_v @ N3 @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ).
% graph.stack_visited
thf(fact_1052_graph_Ois__scc__def,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,S: set_set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl1515522642333523865_set_v @ Successors @ S )
= ( ( S != bot_bot_set_set_v )
& ( sCC_Bl7907073126578335045_set_v @ Successors @ S )
& ! [S3: set_set_v] :
( ( ( ord_le5216385588623774835_set_v @ S @ S3 )
& ( sCC_Bl7907073126578335045_set_v @ Successors @ S3 ) )
=> ( S3 = S ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_1053_graph_Ois__scc__def,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S: set_Product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S )
= ( ( S != bot_bo723834152578015283od_v_v )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S )
& ! [S3: set_Product_prod_v_v] :
( ( ( ord_le7336532860387713383od_v_v @ S @ S3 )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S3 ) )
=> ( S3 = S ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_1054_graph_Ois__scc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S )
= ( ( S != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S )
& ! [S3: set_v] :
( ( ( ord_less_eq_set_v @ S @ S3 )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S3 ) )
=> ( S3 = S ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_1055_graph_Oreachable__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V3: v,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V3 @ W )
=> ( ! [X3: v] :
( ( member_v @ X3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa: v] :
( ( member_v @ Xa @ ( minus_minus_set_v @ ( Successors @ X3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X3 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V3 @ X3 )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Xa @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ) ).
% graph.reachable_visited
thf(fact_1056_graph_Opre__dfs__def,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl36166008131615352t_unit @ Successors @ V3 @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
& ~ ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ ( sCC_Bl1090238580953940555t_unit @ E ) @ V3 )
& ( ( sCC_Bl3795065053823578884t_unit @ E @ V3 )
= bot_bot_set_v )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X2 @ V3 ) ) ) ) ) ).
% graph.pre_dfs_def
thf(fact_1057_graph_Ovisited__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,M2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ M2 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ M2 @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ~ ! [N2: v] :
( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ M2 @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ) ) ) ).
% graph.visited_unexplored
thf(fact_1058_graph_Ostack__class,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N3: v,M2: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( ( member_v @ M2 @ ( sCC_Bl1280885523602775798t_unit @ E @ N3 ) )
=> ( member_v @ M2 @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ) ).
% graph.stack_class
thf(fact_1059_graph_Opre__dfs__pre__dfss,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl36166008131615352t_unit @ Successors @ V3 @ E )
=> ( sCC_Bl1748261141445803503t_unit @ Successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) ) ) ).
% graph.pre_dfs_pre_dfss
thf(fact_1060_pre__dfss__unite__pre__dfss,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl1748261141445803503t_unit @ successors @ V3
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X2: v] : ( if_set_v @ ( X2 = V3 ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ V3 ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ X2 ) )
@ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ).
% pre_dfss_unite_pre_dfss
thf(fact_1061_calculation_I18_J,axiom,
( ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uv: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ N4 ) )
& ( member_v @ N4
@ ( set_v2
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uv: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ) ) ) )
= ( minus_minus_set_v
@ ( sCC_Bl4645233313691564917t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) )
@ ( sCC_Bl157864678168468314t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ) ).
% calculation(18)
thf(fact_1062_pre__dfss__def,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
& ( member_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( ( sCC_Bl8828226123343373779t_unit @ E )
!= nil_v )
& ( member_v @ V3 @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( member_v @ X2 @ ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl1280885523602775798t_unit @ E @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
& ! [X2: v] :
( ( member_v @ X2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X2 @ V3 ) )
& ? [Ns: list_v] :
( ( sCC_Bl9201514103433284750t_unit @ E )
= ( cons_v @ V3 @ Ns ) ) ) ) ).
% pre_dfss_def
thf(fact_1063_unite__S__equal,axiom,
! [E: sCC_Bl1394983891496994913t_unit,W: v,V3: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ N4 ) )
& ( member_v @ N4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) )
= ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E @ N4 ) )
& ( member_v @ N4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) ) ) ) ) ) ) ) ).
% unite_S_equal
thf(fact_1064_set__empty,axiom,
! [Xs: list_set_v] :
( ( ( set_set_v2 @ Xs )
= bot_bot_set_set_v )
= ( Xs = nil_set_v ) ) ).
% set_empty
thf(fact_1065_set__empty,axiom,
! [Xs: list_v] :
( ( ( set_v2 @ Xs )
= bot_bot_set_v )
= ( Xs = nil_v ) ) ).
% set_empty
thf(fact_1066_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_1067_set__empty2,axiom,
! [Xs: list_set_v] :
( ( bot_bot_set_set_v
= ( set_set_v2 @ Xs ) )
= ( Xs = nil_set_v ) ) ).
% set_empty2
thf(fact_1068_set__empty2,axiom,
! [Xs: list_v] :
( ( bot_bot_set_v
= ( set_v2 @ Xs ) )
= ( Xs = nil_v ) ) ).
% set_empty2
thf(fact_1069_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_1070_sub__env__def,axiom,
( sCC_Bl5768913643336123637t_unit
= ( ^ [E4: sCC_Bl1394983891496994913t_unit,E5: sCC_Bl1394983891496994913t_unit] :
( ( ( sCC_Bl1090238580953940555t_unit @ E5 )
= ( sCC_Bl1090238580953940555t_unit @ E4 ) )
& ( ord_less_eq_set_v @ ( sCC_Bl4645233313691564917t_unit @ E4 ) @ ( sCC_Bl4645233313691564917t_unit @ E5 ) )
& ( ord_less_eq_set_v @ ( sCC_Bl157864678168468314t_unit @ E4 ) @ ( sCC_Bl157864678168468314t_unit @ E5 ) )
& ! [V4: v] : ( ord_less_eq_set_v @ ( sCC_Bl3795065053823578884t_unit @ E4 @ V4 ) @ ( sCC_Bl3795065053823578884t_unit @ E5 @ V4 ) )
& ! [V4: v] : ( ord_less_eq_set_v @ ( sCC_Bl1280885523602775798t_unit @ E4 @ V4 ) @ ( sCC_Bl1280885523602775798t_unit @ E5 @ V4 ) )
& ( ord_less_eq_set_v
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E4 @ V4 ) )
& ( member_v @ V4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E4 ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E5 @ V4 ) )
& ( member_v @ V4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E5 ) ) ) ) ) ) ) ) ) ) ).
% sub_env_def
thf(fact_1071_hd__Cons__tl,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( ( cons_v @ ( hd_v @ Xs ) @ ( tl_v @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_1072_list_Ocollapse,axiom,
! [List: list_v] :
( ( List != nil_v )
=> ( ( cons_v @ ( hd_v @ List ) @ ( tl_v @ List ) )
= List ) ) ).
% list.collapse
thf(fact_1073_list__nonempty__induct,axiom,
! [Xs: list_v,P: list_v > $o] :
( ( Xs != nil_v )
=> ( ! [X3: v] : ( P @ ( cons_v @ X3 @ nil_v ) )
=> ( ! [X3: v,Xs2: list_v] :
( ( Xs2 != nil_v )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_v @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_1074_list__induct2_H,axiom,
! [P: list_v > list_v > $o,Xs: list_v,Ys: list_v] :
( ( P @ nil_v @ nil_v )
=> ( ! [X3: v,Xs2: list_v] : ( P @ ( cons_v @ X3 @ Xs2 ) @ nil_v )
=> ( ! [Y3: v,Ys2: list_v] : ( P @ nil_v @ ( cons_v @ Y3 @ Ys2 ) )
=> ( ! [X3: v,Xs2: list_v,Y3: v,Ys2: list_v] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_v @ X3 @ Xs2 ) @ ( cons_v @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1075_neq__Nil__conv,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
= ( ? [Y4: v,Ys3: list_v] :
( Xs
= ( cons_v @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_1076_remdups__adj_Ocases,axiom,
! [X: list_v] :
( ( X != nil_v )
=> ( ! [X3: v] :
( X
!= ( cons_v @ X3 @ nil_v ) )
=> ~ ! [X3: v,Y3: v,Xs2: list_v] :
( X
!= ( cons_v @ X3 @ ( cons_v @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_1077_list_Oexhaust,axiom,
! [Y: list_v] :
( ( Y != nil_v )
=> ~ ! [X212: v,X222: list_v] :
( Y
!= ( cons_v @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_1078_list_OdiscI,axiom,
! [List: list_v,X21: v,X22: list_v] :
( ( List
= ( cons_v @ X21 @ X22 ) )
=> ( List != nil_v ) ) ).
% list.discI
thf(fact_1079_list_Odistinct_I1_J,axiom,
! [X21: v,X22: list_v] :
( nil_v
!= ( cons_v @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_1080_list_Osel_I2_J,axiom,
( ( tl_v @ nil_v )
= nil_v ) ).
% list.sel(2)
thf(fact_1081_distinct_Osimps_I1_J,axiom,
distinct_v @ nil_v ).
% distinct.simps(1)
thf(fact_1082_transpose_Ocases,axiom,
! [X: list_list_v] :
( ( X != nil_list_v )
=> ( ! [Xss: list_list_v] :
( X
!= ( cons_list_v @ nil_v @ Xss ) )
=> ~ ! [X3: v,Xs2: list_v,Xss: list_list_v] :
( X
!= ( cons_list_v @ ( cons_v @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_1083_empty__set,axiom,
( bot_bot_set_set_v
= ( set_set_v2 @ nil_set_v ) ) ).
% empty_set
thf(fact_1084_empty__set,axiom,
( bot_bot_set_v
= ( set_v2 @ nil_v ) ) ).
% empty_set
thf(fact_1085_empty__set,axiom,
( bot_bo723834152578015283od_v_v
= ( set_Product_prod_v_v2 @ nil_Product_prod_v_v ) ) ).
% empty_set
thf(fact_1086_tl__Nil,axiom,
! [Xs: list_v] :
( ( ( tl_v @ Xs )
= nil_v )
= ( ( Xs = nil_v )
| ? [X2: v] :
( Xs
= ( cons_v @ X2 @ nil_v ) ) ) ) ).
% tl_Nil
thf(fact_1087_Nil__tl,axiom,
! [Xs: list_v] :
( ( nil_v
= ( tl_v @ Xs ) )
= ( ( Xs = nil_v )
| ? [X2: v] :
( Xs
= ( cons_v @ X2 @ nil_v ) ) ) ) ).
% Nil_tl
thf(fact_1088_distinct__singleton,axiom,
! [X: v] : ( distinct_v @ ( cons_v @ X @ nil_v ) ) ).
% distinct_singleton
thf(fact_1089_list_Oset__sel_I2_J,axiom,
! [A: list_v,X: v] :
( ( A != nil_v )
=> ( ( member_v @ X @ ( set_v2 @ ( tl_v @ A ) ) )
=> ( member_v @ X @ ( set_v2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_1090_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_1091_hd__in__set,axiom,
! [Xs: list_v] :
( ( Xs != nil_v )
=> ( member_v @ ( hd_v @ Xs ) @ ( set_v2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1092_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_1093_list_Oset__sel_I1_J,axiom,
! [A: list_v] :
( ( A != nil_v )
=> ( member_v @ ( hd_v @ A ) @ ( set_v2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1094_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_1095_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_1096_unfold__congs_I5_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: v > set_v,F: ( v > set_v ) > v > set_v,F2: ( v > set_v ) > v > set_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl3795065053823578884t_unit @ R2 )
= V )
=> ( ! [V2: v > set_v] :
( ( V2 = V )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl48393358579903213t_unit @ F @ R )
= ( sCC_Bl48393358579903213t_unit @ F2 @ R2 ) ) ) ) ) ).
% unfold_congs(5)
thf(fact_1097_fold__congs_I5_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V: v > set_v,F: ( v > set_v ) > v > set_v,F2: ( v > set_v ) > v > set_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl3795065053823578884t_unit @ R2 )
= V )
=> ( ! [V2: v > set_v] :
( ( V = V2 )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( sCC_Bl48393358579903213t_unit @ F @ R )
= ( sCC_Bl48393358579903213t_unit @ F2 @ R2 ) ) ) ) ) ).
% fold_congs(5)
thf(fact_1098_list_Oexhaust__sel,axiom,
! [List: list_v] :
( ( List != nil_v )
=> ( List
= ( cons_v @ ( hd_v @ List ) @ ( tl_v @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_1099_graph_Opre__dfss__unite__pre__dfss,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,V3: set_v,E: sCC_Bl337355980704484737t_unit,W: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl7011918439528173327t_unit @ Successors @ V3 @ E )
=> ( ( member_set_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member_set_v @ W @ ( sCC_Bl1947314421851746340t_unit @ E @ V3 ) )
=> ( ( member_set_v @ W @ ( sCC_Bl2616308362330240149t_unit @ E ) )
=> ( ~ ( member_set_v @ W @ ( sCC_Bl8056216847179350138t_unit @ E ) )
=> ( sCC_Bl7011918439528173327t_unit @ Successors @ V3
@ ( sCC_Bl8982985682160534541t_unit
@ ^ [Uu: set_v > set_set_v,X2: set_v] : ( if_set_set_v @ ( X2 = V3 ) @ ( sup_sup_set_set_v @ ( sCC_Bl1947314421851746340t_unit @ ( sCC_Bl2303001396873455969_set_v @ V3 @ W @ E ) @ V3 ) @ ( insert_set_v @ W @ bot_bot_set_set_v ) ) @ ( sCC_Bl1947314421851746340t_unit @ ( sCC_Bl2303001396873455969_set_v @ V3 @ W @ E ) @ X2 ) )
@ ( sCC_Bl2303001396873455969_set_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ) ).
% graph.pre_dfss_unite_pre_dfss
thf(fact_1100_graph_Opre__dfss__unite__pre__dfss,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V3: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V3 @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V3 ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl3607325323686918683t_unit @ Successors @ V3
@ ( sCC_Bl2958793191457503513t_unit
@ ^ [Uu: product_prod_v_v > set_Product_prod_v_v,X2: product_prod_v_v] : ( if_set4279007504652509325od_v_v @ ( X2 = V3 ) @ ( sup_su414716646722978715od_v_v @ ( sCC_Bl3878977043676959280t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) @ V3 ) @ ( insert1338601472111419319od_v_v @ W @ bot_bo723834152578015283od_v_v ) ) @ ( sCC_Bl3878977043676959280t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) @ X2 ) )
@ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ) ).
% graph.pre_dfss_unite_pre_dfss
thf(fact_1101_graph_Opre__dfss__unite__pre__dfss,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V3 @ E )
=> ( ( member_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl1748261141445803503t_unit @ Successors @ V3
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X2: v] : ( if_set_v @ ( X2 = V3 ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ V3 ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ X2 ) )
@ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ) ).
% graph.pre_dfss_unite_pre_dfss
thf(fact_1102_graph_Ounite__wf__env,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V3: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V3 @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V3 ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl7798947040364291444t_unit @ Successors @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_wf_env
thf(fact_1103_graph_Ounite__wf__env,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V3 @ E )
=> ( ( member_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl9196236973127232072t_unit @ Successors @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_wf_env
thf(fact_1104_graph_Ounite__S__equal,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,V3: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl7798947040364291444t_unit @ Successors @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V3 ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( ( comple5788137035815166516od_v_v
@ ( collec8263177866097347122od_v_v
@ ^ [Uu: set_Product_prod_v_v] :
? [N4: product_prod_v_v] :
( ( Uu
= ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) @ N4 ) )
& ( member7453568604450474000od_v_v @ N4 @ ( set_Product_prod_v_v2 @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) ) ) ) ) ) )
= ( comple5788137035815166516od_v_v
@ ( collec8263177866097347122od_v_v
@ ^ [Uu: set_Product_prod_v_v] :
? [N4: product_prod_v_v] :
( ( Uu
= ( sCC_Bl8440648026628373538t_unit @ E @ N4 ) )
& ( member7453568604450474000od_v_v @ N4 @ ( set_Product_prod_v_v2 @ ( sCC_Bl2021302119412358655t_unit @ E ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_equal
thf(fact_1105_graph_Ounite__S__equal,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,W: v,V3: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) @ N4 ) )
& ( member_v @ N4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) )
= ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E @ N4 ) )
& ( member_v @ N4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_S_equal
thf(fact_1106_graph_Ounite__sub__env,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V3: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V3 @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V3 ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl7963838319573962697t_unit @ E @ ( sCC_Bl4702006153222411093od_v_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_1107_graph_Ounite__sub__env,axiom,
! [Vertices: set_v,Successors: v > set_v,V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V3 @ E )
=> ( ( member_v @ W @ ( Successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5768913643336123637t_unit @ E @ ( sCC_Bloemen_unite_v @ V3 @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_1108_graph_Osub__env__def,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,E2: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5768913643336123637t_unit @ E @ E2 )
= ( ( ( sCC_Bl1090238580953940555t_unit @ E2 )
= ( sCC_Bl1090238580953940555t_unit @ E ) )
& ( ord_less_eq_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl4645233313691564917t_unit @ E2 ) )
& ( ord_less_eq_set_v @ ( sCC_Bl157864678168468314t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E2 ) )
& ! [V4: v] : ( ord_less_eq_set_v @ ( sCC_Bl3795065053823578884t_unit @ E @ V4 ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ V4 ) )
& ! [V4: v] : ( ord_less_eq_set_v @ ( sCC_Bl1280885523602775798t_unit @ E @ V4 ) @ ( sCC_Bl1280885523602775798t_unit @ E2 @ V4 ) )
& ( ord_less_eq_set_v
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E @ V4 ) )
& ( member_v @ V4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [V4: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E2 @ V4 ) )
& ( member_v @ V4 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E2 ) ) ) ) ) ) ) ) ) ) ).
% graph.sub_env_def
thf(fact_1109_pre__dfss__post__dfs__pre__dfss,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V3 @ E )
=> ( ( member_v @ W @ ( successors @ V3 ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ W @ E @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) )
=> ( sCC_Bl1748261141445803503t_unit @ successors @ V3
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X2: v] : ( if_set_v @ ( X2 = V3 ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) @ V3 ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) @ X2 ) )
@ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) ) ) ) ) ) ) ).
% pre_dfss_post_dfs_pre_dfss
thf(fact_1110_reachable__re,axiom,
! [X: v,Y: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y ) ) ).
% reachable_re
thf(fact_1111_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_v @ A22 @ ( successors @ Y3 ) ) ) ) ) ).
% reachable_end.cases
thf(fact_1112_re__refl,axiom,
! [X: v] : ( sCC_Bl770211535891879572_end_v @ successors @ X @ X ) ).
% re_refl
thf(fact_1113_re__succ,axiom,
! [X: v,Y: v,Z2: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y )
=> ( ( member_v @ Z2 @ ( successors @ Y ) )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Z2 ) ) ) ).
% re_succ
thf(fact_1114_reachable__end_Osimps,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A22 )
= ( ? [X2: v] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: v,Y4: v,Z5: v] :
( ( A1 = X2 )
& ( A22 = Z5 )
& ( sCC_Bl770211535891879572_end_v @ successors @ X2 @ Y4 )
& ( member_v @ Z5 @ ( successors @ Y4 ) ) ) ) ) ).
% reachable_end.simps
thf(fact_1115_succ__re,axiom,
! [Y: v,X: v,Z2: v] :
( ( member_v @ Y @ ( successors @ X ) )
=> ( ( sCC_Bl770211535891879572_end_v @ successors @ Y @ Z2 )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Z2 ) ) ) ).
% succ_re
thf(fact_1116_re__reachable,axiom,
! [X: v,Y: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% re_reachable
thf(fact_1117_dfs_Opsimps,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V3 @ E ) ) )
=> ( ( sCC_Bloemen_dfs_v @ successors @ V3 @ E )
= ( if_SCC4926449794303880475t_unit
@ ( V3
= ( hd_v
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) ) ) )
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] :
( sup_sup_set_v
@ ( sCC_Bl157864678168468314t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) )
@ V3 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] :
( sup_sup_set_set_v
@ ( sCC_Bl2536197123907397897t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) )
@ ( insert_set_v
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) )
@ V3 )
@ bot_bot_set_set_v ) )
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) ) ) ) )
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) ) )
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V3 @ bot_bot_set_v ) )
@ E ) ) ) ) ) ) ) ) ).
% dfs.psimps
thf(fact_1118_dfss_Ocases,axiom,
! [X: produc5741669702376414499t_unit] :
~ ! [V2: v,E6: sCC_Bl1394983891496994913t_unit] :
( X
!= ( produc3862955338007567901t_unit @ V2 @ E6 ) ) ).
% dfss.cases
thf(fact_1119__C2_C,axiom,
accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ v2 @ e ) ) ).
% "2"
thf(fact_1120_dfs__dfss_Odomintros_I1_J,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors )
@ ( sum_In5289330923152326972t_unit
@ ( produc3862955338007567901t_unit @ V3
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V3 @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( insert_v @ V3 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ E ) ) ) ) ) )
=> ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V3 @ E ) ) ) ) ).
% dfs_dfss.domintros(1)
thf(fact_1121_dfss_Opsimps,axiom,
! [V3: v,E: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In5289330923152326972t_unit @ ( produc3862955338007567901t_unit @ V3 @ E ) ) )
=> ( ( sCC_Bloemen_dfss_v @ successors @ V3 @ E )
= ( if_SCC4926449794303880475t_unit
@ ( ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) )
= bot_bot_set_v )
@ E
@ ( sCC_Bloemen_dfss_v @ successors @ V3
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X2: v] :
( if_set_v @ ( X2 = V3 )
@ ( sup_sup_set_v
@ ( sCC_Bl3795065053823578884t_unit
@ ( if_SCC4926449794303880475t_unit
@ ( member_v
@ ( fChoice_v
@ ^ [Y4: v] : ( member_v @ Y4 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E ) )
@ E
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v
@ ( fChoice_v
@ ^ [Y4: v] : ( member_v @ Y4 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [Y4: v] : ( member_v @ Y4 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ E )
@ ( sCC_Bloemen_unite_v @ V3
@ ( fChoice_v
@ ^ [Y4: v] : ( member_v @ Y4 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ E ) ) )
@ V3 )
@ ( insert_v
@ ( fChoice_v
@ ^ [Y4: v] : ( member_v @ Y4 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ bot_bot_set_v ) )
@ ( sCC_Bl3795065053823578884t_unit
@ ( if_SCC4926449794303880475t_unit
@ ( member_v
@ ( fChoice_v
@ ^ [Y4: v] : ( member_v @ Y4 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E ) )
@ E
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v
@ ( fChoice_v
@ ^ [Y4: v] : ( member_v @ Y4 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [Y4: v] : ( member_v @ Y4 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ E )
@ ( sCC_Bloemen_unite_v @ V3
@ ( fChoice_v
@ ^ [Y4: v] : ( member_v @ Y4 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ E ) ) )
@ X2 ) )
@ ( if_SCC4926449794303880475t_unit
@ ( member_v
@ ( fChoice_v
@ ^ [X2: v] : ( member_v @ X2 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E ) )
@ E
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v
@ ( fChoice_v
@ ^ [X2: v] : ( member_v @ X2 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [X2: v] : ( member_v @ X2 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ E )
@ ( sCC_Bloemen_unite_v @ V3
@ ( fChoice_v
@ ^ [X2: v] : ( member_v @ X2 @ ( minus_minus_set_v @ ( successors @ V3 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V3 ) ) ) )
@ E ) ) ) ) ) ) ) ) ).
% dfss.psimps
thf(fact_1122_ra__mono,axiom,
! [X: v,Y: v,E7: set_Product_prod_v_v,E8: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E7 )
=> ( ( ord_le7336532860387713383od_v_v @ E8 @ E7 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E8 ) ) ) ).
% ra_mono
thf(fact_1123_ra__refl,axiom,
! [X: v,E7: set_Product_prod_v_v] : ( sCC_Bl4291963740693775144ding_v @ successors @ X @ X @ E7 ) ).
% ra_refl
thf(fact_1124_ra__trans,axiom,
! [X: v,Y: v,E7: set_Product_prod_v_v,Z2: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E7 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ Y @ Z2 @ E7 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z2 @ E7 ) ) ) ).
% ra_trans
thf(fact_1125_ra__reachable,axiom,
! [X: v,Y: v,E7: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E7 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% ra_reachable
thf(fact_1126_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_v @ A22 @ ( successors @ Y3 ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ A22 ) @ A32 ) ) ) ) ) ).
% reachable_avoiding.cases
thf(fact_1127_ra__succ,axiom,
! [X: v,Y: v,E7: set_Product_prod_v_v,Z2: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E7 )
=> ( ( member_v @ Z2 @ ( successors @ Y ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Z2 ) @ E7 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z2 @ E7 ) ) ) ) ).
% ra_succ
thf(fact_1128_reachable__avoiding_Osimps,axiom,
! [A1: v,A22: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A22 @ A32 )
= ( ? [X2: v,E9: set_Product_prod_v_v] :
( ( A1 = X2 )
& ( A22 = X2 )
& ( A32 = E9 ) )
| ? [X2: v,Y4: v,E9: set_Product_prod_v_v,Z5: v] :
( ( A1 = X2 )
& ( A22 = Z5 )
& ( A32 = E9 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ X2 @ Y4 @ E9 )
& ( member_v @ Z5 @ ( successors @ Y4 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y4 @ Z5 ) @ E9 ) ) ) ) ).
% reachable_avoiding.simps
thf(fact_1129_edge__ra,axiom,
! [Y: v,X: v,E7: set_Product_prod_v_v] :
( ( member_v @ Y @ ( successors @ X ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ E7 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E7 ) ) ) ).
% edge_ra
thf(fact_1130_ra__cases,axiom,
! [X: v,Y: v,E7: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E7 )
=> ( ( X = Y )
| ? [Z4: v] :
( ( member_v @ Z4 @ ( successors @ X ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Z4 ) @ E7 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ Z4 @ Y @ E7 ) ) ) ) ).
% ra_cases
thf(fact_1131_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_1132_ra__add__edge,axiom,
! [X: v,Y: v,E7: set_Product_prod_v_v,V3: v,W: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E7 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ ( sup_su414716646722978715od_v_v @ E7 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ V3 @ ( sup_su414716646722978715od_v_v @ E7 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ successors @ W @ Y @ ( sup_su414716646722978715od_v_v @ E7 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V3 @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ).
% ra_add_edge
thf(fact_1133_unite__def,axiom,
( sCC_Bloemen_unite_v
= ( ^ [V4: v,W2: v,E4: sCC_Bl1394983891496994913t_unit] :
( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] :
( dropWhile_v
@ ^ [X2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E4 @ X2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E4 ) )
@ ( sCC_Bl3155122997657187039t_unit
@ ^ [Uu: v > set_v,X2: v] :
( if_set_v
@ ( member_v @ X2
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uv: set_v] :
? [Y4: v] :
( ( Uv
= ( sCC_Bl1280885523602775798t_unit @ E4 @ Y4 ) )
& ( member_v @ Y4
@ ( sup_sup_set_v
@ ( set_v2
@ ( takeWhile_v
@ ^ [Z5: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E4 @ Z5 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E4 ) ) )
@ ( insert_v
@ ( hd_v
@ ( dropWhile_v
@ ^ [Z5: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E4 @ Z5 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E4 ) ) )
@ bot_bot_set_v ) ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uv: set_v] :
? [Y4: v] :
( ( Uv
= ( sCC_Bl1280885523602775798t_unit @ E4 @ Y4 ) )
& ( member_v @ Y4
@ ( sup_sup_set_v
@ ( set_v2
@ ( takeWhile_v
@ ^ [Z5: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E4 @ Z5 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E4 ) ) )
@ ( insert_v
@ ( hd_v
@ ( dropWhile_v
@ ^ [Z5: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E4 @ Z5 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E4 ) ) )
@ bot_bot_set_v ) ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit @ E4 @ X2 ) )
@ E4 ) ) ) ) ).
% unite_def
thf(fact_1134_vfin,axiom,
finite_finite_v @ vertices ).
% vfin
% Helper facts (10)
thf(help_fChoice_1_1_fChoice_001tf__v_T,axiom,
! [P: v > $o] :
( ( P @ ( fChoice_v @ P ) )
= ( ? [X7: v] : ( P @ 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_2_1_If_001t__Set__Oset_It__Set__Oset_Itf__v_J_J_T,axiom,
! [X: set_set_v,Y: set_set_v] :
( ( if_set_set_v @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Set__Oset_Itf__v_J_J_T,axiom,
! [X: set_set_v,Y: set_set_v] :
( ( if_set_set_v @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( if_set4279007504652509325od_v_v @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Oprod_Itf__v_Mtf__v_J_J_T,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( if_set4279007504652509325od_v_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,
! [P: $o] :
( ( P = $true )
| ( P = $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,
! [X3: v] :
( ( member_v @ X3
@ ( set_v2
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) )
=> ! [Xa: v] :
( ( member_v @ Xa
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) )
@ X3 ) )
=> ( ( member_v @ Xa
@ ( set_v2
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) )
=> ( sCC_Bl4022239298816431255edes_v @ Xa @ X3
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl9201514103433284750t_unit @ e2 ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ e2 ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl157864678168468314t_unit @ e2 ) @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] : ( sup_sup_set_set_v @ ( sCC_Bl2536197123907397897t_unit @ e2 ) @ ( insert_set_v @ ( sCC_Bl1280885523602775798t_unit @ e2 @ v2 ) @ bot_bot_set_set_v ) )
@ e2 ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------