TPTP Problem File: SLH0859^1.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_02750_094508__6460312_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1468 ( 648 unt; 186 typ; 0 def)
% Number of atoms : 3555 (1386 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 11998 ( 446 ~; 55 |; 261 &;9701 @)
% ( 0 <=>;1535 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 6 avg)
% Number of types : 21 ( 20 usr)
% Number of type conns : 537 ( 537 >; 0 *; 0 +; 0 <<)
% Number of symbols : 169 ( 166 usr; 16 con; 0-9 aty)
% Number of variables : 3635 ( 240 ^;3300 !; 95 ?;3635 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:54:16.948
%------------------------------------------------------------------------------
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thf(sy_c_member_001t__Set__Oset_Itf__v_J,type,
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thf(sy_v_ea____,type,
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thf(sy_v_successors,type,
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% Relevant facts (1275)
thf(fact_0_predfss,axiom,
sCC_Bl1748261141445803503t_unit @ successors @ va @ ea ).
% predfss
thf(fact_1_dfs__dfss__rel_Ocong,axiom,
sCC_Bl907557413677168252_rel_v = sCC_Bl907557413677168252_rel_v ).
% dfs_dfss_rel.cong
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_local_Owf,axiom,
sCC_Bl9196236973127232072t_unit @ successors @ ea ).
% local.wf
thf(fact_4_pre__dfss__pre__dfs,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( sCC_Bl36166008131615352t_unit @ successors @ W @ E ) ) ) ) ).
% pre_dfss_pre_dfs
thf(fact_5_fold__congs_I4_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V2: set_v,F: set_v > set_v,F2: set_v > set_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl4645233313691564917t_unit @ R2 )
= V2 )
=> ( ! [V3: set_v] :
( ( V2 = V3 )
=> ( ( F @ V3 )
= ( F2 @ V3 ) ) )
=> ( ( sCC_Bl7870604408699998558t_unit @ F @ R )
= ( sCC_Bl7870604408699998558t_unit @ F2 @ R2 ) ) ) ) ) ).
% fold_congs(4)
thf(fact_6_unfold__congs_I4_J,axiom,
! [R: sCC_Bl1394983891496994913t_unit,R2: sCC_Bl1394983891496994913t_unit,V2: set_v,F: set_v > set_v,F2: set_v > set_v] :
( ( R = R2 )
=> ( ( ( sCC_Bl4645233313691564917t_unit @ R2 )
= V2 )
=> ( ! [V3: set_v] :
( ( V3 = V2 )
=> ( ( F @ V3 )
= ( F2 @ V3 ) ) )
=> ( ( sCC_Bl7870604408699998558t_unit @ F @ R )
= ( sCC_Bl7870604408699998558t_unit @ F2 @ R2 ) ) ) ) ) ).
% unfold_congs(4)
thf(fact_7_graph_Opre__dfss_Ocong,axiom,
sCC_Bl1748261141445803503t_unit = sCC_Bl1748261141445803503t_unit ).
% graph.pre_dfss.cong
thf(fact_8_select__convs_I4_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl4645233313691564917t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Visited ) ).
% select_convs(4)
thf(fact_9_dom,axiom,
accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In5289330923152326972t_unit @ ( produc3862955338007567901t_unit @ va @ ea ) ) ).
% dom
thf(fact_10_graph_Opost__dfss_Ocong,axiom,
sCC_Bl6082031138996704384t_unit = sCC_Bl6082031138996704384t_unit ).
% graph.post_dfss.cong
thf(fact_11_dfss_Ocases,axiom,
! [X: produc5741669702376414499t_unit] :
~ ! [V3: v,E4: sCC_Bl1394983891496994913t_unit] :
( X
!= ( produc3862955338007567901t_unit @ V3 @ E4 ) ) ).
% dfss.cases
thf(fact_12_graph_Opre__dfss__pre__dfs,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( sCC_Bl36166008131615352t_unit @ Successors @ W @ E ) ) ) ) ) ).
% graph.pre_dfss_pre_dfs
thf(fact_13_graph__axioms,axiom,
sCC_Bloemen_graph_v @ vertices @ successors ).
% graph_axioms
thf(fact_14_init__env__pre__dfs,axiom,
! [V: v] : ( sCC_Bl36166008131615352t_unit @ successors @ V @ ( sCC_Bl7693227186847904995_env_v @ V ) ) ).
% init_env_pre_dfs
thf(fact_15_ext__inject,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit,Root2: v,S2: v > set_v,Explored2: set_v,Visited2: set_v,Vsuccs2: v > set_v,Sccs2: set_set_v,Stack2: list_v,Cstack2: list_v,More2: product_unit] :
( ( ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More )
= ( sCC_Bl8064756265740546429t_unit @ Root2 @ S2 @ Explored2 @ Visited2 @ Vsuccs2 @ Sccs2 @ Stack2 @ Cstack2 @ More2 ) )
= ( ( Root = Root2 )
& ( S = S2 )
& ( Explored = Explored2 )
& ( Visited = Visited2 )
& ( Vsuccs = Vsuccs2 )
& ( Sccs = Sccs2 )
& ( Stack = Stack2 )
& ( Cstack = Cstack2 )
& ( More = More2 ) ) ) ).
% ext_inject
thf(fact_16_update__convs_I4_J,axiom,
! [Visited2: set_v > set_v,Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl7870604408699998558t_unit @ Visited2 @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ ( Visited2 @ Visited ) @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) ) ).
% update_convs(4)
thf(fact_17_cases__scheme,axiom,
! [R: sCC_Bl1394983891496994913t_unit] :
~ ! [Root3: v,S3: v > set_v,Explored3: set_v,Visited3: set_v,Vsuccs3: v > set_v,Sccs3: set_set_v,Stack3: list_v,Cstack3: list_v,More3: product_unit] :
( R
!= ( sCC_Bl8064756265740546429t_unit @ Root3 @ S3 @ Explored3 @ Visited3 @ Vsuccs3 @ Sccs3 @ Stack3 @ Cstack3 @ More3 ) ) ).
% cases_scheme
thf(fact_18_graph_Odfss_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,X: produc5741669702376414499t_unit] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ~ ! [V3: v,E4: sCC_Bl1394983891496994913t_unit] :
( X
!= ( produc3862955338007567901t_unit @ V3 @ E4 ) ) ) ).
% graph.dfss.cases
thf(fact_19_graph_Owf__env_Ocong,axiom,
sCC_Bl9196236973127232072t_unit = sCC_Bl9196236973127232072t_unit ).
% graph.wf_env.cong
thf(fact_20_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_21_graph_Opre__dfs_Ocong,axiom,
sCC_Bl36166008131615352t_unit = sCC_Bl36166008131615352t_unit ).
% graph.pre_dfs.cong
thf(fact_22_sum_Oinject_I2_J,axiom,
! [X2: produc5741669702376414499t_unit,Y2: produc5741669702376414499t_unit] :
( ( ( sum_In5289330923152326972t_unit @ X2 )
= ( sum_In5289330923152326972t_unit @ Y2 ) )
= ( X2 = Y2 ) ) ).
% sum.inject(2)
thf(fact_23_old_Osum_Oinject_I2_J,axiom,
! [B: produc5741669702376414499t_unit,B2: produc5741669702376414499t_unit] :
( ( ( sum_In5289330923152326972t_unit @ B )
= ( sum_In5289330923152326972t_unit @ B2 ) )
= ( B = B2 ) ) ).
% old.sum.inject(2)
thf(fact_24_S__reflexive,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ).
% S_reflexive
thf(fact_25_prod_Oinject,axiom,
! [X1: v,X2: sCC_Bl1394983891496994913t_unit,Y1: v,Y2: sCC_Bl1394983891496994913t_unit] :
( ( ( produc3862955338007567901t_unit @ X1 @ X2 )
= ( produc3862955338007567901t_unit @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_26_prod_Oinject,axiom,
! [X1: v,X2: v,Y1: v,Y2: v] :
( ( ( product_Pair_v_v @ X1 @ X2 )
= ( product_Pair_v_v @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_27_old_Oprod_Oinject,axiom,
! [A: v,B: sCC_Bl1394983891496994913t_unit,A2: v,B2: sCC_Bl1394983891496994913t_unit] :
( ( ( produc3862955338007567901t_unit @ A @ B )
= ( produc3862955338007567901t_unit @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_28_old_Oprod_Oinject,axiom,
! [A: v,B: v,A2: v,B2: v] :
( ( ( product_Pair_v_v @ A @ B )
= ( product_Pair_v_v @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_29_succ__re,axiom,
! [Y: v,X: v,Z: v] :
( ( member_v @ Y @ ( successors @ X ) )
=> ( ( sCC_Bl770211535891879572_end_v @ successors @ Y @ Z )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Z ) ) ) ).
% succ_re
thf(fact_30_reachable__end_Osimps,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A22 )
= ( ? [X3: v] :
( ( A1 = X3 )
& ( A22 = X3 ) )
| ? [X3: v,Y3: v,Z2: v] :
( ( A1 = X3 )
& ( A22 = Z2 )
& ( sCC_Bl770211535891879572_end_v @ successors @ X3 @ Y3 )
& ( member_v @ Z2 @ ( successors @ Y3 ) ) ) ) ) ).
% reachable_end.simps
thf(fact_31_re__succ,axiom,
! [X: v,Y: v,Z: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y )
=> ( ( member_v @ Z @ ( successors @ Y ) )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Z ) ) ) ).
% re_succ
thf(fact_32_re__refl,axiom,
! [X: v] : ( sCC_Bl770211535891879572_end_v @ successors @ X @ X ) ).
% re_refl
thf(fact_33_reachable__end_Ocases,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y4: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ A1 @ Y4 )
=> ~ ( member_v @ A22 @ ( successors @ Y4 ) ) ) ) ) ).
% reachable_end.cases
thf(fact_34_graph_Oinit__env__pre__dfs,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl36166008131615352t_unit @ Successors @ V @ ( sCC_Bl7693227186847904995_env_v @ V ) ) ) ).
% graph.init_env_pre_dfs
thf(fact_35_graph_Oreachable__end_Ocong,axiom,
sCC_Bl770211535891879572_end_v = sCC_Bl770211535891879572_end_v ).
% graph.reachable_end.cong
thf(fact_36_select__convs_I2_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= S ) ).
% select_convs(2)
thf(fact_37_graph_Ore__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Y )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_38_graph_Ore__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y )
=> ( ( member_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Z ) ) ) ) ).
% graph.re_succ
thf(fact_39_graph_Ore__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ X ) ) ).
% graph.re_refl
thf(fact_40_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A22 )
= ( ? [X3: product_prod_v_v] :
( ( A1 = X3 )
& ( A22 = X3 ) )
| ? [X3: product_prod_v_v,Y3: product_prod_v_v,Z2: product_prod_v_v] :
( ( A1 = X3 )
& ( A22 = Z2 )
& ( sCC_Bl4714988717384592488od_v_v @ Successors @ X3 @ Y3 )
& ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_41_graph_Oreachable__end_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A22 )
= ( ? [X3: v] :
( ( A1 = X3 )
& ( A22 = X3 ) )
| ? [X3: v,Y3: v,Z2: v] :
( ( A1 = X3 )
& ( A22 = Z2 )
& ( sCC_Bl770211535891879572_end_v @ Successors @ X3 @ Y3 )
& ( member_v @ Z2 @ ( Successors @ Y3 ) ) ) ) ) ) ).
% graph.reachable_end.simps
thf(fact_42_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,A1: product_prod_v_v,A22: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y4: product_prod_v_v] :
( ( sCC_Bl4714988717384592488od_v_v @ Successors @ A1 @ Y4 )
=> ~ ( member7453568604450474000od_v_v @ A22 @ ( Successors @ Y4 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_43_graph_Oreachable__end_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y4: v] :
( ( sCC_Bl770211535891879572_end_v @ Successors @ A1 @ Y4 )
=> ~ ( member_v @ A22 @ ( Successors @ Y4 ) ) ) ) ) ) ).
% graph.reachable_end.cases
thf(fact_44_mem__Collect__eq,axiom,
! [A: v,P: v > $o] :
( ( member_v @ A @ ( collect_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_45_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_46_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_47_Collect__mem__eq,axiom,
! [A3: set_v] :
( ( collect_v
@ ^ [X3: v] : ( member_v @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A3: set_Product_prod_v_v] :
( ( collec140062887454715474od_v_v
@ ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A3: set_set_v] :
( ( collect_set_v
@ ^ [X3: set_v] : ( member_set_v @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_50_Collect__cong,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ! [X4: set_v] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_set_v @ P )
= ( collect_set_v @ Q ) ) ) ).
% Collect_cong
thf(fact_51_graph_Osucc__re,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y: product_prod_v_v,X: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl4714988717384592488od_v_v @ Successors @ Y @ Z )
=> ( sCC_Bl4714988717384592488od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_52_graph_Osucc__re,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ Y @ Z )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_re
thf(fact_53_graph_OS__reflexive,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( member_v @ N @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ).
% graph.S_reflexive
thf(fact_54_Pair__inject,axiom,
! [A: v,B: sCC_Bl1394983891496994913t_unit,A2: v,B2: sCC_Bl1394983891496994913t_unit] :
( ( ( produc3862955338007567901t_unit @ A @ B )
= ( produc3862955338007567901t_unit @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_55_Pair__inject,axiom,
! [A: v,B: v,A2: v,B2: v] :
( ( ( product_Pair_v_v @ A @ B )
= ( product_Pair_v_v @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_56_prod__cases,axiom,
! [P: produc5741669702376414499t_unit > $o,P2: produc5741669702376414499t_unit] :
( ! [A4: v,B3: sCC_Bl1394983891496994913t_unit] : ( P @ ( produc3862955338007567901t_unit @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_57_prod__cases,axiom,
! [P: product_prod_v_v > $o,P2: product_prod_v_v] :
( ! [A4: v,B3: v] : ( P @ ( product_Pair_v_v @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_58_surj__pair,axiom,
! [P2: produc5741669702376414499t_unit] :
? [X4: v,Y4: sCC_Bl1394983891496994913t_unit] :
( P2
= ( produc3862955338007567901t_unit @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_59_surj__pair,axiom,
! [P2: product_prod_v_v] :
? [X4: v,Y4: v] :
( P2
= ( product_Pair_v_v @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_60_old_Oprod_Oexhaust,axiom,
! [Y: produc5741669702376414499t_unit] :
~ ! [A4: v,B3: sCC_Bl1394983891496994913t_unit] :
( Y
!= ( produc3862955338007567901t_unit @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_61_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_v_v] :
~ ! [A4: v,B3: v] :
( Y
!= ( product_Pair_v_v @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_62_Inr__inject,axiom,
! [X: produc5741669702376414499t_unit,Y: produc5741669702376414499t_unit] :
( ( ( sum_In5289330923152326972t_unit @ X )
= ( sum_In5289330923152326972t_unit @ Y ) )
=> ( X = Y ) ) ).
% Inr_inject
thf(fact_63_sclosed,axiom,
! [X5: v] :
( ( member_v @ X5 @ vertices )
=> ( ord_less_eq_set_v @ ( successors @ X5 ) @ vertices ) ) ).
% sclosed
thf(fact_64_vfin,axiom,
finite_finite_v @ vertices ).
% vfin
thf(fact_65_reachable__re,axiom,
! [X: v,Y: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y ) ) ).
% reachable_re
thf(fact_66_re__reachable,axiom,
! [X: v,Y: v] :
( ( sCC_Bl770211535891879572_end_v @ successors @ X @ Y )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% re_reachable
thf(fact_67_ra__cases,axiom,
! [X: v,Y: v,E5: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E5 )
=> ( ( X = Y )
| ? [Z3: v] :
( ( member_v @ Z3 @ ( successors @ X ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Z3 ) @ E5 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ Z3 @ Y @ E5 ) ) ) ) ).
% ra_cases
thf(fact_68_edge__ra,axiom,
! [Y: v,X: v,E5: set_Product_prod_v_v] :
( ( member_v @ Y @ ( successors @ X ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ E5 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E5 ) ) ) ).
% edge_ra
thf(fact_69_reachable__avoiding_Osimps,axiom,
! [A1: v,A22: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A22 @ A32 )
= ( ? [X3: v,E6: set_Product_prod_v_v] :
( ( A1 = X3 )
& ( A22 = X3 )
& ( A32 = E6 ) )
| ? [X3: v,Y3: v,E6: set_Product_prod_v_v,Z2: v] :
( ( A1 = X3 )
& ( A22 = Z2 )
& ( A32 = E6 )
& ( sCC_Bl4291963740693775144ding_v @ successors @ X3 @ Y3 @ E6 )
& ( member_v @ Z2 @ ( successors @ Y3 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z2 ) @ E6 ) ) ) ) ).
% reachable_avoiding.simps
thf(fact_70_ra__succ,axiom,
! [X: v,Y: v,E5: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E5 )
=> ( ( member_v @ Z @ ( successors @ Y ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Z ) @ E5 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z @ E5 ) ) ) ) ).
% ra_succ
thf(fact_71_reachable_Ocases,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A22 )
=> ( ( A22 != A1 )
=> ~ ! [Y4: v] :
( ( member_v @ Y4 @ ( successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Y4 @ A22 ) ) ) ) ).
% reachable.cases
thf(fact_72_reachable__refl,axiom,
! [X: v] : ( sCC_Bl649662514949026229able_v @ successors @ X @ X ) ).
% reachable_refl
thf(fact_73_reachable__succ,axiom,
! [Y: v,X: v,Z: v] :
( ( member_v @ Y @ ( successors @ X ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% reachable_succ
thf(fact_74_reachable_Osimps,axiom,
! [A1: v,A22: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ A1 @ A22 )
= ( ? [X3: v] :
( ( A1 = X3 )
& ( A22 = X3 ) )
| ? [X3: v,Y3: v,Z2: v] :
( ( A1 = X3 )
& ( A22 = Z2 )
& ( member_v @ Y3 @ ( successors @ X3 ) )
& ( sCC_Bl649662514949026229able_v @ successors @ Y3 @ Z2 ) ) ) ) ).
% reachable.simps
thf(fact_75_reachable__edge,axiom,
! [Y: v,X: v] :
( ( member_v @ Y @ ( successors @ X ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% reachable_edge
thf(fact_76_reachable__end__induct,axiom,
! [X: v,Y: v,P: v > v > $o] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ! [X4: v] : ( P @ X4 @ X4 )
=> ( ! [X4: v,Y4: v,Z3: v] :
( ( P @ X4 @ Y4 )
=> ( ( member_v @ Z3 @ ( successors @ Y4 ) )
=> ( P @ X4 @ Z3 ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% reachable_end_induct
thf(fact_77_reachable__trans,axiom,
! [X: v,Y: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% reachable_trans
thf(fact_78_succ__reachable,axiom,
! [X: v,Y: v,Z: v] :
( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( member_v @ Z @ ( successors @ Y ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Z ) ) ) ).
% succ_reachable
thf(fact_79_ra__refl,axiom,
! [X: v,E5: set_Product_prod_v_v] : ( sCC_Bl4291963740693775144ding_v @ successors @ X @ X @ E5 ) ).
% ra_refl
thf(fact_80_ra__trans,axiom,
! [X: v,Y: v,E5: set_Product_prod_v_v,Z: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E5 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ Y @ Z @ E5 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Z @ E5 ) ) ) ).
% ra_trans
thf(fact_81_ra__reachable,axiom,
! [X: v,Y: v,E5: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E5 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X @ Y ) ) ).
% ra_reachable
thf(fact_82_reachable__avoiding_Ocases,axiom,
! [A1: v,A22: v,A32: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ A22 @ A32 )
=> ( ( A22 != A1 )
=> ~ ! [Y4: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ A1 @ Y4 @ A32 )
=> ( ( member_v @ A22 @ ( successors @ Y4 ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y4 @ A22 ) @ A32 ) ) ) ) ) ).
% reachable_avoiding.cases
thf(fact_83_sccE,axiom,
! [S4: set_v,X: v,Y: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S4 )
=> ( ( member_v @ X @ S4 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ X )
=> ( member_v @ Y @ S4 ) ) ) ) ) ).
% sccE
thf(fact_84_graph_Oreachable_Ocong,axiom,
sCC_Bl649662514949026229able_v = sCC_Bl649662514949026229able_v ).
% graph.reachable.cong
thf(fact_85_graph_Oreachable__avoiding_Ocong,axiom,
sCC_Bl4291963740693775144ding_v = sCC_Bl4291963740693775144ding_v ).
% graph.reachable_avoiding.cong
thf(fact_86_graph_Ointro,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ Vertices )
=> ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ Vertices )
=> ( ord_le7336532860387713383od_v_v @ ( Successors @ X4 ) @ Vertices ) )
=> ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_87_graph_Ointro,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( finite_finite_v @ Vertices )
=> ( ! [X4: v] :
( ( member_v @ X4 @ Vertices )
=> ( ord_less_eq_set_v @ ( Successors @ X4 ) @ Vertices ) )
=> ( sCC_Bloemen_graph_v @ Vertices @ Successors ) ) ) ).
% graph.intro
thf(fact_88_SCC__Bloemen__Sequential_Ograph__def,axiom,
( sCC_Bl8307124943676871238od_v_v
= ( ^ [Vertices2: set_Product_prod_v_v,Successors2: product_prod_v_v > set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ Vertices2 )
& ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ Vertices2 )
=> ( ord_le7336532860387713383od_v_v @ ( Successors2 @ X3 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_89_SCC__Bloemen__Sequential_Ograph__def,axiom,
( sCC_Bloemen_graph_v
= ( ^ [Vertices2: set_v,Successors2: v > set_v] :
( ( finite_finite_v @ Vertices2 )
& ! [X3: v] :
( ( member_v @ X3 @ Vertices2 )
=> ( ord_less_eq_set_v @ ( Successors2 @ X3 ) @ Vertices2 ) ) ) ) ) ).
% SCC_Bloemen_Sequential.graph_def
thf(fact_90_graph_Ora__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E5: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E5 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y ) ) ) ).
% graph.ra_reachable
thf(fact_91_graph_Ovfin,axiom,
! [Vertices: set_v,Successors: v > set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( finite_finite_v @ Vertices ) ) ).
% graph.vfin
thf(fact_92_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_93_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_94_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_95_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_96_graph_Osucc__reachable,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_97_graph_Osucc__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( member_v @ Z @ ( Successors @ Y ) )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.succ_reachable
thf(fact_98_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 )
=> ~ ! [Y4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y4 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y4 @ A22 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_99_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 )
=> ~ ! [Y4: v] :
( ( member_v @ Y4 @ ( Successors @ A1 ) )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Y4 @ A22 ) ) ) ) ) ).
% graph.reachable.cases
thf(fact_100_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 )
= ( ? [X3: product_prod_v_v] :
( ( A1 = X3 )
& ( A22 = X3 ) )
| ? [X3: product_prod_v_v,Y3: product_prod_v_v,Z2: product_prod_v_v] :
( ( A1 = X3 )
& ( A22 = Z2 )
& ( member7453568604450474000od_v_v @ Y3 @ ( Successors @ X3 ) )
& ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y3 @ Z2 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_101_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 )
= ( ? [X3: v] :
( ( A1 = X3 )
& ( A22 = X3 ) )
| ? [X3: v,Y3: v,Z2: v] :
( ( A1 = X3 )
& ( A22 = Z2 )
& ( member_v @ Y3 @ ( Successors @ X3 ) )
& ( sCC_Bl649662514949026229able_v @ Successors @ Y3 @ Z2 ) ) ) ) ) ).
% graph.reachable.simps
thf(fact_102_graph_Oreachable__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_trans
thf(fact_103_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 )
=> ( ! [X4: product_prod_v_v] : ( P @ X4 @ X4 )
=> ( ! [X4: product_prod_v_v,Y4: product_prod_v_v,Z3: product_prod_v_v] :
( ( P @ X4 @ Y4 )
=> ( ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ Y4 ) )
=> ( P @ X4 @ Z3 ) ) )
=> ( P @ X @ Y ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_104_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 )
=> ( ! [X4: v] : ( P @ X4 @ X4 )
=> ( ! [X4: v,Y4: v,Z3: v] :
( ( P @ X4 @ Y4 )
=> ( ( member_v @ Z3 @ ( Successors @ Y4 ) )
=> ( P @ X4 @ Z3 ) ) )
=> ( P @ X @ Y ) ) ) ) ) ).
% graph.reachable_end_induct
thf(fact_105_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_106_graph_Oreachable__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,Y: product_prod_v_v,X: product_prod_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ Z )
=> ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_107_graph_Oreachable__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y @ ( Successors @ X ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ Z )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Z ) ) ) ) ).
% graph.reachable_succ
thf(fact_108_graph_Ora__trans,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E5: set_Product_prod_v_v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E5 )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ Y @ Z @ E5 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Z @ E5 ) ) ) ) ).
% graph.ra_trans
thf(fact_109_graph_Ora__refl,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,E5: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ X @ E5 ) ) ).
% graph.ra_refl
thf(fact_110_graph_Oedge__ra,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,E5: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( member7453568604450474000od_v_v @ Y @ ( Successors @ X ) )
=> ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Y ) @ E5 )
=> ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y @ E5 ) ) ) ) ).
% graph.edge_ra
thf(fact_111_graph_Oedge__ra,axiom,
! [Vertices: set_v,Successors: v > set_v,Y: v,X: v,E5: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( member_v @ Y @ ( Successors @ X ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Y ) @ E5 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E5 ) ) ) ) ).
% graph.edge_ra
thf(fact_112_graph_Ora__cases,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,E5: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y @ E5 )
=> ( ( X = Y )
| ? [Z3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Z3 @ ( Successors @ X ) )
& ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ X @ Z3 ) @ E5 )
& ( sCC_Bl5370300055464682748od_v_v @ Successors @ Z3 @ Y @ E5 ) ) ) ) ) ).
% graph.ra_cases
thf(fact_113_graph_Ora__cases,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E5: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E5 )
=> ( ( X = Y )
| ? [Z3: v] :
( ( member_v @ Z3 @ ( Successors @ X ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X @ Z3 ) @ E5 )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ Z3 @ Y @ E5 ) ) ) ) ) ).
% graph.ra_cases
thf(fact_114_graph_Oreachable__avoiding_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,A32: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ A22 @ A32 )
=> ( ( A22 != A1 )
=> ~ ! [Y4: product_prod_v_v] :
( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ Y4 @ A32 )
=> ( ( member7453568604450474000od_v_v @ A22 @ ( Successors @ Y4 ) )
=> ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y4 @ A22 ) @ A32 ) ) ) ) ) ) ).
% graph.reachable_avoiding.cases
thf(fact_115_graph_Oreachable__avoiding_Ocases,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v,A32: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ A22 @ A32 )
=> ( ( A22 != A1 )
=> ~ ! [Y4: v] :
( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ Y4 @ A32 )
=> ( ( member_v @ A22 @ ( Successors @ Y4 ) )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y4 @ A22 ) @ A32 ) ) ) ) ) ) ).
% graph.reachable_avoiding.cases
thf(fact_116_graph_Oreachable__avoiding_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,A32: set_Pr2149350503807050951od_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ A1 @ A22 @ A32 )
= ( ? [X3: product_prod_v_v,E6: set_Pr2149350503807050951od_v_v] :
( ( A1 = X3 )
& ( A22 = X3 )
& ( A32 = E6 ) )
| ? [X3: product_prod_v_v,Y3: product_prod_v_v,E6: set_Pr2149350503807050951od_v_v,Z2: product_prod_v_v] :
( ( A1 = X3 )
& ( A22 = Z2 )
& ( A32 = E6 )
& ( sCC_Bl5370300055464682748od_v_v @ Successors @ X3 @ Y3 @ E6 )
& ( member7453568604450474000od_v_v @ Z2 @ ( Successors @ Y3 ) )
& ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y3 @ Z2 ) @ E6 ) ) ) ) ) ).
% graph.reachable_avoiding.simps
thf(fact_117_graph_Oreachable__avoiding_Osimps,axiom,
! [Vertices: set_v,Successors: v > set_v,A1: v,A22: v,A32: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ A1 @ A22 @ A32 )
= ( ? [X3: v,E6: set_Product_prod_v_v] :
( ( A1 = X3 )
& ( A22 = X3 )
& ( A32 = E6 ) )
| ? [X3: v,Y3: v,E6: set_Product_prod_v_v,Z2: v] :
( ( A1 = X3 )
& ( A22 = Z2 )
& ( A32 = E6 )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ X3 @ Y3 @ E6 )
& ( member_v @ Z2 @ ( Successors @ Y3 ) )
& ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y3 @ Z2 ) @ E6 ) ) ) ) ) ).
% graph.reachable_avoiding.simps
thf(fact_118_graph_Ora__succ,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,X: product_prod_v_v,Y: product_prod_v_v,E5: set_Pr2149350503807050951od_v_v,Z: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Y @ E5 )
=> ( ( member7453568604450474000od_v_v @ Z @ ( Successors @ Y ) )
=> ( ~ ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ Y @ Z ) @ E5 )
=> ( sCC_Bl5370300055464682748od_v_v @ Successors @ X @ Z @ E5 ) ) ) ) ) ).
% graph.ra_succ
thf(fact_119_graph_Ora__succ,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E5: set_Product_prod_v_v,Z: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E5 )
=> ( ( member_v @ Z @ ( Successors @ Y ) )
=> ( ~ ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ Y @ Z ) @ E5 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Z @ E5 ) ) ) ) ) ).
% graph.ra_succ
thf(fact_120_graph_Ore__reachable,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y ) ) ) ).
% graph.re_reachable
thf(fact_121_graph_Oreachable__re,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( sCC_Bl770211535891879572_end_v @ Successors @ X @ Y ) ) ) ).
% graph.reachable_re
thf(fact_122_is__subscc__def,axiom,
! [S4: set_v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S4 )
= ( ! [X3: v] :
( ( member_v @ X3 @ S4 )
=> ! [Y3: v] :
( ( member_v @ Y3 @ S4 )
=> ( sCC_Bl649662514949026229able_v @ successors @ X3 @ Y3 ) ) ) ) ) ).
% is_subscc_def
thf(fact_123_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_124_subsetI,axiom,
! [A3: set_v,B4: set_v] :
( ! [X4: v] :
( ( member_v @ X4 @ A3 )
=> ( member_v @ X4 @ B4 ) )
=> ( ord_less_eq_set_v @ A3 @ B4 ) ) ).
% subsetI
thf(fact_125_subsetI,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A3 )
=> ( member7453568604450474000od_v_v @ X4 @ B4 ) )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B4 ) ) ).
% subsetI
thf(fact_126_subset__antisym,axiom,
! [A3: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ A3 @ B4 )
=> ( ( ord_less_eq_set_v @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ).
% subset_antisym
thf(fact_127_subset__antisym,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B4 )
=> ( ( ord_le7336532860387713383od_v_v @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ).
% subset_antisym
thf(fact_128_ra__mono,axiom,
! [X: v,Y: v,E5: set_Product_prod_v_v,E7: set_Product_prod_v_v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E5 )
=> ( ( ord_le7336532860387713383od_v_v @ E7 @ E5 )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E7 ) ) ) ).
% ra_mono
thf(fact_129_order__refl,axiom,
! [X: set_v] : ( ord_less_eq_set_v @ X @ X ) ).
% order_refl
thf(fact_130_order__refl,axiom,
! [X: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X @ X ) ).
% order_refl
thf(fact_131_dual__order_Orefl,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ A @ A ) ).
% dual_order.refl
thf(fact_132_dual__order_Orefl,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ A ) ).
% dual_order.refl
thf(fact_133_finite__subset,axiom,
! [A3: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ A3 @ B4 )
=> ( ( finite_finite_v @ B4 )
=> ( finite_finite_v @ A3 ) ) ) ).
% finite_subset
thf(fact_134_finite__subset,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B4 )
=> ( ( finite3348123685078250256od_v_v @ B4 )
=> ( finite3348123685078250256od_v_v @ A3 ) ) ) ).
% finite_subset
thf(fact_135_infinite__super,axiom,
! [S4: set_v,T: set_v] :
( ( ord_less_eq_set_v @ S4 @ T )
=> ( ~ ( finite_finite_v @ S4 )
=> ~ ( finite_finite_v @ T ) ) ) ).
% infinite_super
thf(fact_136_infinite__super,axiom,
! [S4: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ S4 @ T )
=> ( ~ ( finite3348123685078250256od_v_v @ S4 )
=> ~ ( finite3348123685078250256od_v_v @ T ) ) ) ).
% infinite_super
thf(fact_137_empty__Collect__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ P ) )
= ( ! [X3: product_prod_v_v] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_138_empty__Collect__eq,axiom,
! [P: v > $o] :
( ( bot_bot_set_v
= ( collect_v @ P ) )
= ( ! [X3: v] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_139_empty__Collect__eq,axiom,
! [P: set_v > $o] :
( ( bot_bot_set_set_v
= ( collect_set_v @ P ) )
= ( ! [X3: set_v] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_140_Collect__empty__eq,axiom,
! [P: product_prod_v_v > $o] :
( ( ( collec140062887454715474od_v_v @ P )
= bot_bo723834152578015283od_v_v )
= ( ! [X3: product_prod_v_v] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_141_Collect__empty__eq,axiom,
! [P: v > $o] :
( ( ( collect_v @ P )
= bot_bot_set_v )
= ( ! [X3: v] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_142_Collect__empty__eq,axiom,
! [P: set_v > $o] :
( ( ( collect_set_v @ P )
= bot_bot_set_set_v )
= ( ! [X3: set_v] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_143_all__not__in__conv,axiom,
! [A3: set_Product_prod_v_v] :
( ( ! [X3: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ X3 @ A3 ) )
= ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% all_not_in_conv
thf(fact_144_all__not__in__conv,axiom,
! [A3: set_v] :
( ( ! [X3: v] :
~ ( member_v @ X3 @ A3 ) )
= ( A3 = bot_bot_set_v ) ) ).
% all_not_in_conv
thf(fact_145_all__not__in__conv,axiom,
! [A3: set_set_v] :
( ( ! [X3: set_v] :
~ ( member_set_v @ X3 @ A3 ) )
= ( A3 = bot_bot_set_set_v ) ) ).
% all_not_in_conv
thf(fact_146_empty__iff,axiom,
! [C: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ C @ bot_bo723834152578015283od_v_v ) ).
% empty_iff
thf(fact_147_empty__iff,axiom,
! [C: v] :
~ ( member_v @ C @ bot_bot_set_v ) ).
% empty_iff
thf(fact_148_empty__iff,axiom,
! [C: set_v] :
~ ( member_set_v @ C @ bot_bot_set_set_v ) ).
% empty_iff
thf(fact_149_empty__subsetI,axiom,
! [A3: set_set_v] : ( ord_le5216385588623774835_set_v @ bot_bot_set_set_v @ A3 ) ).
% empty_subsetI
thf(fact_150_empty__subsetI,axiom,
! [A3: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A3 ) ).
% empty_subsetI
thf(fact_151_empty__subsetI,axiom,
! [A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A3 ) ).
% empty_subsetI
thf(fact_152_subset__empty,axiom,
! [A3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ bot_bot_set_set_v )
= ( A3 = bot_bot_set_set_v ) ) ).
% subset_empty
thf(fact_153_subset__empty,axiom,
! [A3: set_v] :
( ( ord_less_eq_set_v @ A3 @ bot_bot_set_v )
= ( A3 = bot_bot_set_v ) ) ).
% subset_empty
thf(fact_154_subset__empty,axiom,
! [A3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
= ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% subset_empty
thf(fact_155_ex__in__conv,axiom,
! [A3: set_Product_prod_v_v] :
( ( ? [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ A3 ) )
= ( A3 != bot_bo723834152578015283od_v_v ) ) ).
% ex_in_conv
thf(fact_156_ex__in__conv,axiom,
! [A3: set_v] :
( ( ? [X3: v] : ( member_v @ X3 @ A3 ) )
= ( A3 != bot_bot_set_v ) ) ).
% ex_in_conv
thf(fact_157_ex__in__conv,axiom,
! [A3: set_set_v] :
( ( ? [X3: set_v] : ( member_set_v @ X3 @ A3 ) )
= ( A3 != bot_bot_set_set_v ) ) ).
% ex_in_conv
thf(fact_158_equals0I,axiom,
! [A3: set_Product_prod_v_v] :
( ! [Y4: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ Y4 @ A3 )
=> ( A3 = bot_bo723834152578015283od_v_v ) ) ).
% equals0I
thf(fact_159_equals0I,axiom,
! [A3: set_v] :
( ! [Y4: v] :
~ ( member_v @ Y4 @ A3 )
=> ( A3 = bot_bot_set_v ) ) ).
% equals0I
thf(fact_160_equals0I,axiom,
! [A3: set_set_v] :
( ! [Y4: set_v] :
~ ( member_set_v @ Y4 @ A3 )
=> ( A3 = bot_bot_set_set_v ) ) ).
% equals0I
thf(fact_161_equals0D,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v] :
( ( A3 = bot_bo723834152578015283od_v_v )
=> ~ ( member7453568604450474000od_v_v @ A @ A3 ) ) ).
% equals0D
thf(fact_162_equals0D,axiom,
! [A3: set_v,A: v] :
( ( A3 = bot_bot_set_v )
=> ~ ( member_v @ A @ A3 ) ) ).
% equals0D
thf(fact_163_equals0D,axiom,
! [A3: set_set_v,A: set_v] :
( ( A3 = bot_bot_set_set_v )
=> ~ ( member_set_v @ A @ A3 ) ) ).
% equals0D
thf(fact_164_emptyE,axiom,
! [A: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ A @ bot_bo723834152578015283od_v_v ) ).
% emptyE
thf(fact_165_emptyE,axiom,
! [A: v] :
~ ( member_v @ A @ bot_bot_set_v ) ).
% emptyE
thf(fact_166_emptyE,axiom,
! [A: set_v] :
~ ( member_set_v @ A @ bot_bot_set_set_v ) ).
% emptyE
thf(fact_167_graph_Ois__subscc_Ocong,axiom,
sCC_Bl5398416737448265317bscc_v = sCC_Bl5398416737448265317bscc_v ).
% graph.is_subscc.cong
thf(fact_168_bot_Oextremum__uniqueI,axiom,
! [A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ bot_bot_set_set_v )
=> ( A = bot_bot_set_set_v ) ) ).
% bot.extremum_uniqueI
thf(fact_169_bot_Oextremum__uniqueI,axiom,
! [A: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ bot_bot_Product_unit )
=> ( A = bot_bot_Product_unit ) ) ).
% bot.extremum_uniqueI
thf(fact_170_bot_Oextremum__uniqueI,axiom,
! [A: set_v] :
( ( ord_less_eq_set_v @ A @ bot_bot_set_v )
=> ( A = bot_bot_set_v ) ) ).
% bot.extremum_uniqueI
thf(fact_171_bot_Oextremum__uniqueI,axiom,
! [A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ bot_bo723834152578015283od_v_v )
=> ( A = bot_bo723834152578015283od_v_v ) ) ).
% bot.extremum_uniqueI
thf(fact_172_bot_Oextremum__unique,axiom,
! [A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ bot_bot_set_set_v )
= ( A = bot_bot_set_set_v ) ) ).
% bot.extremum_unique
thf(fact_173_bot_Oextremum__unique,axiom,
! [A: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ bot_bot_Product_unit )
= ( A = bot_bot_Product_unit ) ) ).
% bot.extremum_unique
thf(fact_174_bot_Oextremum__unique,axiom,
! [A: set_v] :
( ( ord_less_eq_set_v @ A @ bot_bot_set_v )
= ( A = bot_bot_set_v ) ) ).
% bot.extremum_unique
thf(fact_175_bot_Oextremum__unique,axiom,
! [A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ bot_bo723834152578015283od_v_v )
= ( A = bot_bo723834152578015283od_v_v ) ) ).
% bot.extremum_unique
thf(fact_176_bot_Oextremum,axiom,
! [A: set_set_v] : ( ord_le5216385588623774835_set_v @ bot_bot_set_set_v @ A ) ).
% bot.extremum
thf(fact_177_bot_Oextremum,axiom,
! [A: product_unit] : ( ord_le3221252021190050221t_unit @ bot_bot_Product_unit @ A ) ).
% bot.extremum
thf(fact_178_bot_Oextremum,axiom,
! [A: set_v] : ( ord_less_eq_set_v @ bot_bot_set_v @ A ) ).
% bot.extremum
thf(fact_179_bot_Oextremum,axiom,
! [A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ bot_bo723834152578015283od_v_v @ A ) ).
% bot.extremum
thf(fact_180_graph_Ois__scc_Ocong,axiom,
sCC_Bloemen_is_scc_v = sCC_Bloemen_is_scc_v ).
% graph.is_scc.cong
thf(fact_181_infinite__imp__nonempty,axiom,
! [S4: set_Product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ S4 )
=> ( S4 != bot_bo723834152578015283od_v_v ) ) ).
% infinite_imp_nonempty
thf(fact_182_infinite__imp__nonempty,axiom,
! [S4: set_v] :
( ~ ( finite_finite_v @ S4 )
=> ( S4 != bot_bot_set_v ) ) ).
% infinite_imp_nonempty
thf(fact_183_infinite__imp__nonempty,axiom,
! [S4: set_set_v] :
( ~ ( finite_finite_set_v @ S4 )
=> ( S4 != bot_bot_set_set_v ) ) ).
% infinite_imp_nonempty
thf(fact_184_finite_OemptyI,axiom,
finite3348123685078250256od_v_v @ bot_bo723834152578015283od_v_v ).
% finite.emptyI
thf(fact_185_finite_OemptyI,axiom,
finite_finite_v @ bot_bot_set_v ).
% finite.emptyI
thf(fact_186_finite_OemptyI,axiom,
finite_finite_set_v @ bot_bot_set_set_v ).
% finite.emptyI
thf(fact_187_graph_Ois__scc__def,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,S4: set_set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl1515522642333523865_set_v @ Successors @ S4 )
= ( ( S4 != bot_bot_set_set_v )
& ( sCC_Bl7907073126578335045_set_v @ Successors @ S4 )
& ! [S5: set_set_v] :
( ( ( ord_le5216385588623774835_set_v @ S4 @ S5 )
& ( sCC_Bl7907073126578335045_set_v @ Successors @ S5 ) )
=> ( S5 = S4 ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_188_graph_Ois__scc__def,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S4: set_Product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S4 )
= ( ( S4 != bot_bo723834152578015283od_v_v )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S4 )
& ! [S5: set_Product_prod_v_v] :
( ( ( ord_le7336532860387713383od_v_v @ S4 @ S5 )
& ( sCC_Bl2301996248249672505od_v_v @ Successors @ S5 ) )
=> ( S5 = S4 ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_189_graph_Ois__scc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S4: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S4 )
= ( ( S4 != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S4 )
& ! [S5: set_v] :
( ( ( ord_less_eq_set_v @ S4 @ S5 )
& ( sCC_Bl5398416737448265317bscc_v @ Successors @ S5 ) )
=> ( S5 = S4 ) ) ) ) ) ).
% graph.is_scc_def
thf(fact_190_finite__has__minimal,axiom,
! [A3: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ? [X4: set_v] :
( ( member_set_v @ X4 @ A3 )
& ! [Xa: set_v] :
( ( member_set_v @ Xa @ A3 )
=> ( ( ord_less_eq_set_v @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_191_finite__has__minimal,axiom,
! [A3: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ? [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ A3 )
& ! [Xa: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_192_finite__has__maximal,axiom,
! [A3: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ? [X4: set_v] :
( ( member_set_v @ X4 @ A3 )
& ! [Xa: set_v] :
( ( member_set_v @ Xa @ A3 )
=> ( ( ord_less_eq_set_v @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_193_finite__has__maximal,axiom,
! [A3: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ? [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ A3 )
& ! [Xa: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_194_graph_Ora__mono,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E5: set_Product_prod_v_v,E7: set_Product_prod_v_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E5 )
=> ( ( ord_le7336532860387713383od_v_v @ E7 @ E5 )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E7 ) ) ) ) ).
% graph.ra_mono
thf(fact_195_order__antisym__conv,axiom,
! [Y: set_v,X: set_v] :
( ( ord_less_eq_set_v @ Y @ X )
=> ( ( ord_less_eq_set_v @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_196_order__antisym__conv,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( ( ord_le7336532860387713383od_v_v @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_197_ord__le__eq__subst,axiom,
! [A: set_v,B: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_v,Y4: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y4 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_198_ord__le__eq__subst,axiom,
! [A: set_v,B: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_v,Y4: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y4 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_199_ord__le__eq__subst,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y4 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_200_ord__le__eq__subst,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y4 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_201_ord__eq__le__subst,axiom,
! [A: set_v,F: set_v > set_v,B: set_v,C: set_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ! [X4: set_v,Y4: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y4 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_202_ord__eq__le__subst,axiom,
! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B: set_v,C: set_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ! [X4: set_v,Y4: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y4 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_203_ord__eq__le__subst,axiom,
! [A: set_v,F: set_Product_prod_v_v > set_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ! [X4: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y4 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_204_ord__eq__le__subst,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A
= ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ! [X4: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y4 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_205_order__eq__refl,axiom,
! [X: set_v,Y: set_v] :
( ( X = Y )
=> ( ord_less_eq_set_v @ X @ Y ) ) ).
% order_eq_refl
thf(fact_206_order__eq__refl,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( X = Y )
=> ( ord_le7336532860387713383od_v_v @ X @ Y ) ) ).
% order_eq_refl
thf(fact_207_order__subst2,axiom,
! [A: set_v,B: set_v,F: set_v > set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ ( F @ B ) @ C )
=> ( ! [X4: set_v,Y4: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y4 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_208_order__subst2,axiom,
! [A: set_v,B: set_v,F: set_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B ) @ C )
=> ( ! [X4: set_v,Y4: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y4 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_209_order__subst2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_less_eq_set_v @ ( F @ B ) @ C )
=> ( ! [X4: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y4 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_210_order__subst2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ ( F @ B ) @ C )
=> ( ! [X4: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y4 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_211_order__subst1,axiom,
! [A: set_v,F: set_v > set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ! [X4: set_v,Y4: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y4 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_212_order__subst1,axiom,
! [A: set_v,F: set_Product_prod_v_v > set_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_less_eq_set_v @ A @ ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ! [X4: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y4 )
=> ( ord_less_eq_set_v @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_213_order__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_v > set_Product_prod_v_v,B: set_v,C: set_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ! [X4: set_v,Y4: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y4 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_214_order__subst1,axiom,
! [A: set_Product_prod_v_v,F: set_Product_prod_v_v > set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( F @ B ) )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ! [X4: set_Product_prod_v_v,Y4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y4 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le7336532860387713383od_v_v @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_215_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_v,Z4: set_v] : ( Y5 = Z4 ) )
= ( ^ [A5: set_v,B5: set_v] :
( ( ord_less_eq_set_v @ A5 @ B5 )
& ( ord_less_eq_set_v @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_216_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y5 = Z4 ) )
= ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A5 @ B5 )
& ( ord_le7336532860387713383od_v_v @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_217_antisym,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_218_antisym,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_219_dual__order_Otrans,axiom,
! [B: set_v,A: set_v,C: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( ord_less_eq_set_v @ C @ B )
=> ( ord_less_eq_set_v @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_220_dual__order_Otrans,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C @ B )
=> ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_221_dual__order_Oantisym,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( ord_less_eq_set_v @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_222_dual__order_Oantisym,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_223_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_v,Z4: set_v] : ( Y5 = Z4 ) )
= ( ^ [A5: set_v,B5: set_v] :
( ( ord_less_eq_set_v @ B5 @ A5 )
& ( ord_less_eq_set_v @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_224_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y5 = Z4 ) )
= ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B5 @ A5 )
& ( ord_le7336532860387713383od_v_v @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_225_order__trans,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( ord_less_eq_set_v @ Y @ Z )
=> ( ord_less_eq_set_v @ X @ Z ) ) ) ).
% order_trans
thf(fact_226_order__trans,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ Y @ Z )
=> ( ord_le7336532860387713383od_v_v @ X @ Z ) ) ) ).
% order_trans
thf(fact_227_order_Otrans,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% order.trans
thf(fact_228_order_Otrans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% order.trans
thf(fact_229_order__antisym,axiom,
! [X: set_v,Y: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( ord_less_eq_set_v @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_230_order__antisym,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_231_ord__le__eq__trans,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_232_ord__le__eq__trans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( B = C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_233_ord__eq__le__trans,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( A = B )
=> ( ( ord_less_eq_set_v @ B @ C )
=> ( ord_less_eq_set_v @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_234_ord__eq__le__trans,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( A = B )
=> ( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_235_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_v,Z4: set_v] : ( Y5 = Z4 ) )
= ( ^ [X3: set_v,Y3: set_v] :
( ( ord_less_eq_set_v @ X3 @ Y3 )
& ( ord_less_eq_set_v @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_236_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y5 = Z4 ) )
= ( ^ [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X3 @ Y3 )
& ( ord_le7336532860387713383od_v_v @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_237_Collect__mono__iff,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) )
= ( ! [X3: set_v] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_238_Collect__mono__iff,axiom,
! [P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) )
= ( ! [X3: v] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_239_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 ) )
= ( ! [X3: product_prod_v_v] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_240_set__eq__subset,axiom,
( ( ^ [Y5: set_v,Z4: set_v] : ( Y5 = Z4 ) )
= ( ^ [A6: set_v,B6: set_v] :
( ( ord_less_eq_set_v @ A6 @ B6 )
& ( ord_less_eq_set_v @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_241_set__eq__subset,axiom,
( ( ^ [Y5: set_Product_prod_v_v,Z4: set_Product_prod_v_v] : ( Y5 = Z4 ) )
= ( ^ [A6: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A6 @ B6 )
& ( ord_le7336532860387713383od_v_v @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_242_subset__trans,axiom,
! [A3: set_v,B4: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A3 @ B4 )
=> ( ( ord_less_eq_set_v @ B4 @ C2 )
=> ( ord_less_eq_set_v @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_243_subset__trans,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B4 )
=> ( ( ord_le7336532860387713383od_v_v @ B4 @ C2 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_244_Collect__mono,axiom,
! [P: set_v > $o,Q: set_v > $o] :
( ! [X4: set_v] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le5216385588623774835_set_v @ ( collect_set_v @ P ) @ ( collect_set_v @ Q ) ) ) ).
% Collect_mono
thf(fact_245_Collect__mono,axiom,
! [P: v > $o,Q: v > $o] :
( ! [X4: v] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_v @ ( collect_v @ P ) @ ( collect_v @ Q ) ) ) ).
% Collect_mono
thf(fact_246_Collect__mono,axiom,
! [P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ! [X4: product_prod_v_v] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le7336532860387713383od_v_v @ ( collec140062887454715474od_v_v @ P ) @ ( collec140062887454715474od_v_v @ Q ) ) ) ).
% Collect_mono
thf(fact_247_subset__refl,axiom,
! [A3: set_v] : ( ord_less_eq_set_v @ A3 @ A3 ) ).
% subset_refl
thf(fact_248_subset__refl,axiom,
! [A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A3 @ A3 ) ).
% subset_refl
thf(fact_249_subset__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A6: set_v,B6: set_v] :
! [T2: v] :
( ( member_v @ T2 @ A6 )
=> ( member_v @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_250_subset__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A6: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
! [T2: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ T2 @ A6 )
=> ( member7453568604450474000od_v_v @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_251_equalityD2,axiom,
! [A3: set_v,B4: set_v] :
( ( A3 = B4 )
=> ( ord_less_eq_set_v @ B4 @ A3 ) ) ).
% equalityD2
thf(fact_252_equalityD2,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( A3 = B4 )
=> ( ord_le7336532860387713383od_v_v @ B4 @ A3 ) ) ).
% equalityD2
thf(fact_253_equalityD1,axiom,
! [A3: set_v,B4: set_v] :
( ( A3 = B4 )
=> ( ord_less_eq_set_v @ A3 @ B4 ) ) ).
% equalityD1
thf(fact_254_equalityD1,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( A3 = B4 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B4 ) ) ).
% equalityD1
thf(fact_255_subset__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A6: set_v,B6: set_v] :
! [X3: v] :
( ( member_v @ X3 @ A6 )
=> ( member_v @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_256_subset__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A6: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A6 )
=> ( member7453568604450474000od_v_v @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_257_equalityE,axiom,
! [A3: set_v,B4: set_v] :
( ( A3 = B4 )
=> ~ ( ( ord_less_eq_set_v @ A3 @ B4 )
=> ~ ( ord_less_eq_set_v @ B4 @ A3 ) ) ) ).
% equalityE
thf(fact_258_equalityE,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( A3 = B4 )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A3 @ B4 )
=> ~ ( ord_le7336532860387713383od_v_v @ B4 @ A3 ) ) ) ).
% equalityE
thf(fact_259_subsetD,axiom,
! [A3: set_v,B4: set_v,C: v] :
( ( ord_less_eq_set_v @ A3 @ B4 )
=> ( ( member_v @ C @ A3 )
=> ( member_v @ C @ B4 ) ) ) ).
% subsetD
thf(fact_260_subsetD,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v,C: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B4 )
=> ( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( member7453568604450474000od_v_v @ C @ B4 ) ) ) ).
% subsetD
thf(fact_261_in__mono,axiom,
! [A3: set_v,B4: set_v,X: v] :
( ( ord_less_eq_set_v @ A3 @ B4 )
=> ( ( member_v @ X @ A3 )
=> ( member_v @ X @ B4 ) ) ) ).
% in_mono
thf(fact_262_in__mono,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B4 )
=> ( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( member7453568604450474000od_v_v @ X @ B4 ) ) ) ).
% in_mono
thf(fact_263_graph_Ora__empty,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ bot_bo723834152578015283od_v_v )
= ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y ) ) ) ).
% graph.ra_empty
thf(fact_264_graph_Ois__subscc__def,axiom,
! [Vertices: set_v,Successors: v > set_v,S4: set_v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S4 )
= ( ! [X3: v] :
( ( member_v @ X3 @ S4 )
=> ! [Y3: v] :
( ( member_v @ Y3 @ S4 )
=> ( sCC_Bl649662514949026229able_v @ Successors @ X3 @ Y3 ) ) ) ) ) ) ).
% graph.is_subscc_def
thf(fact_265_graph_OsccE,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S4: 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 @ S4 )
=> ( ( member7453568604450474000od_v_v @ X @ S4 )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ X )
=> ( member7453568604450474000od_v_v @ Y @ S4 ) ) ) ) ) ) ).
% graph.sccE
thf(fact_266_graph_OsccE,axiom,
! [Vertices: set_v,Successors: v > set_v,S4: set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S4 )
=> ( ( member_v @ X @ S4 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ X )
=> ( member_v @ Y @ S4 ) ) ) ) ) ) ).
% graph.sccE
thf(fact_267_finite__has__minimal2,axiom,
! [A3: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ A @ A3 )
=> ? [X4: set_v] :
( ( member_set_v @ X4 @ A3 )
& ( ord_less_eq_set_v @ X4 @ A )
& ! [Xa: set_v] :
( ( member_set_v @ Xa @ A3 )
=> ( ( ord_less_eq_set_v @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_268_finite__has__minimal2,axiom,
! [A3: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( member8406446414694345712od_v_v @ A @ A3 )
=> ? [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ A3 )
& ( ord_le7336532860387713383od_v_v @ X4 @ A )
& ! [Xa: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_269_finite__has__maximal2,axiom,
! [A3: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ A @ A3 )
=> ? [X4: set_v] :
( ( member_set_v @ X4 @ A3 )
& ( ord_less_eq_set_v @ A @ X4 )
& ! [Xa: set_v] :
( ( member_set_v @ Xa @ A3 )
=> ( ( ord_less_eq_set_v @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_270_finite__has__maximal2,axiom,
! [A3: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( member8406446414694345712od_v_v @ A @ A3 )
=> ? [X4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X4 @ A3 )
& ( ord_le7336532860387713383od_v_v @ A @ X4 )
& ! [Xa: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ Xa @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_271_rev__finite__subset,axiom,
! [B4: set_v,A3: set_v] :
( ( finite_finite_v @ B4 )
=> ( ( ord_less_eq_set_v @ A3 @ B4 )
=> ( finite_finite_v @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_272_rev__finite__subset,axiom,
! [B4: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ B4 )
=> ( ( ord_le7336532860387713383od_v_v @ A3 @ B4 )
=> ( finite3348123685078250256od_v_v @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_273_is__scc__def,axiom,
! [S4: set_v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S4 )
= ( ( S4 != bot_bot_set_v )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S4 )
& ! [S5: set_v] :
( ( ( ord_less_eq_set_v @ S4 @ S5 )
& ( sCC_Bl5398416737448265317bscc_v @ successors @ S5 ) )
=> ( S5 = S4 ) ) ) ) ).
% is_scc_def
thf(fact_274_subscc__add,axiom,
! [S4: set_v,X: v,Y: v] :
( ( sCC_Bl5398416737448265317bscc_v @ successors @ S4 )
=> ( ( member_v @ X @ S4 )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ Y @ X )
=> ( sCC_Bl5398416737448265317bscc_v @ successors @ ( insert_v @ Y @ S4 ) ) ) ) ) ) ).
% subscc_add
thf(fact_275_scc__partition,axiom,
! [S4: set_v,S6: set_v,X: v] :
( ( sCC_Bloemen_is_scc_v @ successors @ S4 )
=> ( ( sCC_Bloemen_is_scc_v @ successors @ S6 )
=> ( ( member_v @ X @ ( inf_inf_set_v @ S4 @ S6 ) )
=> ( S4 = S6 ) ) ) ) ).
% scc_partition
thf(fact_276_less__by__empty,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( A3 = bot_bo723834152578015283od_v_v )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B4 ) ) ).
% less_by_empty
thf(fact_277_subrelI,axiom,
! [R: set_Pr6425124735969554649t_unit,S7: set_Pr6425124735969554649t_unit] :
( ! [X4: v,Y4: sCC_Bl1394983891496994913t_unit] :
( ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X4 @ Y4 ) @ R )
=> ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ X4 @ Y4 ) @ S7 ) )
=> ( ord_le7290744839000465721t_unit @ R @ S7 ) ) ).
% subrelI
thf(fact_278_subrelI,axiom,
! [R: set_Product_prod_v_v,S7: set_Product_prod_v_v] :
( ! [X4: v,Y4: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X4 @ Y4 ) @ R )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ X4 @ Y4 ) @ S7 ) )
=> ( ord_le7336532860387713383od_v_v @ R @ S7 ) ) ).
% subrelI
thf(fact_279_subset__emptyI,axiom,
! [A3: set_set_v] :
( ! [X4: set_v] :
~ ( member_set_v @ X4 @ A3 )
=> ( ord_le5216385588623774835_set_v @ A3 @ bot_bot_set_set_v ) ) ).
% subset_emptyI
thf(fact_280_subset__emptyI,axiom,
! [A3: set_v] :
( ! [X4: v] :
~ ( member_v @ X4 @ A3 )
=> ( ord_less_eq_set_v @ A3 @ bot_bot_set_v ) ) ).
% subset_emptyI
thf(fact_281_subset__emptyI,axiom,
! [A3: set_Product_prod_v_v] :
( ! [X4: product_prod_v_v] :
~ ( member7453568604450474000od_v_v @ X4 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ bot_bo723834152578015283od_v_v ) ) ).
% subset_emptyI
thf(fact_282_Set_Ois__empty__def,axiom,
( is_emp8964507351669718201od_v_v
= ( ^ [A6: set_Product_prod_v_v] : ( A6 = bot_bo723834152578015283od_v_v ) ) ) ).
% Set.is_empty_def
thf(fact_283_Set_Ois__empty__def,axiom,
( is_empty_v
= ( ^ [A6: set_v] : ( A6 = bot_bot_set_v ) ) ) ).
% Set.is_empty_def
thf(fact_284_Set_Ois__empty__def,axiom,
( is_empty_set_v
= ( ^ [A6: set_set_v] : ( A6 = bot_bot_set_set_v ) ) ) ).
% Set.is_empty_def
thf(fact_285_reachable__visited,axiom,
! [E: sCC_Bl1394983891496994913t_unit,V: v,W: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V @ W )
=> ( ! [X4: v] :
( ( member_v @ X4 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa2: v] :
( ( member_v @ Xa2 @ ( minus_minus_set_v @ ( successors @ X4 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X4 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ successors @ V @ X4 )
=> ~ ( sCC_Bl649662514949026229able_v @ successors @ Xa2 @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ).
% reachable_visited
thf(fact_286_insert__absorb2,axiom,
! [X: v,A3: set_v] :
( ( insert_v @ X @ ( insert_v @ X @ A3 ) )
= ( insert_v @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_287_insert__absorb2,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( insert1338601472111419319od_v_v @ X @ A3 ) )
= ( insert1338601472111419319od_v_v @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_288_insert__absorb2,axiom,
! [X: set_v,A3: set_set_v] :
( ( insert_set_v @ X @ ( insert_set_v @ X @ A3 ) )
= ( insert_set_v @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_289_insert__iff,axiom,
! [A: set_v,B: set_v,A3: set_set_v] :
( ( member_set_v @ A @ ( insert_set_v @ B @ A3 ) )
= ( ( A = B )
| ( member_set_v @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_290_insert__iff,axiom,
! [A: v,B: v,A3: set_v] :
( ( member_v @ A @ ( insert_v @ B @ A3 ) )
= ( ( A = B )
| ( member_v @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_291_insert__iff,axiom,
! [A: product_prod_v_v,B: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ A3 ) )
= ( ( A = B )
| ( member7453568604450474000od_v_v @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_292_insertCI,axiom,
! [A: set_v,B4: set_set_v,B: set_v] :
( ( ~ ( member_set_v @ A @ B4 )
=> ( A = B ) )
=> ( member_set_v @ A @ ( insert_set_v @ B @ B4 ) ) ) ).
% insertCI
thf(fact_293_insertCI,axiom,
! [A: v,B4: set_v,B: v] :
( ( ~ ( member_v @ A @ B4 )
=> ( A = B ) )
=> ( member_v @ A @ ( insert_v @ B @ B4 ) ) ) ).
% insertCI
thf(fact_294_insertCI,axiom,
! [A: product_prod_v_v,B4: set_Product_prod_v_v,B: product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ A @ B4 )
=> ( A = B ) )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ B4 ) ) ) ).
% insertCI
thf(fact_295_Int__iff,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B4 ) )
= ( ( member7453568604450474000od_v_v @ C @ A3 )
& ( member7453568604450474000od_v_v @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_296_Int__iff,axiom,
! [C: v,A3: set_v,B4: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A3 @ B4 ) )
= ( ( member_v @ C @ A3 )
& ( member_v @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_297_IntI,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( ( member7453568604450474000od_v_v @ C @ B4 )
=> ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B4 ) ) ) ) ).
% IntI
thf(fact_298_IntI,axiom,
! [C: v,A3: set_v,B4: set_v] :
( ( member_v @ C @ A3 )
=> ( ( member_v @ C @ B4 )
=> ( member_v @ C @ ( inf_inf_set_v @ A3 @ B4 ) ) ) ) ).
% IntI
thf(fact_299_Diff__idemp,axiom,
! [A3: set_v,B4: set_v] :
( ( minus_minus_set_v @ ( minus_minus_set_v @ A3 @ B4 ) @ B4 )
= ( minus_minus_set_v @ A3 @ B4 ) ) ).
% Diff_idemp
thf(fact_300_Diff__iff,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B4 ) )
= ( ( member7453568604450474000od_v_v @ C @ A3 )
& ~ ( member7453568604450474000od_v_v @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_301_Diff__iff,axiom,
! [C: v,A3: set_v,B4: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A3 @ B4 ) )
= ( ( member_v @ C @ A3 )
& ~ ( member_v @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_302_DiffI,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ C @ B4 )
=> ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B4 ) ) ) ) ).
% DiffI
thf(fact_303_DiffI,axiom,
! [C: v,A3: set_v,B4: set_v] :
( ( member_v @ C @ A3 )
=> ( ~ ( member_v @ C @ B4 )
=> ( member_v @ C @ ( minus_minus_set_v @ A3 @ B4 ) ) ) ) ).
% DiffI
thf(fact_304_singletonI,axiom,
! [A: product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ).
% singletonI
thf(fact_305_singletonI,axiom,
! [A: v] : ( member_v @ A @ ( insert_v @ A @ bot_bot_set_v ) ) ).
% singletonI
thf(fact_306_singletonI,axiom,
! [A: set_v] : ( member_set_v @ A @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) ).
% singletonI
thf(fact_307_finite__insert,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) )
= ( finite3348123685078250256od_v_v @ A3 ) ) ).
% finite_insert
thf(fact_308_finite__insert,axiom,
! [A: set_v,A3: set_set_v] :
( ( finite_finite_set_v @ ( insert_set_v @ A @ A3 ) )
= ( finite_finite_set_v @ A3 ) ) ).
% finite_insert
thf(fact_309_finite__insert,axiom,
! [A: v,A3: set_v] :
( ( finite_finite_v @ ( insert_v @ A @ A3 ) )
= ( finite_finite_v @ A3 ) ) ).
% finite_insert
thf(fact_310_insert__subset,axiom,
! [X: set_v,A3: set_set_v,B4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( insert_set_v @ X @ A3 ) @ B4 )
= ( ( member_set_v @ X @ B4 )
& ( ord_le5216385588623774835_set_v @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_311_insert__subset,axiom,
! [X: v,A3: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ ( insert_v @ X @ A3 ) @ B4 )
= ( ( member_v @ X @ B4 )
& ( ord_less_eq_set_v @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_312_insert__subset,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ B4 )
= ( ( member7453568604450474000od_v_v @ X @ B4 )
& ( ord_le7336532860387713383od_v_v @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_313_finite__Int,axiom,
! [F3: set_v,G: set_v] :
( ( ( finite_finite_v @ F3 )
| ( finite_finite_v @ G ) )
=> ( finite_finite_v @ ( inf_inf_set_v @ F3 @ G ) ) ) ).
% finite_Int
thf(fact_314_Diff__empty,axiom,
! [A3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
= A3 ) ).
% Diff_empty
thf(fact_315_Diff__empty,axiom,
! [A3: set_set_v] :
( ( minus_7228012346218142266_set_v @ A3 @ bot_bot_set_set_v )
= A3 ) ).
% Diff_empty
thf(fact_316_Diff__empty,axiom,
! [A3: set_v] :
( ( minus_minus_set_v @ A3 @ bot_bot_set_v )
= A3 ) ).
% Diff_empty
thf(fact_317_empty__Diff,axiom,
! [A3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ bot_bo723834152578015283od_v_v @ A3 )
= bot_bo723834152578015283od_v_v ) ).
% empty_Diff
thf(fact_318_empty__Diff,axiom,
! [A3: set_set_v] :
( ( minus_7228012346218142266_set_v @ bot_bot_set_set_v @ A3 )
= bot_bot_set_set_v ) ).
% empty_Diff
thf(fact_319_empty__Diff,axiom,
! [A3: set_v] :
( ( minus_minus_set_v @ bot_bot_set_v @ A3 )
= bot_bot_set_v ) ).
% empty_Diff
thf(fact_320_Diff__cancel,axiom,
! [A3: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ A3 )
= bot_bo723834152578015283od_v_v ) ).
% Diff_cancel
thf(fact_321_Diff__cancel,axiom,
! [A3: set_set_v] :
( ( minus_7228012346218142266_set_v @ A3 @ A3 )
= bot_bot_set_set_v ) ).
% Diff_cancel
thf(fact_322_Diff__cancel,axiom,
! [A3: set_v] :
( ( minus_minus_set_v @ A3 @ A3 )
= bot_bot_set_v ) ).
% Diff_cancel
thf(fact_323_Int__subset__iff,axiom,
! [C2: set_v,A3: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A3 @ B4 ) )
= ( ( ord_less_eq_set_v @ C2 @ A3 )
& ( ord_less_eq_set_v @ C2 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_324_Int__subset__iff,axiom,
! [C2: set_Product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A3 @ B4 ) )
= ( ( ord_le7336532860387713383od_v_v @ C2 @ A3 )
& ( ord_le7336532860387713383od_v_v @ C2 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_325_finite__Diff,axiom,
! [A3: set_v,B4: set_v] :
( ( finite_finite_v @ A3 )
=> ( finite_finite_v @ ( minus_minus_set_v @ A3 @ B4 ) ) ) ).
% finite_Diff
thf(fact_326_finite__Diff2,axiom,
! [B4: set_v,A3: set_v] :
( ( finite_finite_v @ B4 )
=> ( ( finite_finite_v @ ( minus_minus_set_v @ A3 @ B4 ) )
= ( finite_finite_v @ A3 ) ) ) ).
% finite_Diff2
thf(fact_327_Int__insert__right__if1,axiom,
! [A: set_v,A3: set_set_v,B4: set_set_v] :
( ( member_set_v @ A @ A3 )
=> ( ( inf_inf_set_set_v @ A3 @ ( insert_set_v @ A @ B4 ) )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ A3 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_328_Int__insert__right__if1,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B4 ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A3 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_329_Int__insert__right__if1,axiom,
! [A: v,A3: set_v,B4: set_v] :
( ( member_v @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v @ A @ B4 ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A3 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_330_Int__insert__right__if0,axiom,
! [A: set_v,A3: set_set_v,B4: set_set_v] :
( ~ ( member_set_v @ A @ A3 )
=> ( ( inf_inf_set_set_v @ A3 @ ( insert_set_v @ A @ B4 ) )
= ( inf_inf_set_set_v @ A3 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_331_Int__insert__right__if0,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B4 ) )
= ( inf_in6271465464967711157od_v_v @ A3 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_332_Int__insert__right__if0,axiom,
! [A: v,A3: set_v,B4: set_v] :
( ~ ( member_v @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v @ A @ B4 ) )
= ( inf_inf_set_v @ A3 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_333_insert__inter__insert,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) @ ( insert1338601472111419319od_v_v @ A @ B4 ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A3 @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_334_insert__inter__insert,axiom,
! [A: set_v,A3: set_set_v,B4: set_set_v] :
( ( inf_inf_set_set_v @ ( insert_set_v @ A @ A3 ) @ ( insert_set_v @ A @ B4 ) )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ A3 @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_335_insert__inter__insert,axiom,
! [A: v,A3: set_v,B4: set_v] :
( ( inf_inf_set_v @ ( insert_v @ A @ A3 ) @ ( insert_v @ A @ B4 ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A3 @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_336_Int__insert__left__if1,axiom,
! [A: set_v,C2: set_set_v,B4: set_set_v] :
( ( member_set_v @ A @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B4 ) @ C2 )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_337_Int__insert__left__if1,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B4 ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_338_Int__insert__left__if1,axiom,
! [A: v,C2: set_v,B4: set_v] :
( ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B4 ) @ C2 )
= ( insert_v @ A @ ( inf_inf_set_v @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_339_Int__insert__left__if0,axiom,
! [A: set_v,C2: set_set_v,B4: set_set_v] :
( ~ ( member_set_v @ A @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B4 ) @ C2 )
= ( inf_inf_set_set_v @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_340_Int__insert__left__if0,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B4 ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_341_Int__insert__left__if0,axiom,
! [A: v,C2: set_v,B4: set_v] :
( ~ ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B4 ) @ C2 )
= ( inf_inf_set_v @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_342_insert__Diff1,axiom,
! [X: set_v,B4: set_set_v,A3: set_set_v] :
( ( member_set_v @ X @ B4 )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X @ A3 ) @ B4 )
= ( minus_7228012346218142266_set_v @ A3 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_343_insert__Diff1,axiom,
! [X: product_prod_v_v,B4: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ B4 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ B4 )
= ( minus_4183494784930505774od_v_v @ A3 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_344_insert__Diff1,axiom,
! [X: v,B4: set_v,A3: set_v] :
( ( member_v @ X @ B4 )
=> ( ( minus_minus_set_v @ ( insert_v @ X @ A3 ) @ B4 )
= ( minus_minus_set_v @ A3 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_345_Diff__insert0,axiom,
! [X: set_v,A3: set_set_v,B4: set_set_v] :
( ~ ( member_set_v @ X @ A3 )
=> ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ B4 ) )
= ( minus_7228012346218142266_set_v @ A3 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_346_Diff__insert0,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ B4 ) )
= ( minus_4183494784930505774od_v_v @ A3 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_347_Diff__insert0,axiom,
! [X: v,A3: set_v,B4: set_v] :
( ~ ( member_v @ X @ A3 )
=> ( ( minus_minus_set_v @ A3 @ ( insert_v @ X @ B4 ) )
= ( minus_minus_set_v @ A3 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_348_singleton__insert__inj__eq_H,axiom,
! [A: set_v,A3: set_set_v,B: set_v] :
( ( ( insert_set_v @ A @ A3 )
= ( insert_set_v @ B @ bot_bot_set_set_v ) )
= ( ( A = B )
& ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ B @ bot_bot_set_set_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_349_singleton__insert__inj__eq_H,axiom,
! [A: v,A3: set_v,B: v] :
( ( ( insert_v @ A @ A3 )
= ( insert_v @ B @ bot_bot_set_v ) )
= ( ( A = B )
& ( ord_less_eq_set_v @ A3 @ ( insert_v @ B @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_350_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ A3 )
= ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) )
= ( ( A = B )
& ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_351_singleton__insert__inj__eq,axiom,
! [B: set_v,A: set_v,A3: set_set_v] :
( ( ( insert_set_v @ B @ bot_bot_set_set_v )
= ( insert_set_v @ A @ A3 ) )
= ( ( A = B )
& ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ B @ bot_bot_set_set_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_352_singleton__insert__inj__eq,axiom,
! [B: v,A: v,A3: set_v] :
( ( ( insert_v @ B @ bot_bot_set_v )
= ( insert_v @ A @ A3 ) )
= ( ( A = B )
& ( ord_less_eq_set_v @ A3 @ ( insert_v @ B @ bot_bot_set_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_353_singleton__insert__inj__eq,axiom,
! [B: product_prod_v_v,A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ A @ A3 ) )
= ( ( A = B )
& ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_354_insert__disjoint_I1_J,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) @ B4 )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B4 )
& ( ( inf_in6271465464967711157od_v_v @ A3 @ B4 )
= bot_bo723834152578015283od_v_v ) ) ) ).
% insert_disjoint(1)
thf(fact_355_insert__disjoint_I1_J,axiom,
! [A: v,A3: set_v,B4: set_v] :
( ( ( inf_inf_set_v @ ( insert_v @ A @ A3 ) @ B4 )
= bot_bot_set_v )
= ( ~ ( member_v @ A @ B4 )
& ( ( inf_inf_set_v @ A3 @ B4 )
= bot_bot_set_v ) ) ) ).
% insert_disjoint(1)
thf(fact_356_insert__disjoint_I1_J,axiom,
! [A: set_v,A3: set_set_v,B4: set_set_v] :
( ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ A3 ) @ B4 )
= bot_bot_set_set_v )
= ( ~ ( member_set_v @ A @ B4 )
& ( ( inf_inf_set_set_v @ A3 @ B4 )
= bot_bot_set_set_v ) ) ) ).
% insert_disjoint(1)
thf(fact_357_insert__disjoint_I2_J,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) @ B4 ) )
= ( ~ ( member7453568604450474000od_v_v @ A @ B4 )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A3 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_358_insert__disjoint_I2_J,axiom,
! [A: v,A3: set_v,B4: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ ( insert_v @ A @ A3 ) @ B4 ) )
= ( ~ ( member_v @ A @ B4 )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A3 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_359_insert__disjoint_I2_J,axiom,
! [A: set_v,A3: set_set_v,B4: set_set_v] :
( ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ ( insert_set_v @ A @ A3 ) @ B4 ) )
= ( ~ ( member_set_v @ A @ B4 )
& ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A3 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_360_disjoint__insert_I1_J,axiom,
! [B4: set_Product_prod_v_v,A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ B4 @ ( insert1338601472111419319od_v_v @ A @ A3 ) )
= bot_bo723834152578015283od_v_v )
= ( ~ ( member7453568604450474000od_v_v @ A @ B4 )
& ( ( inf_in6271465464967711157od_v_v @ B4 @ A3 )
= bot_bo723834152578015283od_v_v ) ) ) ).
% disjoint_insert(1)
thf(fact_361_disjoint__insert_I1_J,axiom,
! [B4: set_v,A: v,A3: set_v] :
( ( ( inf_inf_set_v @ B4 @ ( insert_v @ A @ A3 ) )
= bot_bot_set_v )
= ( ~ ( member_v @ A @ B4 )
& ( ( inf_inf_set_v @ B4 @ A3 )
= bot_bot_set_v ) ) ) ).
% disjoint_insert(1)
thf(fact_362_disjoint__insert_I1_J,axiom,
! [B4: set_set_v,A: set_v,A3: set_set_v] :
( ( ( inf_inf_set_set_v @ B4 @ ( insert_set_v @ A @ A3 ) )
= bot_bot_set_set_v )
= ( ~ ( member_set_v @ A @ B4 )
& ( ( inf_inf_set_set_v @ B4 @ A3 )
= bot_bot_set_set_v ) ) ) ).
% disjoint_insert(1)
thf(fact_363_disjoint__insert_I2_J,axiom,
! [A3: set_Product_prod_v_v,B: product_prod_v_v,B4: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B @ B4 ) ) )
= ( ~ ( member7453568604450474000od_v_v @ B @ A3 )
& ( bot_bo723834152578015283od_v_v
= ( inf_in6271465464967711157od_v_v @ A3 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_364_disjoint__insert_I2_J,axiom,
! [A3: set_v,B: v,B4: set_v] :
( ( bot_bot_set_v
= ( inf_inf_set_v @ A3 @ ( insert_v @ B @ B4 ) ) )
= ( ~ ( member_v @ B @ A3 )
& ( bot_bot_set_v
= ( inf_inf_set_v @ A3 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_365_disjoint__insert_I2_J,axiom,
! [A3: set_set_v,B: set_v,B4: set_set_v] :
( ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A3 @ ( insert_set_v @ B @ B4 ) ) )
= ( ~ ( member_set_v @ B @ A3 )
& ( bot_bot_set_set_v
= ( inf_inf_set_set_v @ A3 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_366_Diff__eq__empty__iff,axiom,
! [A3: set_set_v,B4: set_set_v] :
( ( ( minus_7228012346218142266_set_v @ A3 @ B4 )
= bot_bot_set_set_v )
= ( ord_le5216385588623774835_set_v @ A3 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_367_Diff__eq__empty__iff,axiom,
! [A3: set_v,B4: set_v] :
( ( ( minus_minus_set_v @ A3 @ B4 )
= bot_bot_set_v )
= ( ord_less_eq_set_v @ A3 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_368_Diff__eq__empty__iff,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ( minus_4183494784930505774od_v_v @ A3 @ B4 )
= bot_bo723834152578015283od_v_v )
= ( ord_le7336532860387713383od_v_v @ A3 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_369_insert__Diff__single,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= ( insert1338601472111419319od_v_v @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_370_insert__Diff__single,axiom,
! [A: set_v,A3: set_set_v] :
( ( insert_set_v @ A @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) )
= ( insert_set_v @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_371_insert__Diff__single,axiom,
! [A: v,A3: set_v] :
( ( insert_v @ A @ ( minus_minus_set_v @ A3 @ ( insert_v @ A @ bot_bot_set_v ) ) )
= ( insert_v @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_372_finite__Diff__insert,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v,B4: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B4 ) ) )
= ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_373_finite__Diff__insert,axiom,
! [A3: set_set_v,A: set_v,B4: set_set_v] :
( ( finite_finite_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ A @ B4 ) ) )
= ( finite_finite_set_v @ ( minus_7228012346218142266_set_v @ A3 @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_374_finite__Diff__insert,axiom,
! [A3: set_v,A: v,B4: set_v] :
( ( finite_finite_v @ ( minus_minus_set_v @ A3 @ ( insert_v @ A @ B4 ) ) )
= ( finite_finite_v @ ( minus_minus_set_v @ A3 @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_375_Diff__disjoint,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B4 @ A3 ) )
= bot_bo723834152578015283od_v_v ) ).
% Diff_disjoint
thf(fact_376_Diff__disjoint,axiom,
! [A3: set_set_v,B4: set_set_v] :
( ( inf_inf_set_set_v @ A3 @ ( minus_7228012346218142266_set_v @ B4 @ A3 ) )
= bot_bot_set_set_v ) ).
% Diff_disjoint
thf(fact_377_Diff__disjoint,axiom,
! [A3: set_v,B4: set_v] :
( ( inf_inf_set_v @ A3 @ ( minus_minus_set_v @ B4 @ A3 ) )
= bot_bot_set_v ) ).
% Diff_disjoint
thf(fact_378_subset__Diff__insert,axiom,
! [A3: set_set_v,B4: set_set_v,X: set_v,C2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ ( minus_7228012346218142266_set_v @ B4 @ ( insert_set_v @ X @ C2 ) ) )
= ( ( ord_le5216385588623774835_set_v @ A3 @ ( minus_7228012346218142266_set_v @ B4 @ C2 ) )
& ~ ( member_set_v @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_379_subset__Diff__insert,axiom,
! [A3: set_v,B4: set_v,X: v,C2: set_v] :
( ( ord_less_eq_set_v @ A3 @ ( minus_minus_set_v @ B4 @ ( insert_v @ X @ C2 ) ) )
= ( ( ord_less_eq_set_v @ A3 @ ( minus_minus_set_v @ B4 @ C2 ) )
& ~ ( member_v @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_380_subset__Diff__insert,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v,X: product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B4 @ ( insert1338601472111419319od_v_v @ X @ C2 ) ) )
= ( ( ord_le7336532860387713383od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B4 @ C2 ) )
& ~ ( member7453568604450474000od_v_v @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_381_mk__disjoint__insert,axiom,
! [A: set_v,A3: set_set_v] :
( ( member_set_v @ A @ A3 )
=> ? [B7: set_set_v] :
( ( A3
= ( insert_set_v @ A @ B7 ) )
& ~ ( member_set_v @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_382_mk__disjoint__insert,axiom,
! [A: v,A3: set_v] :
( ( member_v @ A @ A3 )
=> ? [B7: set_v] :
( ( A3
= ( insert_v @ A @ B7 ) )
& ~ ( member_v @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_383_mk__disjoint__insert,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ? [B7: set_Product_prod_v_v] :
( ( A3
= ( insert1338601472111419319od_v_v @ A @ B7 ) )
& ~ ( member7453568604450474000od_v_v @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_384_Diff__insert__absorb,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_385_Diff__insert__absorb,axiom,
! [X: set_v,A3: set_set_v] :
( ~ ( member_set_v @ X @ A3 )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X @ A3 ) @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_386_Diff__insert__absorb,axiom,
! [X: v,A3: set_v] :
( ~ ( member_v @ X @ A3 )
=> ( ( minus_minus_set_v @ ( insert_v @ X @ A3 ) @ ( insert_v @ X @ bot_bot_set_v ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_387_Int__Diff__disjoint,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B4 ) @ ( minus_4183494784930505774od_v_v @ A3 @ B4 ) )
= bot_bo723834152578015283od_v_v ) ).
% Int_Diff_disjoint
thf(fact_388_Int__Diff__disjoint,axiom,
! [A3: set_set_v,B4: set_set_v] :
( ( inf_inf_set_set_v @ ( inf_inf_set_set_v @ A3 @ B4 ) @ ( minus_7228012346218142266_set_v @ A3 @ B4 ) )
= bot_bot_set_set_v ) ).
% Int_Diff_disjoint
thf(fact_389_Int__Diff__disjoint,axiom,
! [A3: set_v,B4: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A3 @ B4 ) @ ( minus_minus_set_v @ A3 @ B4 ) )
= bot_bot_set_v ) ).
% Int_Diff_disjoint
thf(fact_390_Diff__Int__distrib2,axiom,
! [A3: set_v,B4: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( minus_minus_set_v @ A3 @ B4 ) @ C2 )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ C2 ) @ ( inf_inf_set_v @ B4 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_391_bot__empty__eq,axiom,
( bot_bo8461541820394803818_v_v_o
= ( ^ [X3: product_prod_v_v] : ( member7453568604450474000od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) ) ) ).
% bot_empty_eq
thf(fact_392_bot__empty__eq,axiom,
( bot_bot_v_o
= ( ^ [X3: v] : ( member_v @ X3 @ bot_bot_set_v ) ) ) ).
% bot_empty_eq
thf(fact_393_bot__empty__eq,axiom,
( bot_bot_set_v_o
= ( ^ [X3: set_v] : ( member_set_v @ X3 @ bot_bot_set_set_v ) ) ) ).
% bot_empty_eq
thf(fact_394_Int__left__commute,axiom,
! [A3: set_v,B4: set_v,C2: set_v] :
( ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ B4 @ C2 ) )
= ( inf_inf_set_v @ B4 @ ( inf_inf_set_v @ A3 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_395_Int__insert__right,axiom,
! [A: set_v,A3: set_set_v,B4: set_set_v] :
( ( ( member_set_v @ A @ A3 )
=> ( ( inf_inf_set_set_v @ A3 @ ( insert_set_v @ A @ B4 ) )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ A3 @ B4 ) ) ) )
& ( ~ ( member_set_v @ A @ A3 )
=> ( ( inf_inf_set_set_v @ A3 @ ( insert_set_v @ A @ B4 ) )
= ( inf_inf_set_set_v @ A3 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_396_Int__insert__right,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B4 ) )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ A3 @ B4 ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B4 ) )
= ( inf_in6271465464967711157od_v_v @ A3 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_397_Int__insert__right,axiom,
! [A: v,A3: set_v,B4: set_v] :
( ( ( member_v @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v @ A @ B4 ) )
= ( insert_v @ A @ ( inf_inf_set_v @ A3 @ B4 ) ) ) )
& ( ~ ( member_v @ A @ A3 )
=> ( ( inf_inf_set_v @ A3 @ ( insert_v @ A @ B4 ) )
= ( inf_inf_set_v @ A3 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_398_Diff__Int__distrib,axiom,
! [C2: set_v,A3: set_v,B4: set_v] :
( ( inf_inf_set_v @ C2 @ ( minus_minus_set_v @ A3 @ B4 ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ C2 @ A3 ) @ ( inf_inf_set_v @ C2 @ B4 ) ) ) ).
% Diff_Int_distrib
thf(fact_399_Int__left__absorb,axiom,
! [A3: set_v,B4: set_v] :
( ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ A3 @ B4 ) )
= ( inf_inf_set_v @ A3 @ B4 ) ) ).
% Int_left_absorb
thf(fact_400_Int__insert__left,axiom,
! [A: set_v,C2: set_set_v,B4: set_set_v] :
( ( ( member_set_v @ A @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B4 ) @ C2 )
= ( insert_set_v @ A @ ( inf_inf_set_set_v @ B4 @ C2 ) ) ) )
& ( ~ ( member_set_v @ A @ C2 )
=> ( ( inf_inf_set_set_v @ ( insert_set_v @ A @ B4 ) @ C2 )
= ( inf_inf_set_set_v @ B4 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_401_Int__insert__left,axiom,
! [A: product_prod_v_v,C2: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B4 ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B4 @ C2 ) ) ) )
& ( ~ ( member7453568604450474000od_v_v @ A @ C2 )
=> ( ( inf_in6271465464967711157od_v_v @ ( insert1338601472111419319od_v_v @ A @ B4 ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ B4 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_402_Int__insert__left,axiom,
! [A: v,C2: set_v,B4: set_v] :
( ( ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B4 ) @ C2 )
= ( insert_v @ A @ ( inf_inf_set_v @ B4 @ C2 ) ) ) )
& ( ~ ( member_v @ A @ C2 )
=> ( ( inf_inf_set_v @ ( insert_v @ A @ B4 ) @ C2 )
= ( inf_inf_set_v @ B4 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_403_insert__commute,axiom,
! [X: v,Y: v,A3: set_v] :
( ( insert_v @ X @ ( insert_v @ Y @ A3 ) )
= ( insert_v @ Y @ ( insert_v @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_404_insert__commute,axiom,
! [X: product_prod_v_v,Y: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ X @ ( insert1338601472111419319od_v_v @ Y @ A3 ) )
= ( insert1338601472111419319od_v_v @ Y @ ( insert1338601472111419319od_v_v @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_405_insert__commute,axiom,
! [X: set_v,Y: set_v,A3: set_set_v] :
( ( insert_set_v @ X @ ( insert_set_v @ Y @ A3 ) )
= ( insert_set_v @ Y @ ( insert_set_v @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_406_insert__Diff__if,axiom,
! [X: set_v,B4: set_set_v,A3: set_set_v] :
( ( ( member_set_v @ X @ B4 )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X @ A3 ) @ B4 )
= ( minus_7228012346218142266_set_v @ A3 @ B4 ) ) )
& ( ~ ( member_set_v @ X @ B4 )
=> ( ( minus_7228012346218142266_set_v @ ( insert_set_v @ X @ A3 ) @ B4 )
= ( insert_set_v @ X @ ( minus_7228012346218142266_set_v @ A3 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_407_insert__Diff__if,axiom,
! [X: product_prod_v_v,B4: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ( member7453568604450474000od_v_v @ X @ B4 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ B4 )
= ( minus_4183494784930505774od_v_v @ A3 @ B4 ) ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ B4 )
=> ( ( minus_4183494784930505774od_v_v @ ( insert1338601472111419319od_v_v @ X @ A3 ) @ B4 )
= ( insert1338601472111419319od_v_v @ X @ ( minus_4183494784930505774od_v_v @ A3 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_408_insert__Diff__if,axiom,
! [X: v,B4: set_v,A3: set_v] :
( ( ( member_v @ X @ B4 )
=> ( ( minus_minus_set_v @ ( insert_v @ X @ A3 ) @ B4 )
= ( minus_minus_set_v @ A3 @ B4 ) ) )
& ( ~ ( member_v @ X @ B4 )
=> ( ( minus_minus_set_v @ ( insert_v @ X @ A3 ) @ B4 )
= ( insert_v @ X @ ( minus_minus_set_v @ A3 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_409_insert__eq__iff,axiom,
! [A: set_v,A3: set_set_v,B: set_v,B4: set_set_v] :
( ~ ( member_set_v @ A @ A3 )
=> ( ~ ( member_set_v @ B @ B4 )
=> ( ( ( insert_set_v @ A @ A3 )
= ( insert_set_v @ B @ B4 ) )
= ( ( ( A = B )
=> ( A3 = B4 ) )
& ( ( A != B )
=> ? [C3: set_set_v] :
( ( A3
= ( insert_set_v @ B @ C3 ) )
& ~ ( member_set_v @ B @ C3 )
& ( B4
= ( insert_set_v @ A @ C3 ) )
& ~ ( member_set_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_410_insert__eq__iff,axiom,
! [A: v,A3: set_v,B: v,B4: set_v] :
( ~ ( member_v @ A @ A3 )
=> ( ~ ( member_v @ B @ B4 )
=> ( ( ( insert_v @ A @ A3 )
= ( insert_v @ B @ B4 ) )
= ( ( ( A = B )
=> ( A3 = B4 ) )
& ( ( A != B )
=> ? [C3: set_v] :
( ( A3
= ( insert_v @ B @ C3 ) )
& ~ ( member_v @ B @ C3 )
& ( B4
= ( insert_v @ A @ C3 ) )
& ~ ( member_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_411_insert__eq__iff,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v,B: product_prod_v_v,B4: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ B @ B4 )
=> ( ( ( insert1338601472111419319od_v_v @ A @ A3 )
= ( insert1338601472111419319od_v_v @ B @ B4 ) )
= ( ( ( A = B )
=> ( A3 = B4 ) )
& ( ( A != B )
=> ? [C3: set_Product_prod_v_v] :
( ( A3
= ( insert1338601472111419319od_v_v @ B @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ B @ C3 )
& ( B4
= ( insert1338601472111419319od_v_v @ A @ C3 ) )
& ~ ( member7453568604450474000od_v_v @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_412_insert__absorb,axiom,
! [A: set_v,A3: set_set_v] :
( ( member_set_v @ A @ A3 )
=> ( ( insert_set_v @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_413_insert__absorb,axiom,
! [A: v,A3: set_v] :
( ( member_v @ A @ A3 )
=> ( ( insert_v @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_414_insert__absorb,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( insert1338601472111419319od_v_v @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_415_Diff__Diff__Int,axiom,
! [A3: set_v,B4: set_v] :
( ( minus_minus_set_v @ A3 @ ( minus_minus_set_v @ A3 @ B4 ) )
= ( inf_inf_set_v @ A3 @ B4 ) ) ).
% Diff_Diff_Int
thf(fact_416_insert__ident,axiom,
! [X: set_v,A3: set_set_v,B4: set_set_v] :
( ~ ( member_set_v @ X @ A3 )
=> ( ~ ( member_set_v @ X @ B4 )
=> ( ( ( insert_set_v @ X @ A3 )
= ( insert_set_v @ X @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_417_insert__ident,axiom,
! [X: v,A3: set_v,B4: set_v] :
( ~ ( member_v @ X @ A3 )
=> ( ~ ( member_v @ X @ B4 )
=> ( ( ( insert_v @ X @ A3 )
= ( insert_v @ X @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_418_insert__ident,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ X @ B4 )
=> ( ( ( insert1338601472111419319od_v_v @ X @ A3 )
= ( insert1338601472111419319od_v_v @ X @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_419_Diff__insert2,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v,B4: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B4 ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_420_Diff__insert2,axiom,
! [A3: set_set_v,A: set_v,B4: set_set_v] :
( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ A @ B4 ) )
= ( minus_7228012346218142266_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_421_Diff__insert2,axiom,
! [A3: set_v,A: v,B4: set_v] :
( ( minus_minus_set_v @ A3 @ ( insert_v @ A @ B4 ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v @ A @ bot_bot_set_v ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_422_insert__Diff,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ A3 )
=> ( ( insert1338601472111419319od_v_v @ A @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_423_insert__Diff,axiom,
! [A: set_v,A3: set_set_v] :
( ( member_set_v @ A @ A3 )
=> ( ( insert_set_v @ A @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_424_insert__Diff,axiom,
! [A: v,A3: set_v] :
( ( member_v @ A @ A3 )
=> ( ( insert_v @ A @ ( minus_minus_set_v @ A3 @ ( insert_v @ A @ bot_bot_set_v ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_425_Int__commute,axiom,
( inf_inf_set_v
= ( ^ [A6: set_v,B6: set_v] : ( inf_inf_set_v @ B6 @ A6 ) ) ) ).
% Int_commute
thf(fact_426_Diff__insert,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v,B4: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B4 ) )
= ( minus_4183494784930505774od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B4 ) @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ).
% Diff_insert
thf(fact_427_Diff__insert,axiom,
! [A3: set_set_v,A: set_v,B4: set_set_v] :
( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ A @ B4 ) )
= ( minus_7228012346218142266_set_v @ ( minus_7228012346218142266_set_v @ A3 @ B4 ) @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) ) ).
% Diff_insert
thf(fact_428_Diff__insert,axiom,
! [A3: set_v,A: v,B4: set_v] :
( ( minus_minus_set_v @ A3 @ ( insert_v @ A @ B4 ) )
= ( minus_minus_set_v @ ( minus_minus_set_v @ A3 @ B4 ) @ ( insert_v @ A @ bot_bot_set_v ) ) ) ).
% Diff_insert
thf(fact_429_Set_Oset__insert,axiom,
! [X: set_v,A3: set_set_v] :
( ( member_set_v @ X @ A3 )
=> ~ ! [B7: set_set_v] :
( ( A3
= ( insert_set_v @ X @ B7 ) )
=> ( member_set_v @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_430_Set_Oset__insert,axiom,
! [X: v,A3: set_v] :
( ( member_v @ X @ A3 )
=> ~ ! [B7: set_v] :
( ( A3
= ( insert_v @ X @ B7 ) )
=> ( member_v @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_431_Set_Oset__insert,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A3 )
=> ~ ! [B7: set_Product_prod_v_v] :
( ( A3
= ( insert1338601472111419319od_v_v @ X @ B7 ) )
=> ( member7453568604450474000od_v_v @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_432_Int__absorb,axiom,
! [A3: set_v] :
( ( inf_inf_set_v @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_433_Int__assoc,axiom,
! [A3: set_v,B4: set_v,C2: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A3 @ B4 ) @ C2 )
= ( inf_inf_set_v @ A3 @ ( inf_inf_set_v @ B4 @ C2 ) ) ) ).
% Int_assoc
thf(fact_434_Diff__triv,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A3 @ B4 )
= bot_bo723834152578015283od_v_v )
=> ( ( minus_4183494784930505774od_v_v @ A3 @ B4 )
= A3 ) ) ).
% Diff_triv
thf(fact_435_Diff__triv,axiom,
! [A3: set_set_v,B4: set_set_v] :
( ( ( inf_inf_set_set_v @ A3 @ B4 )
= bot_bot_set_set_v )
=> ( ( minus_7228012346218142266_set_v @ A3 @ B4 )
= A3 ) ) ).
% Diff_triv
thf(fact_436_Diff__triv,axiom,
! [A3: set_v,B4: set_v] :
( ( ( inf_inf_set_v @ A3 @ B4 )
= bot_bot_set_v )
=> ( ( minus_minus_set_v @ A3 @ B4 )
= A3 ) ) ).
% Diff_triv
thf(fact_437_Diff__Int2,axiom,
! [A3: set_v,C2: set_v,B4: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ C2 ) @ ( inf_inf_set_v @ B4 @ C2 ) )
= ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ C2 ) @ B4 ) ) ).
% Diff_Int2
thf(fact_438_insertI2,axiom,
! [A: set_v,B4: set_set_v,B: set_v] :
( ( member_set_v @ A @ B4 )
=> ( member_set_v @ A @ ( insert_set_v @ B @ B4 ) ) ) ).
% insertI2
thf(fact_439_insertI2,axiom,
! [A: v,B4: set_v,B: v] :
( ( member_v @ A @ B4 )
=> ( member_v @ A @ ( insert_v @ B @ B4 ) ) ) ).
% insertI2
thf(fact_440_insertI2,axiom,
! [A: product_prod_v_v,B4: set_Product_prod_v_v,B: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ B4 )
=> ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ B4 ) ) ) ).
% insertI2
thf(fact_441_insertI1,axiom,
! [A: set_v,B4: set_set_v] : ( member_set_v @ A @ ( insert_set_v @ A @ B4 ) ) ).
% insertI1
thf(fact_442_insertI1,axiom,
! [A: v,B4: set_v] : ( member_v @ A @ ( insert_v @ A @ B4 ) ) ).
% insertI1
thf(fact_443_insertI1,axiom,
! [A: product_prod_v_v,B4: set_Product_prod_v_v] : ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ A @ B4 ) ) ).
% insertI1
thf(fact_444_Int__Diff,axiom,
! [A3: set_v,B4: set_v,C2: set_v] :
( ( minus_minus_set_v @ ( inf_inf_set_v @ A3 @ B4 ) @ C2 )
= ( inf_inf_set_v @ A3 @ ( minus_minus_set_v @ B4 @ C2 ) ) ) ).
% Int_Diff
thf(fact_445_insertE,axiom,
! [A: set_v,B: set_v,A3: set_set_v] :
( ( member_set_v @ A @ ( insert_set_v @ B @ A3 ) )
=> ( ( A != B )
=> ( member_set_v @ A @ A3 ) ) ) ).
% insertE
thf(fact_446_insertE,axiom,
! [A: v,B: v,A3: set_v] :
( ( member_v @ A @ ( insert_v @ B @ A3 ) )
=> ( ( A != B )
=> ( member_v @ A @ A3 ) ) ) ).
% insertE
thf(fact_447_insertE,axiom,
! [A: product_prod_v_v,B: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ A3 ) )
=> ( ( A != B )
=> ( member7453568604450474000od_v_v @ A @ A3 ) ) ) ).
% insertE
thf(fact_448_DiffD2,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B4 ) )
=> ~ ( member7453568604450474000od_v_v @ C @ B4 ) ) ).
% DiffD2
thf(fact_449_DiffD2,axiom,
! [C: v,A3: set_v,B4: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A3 @ B4 ) )
=> ~ ( member_v @ C @ B4 ) ) ).
% DiffD2
thf(fact_450_DiffD1,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B4 ) )
=> ( member7453568604450474000od_v_v @ C @ A3 ) ) ).
% DiffD1
thf(fact_451_DiffD1,axiom,
! [C: v,A3: set_v,B4: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A3 @ B4 ) )
=> ( member_v @ C @ A3 ) ) ).
% DiffD1
thf(fact_452_IntD2,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B4 ) )
=> ( member7453568604450474000od_v_v @ C @ B4 ) ) ).
% IntD2
thf(fact_453_IntD2,axiom,
! [C: v,A3: set_v,B4: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A3 @ B4 ) )
=> ( member_v @ C @ B4 ) ) ).
% IntD2
thf(fact_454_IntD1,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B4 ) )
=> ( member7453568604450474000od_v_v @ C @ A3 ) ) ).
% IntD1
thf(fact_455_IntD1,axiom,
! [C: v,A3: set_v,B4: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A3 @ B4 ) )
=> ( member_v @ C @ A3 ) ) ).
% IntD1
thf(fact_456_DiffE,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( minus_4183494784930505774od_v_v @ A3 @ B4 ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( member7453568604450474000od_v_v @ C @ B4 ) ) ) ).
% DiffE
thf(fact_457_DiffE,axiom,
! [C: v,A3: set_v,B4: set_v] :
( ( member_v @ C @ ( minus_minus_set_v @ A3 @ B4 ) )
=> ~ ( ( member_v @ C @ A3 )
=> ( member_v @ C @ B4 ) ) ) ).
% DiffE
thf(fact_458_IntE,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( inf_in6271465464967711157od_v_v @ A3 @ B4 ) )
=> ~ ( ( member7453568604450474000od_v_v @ C @ A3 )
=> ~ ( member7453568604450474000od_v_v @ C @ B4 ) ) ) ).
% IntE
thf(fact_459_IntE,axiom,
! [C: v,A3: set_v,B4: set_v] :
( ( member_v @ C @ ( inf_inf_set_v @ A3 @ B4 ) )
=> ~ ( ( member_v @ C @ A3 )
=> ~ ( member_v @ C @ B4 ) ) ) ).
% IntE
thf(fact_460_insert__subsetI,axiom,
! [X: set_v,A3: set_set_v,X6: set_set_v] :
( ( member_set_v @ X @ A3 )
=> ( ( ord_le5216385588623774835_set_v @ X6 @ A3 )
=> ( ord_le5216385588623774835_set_v @ ( insert_set_v @ X @ X6 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_461_insert__subsetI,axiom,
! [X: v,A3: set_v,X6: set_v] :
( ( member_v @ X @ A3 )
=> ( ( ord_less_eq_set_v @ X6 @ A3 )
=> ( ord_less_eq_set_v @ ( insert_v @ X @ X6 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_462_insert__subsetI,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,X6: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ X6 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ X @ X6 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_463_finite__empty__induct,axiom,
! [A3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ A3 )
=> ( ( P @ A3 )
=> ( ! [A4: product_prod_v_v,A7: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A7 )
=> ( ( member7453568604450474000od_v_v @ A4 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_4183494784930505774od_v_v @ A7 @ ( insert1338601472111419319od_v_v @ A4 @ bot_bo723834152578015283od_v_v ) ) ) ) ) )
=> ( P @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% finite_empty_induct
thf(fact_464_finite__empty__induct,axiom,
! [A3: set_set_v,P: set_set_v > $o] :
( ( finite_finite_set_v @ A3 )
=> ( ( P @ A3 )
=> ( ! [A4: set_v,A7: set_set_v] :
( ( finite_finite_set_v @ A7 )
=> ( ( member_set_v @ A4 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_7228012346218142266_set_v @ A7 @ ( insert_set_v @ A4 @ bot_bot_set_set_v ) ) ) ) ) )
=> ( P @ bot_bot_set_set_v ) ) ) ) ).
% finite_empty_induct
thf(fact_465_finite__empty__induct,axiom,
! [A3: set_v,P: set_v > $o] :
( ( finite_finite_v @ A3 )
=> ( ( P @ A3 )
=> ( ! [A4: v,A7: set_v] :
( ( finite_finite_v @ A7 )
=> ( ( member_v @ A4 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus_set_v @ A7 @ ( insert_v @ A4 @ bot_bot_set_v ) ) ) ) ) )
=> ( P @ bot_bot_set_v ) ) ) ) ).
% finite_empty_induct
thf(fact_466_infinite__coinduct,axiom,
! [X6: set_Product_prod_v_v > $o,A3: set_Product_prod_v_v] :
( ( X6 @ A3 )
=> ( ! [A7: set_Product_prod_v_v] :
( ( X6 @ A7 )
=> ? [X5: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X5 @ A7 )
& ( ( X6 @ ( minus_4183494784930505774od_v_v @ A7 @ ( insert1338601472111419319od_v_v @ X5 @ bot_bo723834152578015283od_v_v ) ) )
| ~ ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ A7 @ ( insert1338601472111419319od_v_v @ X5 @ bot_bo723834152578015283od_v_v ) ) ) ) ) )
=> ~ ( finite3348123685078250256od_v_v @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_467_infinite__coinduct,axiom,
! [X6: set_set_v > $o,A3: set_set_v] :
( ( X6 @ A3 )
=> ( ! [A7: set_set_v] :
( ( X6 @ A7 )
=> ? [X5: set_v] :
( ( member_set_v @ X5 @ A7 )
& ( ( X6 @ ( minus_7228012346218142266_set_v @ A7 @ ( insert_set_v @ X5 @ bot_bot_set_set_v ) ) )
| ~ ( finite_finite_set_v @ ( minus_7228012346218142266_set_v @ A7 @ ( insert_set_v @ X5 @ bot_bot_set_set_v ) ) ) ) ) )
=> ~ ( finite_finite_set_v @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_468_infinite__coinduct,axiom,
! [X6: set_v > $o,A3: set_v] :
( ( X6 @ A3 )
=> ( ! [A7: set_v] :
( ( X6 @ A7 )
=> ? [X5: v] :
( ( member_v @ X5 @ A7 )
& ( ( X6 @ ( minus_minus_set_v @ A7 @ ( insert_v @ X5 @ bot_bot_set_v ) ) )
| ~ ( finite_finite_v @ ( minus_minus_set_v @ A7 @ ( insert_v @ X5 @ bot_bot_set_v ) ) ) ) ) )
=> ~ ( finite_finite_v @ A3 ) ) ) ).
% infinite_coinduct
thf(fact_469_infinite__remove,axiom,
! [S4: set_Product_prod_v_v,A: product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ S4 )
=> ~ ( finite3348123685078250256od_v_v @ ( minus_4183494784930505774od_v_v @ S4 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% infinite_remove
thf(fact_470_infinite__remove,axiom,
! [S4: set_set_v,A: set_v] :
( ~ ( finite_finite_set_v @ S4 )
=> ~ ( finite_finite_set_v @ ( minus_7228012346218142266_set_v @ S4 @ ( insert_set_v @ A @ bot_bot_set_set_v ) ) ) ) ).
% infinite_remove
thf(fact_471_infinite__remove,axiom,
! [S4: set_v,A: v] :
( ~ ( finite_finite_v @ S4 )
=> ~ ( finite_finite_v @ ( minus_minus_set_v @ S4 @ ( insert_v @ A @ bot_bot_set_v ) ) ) ) ).
% infinite_remove
thf(fact_472_Diff__single__insert,axiom,
! [A3: set_set_v,X: set_v,B4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) @ B4 )
=> ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_473_Diff__single__insert,axiom,
! [A3: set_v,X: v,B4: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v @ X @ bot_bot_set_v ) ) @ B4 )
=> ( ord_less_eq_set_v @ A3 @ ( insert_v @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_474_Diff__single__insert,axiom,
! [A3: set_Product_prod_v_v,X: product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B4 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_475_subset__insert__iff,axiom,
! [A3: set_set_v,X: set_v,B4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ X @ B4 ) )
= ( ( ( member_set_v @ X @ A3 )
=> ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) @ B4 ) )
& ( ~ ( member_set_v @ X @ A3 )
=> ( ord_le5216385588623774835_set_v @ A3 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_476_subset__insert__iff,axiom,
! [A3: set_v,X: v,B4: set_v] :
( ( ord_less_eq_set_v @ A3 @ ( insert_v @ X @ B4 ) )
= ( ( ( member_v @ X @ A3 )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ ( insert_v @ X @ bot_bot_set_v ) ) @ B4 ) )
& ( ~ ( member_v @ X @ A3 )
=> ( ord_less_eq_set_v @ A3 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_477_subset__insert__iff,axiom,
! [A3: set_Product_prod_v_v,X: product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ B4 ) )
= ( ( ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) @ B4 ) )
& ( ~ ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_478_bot__set__def,axiom,
( bot_bo723834152578015283od_v_v
= ( collec140062887454715474od_v_v @ bot_bo8461541820394803818_v_v_o ) ) ).
% bot_set_def
thf(fact_479_bot__set__def,axiom,
( bot_bot_set_v
= ( collect_v @ bot_bot_v_o ) ) ).
% bot_set_def
thf(fact_480_bot__set__def,axiom,
( bot_bot_set_set_v
= ( collect_set_v @ bot_bot_set_v_o ) ) ).
% bot_set_def
thf(fact_481_Int__emptyI,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A3 )
=> ~ ( member7453568604450474000od_v_v @ X4 @ B4 ) )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ B4 )
= bot_bo723834152578015283od_v_v ) ) ).
% Int_emptyI
thf(fact_482_Int__emptyI,axiom,
! [A3: set_v,B4: set_v] :
( ! [X4: v] :
( ( member_v @ X4 @ A3 )
=> ~ ( member_v @ X4 @ B4 ) )
=> ( ( inf_inf_set_v @ A3 @ B4 )
= bot_bot_set_v ) ) ).
% Int_emptyI
thf(fact_483_Int__emptyI,axiom,
! [A3: set_set_v,B4: set_set_v] :
( ! [X4: set_v] :
( ( member_set_v @ X4 @ A3 )
=> ~ ( member_set_v @ X4 @ B4 ) )
=> ( ( inf_inf_set_set_v @ A3 @ B4 )
= bot_bot_set_set_v ) ) ).
% Int_emptyI
thf(fact_484_singletonD,axiom,
! [B: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_485_singletonD,axiom,
! [B: v,A: v] :
( ( member_v @ B @ ( insert_v @ A @ bot_bot_set_v ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_486_singletonD,axiom,
! [B: set_v,A: set_v] :
( ( member_set_v @ B @ ( insert_set_v @ A @ bot_bot_set_set_v ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_487_disjoint__iff,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A3 @ B4 )
= bot_bo723834152578015283od_v_v )
= ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ~ ( member7453568604450474000od_v_v @ X3 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_488_disjoint__iff,axiom,
! [A3: set_v,B4: set_v] :
( ( ( inf_inf_set_v @ A3 @ B4 )
= bot_bot_set_v )
= ( ! [X3: v] :
( ( member_v @ X3 @ A3 )
=> ~ ( member_v @ X3 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_489_disjoint__iff,axiom,
! [A3: set_set_v,B4: set_set_v] :
( ( ( inf_inf_set_set_v @ A3 @ B4 )
= bot_bot_set_set_v )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
=> ~ ( member_set_v @ X3 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_490_singleton__iff,axiom,
! [B: product_prod_v_v,A: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ B @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_491_singleton__iff,axiom,
! [B: v,A: v] :
( ( member_v @ B @ ( insert_v @ A @ bot_bot_set_v ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_492_singleton__iff,axiom,
! [B: set_v,A: set_v] :
( ( member_set_v @ B @ ( insert_set_v @ A @ bot_bot_set_set_v ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_493_Int__empty__left,axiom,
! [B4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ bot_bo723834152578015283od_v_v @ B4 )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_left
thf(fact_494_Int__empty__left,axiom,
! [B4: set_v] :
( ( inf_inf_set_v @ bot_bot_set_v @ B4 )
= bot_bot_set_v ) ).
% Int_empty_left
thf(fact_495_Int__empty__left,axiom,
! [B4: set_set_v] :
( ( inf_inf_set_set_v @ bot_bot_set_set_v @ B4 )
= bot_bot_set_set_v ) ).
% Int_empty_left
thf(fact_496_Int__empty__right,axiom,
! [A3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Int_empty_right
thf(fact_497_Int__empty__right,axiom,
! [A3: set_v] :
( ( inf_inf_set_v @ A3 @ bot_bot_set_v )
= bot_bot_set_v ) ).
% Int_empty_right
thf(fact_498_Int__empty__right,axiom,
! [A3: set_set_v] :
( ( inf_inf_set_set_v @ A3 @ bot_bot_set_set_v )
= bot_bot_set_set_v ) ).
% Int_empty_right
thf(fact_499_doubleton__eq__iff,axiom,
! [A: product_prod_v_v,B: product_prod_v_v,C: product_prod_v_v,D: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) )
= ( insert1338601472111419319od_v_v @ C @ ( insert1338601472111419319od_v_v @ D @ bot_bo723834152578015283od_v_v ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_500_doubleton__eq__iff,axiom,
! [A: v,B: v,C: v,D: v] :
( ( ( insert_v @ A @ ( insert_v @ B @ bot_bot_set_v ) )
= ( insert_v @ C @ ( insert_v @ D @ bot_bot_set_v ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_501_doubleton__eq__iff,axiom,
! [A: set_v,B: set_v,C: set_v,D: set_v] :
( ( ( insert_set_v @ A @ ( insert_set_v @ B @ bot_bot_set_set_v ) )
= ( insert_set_v @ C @ ( insert_set_v @ D @ bot_bot_set_set_v ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_502_insert__not__empty,axiom,
! [A: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( insert1338601472111419319od_v_v @ A @ A3 )
!= bot_bo723834152578015283od_v_v ) ).
% insert_not_empty
thf(fact_503_insert__not__empty,axiom,
! [A: v,A3: set_v] :
( ( insert_v @ A @ A3 )
!= bot_bot_set_v ) ).
% insert_not_empty
thf(fact_504_insert__not__empty,axiom,
! [A: set_v,A3: set_set_v] :
( ( insert_set_v @ A @ A3 )
!= bot_bot_set_set_v ) ).
% insert_not_empty
thf(fact_505_singleton__inject,axiom,
! [A: product_prod_v_v,B: product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v )
= ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_506_singleton__inject,axiom,
! [A: v,B: v] :
( ( ( insert_v @ A @ bot_bot_set_v )
= ( insert_v @ B @ bot_bot_set_v ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_507_singleton__inject,axiom,
! [A: set_v,B: set_v] :
( ( ( insert_set_v @ A @ bot_bot_set_set_v )
= ( insert_set_v @ B @ bot_bot_set_set_v ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_508_disjoint__iff__not__equal,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ( inf_in6271465464967711157od_v_v @ A3 @ B4 )
= bot_bo723834152578015283od_v_v )
= ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ! [Y3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ Y3 @ B4 )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_509_disjoint__iff__not__equal,axiom,
! [A3: set_v,B4: set_v] :
( ( ( inf_inf_set_v @ A3 @ B4 )
= bot_bot_set_v )
= ( ! [X3: v] :
( ( member_v @ X3 @ A3 )
=> ! [Y3: v] :
( ( member_v @ Y3 @ B4 )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_510_disjoint__iff__not__equal,axiom,
! [A3: set_set_v,B4: set_set_v] :
( ( ( inf_inf_set_set_v @ A3 @ B4 )
= bot_bot_set_set_v )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
=> ! [Y3: set_v] :
( ( member_set_v @ Y3 @ B4 )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_511_finite_OinsertI,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A3 )
=> ( finite3348123685078250256od_v_v @ ( insert1338601472111419319od_v_v @ A @ A3 ) ) ) ).
% finite.insertI
thf(fact_512_finite_OinsertI,axiom,
! [A3: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( finite_finite_set_v @ ( insert_set_v @ A @ A3 ) ) ) ).
% finite.insertI
thf(fact_513_finite_OinsertI,axiom,
! [A3: set_v,A: v] :
( ( finite_finite_v @ A3 )
=> ( finite_finite_v @ ( insert_v @ A @ A3 ) ) ) ).
% finite.insertI
thf(fact_514_Int__mono,axiom,
! [A3: set_v,C2: set_v,B4: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A3 @ C2 )
=> ( ( ord_less_eq_set_v @ B4 @ D2 )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B4 ) @ ( inf_inf_set_v @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_515_Int__mono,axiom,
! [A3: set_Product_prod_v_v,C2: set_Product_prod_v_v,B4: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B4 @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B4 ) @ ( inf_in6271465464967711157od_v_v @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_516_Int__lower1,axiom,
! [A3: set_v,B4: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B4 ) @ A3 ) ).
% Int_lower1
thf(fact_517_Int__lower1,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B4 ) @ A3 ) ).
% Int_lower1
thf(fact_518_Int__lower2,axiom,
! [A3: set_v,B4: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_519_Int__lower2,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_520_Int__absorb1,axiom,
! [B4: set_v,A3: set_v] :
( ( ord_less_eq_set_v @ B4 @ A3 )
=> ( ( inf_inf_set_v @ A3 @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_521_Int__absorb1,axiom,
! [B4: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B4 @ A3 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_522_Int__absorb2,axiom,
! [A3: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ A3 @ B4 )
=> ( ( inf_inf_set_v @ A3 @ B4 )
= A3 ) ) ).
% Int_absorb2
thf(fact_523_Int__absorb2,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B4 )
=> ( ( inf_in6271465464967711157od_v_v @ A3 @ B4 )
= A3 ) ) ).
% Int_absorb2
thf(fact_524_insert__mono,axiom,
! [C2: set_set_v,D2: set_set_v,A: set_v] :
( ( ord_le5216385588623774835_set_v @ C2 @ D2 )
=> ( ord_le5216385588623774835_set_v @ ( insert_set_v @ A @ C2 ) @ ( insert_set_v @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_525_insert__mono,axiom,
! [C2: set_v,D2: set_v,A: v] :
( ( ord_less_eq_set_v @ C2 @ D2 )
=> ( ord_less_eq_set_v @ ( insert_v @ A @ C2 ) @ ( insert_v @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_526_insert__mono,axiom,
! [C2: set_Product_prod_v_v,D2: set_Product_prod_v_v,A: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( insert1338601472111419319od_v_v @ A @ C2 ) @ ( insert1338601472111419319od_v_v @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_527_Int__greatest,axiom,
! [C2: set_v,A3: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ C2 @ A3 )
=> ( ( ord_less_eq_set_v @ C2 @ B4 )
=> ( ord_less_eq_set_v @ C2 @ ( inf_inf_set_v @ A3 @ B4 ) ) ) ) ).
% Int_greatest
thf(fact_528_Int__greatest,axiom,
! [C2: set_Product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ C2 @ B4 )
=> ( ord_le7336532860387713383od_v_v @ C2 @ ( inf_in6271465464967711157od_v_v @ A3 @ B4 ) ) ) ) ).
% Int_greatest
thf(fact_529_subset__insert,axiom,
! [X: set_v,A3: set_set_v,B4: set_set_v] :
( ~ ( member_set_v @ X @ A3 )
=> ( ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ X @ B4 ) )
= ( ord_le5216385588623774835_set_v @ A3 @ B4 ) ) ) ).
% subset_insert
thf(fact_530_subset__insert,axiom,
! [X: v,A3: set_v,B4: set_v] :
( ~ ( member_v @ X @ A3 )
=> ( ( ord_less_eq_set_v @ A3 @ ( insert_v @ X @ B4 ) )
= ( ord_less_eq_set_v @ A3 @ B4 ) ) ) ).
% subset_insert
thf(fact_531_subset__insert,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ~ ( member7453568604450474000od_v_v @ X @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ B4 ) )
= ( ord_le7336532860387713383od_v_v @ A3 @ B4 ) ) ) ).
% subset_insert
thf(fact_532_subset__insertI,axiom,
! [B4: set_set_v,A: set_v] : ( ord_le5216385588623774835_set_v @ B4 @ ( insert_set_v @ A @ B4 ) ) ).
% subset_insertI
thf(fact_533_subset__insertI,axiom,
! [B4: set_v,A: v] : ( ord_less_eq_set_v @ B4 @ ( insert_v @ A @ B4 ) ) ).
% subset_insertI
thf(fact_534_subset__insertI,axiom,
! [B4: set_Product_prod_v_v,A: product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B4 @ ( insert1338601472111419319od_v_v @ A @ B4 ) ) ).
% subset_insertI
thf(fact_535_subset__insertI2,axiom,
! [A3: set_set_v,B4: set_set_v,B: set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ B4 )
=> ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_536_subset__insertI2,axiom,
! [A3: set_v,B4: set_v,B: v] :
( ( ord_less_eq_set_v @ A3 @ B4 )
=> ( ord_less_eq_set_v @ A3 @ ( insert_v @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_537_subset__insertI2,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v,B: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B4 )
=> ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_538_Int__Collect__mono,axiom,
! [A3: set_set_v,B4: set_set_v,P: set_v > $o,Q: set_v > $o] :
( ( ord_le5216385588623774835_set_v @ A3 @ B4 )
=> ( ! [X4: set_v] :
( ( member_set_v @ X4 @ A3 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le5216385588623774835_set_v @ ( inf_inf_set_set_v @ A3 @ ( collect_set_v @ P ) ) @ ( inf_inf_set_set_v @ B4 @ ( collect_set_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_539_Int__Collect__mono,axiom,
! [A3: set_v,B4: set_v,P: v > $o,Q: v > $o] :
( ( ord_less_eq_set_v @ A3 @ B4 )
=> ( ! [X4: v] :
( ( member_v @ X4 @ A3 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A3 @ ( collect_v @ P ) ) @ ( inf_inf_set_v @ B4 @ ( collect_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_540_Int__Collect__mono,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v,P: product_prod_v_v > $o,Q: product_prod_v_v > $o] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B4 )
=> ( ! [X4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A3 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ ( collec140062887454715474od_v_v @ P ) ) @ ( inf_in6271465464967711157od_v_v @ B4 @ ( collec140062887454715474od_v_v @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_541_Diff__infinite__finite,axiom,
! [T: set_v,S4: set_v] :
( ( finite_finite_v @ T )
=> ( ~ ( finite_finite_v @ S4 )
=> ~ ( finite_finite_v @ ( minus_minus_set_v @ S4 @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_542_Diff__mono,axiom,
! [A3: set_v,C2: set_v,D2: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ A3 @ C2 )
=> ( ( ord_less_eq_set_v @ D2 @ B4 )
=> ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ B4 ) @ ( minus_minus_set_v @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_543_Diff__mono,axiom,
! [A3: set_Product_prod_v_v,C2: set_Product_prod_v_v,D2: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ D2 @ B4 )
=> ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B4 ) @ ( minus_4183494784930505774od_v_v @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_544_Diff__subset,axiom,
! [A3: set_v,B4: set_v] : ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ B4 ) @ A3 ) ).
% Diff_subset
thf(fact_545_Diff__subset,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B4 ) @ A3 ) ).
% Diff_subset
thf(fact_546_double__diff,axiom,
! [A3: set_v,B4: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ A3 @ B4 )
=> ( ( ord_less_eq_set_v @ B4 @ C2 )
=> ( ( minus_minus_set_v @ B4 @ ( minus_minus_set_v @ C2 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_547_double__diff,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B4 )
=> ( ( ord_le7336532860387713383od_v_v @ B4 @ C2 )
=> ( ( minus_4183494784930505774od_v_v @ B4 @ ( minus_4183494784930505774od_v_v @ C2 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_548_finite__remove__induct,axiom,
! [B4: set_set_v,P: set_set_v > $o] :
( ( finite_finite_set_v @ B4 )
=> ( ( P @ bot_bot_set_set_v )
=> ( ! [A7: set_set_v] :
( ( finite_finite_set_v @ A7 )
=> ( ( A7 != bot_bot_set_set_v )
=> ( ( ord_le5216385588623774835_set_v @ A7 @ B4 )
=> ( ! [X5: set_v] :
( ( member_set_v @ X5 @ A7 )
=> ( P @ ( minus_7228012346218142266_set_v @ A7 @ ( insert_set_v @ X5 @ bot_bot_set_set_v ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_549_finite__remove__induct,axiom,
! [B4: set_v,P: set_v > $o] :
( ( finite_finite_v @ B4 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [A7: set_v] :
( ( finite_finite_v @ A7 )
=> ( ( A7 != bot_bot_set_v )
=> ( ( ord_less_eq_set_v @ A7 @ B4 )
=> ( ! [X5: v] :
( ( member_v @ X5 @ A7 )
=> ( P @ ( minus_minus_set_v @ A7 @ ( insert_v @ X5 @ bot_bot_set_v ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_550_finite__remove__induct,axiom,
! [B4: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ B4 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [A7: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A7 )
=> ( ( A7 != bot_bo723834152578015283od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ A7 @ B4 )
=> ( ! [X5: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X5 @ A7 )
=> ( P @ ( minus_4183494784930505774od_v_v @ A7 @ ( insert1338601472111419319od_v_v @ X5 @ bot_bo723834152578015283od_v_v ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_551_remove__induct,axiom,
! [P: set_set_v > $o,B4: set_set_v] :
( ( P @ bot_bot_set_set_v )
=> ( ( ~ ( finite_finite_set_v @ B4 )
=> ( P @ B4 ) )
=> ( ! [A7: set_set_v] :
( ( finite_finite_set_v @ A7 )
=> ( ( A7 != bot_bot_set_set_v )
=> ( ( ord_le5216385588623774835_set_v @ A7 @ B4 )
=> ( ! [X5: set_v] :
( ( member_set_v @ X5 @ A7 )
=> ( P @ ( minus_7228012346218142266_set_v @ A7 @ ( insert_set_v @ X5 @ bot_bot_set_set_v ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_552_remove__induct,axiom,
! [P: set_v > $o,B4: set_v] :
( ( P @ bot_bot_set_v )
=> ( ( ~ ( finite_finite_v @ B4 )
=> ( P @ B4 ) )
=> ( ! [A7: set_v] :
( ( finite_finite_v @ A7 )
=> ( ( A7 != bot_bot_set_v )
=> ( ( ord_less_eq_set_v @ A7 @ B4 )
=> ( ! [X5: v] :
( ( member_v @ X5 @ A7 )
=> ( P @ ( minus_minus_set_v @ A7 @ ( insert_v @ X5 @ bot_bot_set_v ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_553_remove__induct,axiom,
! [P: set_Product_prod_v_v > $o,B4: set_Product_prod_v_v] :
( ( P @ bot_bo723834152578015283od_v_v )
=> ( ( ~ ( finite3348123685078250256od_v_v @ B4 )
=> ( P @ B4 ) )
=> ( ! [A7: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A7 )
=> ( ( A7 != bot_bo723834152578015283od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ A7 @ B4 )
=> ( ! [X5: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X5 @ A7 )
=> ( P @ ( minus_4183494784930505774od_v_v @ A7 @ ( insert1338601472111419319od_v_v @ X5 @ bot_bo723834152578015283od_v_v ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_554_select__convs_I5_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Vsuccs ) ).
% select_convs(5)
thf(fact_555_infinite__finite__induct,axiom,
! [P: set_Product_prod_v_v > $o,A3: set_Product_prod_v_v] :
( ! [A7: set_Product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [X4: product_prod_v_v,F4: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F4 )
=> ( ~ ( member7453568604450474000od_v_v @ X4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert1338601472111419319od_v_v @ X4 @ F4 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_556_infinite__finite__induct,axiom,
! [P: set_v > $o,A3: set_v] :
( ! [A7: set_v] :
( ~ ( finite_finite_v @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_v )
=> ( ! [X4: v,F4: set_v] :
( ( finite_finite_v @ F4 )
=> ( ~ ( member_v @ X4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_v @ X4 @ F4 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_557_infinite__finite__induct,axiom,
! [P: set_set_v > $o,A3: set_set_v] :
( ! [A7: set_set_v] :
( ~ ( finite_finite_set_v @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_set_v )
=> ( ! [X4: set_v,F4: set_set_v] :
( ( finite_finite_set_v @ F4 )
=> ( ~ ( member_set_v @ X4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_set_v @ X4 @ F4 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% infinite_finite_induct
thf(fact_558_finite__ne__induct,axiom,
! [F3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( F3 != bot_bo723834152578015283od_v_v )
=> ( ! [X4: product_prod_v_v] : ( P @ ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) )
=> ( ! [X4: product_prod_v_v,F4: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F4 )
=> ( ( F4 != bot_bo723834152578015283od_v_v )
=> ( ~ ( member7453568604450474000od_v_v @ X4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert1338601472111419319od_v_v @ X4 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_559_finite__ne__induct,axiom,
! [F3: set_v,P: set_v > $o] :
( ( finite_finite_v @ F3 )
=> ( ( F3 != bot_bot_set_v )
=> ( ! [X4: v] : ( P @ ( insert_v @ X4 @ bot_bot_set_v ) )
=> ( ! [X4: v,F4: set_v] :
( ( finite_finite_v @ F4 )
=> ( ( F4 != bot_bot_set_v )
=> ( ~ ( member_v @ X4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_v @ X4 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_560_finite__ne__induct,axiom,
! [F3: set_set_v,P: set_set_v > $o] :
( ( finite_finite_set_v @ F3 )
=> ( ( F3 != bot_bot_set_set_v )
=> ( ! [X4: set_v] : ( P @ ( insert_set_v @ X4 @ bot_bot_set_set_v ) )
=> ( ! [X4: set_v,F4: set_set_v] :
( ( finite_finite_set_v @ F4 )
=> ( ( F4 != bot_bot_set_set_v )
=> ( ~ ( member_set_v @ X4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_set_v @ X4 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_561_finite__induct,axiom,
! [F3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [X4: product_prod_v_v,F4: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F4 )
=> ( ~ ( member7453568604450474000od_v_v @ X4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert1338601472111419319od_v_v @ X4 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_562_finite__induct,axiom,
! [F3: set_v,P: set_v > $o] :
( ( finite_finite_v @ F3 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [X4: v,F4: set_v] :
( ( finite_finite_v @ F4 )
=> ( ~ ( member_v @ X4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_v @ X4 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_563_finite__induct,axiom,
! [F3: set_set_v,P: set_set_v > $o] :
( ( finite_finite_set_v @ F3 )
=> ( ( P @ bot_bot_set_set_v )
=> ( ! [X4: set_v,F4: set_set_v] :
( ( finite_finite_set_v @ F4 )
=> ( ~ ( member_set_v @ X4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_set_v @ X4 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_564_finite_Osimps,axiom,
( finite3348123685078250256od_v_v
= ( ^ [A5: set_Product_prod_v_v] :
( ( A5 = bot_bo723834152578015283od_v_v )
| ? [A6: set_Product_prod_v_v,B5: product_prod_v_v] :
( ( A5
= ( insert1338601472111419319od_v_v @ B5 @ A6 ) )
& ( finite3348123685078250256od_v_v @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_565_finite_Osimps,axiom,
( finite_finite_v
= ( ^ [A5: set_v] :
( ( A5 = bot_bot_set_v )
| ? [A6: set_v,B5: v] :
( ( A5
= ( insert_v @ B5 @ A6 ) )
& ( finite_finite_v @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_566_finite_Osimps,axiom,
( finite_finite_set_v
= ( ^ [A5: set_set_v] :
( ( A5 = bot_bot_set_set_v )
| ? [A6: set_set_v,B5: set_v] :
( ( A5
= ( insert_set_v @ B5 @ A6 ) )
& ( finite_finite_set_v @ A6 ) ) ) ) ) ).
% finite.simps
thf(fact_567_finite_Ocases,axiom,
! [A: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ A )
=> ( ( A != bot_bo723834152578015283od_v_v )
=> ~ ! [A7: set_Product_prod_v_v] :
( ? [A4: product_prod_v_v] :
( A
= ( insert1338601472111419319od_v_v @ A4 @ A7 ) )
=> ~ ( finite3348123685078250256od_v_v @ A7 ) ) ) ) ).
% finite.cases
thf(fact_568_finite_Ocases,axiom,
! [A: set_v] :
( ( finite_finite_v @ A )
=> ( ( A != bot_bot_set_v )
=> ~ ! [A7: set_v] :
( ? [A4: v] :
( A
= ( insert_v @ A4 @ A7 ) )
=> ~ ( finite_finite_v @ A7 ) ) ) ) ).
% finite.cases
thf(fact_569_finite_Ocases,axiom,
! [A: set_set_v] :
( ( finite_finite_set_v @ A )
=> ( ( A != bot_bot_set_set_v )
=> ~ ! [A7: set_set_v] :
( ? [A4: set_v] :
( A
= ( insert_set_v @ A4 @ A7 ) )
=> ~ ( finite_finite_set_v @ A7 ) ) ) ) ).
% finite.cases
thf(fact_570_subset__singleton__iff,axiom,
! [X6: set_set_v,A: set_v] :
( ( ord_le5216385588623774835_set_v @ X6 @ ( insert_set_v @ A @ bot_bot_set_set_v ) )
= ( ( X6 = bot_bot_set_set_v )
| ( X6
= ( insert_set_v @ A @ bot_bot_set_set_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_571_subset__singleton__iff,axiom,
! [X6: set_v,A: v] :
( ( ord_less_eq_set_v @ X6 @ ( insert_v @ A @ bot_bot_set_v ) )
= ( ( X6 = bot_bot_set_v )
| ( X6
= ( insert_v @ A @ bot_bot_set_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_572_subset__singleton__iff,axiom,
! [X6: set_Product_prod_v_v,A: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X6 @ ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) )
= ( ( X6 = bot_bo723834152578015283od_v_v )
| ( X6
= ( insert1338601472111419319od_v_v @ A @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singleton_iff
thf(fact_573_subset__singletonD,axiom,
! [A3: set_set_v,X: set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
=> ( ( A3 = bot_bot_set_set_v )
| ( A3
= ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ).
% subset_singletonD
thf(fact_574_subset__singletonD,axiom,
! [A3: set_v,X: v] :
( ( ord_less_eq_set_v @ A3 @ ( insert_v @ X @ bot_bot_set_v ) )
=> ( ( A3 = bot_bot_set_v )
| ( A3
= ( insert_v @ X @ bot_bot_set_v ) ) ) ) ).
% subset_singletonD
thf(fact_575_subset__singletonD,axiom,
! [A3: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
=> ( ( A3 = bot_bo723834152578015283od_v_v )
| ( A3
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% subset_singletonD
thf(fact_576_graph_Oreachable__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,V: v,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V @ W )
=> ( ! [X4: v] :
( ( member_v @ X4 @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ! [Xa2: v] :
( ( member_v @ Xa2 @ ( minus_minus_set_v @ ( Successors @ X4 ) @ ( sCC_Bl3795065053823578884t_unit @ E @ X4 ) ) )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ V @ X4 )
=> ~ ( sCC_Bl649662514949026229able_v @ Successors @ Xa2 @ W ) ) ) )
=> ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ) ) ).
% graph.reachable_visited
thf(fact_577_graph_Oscc__partition,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S4: set_Product_prod_v_v,S6: set_Product_prod_v_v,X: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S4 )
=> ( ( sCC_Bl6242042402218619277od_v_v @ Successors @ S6 )
=> ( ( member7453568604450474000od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ S4 @ S6 ) )
=> ( S4 = S6 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_578_graph_Oscc__partition,axiom,
! [Vertices: set_v,Successors: v > set_v,S4: set_v,S6: set_v,X: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S4 )
=> ( ( sCC_Bloemen_is_scc_v @ Successors @ S6 )
=> ( ( member_v @ X @ ( inf_inf_set_v @ S4 @ S6 ) )
=> ( S4 = S6 ) ) ) ) ) ).
% graph.scc_partition
thf(fact_579_finite__subset__induct,axiom,
! [F3: set_set_v,A3: set_set_v,P: set_set_v > $o] :
( ( finite_finite_set_v @ F3 )
=> ( ( ord_le5216385588623774835_set_v @ F3 @ A3 )
=> ( ( P @ bot_bot_set_set_v )
=> ( ! [A4: set_v,F4: set_set_v] :
( ( finite_finite_set_v @ F4 )
=> ( ( member_set_v @ A4 @ A3 )
=> ( ~ ( member_set_v @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_set_v @ A4 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_580_finite__subset__induct,axiom,
! [F3: set_v,A3: set_v,P: set_v > $o] :
( ( finite_finite_v @ F3 )
=> ( ( ord_less_eq_set_v @ F3 @ A3 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [A4: v,F4: set_v] :
( ( finite_finite_v @ F4 )
=> ( ( member_v @ A4 @ A3 )
=> ( ~ ( member_v @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_v @ A4 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_581_finite__subset__induct,axiom,
! [F3: set_Product_prod_v_v,A3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( ord_le7336532860387713383od_v_v @ F3 @ A3 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [A4: product_prod_v_v,F4: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F4 )
=> ( ( member7453568604450474000od_v_v @ A4 @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert1338601472111419319od_v_v @ A4 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_582_finite__subset__induct_H,axiom,
! [F3: set_set_v,A3: set_set_v,P: set_set_v > $o] :
( ( finite_finite_set_v @ F3 )
=> ( ( ord_le5216385588623774835_set_v @ F3 @ A3 )
=> ( ( P @ bot_bot_set_set_v )
=> ( ! [A4: set_v,F4: set_set_v] :
( ( finite_finite_set_v @ F4 )
=> ( ( member_set_v @ A4 @ A3 )
=> ( ( ord_le5216385588623774835_set_v @ F4 @ A3 )
=> ( ~ ( member_set_v @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_set_v @ A4 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_583_finite__subset__induct_H,axiom,
! [F3: set_v,A3: set_v,P: set_v > $o] :
( ( finite_finite_v @ F3 )
=> ( ( ord_less_eq_set_v @ F3 @ A3 )
=> ( ( P @ bot_bot_set_v )
=> ( ! [A4: v,F4: set_v] :
( ( finite_finite_v @ F4 )
=> ( ( member_v @ A4 @ A3 )
=> ( ( ord_less_eq_set_v @ F4 @ A3 )
=> ( ~ ( member_v @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_v @ A4 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_584_finite__subset__induct_H,axiom,
! [F3: set_Product_prod_v_v,A3: set_Product_prod_v_v,P: set_Product_prod_v_v > $o] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( ord_le7336532860387713383od_v_v @ F3 @ A3 )
=> ( ( P @ bot_bo723834152578015283od_v_v )
=> ( ! [A4: product_prod_v_v,F4: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F4 )
=> ( ( member7453568604450474000od_v_v @ A4 @ A3 )
=> ( ( ord_le7336532860387713383od_v_v @ F4 @ A3 )
=> ( ~ ( member7453568604450474000od_v_v @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert1338601472111419319od_v_v @ A4 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_585_ssubst__Pair__rhs,axiom,
! [R: v,S7: sCC_Bl1394983891496994913t_unit,R3: set_Pr6425124735969554649t_unit,S8: sCC_Bl1394983891496994913t_unit] :
( ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ R @ S7 ) @ R3 )
=> ( ( S8 = S7 )
=> ( member7924940910754673978t_unit @ ( produc3862955338007567901t_unit @ R @ S8 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_586_ssubst__Pair__rhs,axiom,
! [R: v,S7: v,R3: set_Product_prod_v_v,S8: v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ R @ S7 ) @ R3 )
=> ( ( S8 = S7 )
=> ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ R @ S8 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_587_graph_Osubscc__add,axiom,
! [Vertices: set_set_v,Successors: set_v > set_set_v,S4: set_set_v,X: set_v,Y: set_v] :
( ( sCC_Bl5810666556806954322_set_v @ Vertices @ Successors )
=> ( ( sCC_Bl7907073126578335045_set_v @ Successors @ S4 )
=> ( ( member_set_v @ X @ S4 )
=> ( ( sCC_Bl7354734129683093653_set_v @ Successors @ X @ Y )
=> ( ( sCC_Bl7354734129683093653_set_v @ Successors @ Y @ X )
=> ( sCC_Bl7907073126578335045_set_v @ Successors @ ( insert_set_v @ Y @ S4 ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_588_graph_Osubscc__add,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,S4: 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 @ S4 )
=> ( ( member7453568604450474000od_v_v @ X @ S4 )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ X @ Y )
=> ( ( sCC_Bl4981926079593201289od_v_v @ Successors @ Y @ X )
=> ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( insert1338601472111419319od_v_v @ Y @ S4 ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_589_graph_Osubscc__add,axiom,
! [Vertices: set_v,Successors: v > set_v,S4: set_v,X: v,Y: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl5398416737448265317bscc_v @ Successors @ S4 )
=> ( ( member_v @ X @ S4 )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ X @ Y )
=> ( ( sCC_Bl649662514949026229able_v @ Successors @ Y @ X )
=> ( sCC_Bl5398416737448265317bscc_v @ Successors @ ( insert_v @ Y @ S4 ) ) ) ) ) ) ) ).
% graph.subscc_add
thf(fact_590_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_591_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_592_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_593_inf__bot__left,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X )
= bot_bot_Product_unit ) ).
% inf_bot_left
thf(fact_594_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_595_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_596_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_597_inf__bot__right,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ X @ bot_bot_Product_unit )
= bot_bot_Product_unit ) ).
% inf_bot_right
thf(fact_598_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_599_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_600_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_601_boolean__algebra_Oconj__zero__left,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X )
= bot_bot_Product_unit ) ).
% boolean_algebra.conj_zero_left
thf(fact_602_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_603_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_604_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_605_boolean__algebra_Oconj__zero__right,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ X @ bot_bot_Product_unit )
= bot_bot_Product_unit ) ).
% boolean_algebra.conj_zero_right
thf(fact_606_le__inf__iff,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
= ( ( ord_le3221252021190050221t_unit @ X @ Y )
& ( ord_le3221252021190050221t_unit @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_607_le__inf__iff,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( ( ord_less_eq_set_v @ X @ Y )
& ( ord_less_eq_set_v @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_608_le__inf__iff,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) )
= ( ( ord_le7336532860387713383od_v_v @ X @ Y )
& ( ord_le7336532860387713383od_v_v @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_609_inf_Obounded__iff,axiom,
! [A: product_unit,B: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ ( inf_inf_Product_unit @ B @ C ) )
= ( ( ord_le3221252021190050221t_unit @ A @ B )
& ( ord_le3221252021190050221t_unit @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_610_inf_Obounded__iff,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B @ C ) )
= ( ( ord_less_eq_set_v @ A @ B )
& ( ord_less_eq_set_v @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_611_inf_Obounded__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C ) )
= ( ( ord_le7336532860387713383od_v_v @ A @ B )
& ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_612_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_613_diff__shunt__var,axiom,
! [X: product_unit,Y: product_unit] :
( ( ( minus_3524152463667985524t_unit @ X @ Y )
= bot_bot_Product_unit )
= ( ord_le3221252021190050221t_unit @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_614_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_615_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_616_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_617_inf__right__idem,axiom,
! [X: product_unit,Y: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Y )
= ( inf_inf_Product_unit @ X @ Y ) ) ).
% inf_right_idem
thf(fact_618_inf_Oright__idem,axiom,
! [A: set_v,B: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B ) @ B )
= ( inf_inf_set_v @ A @ B ) ) ).
% inf.right_idem
thf(fact_619_inf_Oright__idem,axiom,
! [A: product_unit,B: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ A @ B ) @ B )
= ( inf_inf_Product_unit @ A @ B ) ) ).
% inf.right_idem
thf(fact_620_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_621_inf__left__idem,axiom,
! [X: product_unit,Y: product_unit] :
( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ X @ Y ) )
= ( inf_inf_Product_unit @ X @ Y ) ) ).
% inf_left_idem
thf(fact_622_inf_Oleft__idem,axiom,
! [A: set_v,B: set_v] :
( ( inf_inf_set_v @ A @ ( inf_inf_set_v @ A @ B ) )
= ( inf_inf_set_v @ A @ B ) ) ).
% inf.left_idem
thf(fact_623_inf_Oleft__idem,axiom,
! [A: product_unit,B: product_unit] :
( ( inf_inf_Product_unit @ A @ ( inf_inf_Product_unit @ A @ B ) )
= ( inf_inf_Product_unit @ A @ B ) ) ).
% inf.left_idem
thf(fact_624_inf__idem,axiom,
! [X: set_v] :
( ( inf_inf_set_v @ X @ X )
= X ) ).
% inf_idem
thf(fact_625_inf__idem,axiom,
! [X: product_unit] :
( ( inf_inf_Product_unit @ X @ X )
= X ) ).
% inf_idem
thf(fact_626_inf_Oidem,axiom,
! [A: set_v] :
( ( inf_inf_set_v @ A @ A )
= A ) ).
% inf.idem
thf(fact_627_inf_Oidem,axiom,
! [A: product_unit] :
( ( inf_inf_Product_unit @ A @ A )
= A ) ).
% inf.idem
thf(fact_628_inf__left__commute,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ Y @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_629_inf__left__commute,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
= ( inf_inf_Product_unit @ Y @ ( inf_inf_Product_unit @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_630_inf_Oleft__commute,axiom,
! [B: set_v,A: set_v,C: set_v] :
( ( inf_inf_set_v @ B @ ( inf_inf_set_v @ A @ C ) )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_631_inf_Oleft__commute,axiom,
! [B: product_unit,A: product_unit,C: product_unit] :
( ( inf_inf_Product_unit @ B @ ( inf_inf_Product_unit @ A @ C ) )
= ( inf_inf_Product_unit @ A @ ( inf_inf_Product_unit @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_632_boolean__algebra__cancel_Oinf2,axiom,
! [B4: set_v,K: set_v,B: set_v,A: set_v] :
( ( B4
= ( inf_inf_set_v @ K @ B ) )
=> ( ( inf_inf_set_v @ A @ B4 )
= ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_633_boolean__algebra__cancel_Oinf2,axiom,
! [B4: product_unit,K: product_unit,B: product_unit,A: product_unit] :
( ( B4
= ( inf_inf_Product_unit @ K @ B ) )
=> ( ( inf_inf_Product_unit @ A @ B4 )
= ( inf_inf_Product_unit @ K @ ( inf_inf_Product_unit @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_634_boolean__algebra__cancel_Oinf1,axiom,
! [A3: set_v,K: set_v,A: set_v,B: set_v] :
( ( A3
= ( inf_inf_set_v @ K @ A ) )
=> ( ( inf_inf_set_v @ A3 @ B )
= ( inf_inf_set_v @ K @ ( inf_inf_set_v @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_635_boolean__algebra__cancel_Oinf1,axiom,
! [A3: product_unit,K: product_unit,A: product_unit,B: product_unit] :
( ( A3
= ( inf_inf_Product_unit @ K @ A ) )
=> ( ( inf_inf_Product_unit @ A3 @ B )
= ( inf_inf_Product_unit @ K @ ( inf_inf_Product_unit @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_636_inf__commute,axiom,
( inf_inf_set_v
= ( ^ [X3: set_v,Y3: set_v] : ( inf_inf_set_v @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_637_inf__commute,axiom,
( inf_inf_Product_unit
= ( ^ [X3: product_unit,Y3: product_unit] : ( inf_inf_Product_unit @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_638_inf_Ocommute,axiom,
( inf_inf_set_v
= ( ^ [A5: set_v,B5: set_v] : ( inf_inf_set_v @ B5 @ A5 ) ) ) ).
% inf.commute
thf(fact_639_inf_Ocommute,axiom,
( inf_inf_Product_unit
= ( ^ [A5: product_unit,B5: product_unit] : ( inf_inf_Product_unit @ B5 @ A5 ) ) ) ).
% inf.commute
thf(fact_640_inf__assoc,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y ) @ Z )
= ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_641_inf__assoc,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Z )
= ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_642_inf_Oassoc,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ A @ B ) @ C )
= ( inf_inf_set_v @ A @ ( inf_inf_set_v @ B @ C ) ) ) ).
% inf.assoc
thf(fact_643_inf_Oassoc,axiom,
! [A: product_unit,B: product_unit,C: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ A @ B ) @ C )
= ( inf_inf_Product_unit @ A @ ( inf_inf_Product_unit @ B @ C ) ) ) ).
% inf.assoc
thf(fact_644_inf__sup__aci_I1_J,axiom,
( inf_inf_set_v
= ( ^ [X3: set_v,Y3: set_v] : ( inf_inf_set_v @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_645_inf__sup__aci_I1_J,axiom,
( inf_inf_Product_unit
= ( ^ [X3: product_unit,Y3: product_unit] : ( inf_inf_Product_unit @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_646_inf__sup__aci_I2_J,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ ( inf_inf_set_v @ X @ Y ) @ Z )
= ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_647_inf__sup__aci_I2_J,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Z )
= ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_648_inf__sup__aci_I3_J,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ Y @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_649_inf__sup__aci_I3_J,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
= ( inf_inf_Product_unit @ Y @ ( inf_inf_Product_unit @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_650_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_651_inf__sup__aci_I4_J,axiom,
! [X: product_unit,Y: product_unit] :
( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ X @ Y ) )
= ( inf_inf_Product_unit @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_652_inf__sup__ord_I2_J,axiom,
! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_653_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_654_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_655_inf__sup__ord_I1_J,axiom,
! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_656_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_657_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_658_inf__le1,axiom,
! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_659_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_660_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_661_inf__le2,axiom,
! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_662_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_663_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_664_le__infE,axiom,
! [X: product_unit,A: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ ( inf_inf_Product_unit @ A @ B ) )
=> ~ ( ( ord_le3221252021190050221t_unit @ X @ A )
=> ~ ( ord_le3221252021190050221t_unit @ X @ B ) ) ) ).
% le_infE
thf(fact_665_le__infE,axiom,
! [X: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ A @ B ) )
=> ~ ( ( ord_less_eq_set_v @ X @ A )
=> ~ ( ord_less_eq_set_v @ X @ B ) ) ) ).
% le_infE
thf(fact_666_le__infE,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A @ B ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ~ ( ord_le7336532860387713383od_v_v @ X @ B ) ) ) ).
% le_infE
thf(fact_667_le__infI,axiom,
! [X: product_unit,A: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ A )
=> ( ( ord_le3221252021190050221t_unit @ X @ B )
=> ( ord_le3221252021190050221t_unit @ X @ ( inf_inf_Product_unit @ A @ B ) ) ) ) ).
% le_infI
thf(fact_668_le__infI,axiom,
! [X: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ X @ A )
=> ( ( ord_less_eq_set_v @ X @ B )
=> ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ A @ B ) ) ) ) ).
% le_infI
thf(fact_669_le__infI,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ( ( ord_le7336532860387713383od_v_v @ X @ B )
=> ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ) ).
% le_infI
thf(fact_670_inf__mono,axiom,
! [A: product_unit,C: product_unit,B: product_unit,D: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ C )
=> ( ( ord_le3221252021190050221t_unit @ B @ D )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ ( inf_inf_Product_unit @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_671_inf__mono,axiom,
! [A: set_v,C: set_v,B: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ( ord_less_eq_set_v @ B @ D )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ ( inf_inf_set_v @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_672_inf__mono,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ ( inf_in6271465464967711157od_v_v @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_673_le__infI1,axiom,
! [A: product_unit,X: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ X )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_674_le__infI1,axiom,
! [A: set_v,X: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ X )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_675_le__infI1,axiom,
! [A: set_Product_prod_v_v,X: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_676_le__infI2,axiom,
! [B: product_unit,X: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ B @ X )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_677_le__infI2,axiom,
! [B: set_v,X: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ X )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_678_le__infI2,axiom,
! [B: set_Product_prod_v_v,X: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ X )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_679_inf_OorderE,axiom,
! [A: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ B )
=> ( A
= ( inf_inf_Product_unit @ A @ B ) ) ) ).
% inf.orderE
thf(fact_680_inf_OorderE,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( A
= ( inf_inf_set_v @ A @ B ) ) ) ).
% inf.orderE
thf(fact_681_inf_OorderE,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( A
= ( inf_in6271465464967711157od_v_v @ A @ B ) ) ) ).
% inf.orderE
thf(fact_682_inf_OorderI,axiom,
! [A: product_unit,B: product_unit] :
( ( A
= ( inf_inf_Product_unit @ A @ B ) )
=> ( ord_le3221252021190050221t_unit @ A @ B ) ) ).
% inf.orderI
thf(fact_683_inf_OorderI,axiom,
! [A: set_v,B: set_v] :
( ( A
= ( inf_inf_set_v @ A @ B ) )
=> ( ord_less_eq_set_v @ A @ B ) ) ).
% inf.orderI
thf(fact_684_inf_OorderI,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A
= ( inf_in6271465464967711157od_v_v @ A @ B ) )
=> ( ord_le7336532860387713383od_v_v @ A @ B ) ) ).
% inf.orderI
thf(fact_685_inf__unique,axiom,
! [F: product_unit > product_unit > product_unit,X: product_unit,Y: product_unit] :
( ! [X4: product_unit,Y4: product_unit] : ( ord_le3221252021190050221t_unit @ ( F @ X4 @ Y4 ) @ X4 )
=> ( ! [X4: product_unit,Y4: product_unit] : ( ord_le3221252021190050221t_unit @ ( F @ X4 @ Y4 ) @ Y4 )
=> ( ! [X4: product_unit,Y4: product_unit,Z3: product_unit] :
( ( ord_le3221252021190050221t_unit @ X4 @ Y4 )
=> ( ( ord_le3221252021190050221t_unit @ X4 @ Z3 )
=> ( ord_le3221252021190050221t_unit @ X4 @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_Product_unit @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_686_inf__unique,axiom,
! [F: set_v > set_v > set_v,X: set_v,Y: set_v] :
( ! [X4: set_v,Y4: set_v] : ( ord_less_eq_set_v @ ( F @ X4 @ Y4 ) @ X4 )
=> ( ! [X4: set_v,Y4: set_v] : ( ord_less_eq_set_v @ ( F @ X4 @ Y4 ) @ Y4 )
=> ( ! [X4: set_v,Y4: set_v,Z3: set_v] :
( ( ord_less_eq_set_v @ X4 @ Y4 )
=> ( ( ord_less_eq_set_v @ X4 @ Z3 )
=> ( ord_less_eq_set_v @ X4 @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_set_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_687_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] :
( ! [X4: set_Product_prod_v_v,Y4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X4 @ Y4 ) @ X4 )
=> ( ! [X4: set_Product_prod_v_v,Y4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( F @ X4 @ Y4 ) @ Y4 )
=> ( ! [X4: set_Product_prod_v_v,Y4: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X4 @ Y4 )
=> ( ( ord_le7336532860387713383od_v_v @ X4 @ Z3 )
=> ( ord_le7336532860387713383od_v_v @ X4 @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_688_le__iff__inf,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [X3: product_unit,Y3: product_unit] :
( ( inf_inf_Product_unit @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_689_le__iff__inf,axiom,
( ord_less_eq_set_v
= ( ^ [X3: set_v,Y3: set_v] :
( ( inf_inf_set_v @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_690_le__iff__inf,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_691_inf_Oabsorb1,axiom,
! [A: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ B )
=> ( ( inf_inf_Product_unit @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_692_inf_Oabsorb1,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( inf_inf_set_v @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_693_inf_Oabsorb1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_694_inf_Oabsorb2,axiom,
! [B: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ B @ A )
=> ( ( inf_inf_Product_unit @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_695_inf_Oabsorb2,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( inf_inf_set_v @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_696_inf_Oabsorb2,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( inf_in6271465464967711157od_v_v @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_697_inf__absorb1,axiom,
! [X: product_unit,Y: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ Y )
=> ( ( inf_inf_Product_unit @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_698_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_699_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_700_inf__absorb2,axiom,
! [Y: product_unit,X: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y @ X )
=> ( ( inf_inf_Product_unit @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_701_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_702_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_703_inf_OboundedE,axiom,
! [A: product_unit,B: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ ( inf_inf_Product_unit @ B @ C ) )
=> ~ ( ( ord_le3221252021190050221t_unit @ A @ B )
=> ~ ( ord_le3221252021190050221t_unit @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_704_inf_OboundedE,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B @ C ) )
=> ~ ( ( ord_less_eq_set_v @ A @ B )
=> ~ ( ord_less_eq_set_v @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_705_inf_OboundedE,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C ) )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ~ ( ord_le7336532860387713383od_v_v @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_706_inf_OboundedI,axiom,
! [A: product_unit,B: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ B )
=> ( ( ord_le3221252021190050221t_unit @ A @ C )
=> ( ord_le3221252021190050221t_unit @ A @ ( inf_inf_Product_unit @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_707_inf_OboundedI,axiom,
! [A: set_v,B: set_v,C: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( ord_less_eq_set_v @ A @ C )
=> ( ord_less_eq_set_v @ A @ ( inf_inf_set_v @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_708_inf_OboundedI,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ord_le7336532860387713383od_v_v @ A @ ( inf_in6271465464967711157od_v_v @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_709_inf__greatest,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ Y )
=> ( ( ord_le3221252021190050221t_unit @ X @ Z )
=> ( ord_le3221252021190050221t_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_710_inf__greatest,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ X @ Y )
=> ( ( ord_less_eq_set_v @ X @ Z )
=> ( ord_less_eq_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_711_inf__greatest,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ Y )
=> ( ( ord_le7336532860387713383od_v_v @ X @ Z )
=> ( ord_le7336532860387713383od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_712_inf_Oorder__iff,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [A5: product_unit,B5: product_unit] :
( A5
= ( inf_inf_Product_unit @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_713_inf_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B5: set_v] :
( A5
= ( inf_inf_set_v @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_714_inf_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( A5
= ( inf_in6271465464967711157od_v_v @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_715_inf_Ocobounded1,axiom,
! [A: product_unit,B: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_716_inf_Ocobounded1,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_717_inf_Ocobounded1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_718_inf_Ocobounded2,axiom,
! [A: product_unit,B: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_719_inf_Ocobounded2,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_720_inf_Ocobounded2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_721_inf_Oabsorb__iff1,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [A5: product_unit,B5: product_unit] :
( ( inf_inf_Product_unit @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_722_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B5: set_v] :
( ( inf_inf_set_v @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_723_inf_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_724_inf_Oabsorb__iff2,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [B5: product_unit,A5: product_unit] :
( ( inf_inf_Product_unit @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_725_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [B5: set_v,A5: set_v] :
( ( inf_inf_set_v @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_726_inf_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B5: set_Product_prod_v_v,A5: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_727_inf_OcoboundedI1,axiom,
! [A: product_unit,C: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ C )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_728_inf_OcoboundedI1,axiom,
! [A: set_v,C: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_729_inf_OcoboundedI1,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_730_inf_OcoboundedI2,axiom,
! [B: product_unit,C: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ B @ C )
=> ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_731_inf_OcoboundedI2,axiom,
! [B: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ C )
=> ( ord_less_eq_set_v @ ( inf_inf_set_v @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_732_inf_OcoboundedI2,axiom,
! [B: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ C )
=> ( ord_le7336532860387713383od_v_v @ ( inf_in6271465464967711157od_v_v @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_733_ra__add__edge,axiom,
! [X: v,Y: v,E5: set_Product_prod_v_v,V: v,W: v] :
( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E5 )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ ( sup_su414716646722978715od_v_v @ E5 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ V @ ( sup_su414716646722978715od_v_v @ E5 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ successors @ W @ Y @ ( sup_su414716646722978715od_v_v @ E5 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ).
% ra_add_edge
thf(fact_734_the__elem__eq,axiom,
! [X: product_prod_v_v] :
( ( the_el5392834299063928540od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= X ) ).
% the_elem_eq
thf(fact_735_the__elem__eq,axiom,
! [X: v] :
( ( the_elem_v @ ( insert_v @ X @ bot_bot_set_v ) )
= X ) ).
% the_elem_eq
thf(fact_736_the__elem__eq,axiom,
! [X: set_v] :
( ( the_elem_set_v @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= X ) ).
% the_elem_eq
thf(fact_737_Collect__empty__eq__bot,axiom,
! [P: product_prod_v_v > $o] :
( ( ( collec140062887454715474od_v_v @ P )
= bot_bo723834152578015283od_v_v )
= ( P = bot_bo8461541820394803818_v_v_o ) ) ).
% Collect_empty_eq_bot
thf(fact_738_Collect__empty__eq__bot,axiom,
! [P: v > $o] :
( ( ( collect_v @ P )
= bot_bot_set_v )
= ( P = bot_bot_v_o ) ) ).
% Collect_empty_eq_bot
thf(fact_739_Collect__empty__eq__bot,axiom,
! [P: set_v > $o] :
( ( ( collect_set_v @ P )
= bot_bot_set_set_v )
= ( P = bot_bot_set_v_o ) ) ).
% Collect_empty_eq_bot
thf(fact_740_is__singletonI,axiom,
! [X: product_prod_v_v] : ( is_sin9198872032823709915od_v_v @ ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ).
% is_singletonI
thf(fact_741_is__singletonI,axiom,
! [X: v] : ( is_singleton_v @ ( insert_v @ X @ bot_bot_set_v ) ) ).
% is_singletonI
thf(fact_742_is__singletonI,axiom,
! [X: set_v] : ( is_singleton_set_v @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ).
% is_singletonI
thf(fact_743_Un__iff,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B4 ) )
= ( ( member7453568604450474000od_v_v @ C @ A3 )
| ( member7453568604450474000od_v_v @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_744_Un__iff,axiom,
! [C: v,A3: set_v,B4: set_v] :
( ( member_v @ C @ ( sup_sup_set_v @ A3 @ B4 ) )
= ( ( member_v @ C @ A3 )
| ( member_v @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_745_Un__iff,axiom,
! [C: set_v,A3: set_set_v,B4: set_set_v] :
( ( member_set_v @ C @ ( sup_sup_set_set_v @ A3 @ B4 ) )
= ( ( member_set_v @ C @ A3 )
| ( member_set_v @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_746_UnCI,axiom,
! [C: product_prod_v_v,B4: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ~ ( member7453568604450474000od_v_v @ C @ B4 )
=> ( member7453568604450474000od_v_v @ C @ A3 ) )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B4 ) ) ) ).
% UnCI
thf(fact_747_UnCI,axiom,
! [C: v,B4: set_v,A3: set_v] :
( ( ~ ( member_v @ C @ B4 )
=> ( member_v @ C @ A3 ) )
=> ( member_v @ C @ ( sup_sup_set_v @ A3 @ B4 ) ) ) ).
% UnCI
thf(fact_748_UnCI,axiom,
! [C: set_v,B4: set_set_v,A3: set_set_v] :
( ( ~ ( member_set_v @ C @ B4 )
=> ( member_set_v @ C @ A3 ) )
=> ( member_set_v @ C @ ( sup_sup_set_set_v @ A3 @ B4 ) ) ) ).
% UnCI
thf(fact_749_sup_Obounded__iff,axiom,
! [B: set_set_v,C: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ B @ C ) @ A )
= ( ( ord_le5216385588623774835_set_v @ B @ A )
& ( ord_le5216385588623774835_set_v @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_750_sup_Obounded__iff,axiom,
! [B: product_unit,C: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ B @ C ) @ A )
= ( ( ord_le3221252021190050221t_unit @ B @ A )
& ( ord_le3221252021190050221t_unit @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_751_sup_Obounded__iff,axiom,
! [B: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B @ C ) @ A )
= ( ( ord_less_eq_set_v @ B @ A )
& ( ord_less_eq_set_v @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_752_sup_Obounded__iff,axiom,
! [B: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C ) @ A )
= ( ( ord_le7336532860387713383od_v_v @ B @ A )
& ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_753_le__sup__iff,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ X @ Y ) @ Z )
= ( ( ord_le5216385588623774835_set_v @ X @ Z )
& ( ord_le5216385588623774835_set_v @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_754_le__sup__iff,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ X @ Y ) @ Z )
= ( ( ord_le3221252021190050221t_unit @ X @ Z )
& ( ord_le3221252021190050221t_unit @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_755_le__sup__iff,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_v @ X @ Z )
& ( ord_less_eq_set_v @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_756_le__sup__iff,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ Z )
= ( ( ord_le7336532860387713383od_v_v @ X @ Z )
& ( ord_le7336532860387713383od_v_v @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_757_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_758_sup__bot_Oright__neutral,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ A @ bot_bot_set_v )
= A ) ).
% sup_bot.right_neutral
thf(fact_759_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_760_sup__bot_Oright__neutral,axiom,
! [A: product_unit] :
( ( sup_sup_Product_unit @ A @ bot_bot_Product_unit )
= A ) ).
% sup_bot.right_neutral
thf(fact_761_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( bot_bo723834152578015283od_v_v
= ( sup_su414716646722978715od_v_v @ A @ B ) )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_762_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_v,B: set_v] :
( ( bot_bot_set_v
= ( sup_sup_set_v @ A @ B ) )
= ( ( A = bot_bot_set_v )
& ( B = bot_bot_set_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_763_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_set_v,B: set_set_v] :
( ( bot_bot_set_set_v
= ( sup_sup_set_set_v @ A @ B ) )
= ( ( A = bot_bot_set_set_v )
& ( B = bot_bot_set_set_v ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_764_sup__bot_Oneutr__eq__iff,axiom,
! [A: product_unit,B: product_unit] :
( ( bot_bot_Product_unit
= ( sup_sup_Product_unit @ A @ B ) )
= ( ( A = bot_bot_Product_unit )
& ( B = bot_bot_Product_unit ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_765_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_766_sup__bot_Oleft__neutral,axiom,
! [A: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_767_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_768_sup__bot_Oleft__neutral,axiom,
! [A: product_unit] :
( ( sup_sup_Product_unit @ bot_bot_Product_unit @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_769_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A @ B )
= bot_bo723834152578015283od_v_v )
= ( ( A = bot_bo723834152578015283od_v_v )
& ( B = bot_bo723834152578015283od_v_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_770_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_v,B: set_v] :
( ( ( sup_sup_set_v @ A @ B )
= bot_bot_set_v )
= ( ( A = bot_bot_set_v )
& ( B = bot_bot_set_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_771_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_set_v,B: set_set_v] :
( ( ( sup_sup_set_set_v @ A @ B )
= bot_bot_set_set_v )
= ( ( A = bot_bot_set_set_v )
& ( B = bot_bot_set_set_v ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_772_sup__bot_Oeq__neutr__iff,axiom,
! [A: product_unit,B: product_unit] :
( ( ( sup_sup_Product_unit @ A @ B )
= bot_bot_Product_unit )
= ( ( A = bot_bot_Product_unit )
& ( B = bot_bot_Product_unit ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_773_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_774_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_775_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_776_sup__eq__bot__iff,axiom,
! [X: product_unit,Y: product_unit] :
( ( ( sup_sup_Product_unit @ X @ Y )
= bot_bot_Product_unit )
= ( ( X = bot_bot_Product_unit )
& ( Y = bot_bot_Product_unit ) ) ) ).
% sup_eq_bot_iff
thf(fact_777_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_778_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_779_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_780_bot__eq__sup__iff,axiom,
! [X: product_unit,Y: product_unit] :
( ( bot_bot_Product_unit
= ( sup_sup_Product_unit @ X @ Y ) )
= ( ( X = bot_bot_Product_unit )
& ( Y = bot_bot_Product_unit ) ) ) ).
% bot_eq_sup_iff
thf(fact_781_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_782_sup__bot__right,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ X @ bot_bot_set_v )
= X ) ).
% sup_bot_right
thf(fact_783_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_784_sup__bot__right,axiom,
! [X: product_unit] :
( ( sup_sup_Product_unit @ X @ bot_bot_Product_unit )
= X ) ).
% sup_bot_right
thf(fact_785_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_786_sup__bot__left,axiom,
! [X: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ X )
= X ) ).
% sup_bot_left
thf(fact_787_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_788_sup__bot__left,axiom,
! [X: product_unit] :
( ( sup_sup_Product_unit @ bot_bot_Product_unit @ X )
= X ) ).
% sup_bot_left
thf(fact_789_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_790_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_791_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_792_sup__inf__absorb,axiom,
! [X: product_unit,Y: product_unit] :
( ( sup_sup_Product_unit @ X @ ( inf_inf_Product_unit @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_793_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_794_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_795_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_796_inf__sup__absorb,axiom,
! [X: product_unit,Y: product_unit] :
( ( inf_inf_Product_unit @ X @ ( sup_sup_Product_unit @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_797_Un__empty,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A3 @ B4 )
= bot_bo723834152578015283od_v_v )
= ( ( A3 = bot_bo723834152578015283od_v_v )
& ( B4 = bot_bo723834152578015283od_v_v ) ) ) ).
% Un_empty
thf(fact_798_Un__empty,axiom,
! [A3: set_v,B4: set_v] :
( ( ( sup_sup_set_v @ A3 @ B4 )
= bot_bot_set_v )
= ( ( A3 = bot_bot_set_v )
& ( B4 = bot_bot_set_v ) ) ) ).
% Un_empty
thf(fact_799_Un__empty,axiom,
! [A3: set_set_v,B4: set_set_v] :
( ( ( sup_sup_set_set_v @ A3 @ B4 )
= bot_bot_set_set_v )
= ( ( A3 = bot_bot_set_set_v )
& ( B4 = bot_bot_set_set_v ) ) ) ).
% Un_empty
thf(fact_800_finite__Un,axiom,
! [F3: set_Product_prod_v_v,G: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ F3 @ G ) )
= ( ( finite3348123685078250256od_v_v @ F3 )
& ( finite3348123685078250256od_v_v @ G ) ) ) ).
% finite_Un
thf(fact_801_finite__Un,axiom,
! [F3: set_v,G: set_v] :
( ( finite_finite_v @ ( sup_sup_set_v @ F3 @ G ) )
= ( ( finite_finite_v @ F3 )
& ( finite_finite_v @ G ) ) ) ).
% finite_Un
thf(fact_802_finite__Un,axiom,
! [F3: set_set_v,G: set_set_v] :
( ( finite_finite_set_v @ ( sup_sup_set_set_v @ F3 @ G ) )
= ( ( finite_finite_set_v @ F3 )
& ( finite_finite_set_v @ G ) ) ) ).
% finite_Un
thf(fact_803_Un__subset__iff,axiom,
! [A3: set_set_v,B4: set_set_v,C2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A3 @ B4 ) @ C2 )
= ( ( ord_le5216385588623774835_set_v @ A3 @ C2 )
& ( ord_le5216385588623774835_set_v @ B4 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_804_Un__subset__iff,axiom,
! [A3: set_v,B4: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A3 @ B4 ) @ C2 )
= ( ( ord_less_eq_set_v @ A3 @ C2 )
& ( ord_less_eq_set_v @ B4 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_805_Un__subset__iff,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B4 ) @ C2 )
= ( ( ord_le7336532860387713383od_v_v @ A3 @ C2 )
& ( ord_le7336532860387713383od_v_v @ B4 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_806_Un__insert__right,axiom,
! [A3: set_Product_prod_v_v,A: product_prod_v_v,B4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( insert1338601472111419319od_v_v @ A @ B4 ) )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A3 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_807_Un__insert__right,axiom,
! [A3: set_v,A: v,B4: set_v] :
( ( sup_sup_set_v @ A3 @ ( insert_v @ A @ B4 ) )
= ( insert_v @ A @ ( sup_sup_set_v @ A3 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_808_Un__insert__right,axiom,
! [A3: set_set_v,A: set_v,B4: set_set_v] :
( ( sup_sup_set_set_v @ A3 @ ( insert_set_v @ A @ B4 ) )
= ( insert_set_v @ A @ ( sup_sup_set_set_v @ A3 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_809_Un__insert__left,axiom,
! [A: product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A @ B4 ) @ C2 )
= ( insert1338601472111419319od_v_v @ A @ ( sup_su414716646722978715od_v_v @ B4 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_810_Un__insert__left,axiom,
! [A: v,B4: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( insert_v @ A @ B4 ) @ C2 )
= ( insert_v @ A @ ( sup_sup_set_v @ B4 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_811_Un__insert__left,axiom,
! [A: set_v,B4: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ ( insert_set_v @ A @ B4 ) @ C2 )
= ( insert_set_v @ A @ ( sup_sup_set_set_v @ B4 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_812_Un__Int__eq_I1_J,axiom,
! [S4: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S4 @ T ) @ S4 )
= S4 ) ).
% Un_Int_eq(1)
thf(fact_813_Un__Int__eq_I1_J,axiom,
! [S4: set_v,T: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S4 @ T ) @ S4 )
= S4 ) ).
% Un_Int_eq(1)
thf(fact_814_Un__Int__eq_I1_J,axiom,
! [S4: set_set_v,T: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ S4 @ T ) @ S4 )
= S4 ) ).
% Un_Int_eq(1)
thf(fact_815_Un__Int__eq_I2_J,axiom,
! [S4: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ S4 @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_816_Un__Int__eq_I2_J,axiom,
! [S4: set_v,T: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ S4 @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_817_Un__Int__eq_I2_J,axiom,
! [S4: set_set_v,T: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ S4 @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_818_Un__Int__eq_I3_J,axiom,
! [S4: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ S4 @ ( sup_su414716646722978715od_v_v @ S4 @ T ) )
= S4 ) ).
% Un_Int_eq(3)
thf(fact_819_Un__Int__eq_I3_J,axiom,
! [S4: set_v,T: set_v] :
( ( inf_inf_set_v @ S4 @ ( sup_sup_set_v @ S4 @ T ) )
= S4 ) ).
% Un_Int_eq(3)
thf(fact_820_Un__Int__eq_I3_J,axiom,
! [S4: set_set_v,T: set_set_v] :
( ( inf_inf_set_set_v @ S4 @ ( sup_sup_set_set_v @ S4 @ T ) )
= S4 ) ).
% Un_Int_eq(3)
thf(fact_821_Un__Int__eq_I4_J,axiom,
! [T: set_Product_prod_v_v,S4: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ T @ ( sup_su414716646722978715od_v_v @ S4 @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_822_Un__Int__eq_I4_J,axiom,
! [T: set_v,S4: set_v] :
( ( inf_inf_set_v @ T @ ( sup_sup_set_v @ S4 @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_823_Un__Int__eq_I4_J,axiom,
! [T: set_set_v,S4: set_set_v] :
( ( inf_inf_set_set_v @ T @ ( sup_sup_set_set_v @ S4 @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_824_Int__Un__eq_I1_J,axiom,
! [S4: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S4 @ T ) @ S4 )
= S4 ) ).
% Int_Un_eq(1)
thf(fact_825_Int__Un__eq_I1_J,axiom,
! [S4: set_v,T: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S4 @ T ) @ S4 )
= S4 ) ).
% Int_Un_eq(1)
thf(fact_826_Int__Un__eq_I1_J,axiom,
! [S4: set_set_v,T: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ S4 @ T ) @ S4 )
= S4 ) ).
% Int_Un_eq(1)
thf(fact_827_Int__Un__eq_I2_J,axiom,
! [S4: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ S4 @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_828_Int__Un__eq_I2_J,axiom,
! [S4: set_v,T: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ S4 @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_829_Int__Un__eq_I2_J,axiom,
! [S4: set_set_v,T: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ S4 @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_830_Int__Un__eq_I3_J,axiom,
! [S4: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ S4 @ ( inf_in6271465464967711157od_v_v @ S4 @ T ) )
= S4 ) ).
% Int_Un_eq(3)
thf(fact_831_Int__Un__eq_I3_J,axiom,
! [S4: set_v,T: set_v] :
( ( sup_sup_set_v @ S4 @ ( inf_inf_set_v @ S4 @ T ) )
= S4 ) ).
% Int_Un_eq(3)
thf(fact_832_Int__Un__eq_I3_J,axiom,
! [S4: set_set_v,T: set_set_v] :
( ( sup_sup_set_set_v @ S4 @ ( inf_inf_set_set_v @ S4 @ T ) )
= S4 ) ).
% Int_Un_eq(3)
thf(fact_833_Int__Un__eq_I4_J,axiom,
! [T: set_Product_prod_v_v,S4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ T @ ( inf_in6271465464967711157od_v_v @ S4 @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_834_Int__Un__eq_I4_J,axiom,
! [T: set_v,S4: set_v] :
( ( sup_sup_set_v @ T @ ( inf_inf_set_v @ S4 @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_835_Int__Un__eq_I4_J,axiom,
! [T: set_set_v,S4: set_set_v] :
( ( sup_sup_set_set_v @ T @ ( inf_inf_set_set_v @ S4 @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_836_Un__Diff__cancel2,axiom,
! [B4: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ B4 @ A3 ) @ A3 )
= ( sup_su414716646722978715od_v_v @ B4 @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_837_Un__Diff__cancel2,axiom,
! [B4: set_set_v,A3: set_set_v] :
( ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ B4 @ A3 ) @ A3 )
= ( sup_sup_set_set_v @ B4 @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_838_Un__Diff__cancel2,axiom,
! [B4: set_v,A3: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ B4 @ A3 ) @ A3 )
= ( sup_sup_set_v @ B4 @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_839_Un__Diff__cancel,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B4 @ A3 ) )
= ( sup_su414716646722978715od_v_v @ A3 @ B4 ) ) ).
% Un_Diff_cancel
thf(fact_840_Un__Diff__cancel,axiom,
! [A3: set_set_v,B4: set_set_v] :
( ( sup_sup_set_set_v @ A3 @ ( minus_7228012346218142266_set_v @ B4 @ A3 ) )
= ( sup_sup_set_set_v @ A3 @ B4 ) ) ).
% Un_Diff_cancel
thf(fact_841_Un__Diff__cancel,axiom,
! [A3: set_v,B4: set_v] :
( ( sup_sup_set_v @ A3 @ ( minus_minus_set_v @ B4 @ A3 ) )
= ( sup_sup_set_v @ A3 @ B4 ) ) ).
% Un_Diff_cancel
thf(fact_842_Un__left__commute,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B4 @ C2 ) )
= ( sup_su414716646722978715od_v_v @ B4 @ ( sup_su414716646722978715od_v_v @ A3 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_843_Un__left__commute,axiom,
! [A3: set_v,B4: set_v,C2: set_v] :
( ( sup_sup_set_v @ A3 @ ( sup_sup_set_v @ B4 @ C2 ) )
= ( sup_sup_set_v @ B4 @ ( sup_sup_set_v @ A3 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_844_Un__left__commute,axiom,
! [A3: set_set_v,B4: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ A3 @ ( sup_sup_set_set_v @ B4 @ C2 ) )
= ( sup_sup_set_set_v @ B4 @ ( sup_sup_set_set_v @ A3 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_845_Un__left__absorb,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ A3 @ B4 ) )
= ( sup_su414716646722978715od_v_v @ A3 @ B4 ) ) ).
% Un_left_absorb
thf(fact_846_Un__left__absorb,axiom,
! [A3: set_v,B4: set_v] :
( ( sup_sup_set_v @ A3 @ ( sup_sup_set_v @ A3 @ B4 ) )
= ( sup_sup_set_v @ A3 @ B4 ) ) ).
% Un_left_absorb
thf(fact_847_Un__left__absorb,axiom,
! [A3: set_set_v,B4: set_set_v] :
( ( sup_sup_set_set_v @ A3 @ ( sup_sup_set_set_v @ A3 @ B4 ) )
= ( sup_sup_set_set_v @ A3 @ B4 ) ) ).
% Un_left_absorb
thf(fact_848_Un__commute,axiom,
( sup_su414716646722978715od_v_v
= ( ^ [A6: set_Product_prod_v_v,B6: set_Product_prod_v_v] : ( sup_su414716646722978715od_v_v @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_849_Un__commute,axiom,
( sup_sup_set_v
= ( ^ [A6: set_v,B6: set_v] : ( sup_sup_set_v @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_850_Un__commute,axiom,
( sup_sup_set_set_v
= ( ^ [A6: set_set_v,B6: set_set_v] : ( sup_sup_set_set_v @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_851_Un__absorb,axiom,
! [A3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_852_Un__absorb,axiom,
! [A3: set_v] :
( ( sup_sup_set_v @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_853_Un__absorb,axiom,
! [A3: set_set_v] :
( ( sup_sup_set_set_v @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_854_Un__assoc,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B4 ) @ C2 )
= ( sup_su414716646722978715od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B4 @ C2 ) ) ) ).
% Un_assoc
thf(fact_855_Un__assoc,axiom,
! [A3: set_v,B4: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ A3 @ B4 ) @ C2 )
= ( sup_sup_set_v @ A3 @ ( sup_sup_set_v @ B4 @ C2 ) ) ) ).
% Un_assoc
thf(fact_856_Un__assoc,axiom,
! [A3: set_set_v,B4: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ A3 @ B4 ) @ C2 )
= ( sup_sup_set_set_v @ A3 @ ( sup_sup_set_set_v @ B4 @ C2 ) ) ) ).
% Un_assoc
thf(fact_857_ball__Un,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ A3 @ B4 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
=> ( P @ X3 ) )
& ! [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ B4 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_858_ball__Un,axiom,
! [A3: set_v,B4: set_v,P: v > $o] :
( ( ! [X3: v] :
( ( member_v @ X3 @ ( sup_sup_set_v @ A3 @ B4 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: v] :
( ( member_v @ X3 @ A3 )
=> ( P @ X3 ) )
& ! [X3: v] :
( ( member_v @ X3 @ B4 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_859_ball__Un,axiom,
! [A3: set_set_v,B4: set_set_v,P: set_v > $o] :
( ( ! [X3: set_v] :
( ( member_set_v @ X3 @ ( sup_sup_set_set_v @ A3 @ B4 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
=> ( P @ X3 ) )
& ! [X3: set_v] :
( ( member_set_v @ X3 @ B4 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_860_bex__Un,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v,P: product_prod_v_v > $o] :
( ( ? [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ ( sup_su414716646722978715od_v_v @ A3 @ B4 ) )
& ( P @ X3 ) ) )
= ( ? [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ A3 )
& ( P @ X3 ) )
| ? [X3: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X3 @ B4 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_861_bex__Un,axiom,
! [A3: set_v,B4: set_v,P: v > $o] :
( ( ? [X3: v] :
( ( member_v @ X3 @ ( sup_sup_set_v @ A3 @ B4 ) )
& ( P @ X3 ) ) )
= ( ? [X3: v] :
( ( member_v @ X3 @ A3 )
& ( P @ X3 ) )
| ? [X3: v] :
( ( member_v @ X3 @ B4 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_862_bex__Un,axiom,
! [A3: set_set_v,B4: set_set_v,P: set_v > $o] :
( ( ? [X3: set_v] :
( ( member_set_v @ X3 @ ( sup_sup_set_set_v @ A3 @ B4 ) )
& ( P @ X3 ) ) )
= ( ? [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
& ( P @ X3 ) )
| ? [X3: set_v] :
( ( member_set_v @ X3 @ B4 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_863_UnI2,axiom,
! [C: product_prod_v_v,B4: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ B4 )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B4 ) ) ) ).
% UnI2
thf(fact_864_UnI2,axiom,
! [C: v,B4: set_v,A3: set_v] :
( ( member_v @ C @ B4 )
=> ( member_v @ C @ ( sup_sup_set_v @ A3 @ B4 ) ) ) ).
% UnI2
thf(fact_865_UnI2,axiom,
! [C: set_v,B4: set_set_v,A3: set_set_v] :
( ( member_set_v @ C @ B4 )
=> ( member_set_v @ C @ ( sup_sup_set_set_v @ A3 @ B4 ) ) ) ).
% UnI2
thf(fact_866_UnI1,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ A3 )
=> ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B4 ) ) ) ).
% UnI1
thf(fact_867_UnI1,axiom,
! [C: v,A3: set_v,B4: set_v] :
( ( member_v @ C @ A3 )
=> ( member_v @ C @ ( sup_sup_set_v @ A3 @ B4 ) ) ) ).
% UnI1
thf(fact_868_UnI1,axiom,
! [C: set_v,A3: set_set_v,B4: set_set_v] :
( ( member_set_v @ C @ A3 )
=> ( member_set_v @ C @ ( sup_sup_set_set_v @ A3 @ B4 ) ) ) ).
% UnI1
thf(fact_869_UnE,axiom,
! [C: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A3 @ B4 ) )
=> ( ~ ( member7453568604450474000od_v_v @ C @ A3 )
=> ( member7453568604450474000od_v_v @ C @ B4 ) ) ) ).
% UnE
thf(fact_870_UnE,axiom,
! [C: v,A3: set_v,B4: set_v] :
( ( member_v @ C @ ( sup_sup_set_v @ A3 @ B4 ) )
=> ( ~ ( member_v @ C @ A3 )
=> ( member_v @ C @ B4 ) ) ) ).
% UnE
thf(fact_871_UnE,axiom,
! [C: set_v,A3: set_set_v,B4: set_set_v] :
( ( member_set_v @ C @ ( sup_sup_set_set_v @ A3 @ B4 ) )
=> ( ~ ( member_set_v @ C @ A3 )
=> ( member_set_v @ C @ B4 ) ) ) ).
% UnE
thf(fact_872_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_873_inf__sup__ord_I4_J,axiom,
! [Y: product_unit,X: product_unit] : ( ord_le3221252021190050221t_unit @ Y @ ( sup_sup_Product_unit @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_874_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_875_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_876_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_877_inf__sup__ord_I3_J,axiom,
! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ X @ ( sup_sup_Product_unit @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_878_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_879_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_880_le__supE,axiom,
! [A: set_set_v,B: set_set_v,X: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A @ B ) @ X )
=> ~ ( ( ord_le5216385588623774835_set_v @ A @ X )
=> ~ ( ord_le5216385588623774835_set_v @ B @ X ) ) ) ).
% le_supE
thf(fact_881_le__supE,axiom,
! [A: product_unit,B: product_unit,X: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ A @ B ) @ X )
=> ~ ( ( ord_le3221252021190050221t_unit @ A @ X )
=> ~ ( ord_le3221252021190050221t_unit @ B @ X ) ) ) ).
% le_supE
thf(fact_882_le__supE,axiom,
! [A: set_v,B: set_v,X: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_set_v @ A @ X )
=> ~ ( ord_less_eq_set_v @ B @ X ) ) ) ).
% le_supE
thf(fact_883_le__supE,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ X )
=> ~ ( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ~ ( ord_le7336532860387713383od_v_v @ B @ X ) ) ) ).
% le_supE
thf(fact_884_le__supI,axiom,
! [A: set_set_v,X: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ X )
=> ( ( ord_le5216385588623774835_set_v @ B @ X )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_885_le__supI,axiom,
! [A: product_unit,X: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ X )
=> ( ( ord_le3221252021190050221t_unit @ B @ X )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_886_le__supI,axiom,
! [A: set_v,X: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ X )
=> ( ( ord_less_eq_set_v @ B @ X )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_887_le__supI,axiom,
! [A: set_Product_prod_v_v,X: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ X )
=> ( ( ord_le7336532860387713383od_v_v @ B @ X )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_888_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_889_sup__ge1,axiom,
! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ X @ ( sup_sup_Product_unit @ X @ Y ) ) ).
% sup_ge1
thf(fact_890_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_891_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_892_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_893_sup__ge2,axiom,
! [Y: product_unit,X: product_unit] : ( ord_le3221252021190050221t_unit @ Y @ ( sup_sup_Product_unit @ X @ Y ) ) ).
% sup_ge2
thf(fact_894_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_895_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_896_le__supI1,axiom,
! [X: set_set_v,A: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ X @ A )
=> ( ord_le5216385588623774835_set_v @ X @ ( sup_sup_set_set_v @ A @ B ) ) ) ).
% le_supI1
thf(fact_897_le__supI1,axiom,
! [X: product_unit,A: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ A )
=> ( ord_le3221252021190050221t_unit @ X @ ( sup_sup_Product_unit @ A @ B ) ) ) ).
% le_supI1
thf(fact_898_le__supI1,axiom,
! [X: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ X @ A )
=> ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ A @ B ) ) ) ).
% le_supI1
thf(fact_899_le__supI1,axiom,
! [X: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ A )
=> ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% le_supI1
thf(fact_900_le__supI2,axiom,
! [X: set_set_v,B: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ X @ B )
=> ( ord_le5216385588623774835_set_v @ X @ ( sup_sup_set_set_v @ A @ B ) ) ) ).
% le_supI2
thf(fact_901_le__supI2,axiom,
! [X: product_unit,B: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ B )
=> ( ord_le3221252021190050221t_unit @ X @ ( sup_sup_Product_unit @ A @ B ) ) ) ).
% le_supI2
thf(fact_902_le__supI2,axiom,
! [X: set_v,B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ X @ B )
=> ( ord_less_eq_set_v @ X @ ( sup_sup_set_v @ A @ B ) ) ) ).
% le_supI2
thf(fact_903_le__supI2,axiom,
! [X: set_Product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ X @ B )
=> ( ord_le7336532860387713383od_v_v @ X @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% le_supI2
thf(fact_904_sup_Omono,axiom,
! [C: set_set_v,A: set_set_v,D: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C @ A )
=> ( ( ord_le5216385588623774835_set_v @ D @ B )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ C @ D ) @ ( sup_sup_set_set_v @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_905_sup_Omono,axiom,
! [C: product_unit,A: product_unit,D: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ C @ A )
=> ( ( ord_le3221252021190050221t_unit @ D @ B )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ C @ D ) @ ( sup_sup_Product_unit @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_906_sup_Omono,axiom,
! [C: set_v,A: set_v,D: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C @ A )
=> ( ( ord_less_eq_set_v @ D @ B )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ C @ D ) @ ( sup_sup_set_v @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_907_sup_Omono,axiom,
! [C: set_Product_prod_v_v,A: set_Product_prod_v_v,D: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ( ord_le7336532860387713383od_v_v @ D @ B )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ C @ D ) @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_908_sup__mono,axiom,
! [A: set_set_v,C: set_set_v,B: set_set_v,D: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ C )
=> ( ( ord_le5216385588623774835_set_v @ B @ D )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A @ B ) @ ( sup_sup_set_set_v @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_909_sup__mono,axiom,
! [A: product_unit,C: product_unit,B: product_unit,D: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ C )
=> ( ( ord_le3221252021190050221t_unit @ B @ D )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ A @ B ) @ ( sup_sup_Product_unit @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_910_sup__mono,axiom,
! [A: set_v,C: set_v,B: set_v,D: set_v] :
( ( ord_less_eq_set_v @ A @ C )
=> ( ( ord_less_eq_set_v @ B @ D )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A @ B ) @ ( sup_sup_set_v @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_911_sup__mono,axiom,
! [A: set_Product_prod_v_v,C: set_Product_prod_v_v,B: set_Product_prod_v_v,D: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ C )
=> ( ( ord_le7336532860387713383od_v_v @ B @ D )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A @ B ) @ ( sup_su414716646722978715od_v_v @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_912_sup__least,axiom,
! [Y: set_set_v,X: set_set_v,Z: set_set_v] :
( ( ord_le5216385588623774835_set_v @ Y @ X )
=> ( ( ord_le5216385588623774835_set_v @ Z @ X )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_913_sup__least,axiom,
! [Y: product_unit,X: product_unit,Z: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y @ X )
=> ( ( ord_le3221252021190050221t_unit @ Z @ X )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_914_sup__least,axiom,
! [Y: set_v,X: set_v,Z: set_v] :
( ( ord_less_eq_set_v @ Y @ X )
=> ( ( ord_less_eq_set_v @ Z @ X )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_915_sup__least,axiom,
! [Y: set_Product_prod_v_v,X: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y @ X )
=> ( ( ord_le7336532860387713383od_v_v @ Z @ X )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_916_le__iff__sup,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [X3: set_set_v,Y3: set_set_v] :
( ( sup_sup_set_set_v @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_917_le__iff__sup,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [X3: product_unit,Y3: product_unit] :
( ( sup_sup_Product_unit @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_918_le__iff__sup,axiom,
( ord_less_eq_set_v
= ( ^ [X3: set_v,Y3: set_v] :
( ( sup_sup_set_v @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_919_le__iff__sup,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [X3: set_Product_prod_v_v,Y3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_920_sup_OorderE,axiom,
! [B: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B @ A )
=> ( A
= ( sup_sup_set_set_v @ A @ B ) ) ) ).
% sup.orderE
thf(fact_921_sup_OorderE,axiom,
! [B: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ B @ A )
=> ( A
= ( sup_sup_Product_unit @ A @ B ) ) ) ).
% sup.orderE
thf(fact_922_sup_OorderE,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( A
= ( sup_sup_set_v @ A @ B ) ) ) ).
% sup.orderE
thf(fact_923_sup_OorderE,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( A
= ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% sup.orderE
thf(fact_924_sup_OorderI,axiom,
! [A: set_set_v,B: set_set_v] :
( ( A
= ( sup_sup_set_set_v @ A @ B ) )
=> ( ord_le5216385588623774835_set_v @ B @ A ) ) ).
% sup.orderI
thf(fact_925_sup_OorderI,axiom,
! [A: product_unit,B: product_unit] :
( ( A
= ( sup_sup_Product_unit @ A @ B ) )
=> ( ord_le3221252021190050221t_unit @ B @ A ) ) ).
% sup.orderI
thf(fact_926_sup_OorderI,axiom,
! [A: set_v,B: set_v] :
( ( A
= ( sup_sup_set_v @ A @ B ) )
=> ( ord_less_eq_set_v @ B @ A ) ) ).
% sup.orderI
thf(fact_927_sup_OorderI,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( A
= ( sup_su414716646722978715od_v_v @ A @ B ) )
=> ( ord_le7336532860387713383od_v_v @ B @ A ) ) ).
% sup.orderI
thf(fact_928_sup__unique,axiom,
! [F: set_set_v > set_set_v > set_set_v,X: set_set_v,Y: set_set_v] :
( ! [X4: set_set_v,Y4: set_set_v] : ( ord_le5216385588623774835_set_v @ X4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_set_v,Y4: set_set_v] : ( ord_le5216385588623774835_set_v @ Y4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_set_v,Y4: set_set_v,Z3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ Y4 @ X4 )
=> ( ( ord_le5216385588623774835_set_v @ Z3 @ X4 )
=> ( ord_le5216385588623774835_set_v @ ( F @ Y4 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_set_set_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_929_sup__unique,axiom,
! [F: product_unit > product_unit > product_unit,X: product_unit,Y: product_unit] :
( ! [X4: product_unit,Y4: product_unit] : ( ord_le3221252021190050221t_unit @ X4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: product_unit,Y4: product_unit] : ( ord_le3221252021190050221t_unit @ Y4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: product_unit,Y4: product_unit,Z3: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y4 @ X4 )
=> ( ( ord_le3221252021190050221t_unit @ Z3 @ X4 )
=> ( ord_le3221252021190050221t_unit @ ( F @ Y4 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_Product_unit @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_930_sup__unique,axiom,
! [F: set_v > set_v > set_v,X: set_v,Y: set_v] :
( ! [X4: set_v,Y4: set_v] : ( ord_less_eq_set_v @ X4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_v,Y4: set_v] : ( ord_less_eq_set_v @ Y4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_v,Y4: set_v,Z3: set_v] :
( ( ord_less_eq_set_v @ Y4 @ X4 )
=> ( ( ord_less_eq_set_v @ Z3 @ X4 )
=> ( ord_less_eq_set_v @ ( F @ Y4 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_set_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_931_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] :
( ! [X4: set_Product_prod_v_v,Y4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ X4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_Product_prod_v_v,Y4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ Y4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_Product_prod_v_v,Y4: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ Y4 @ X4 )
=> ( ( ord_le7336532860387713383od_v_v @ Z3 @ X4 )
=> ( ord_le7336532860387713383od_v_v @ ( F @ Y4 @ Z3 ) @ X4 ) ) )
=> ( ( sup_su414716646722978715od_v_v @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_932_sup_Oabsorb1,axiom,
! [B: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B @ A )
=> ( ( sup_sup_set_set_v @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_933_sup_Oabsorb1,axiom,
! [B: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ B @ A )
=> ( ( sup_sup_Product_unit @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_934_sup_Oabsorb1,axiom,
! [B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( sup_sup_set_v @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_935_sup_Oabsorb1,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( sup_su414716646722978715od_v_v @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_936_sup_Oabsorb2,axiom,
! [A: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A @ B )
=> ( ( sup_sup_set_set_v @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_937_sup_Oabsorb2,axiom,
! [A: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ A @ B )
=> ( ( sup_sup_Product_unit @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_938_sup_Oabsorb2,axiom,
! [A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ A @ B )
=> ( ( sup_sup_set_v @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_939_sup_Oabsorb2,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A @ B )
=> ( ( sup_su414716646722978715od_v_v @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_940_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_941_sup__absorb1,axiom,
! [Y: product_unit,X: product_unit] :
( ( ord_le3221252021190050221t_unit @ Y @ X )
=> ( ( sup_sup_Product_unit @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_942_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_943_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_944_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_945_sup__absorb2,axiom,
! [X: product_unit,Y: product_unit] :
( ( ord_le3221252021190050221t_unit @ X @ Y )
=> ( ( sup_sup_Product_unit @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_946_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_947_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_948_sup_OboundedE,axiom,
! [B: set_set_v,C: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ B @ C ) @ A )
=> ~ ( ( ord_le5216385588623774835_set_v @ B @ A )
=> ~ ( ord_le5216385588623774835_set_v @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_949_sup_OboundedE,axiom,
! [B: product_unit,C: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ B @ C ) @ A )
=> ~ ( ( ord_le3221252021190050221t_unit @ B @ A )
=> ~ ( ord_le3221252021190050221t_unit @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_950_sup_OboundedE,axiom,
! [B: set_v,C: set_v,A: set_v] :
( ( ord_less_eq_set_v @ ( sup_sup_set_v @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_v @ B @ A )
=> ~ ( ord_less_eq_set_v @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_951_sup_OboundedE,axiom,
! [B: set_Product_prod_v_v,C: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C ) @ A )
=> ~ ( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ~ ( ord_le7336532860387713383od_v_v @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_952_sup_OboundedI,axiom,
! [B: set_set_v,A: set_set_v,C: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B @ A )
=> ( ( ord_le5216385588623774835_set_v @ C @ A )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_953_sup_OboundedI,axiom,
! [B: product_unit,A: product_unit,C: product_unit] :
( ( ord_le3221252021190050221t_unit @ B @ A )
=> ( ( ord_le3221252021190050221t_unit @ C @ A )
=> ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_954_sup_OboundedI,axiom,
! [B: set_v,A: set_v,C: set_v] :
( ( ord_less_eq_set_v @ B @ A )
=> ( ( ord_less_eq_set_v @ C @ A )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_955_sup_OboundedI,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v,C: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B @ A )
=> ( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_956_sup_Oorder__iff,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [B5: set_set_v,A5: set_set_v] :
( A5
= ( sup_sup_set_set_v @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_957_sup_Oorder__iff,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [B5: product_unit,A5: product_unit] :
( A5
= ( sup_sup_Product_unit @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_958_sup_Oorder__iff,axiom,
( ord_less_eq_set_v
= ( ^ [B5: set_v,A5: set_v] :
( A5
= ( sup_sup_set_v @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_959_sup_Oorder__iff,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B5: set_Product_prod_v_v,A5: set_Product_prod_v_v] :
( A5
= ( sup_su414716646722978715od_v_v @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_960_sup_Ocobounded1,axiom,
! [A: set_set_v,B: set_set_v] : ( ord_le5216385588623774835_set_v @ A @ ( sup_sup_set_set_v @ A @ B ) ) ).
% sup.cobounded1
thf(fact_961_sup_Ocobounded1,axiom,
! [A: product_unit,B: product_unit] : ( ord_le3221252021190050221t_unit @ A @ ( sup_sup_Product_unit @ A @ B ) ) ).
% sup.cobounded1
thf(fact_962_sup_Ocobounded1,axiom,
! [A: set_v,B: set_v] : ( ord_less_eq_set_v @ A @ ( sup_sup_set_v @ A @ B ) ) ).
% sup.cobounded1
thf(fact_963_sup_Ocobounded1,axiom,
! [A: set_Product_prod_v_v,B: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% sup.cobounded1
thf(fact_964_sup_Ocobounded2,axiom,
! [B: set_set_v,A: set_set_v] : ( ord_le5216385588623774835_set_v @ B @ ( sup_sup_set_set_v @ A @ B ) ) ).
% sup.cobounded2
thf(fact_965_sup_Ocobounded2,axiom,
! [B: product_unit,A: product_unit] : ( ord_le3221252021190050221t_unit @ B @ ( sup_sup_Product_unit @ A @ B ) ) ).
% sup.cobounded2
thf(fact_966_sup_Ocobounded2,axiom,
! [B: set_v,A: set_v] : ( ord_less_eq_set_v @ B @ ( sup_sup_set_v @ A @ B ) ) ).
% sup.cobounded2
thf(fact_967_sup_Ocobounded2,axiom,
! [B: set_Product_prod_v_v,A: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ).
% sup.cobounded2
thf(fact_968_sup_Oabsorb__iff1,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [B5: set_set_v,A5: set_set_v] :
( ( sup_sup_set_set_v @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_969_sup_Oabsorb__iff1,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [B5: product_unit,A5: product_unit] :
( ( sup_sup_Product_unit @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_970_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_v
= ( ^ [B5: set_v,A5: set_v] :
( ( sup_sup_set_v @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_971_sup_Oabsorb__iff1,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [B5: set_Product_prod_v_v,A5: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_972_sup_Oabsorb__iff2,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [A5: set_set_v,B5: set_set_v] :
( ( sup_sup_set_set_v @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_973_sup_Oabsorb__iff2,axiom,
( ord_le3221252021190050221t_unit
= ( ^ [A5: product_unit,B5: product_unit] :
( ( sup_sup_Product_unit @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_974_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_v
= ( ^ [A5: set_v,B5: set_v] :
( ( sup_sup_set_v @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_975_sup_Oabsorb__iff2,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A5: set_Product_prod_v_v,B5: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_976_sup_OcoboundedI1,axiom,
! [C: set_set_v,A: set_set_v,B: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C @ A )
=> ( ord_le5216385588623774835_set_v @ C @ ( sup_sup_set_set_v @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_977_sup_OcoboundedI1,axiom,
! [C: product_unit,A: product_unit,B: product_unit] :
( ( ord_le3221252021190050221t_unit @ C @ A )
=> ( ord_le3221252021190050221t_unit @ C @ ( sup_sup_Product_unit @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_978_sup_OcoboundedI1,axiom,
! [C: set_v,A: set_v,B: set_v] :
( ( ord_less_eq_set_v @ C @ A )
=> ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_979_sup_OcoboundedI1,axiom,
! [C: set_Product_prod_v_v,A: set_Product_prod_v_v,B: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ A )
=> ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_980_sup_OcoboundedI2,axiom,
! [C: set_set_v,B: set_set_v,A: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C @ B )
=> ( ord_le5216385588623774835_set_v @ C @ ( sup_sup_set_set_v @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_981_sup_OcoboundedI2,axiom,
! [C: product_unit,B: product_unit,A: product_unit] :
( ( ord_le3221252021190050221t_unit @ C @ B )
=> ( ord_le3221252021190050221t_unit @ C @ ( sup_sup_Product_unit @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_982_sup_OcoboundedI2,axiom,
! [C: set_v,B: set_v,A: set_v] :
( ( ord_less_eq_set_v @ C @ B )
=> ( ord_less_eq_set_v @ C @ ( sup_sup_set_v @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_983_sup_OcoboundedI2,axiom,
! [C: set_Product_prod_v_v,B: set_Product_prod_v_v,A: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C @ B )
=> ( ord_le7336532860387713383od_v_v @ C @ ( sup_su414716646722978715od_v_v @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_984_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_985_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_986_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_987_boolean__algebra_Odisj__zero__right,axiom,
! [X: product_unit] :
( ( sup_sup_Product_unit @ X @ bot_bot_Product_unit )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_988_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) @ X )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ X ) @ ( sup_su414716646722978715od_v_v @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_989_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_v,Z: set_v,X: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ Z ) @ X )
= ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ X ) @ ( sup_sup_set_v @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_990_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_set_v,Z: set_set_v,X: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y @ Z ) @ X )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y @ X ) @ ( sup_sup_set_set_v @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_991_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: product_unit,Z: product_unit,X: product_unit] :
( ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y @ Z ) @ X )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y @ X ) @ ( sup_sup_Product_unit @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_992_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ Z ) @ X )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ X ) @ ( inf_in6271465464967711157od_v_v @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_993_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_v,Z: set_v,X: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ Z ) @ X )
= ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ X ) @ ( inf_inf_set_v @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_994_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_set_v,Z: set_set_v,X: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y @ Z ) @ X )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y @ X ) @ ( inf_inf_set_set_v @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_995_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: product_unit,Z: product_unit,X: product_unit] :
( ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y @ Z ) @ X )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y @ X ) @ ( inf_inf_Product_unit @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_996_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_997_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_998_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] :
( ( sup_sup_set_set_v @ X @ ( inf_inf_set_set_v @ Y @ Z ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X @ Y ) @ ( sup_sup_set_set_v @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_999_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( sup_sup_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X @ Y ) @ ( sup_sup_Product_unit @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_1000_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1001_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y ) @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1002_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] :
( ( inf_inf_set_set_v @ X @ ( sup_sup_set_set_v @ Y @ Z ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X @ Y ) @ ( inf_inf_set_set_v @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1003_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( inf_inf_Product_unit @ X @ ( sup_sup_Product_unit @ Y @ Z ) )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ ( inf_inf_Product_unit @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_1004_sup__inf__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) @ X )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ X ) @ ( sup_su414716646722978715od_v_v @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_1005_sup__inf__distrib2,axiom,
! [Y: set_v,Z: set_v,X: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ Z ) @ X )
= ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ X ) @ ( sup_sup_set_v @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_1006_sup__inf__distrib2,axiom,
! [Y: set_set_v,Z: set_set_v,X: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y @ Z ) @ X )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y @ X ) @ ( sup_sup_set_set_v @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_1007_sup__inf__distrib2,axiom,
! [Y: product_unit,Z: product_unit,X: product_unit] :
( ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y @ Z ) @ X )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y @ X ) @ ( sup_sup_Product_unit @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_1008_sup__inf__distrib1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_1009_sup__inf__distrib1,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_1010_sup__inf__distrib1,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] :
( ( sup_sup_set_set_v @ X @ ( inf_inf_set_set_v @ Y @ Z ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X @ Y ) @ ( sup_sup_set_set_v @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_1011_sup__inf__distrib1,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( sup_sup_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X @ Y ) @ ( sup_sup_Product_unit @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_1012_inf__sup__distrib2,axiom,
! [Y: set_Product_prod_v_v,Z: set_Product_prod_v_v,X: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ Y @ Z ) @ X )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ Y @ X ) @ ( inf_in6271465464967711157od_v_v @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_1013_inf__sup__distrib2,axiom,
! [Y: set_v,Z: set_v,X: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ Y @ Z ) @ X )
= ( sup_sup_set_v @ ( inf_inf_set_v @ Y @ X ) @ ( inf_inf_set_v @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_1014_inf__sup__distrib2,axiom,
! [Y: set_set_v,Z: set_set_v,X: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ Y @ Z ) @ X )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ Y @ X ) @ ( inf_inf_set_set_v @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_1015_inf__sup__distrib2,axiom,
! [Y: product_unit,Z: product_unit,X: product_unit] :
( ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y @ Z ) @ X )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y @ X ) @ ( inf_inf_Product_unit @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_1016_inf__sup__distrib1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_1017_inf__sup__distrib1,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y ) @ ( inf_inf_set_v @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_1018_inf__sup__distrib1,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] :
( ( inf_inf_set_set_v @ X @ ( sup_sup_set_set_v @ Y @ Z ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X @ Y ) @ ( inf_inf_set_set_v @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_1019_inf__sup__distrib1,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ( inf_inf_Product_unit @ X @ ( sup_sup_Product_unit @ Y @ Z ) )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ ( inf_inf_Product_unit @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_1020_distrib__imp2,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ! [X4: set_Product_prod_v_v,Y4: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ X4 @ ( inf_in6271465464967711157od_v_v @ Y4 @ Z3 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X4 @ Y4 ) @ ( sup_su414716646722978715od_v_v @ X4 @ Z3 ) ) )
=> ( ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_1021_distrib__imp2,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ! [X4: set_v,Y4: set_v,Z3: set_v] :
( ( sup_sup_set_v @ X4 @ ( inf_inf_set_v @ Y4 @ Z3 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X4 @ Y4 ) @ ( sup_sup_set_v @ X4 @ Z3 ) ) )
=> ( ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y ) @ ( inf_inf_set_v @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_1022_distrib__imp2,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] :
( ! [X4: set_set_v,Y4: set_set_v,Z3: set_set_v] :
( ( sup_sup_set_set_v @ X4 @ ( inf_inf_set_set_v @ Y4 @ Z3 ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X4 @ Y4 ) @ ( sup_sup_set_set_v @ X4 @ Z3 ) ) )
=> ( ( inf_inf_set_set_v @ X @ ( sup_sup_set_set_v @ Y @ Z ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X @ Y ) @ ( inf_inf_set_set_v @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_1023_distrib__imp2,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ! [X4: product_unit,Y4: product_unit,Z3: product_unit] :
( ( sup_sup_Product_unit @ X4 @ ( inf_inf_Product_unit @ Y4 @ Z3 ) )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X4 @ Y4 ) @ ( sup_sup_Product_unit @ X4 @ Z3 ) ) )
=> ( ( inf_inf_Product_unit @ X @ ( sup_sup_Product_unit @ Y @ Z ) )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ ( inf_inf_Product_unit @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_1024_distrib__imp1,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] :
( ! [X4: set_Product_prod_v_v,Y4: set_Product_prod_v_v,Z3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ X4 @ ( sup_su414716646722978715od_v_v @ Y4 @ Z3 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X4 @ Y4 ) @ ( inf_in6271465464967711157od_v_v @ X4 @ Z3 ) ) )
=> ( ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_1025_distrib__imp1,axiom,
! [X: set_v,Y: set_v,Z: set_v] :
( ! [X4: set_v,Y4: set_v,Z3: set_v] :
( ( inf_inf_set_v @ X4 @ ( sup_sup_set_v @ Y4 @ Z3 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ X4 @ Y4 ) @ ( inf_inf_set_v @ X4 @ Z3 ) ) )
=> ( ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y ) @ ( sup_sup_set_v @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_1026_distrib__imp1,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] :
( ! [X4: set_set_v,Y4: set_set_v,Z3: set_set_v] :
( ( inf_inf_set_set_v @ X4 @ ( sup_sup_set_set_v @ Y4 @ Z3 ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ X4 @ Y4 ) @ ( inf_inf_set_set_v @ X4 @ Z3 ) ) )
=> ( ( sup_sup_set_set_v @ X @ ( inf_inf_set_set_v @ Y @ Z ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X @ Y ) @ ( sup_sup_set_set_v @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_1027_distrib__imp1,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] :
( ! [X4: product_unit,Y4: product_unit,Z3: product_unit] :
( ( inf_inf_Product_unit @ X4 @ ( sup_sup_Product_unit @ Y4 @ Z3 ) )
= ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X4 @ Y4 ) @ ( inf_inf_Product_unit @ X4 @ Z3 ) ) )
=> ( ( sup_sup_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
= ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X @ Y ) @ ( sup_sup_Product_unit @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_1028_Un__empty__left,axiom,
! [B4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ bot_bo723834152578015283od_v_v @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_1029_Un__empty__left,axiom,
! [B4: set_v] :
( ( sup_sup_set_v @ bot_bot_set_v @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_1030_Un__empty__left,axiom,
! [B4: set_set_v] :
( ( sup_sup_set_set_v @ bot_bot_set_set_v @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_1031_Un__empty__right,axiom,
! [A3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ bot_bo723834152578015283od_v_v )
= A3 ) ).
% Un_empty_right
thf(fact_1032_Un__empty__right,axiom,
! [A3: set_v] :
( ( sup_sup_set_v @ A3 @ bot_bot_set_v )
= A3 ) ).
% Un_empty_right
thf(fact_1033_Un__empty__right,axiom,
! [A3: set_set_v] :
( ( sup_sup_set_set_v @ A3 @ bot_bot_set_set_v )
= A3 ) ).
% Un_empty_right
thf(fact_1034_finite__UnI,axiom,
! [F3: set_Product_prod_v_v,G: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ F3 )
=> ( ( finite3348123685078250256od_v_v @ G )
=> ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ F3 @ G ) ) ) ) ).
% finite_UnI
thf(fact_1035_finite__UnI,axiom,
! [F3: set_v,G: set_v] :
( ( finite_finite_v @ F3 )
=> ( ( finite_finite_v @ G )
=> ( finite_finite_v @ ( sup_sup_set_v @ F3 @ G ) ) ) ) ).
% finite_UnI
thf(fact_1036_finite__UnI,axiom,
! [F3: set_set_v,G: set_set_v] :
( ( finite_finite_set_v @ F3 )
=> ( ( finite_finite_set_v @ G )
=> ( finite_finite_set_v @ ( sup_sup_set_set_v @ F3 @ G ) ) ) ) ).
% finite_UnI
thf(fact_1037_Un__infinite,axiom,
! [S4: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ~ ( finite3348123685078250256od_v_v @ S4 )
=> ~ ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ S4 @ T ) ) ) ).
% Un_infinite
thf(fact_1038_Un__infinite,axiom,
! [S4: set_v,T: set_v] :
( ~ ( finite_finite_v @ S4 )
=> ~ ( finite_finite_v @ ( sup_sup_set_v @ S4 @ T ) ) ) ).
% Un_infinite
thf(fact_1039_Un__infinite,axiom,
! [S4: set_set_v,T: set_set_v] :
( ~ ( finite_finite_set_v @ S4 )
=> ~ ( finite_finite_set_v @ ( sup_sup_set_set_v @ S4 @ T ) ) ) ).
% Un_infinite
thf(fact_1040_infinite__Un,axiom,
! [S4: set_Product_prod_v_v,T: set_Product_prod_v_v] :
( ( ~ ( finite3348123685078250256od_v_v @ ( sup_su414716646722978715od_v_v @ S4 @ T ) ) )
= ( ~ ( finite3348123685078250256od_v_v @ S4 )
| ~ ( finite3348123685078250256od_v_v @ T ) ) ) ).
% infinite_Un
thf(fact_1041_infinite__Un,axiom,
! [S4: set_v,T: set_v] :
( ( ~ ( finite_finite_v @ ( sup_sup_set_v @ S4 @ T ) ) )
= ( ~ ( finite_finite_v @ S4 )
| ~ ( finite_finite_v @ T ) ) ) ).
% infinite_Un
thf(fact_1042_infinite__Un,axiom,
! [S4: set_set_v,T: set_set_v] :
( ( ~ ( finite_finite_set_v @ ( sup_sup_set_set_v @ S4 @ T ) ) )
= ( ~ ( finite_finite_set_v @ S4 )
| ~ ( finite_finite_set_v @ T ) ) ) ).
% infinite_Un
thf(fact_1043_Un__mono,axiom,
! [A3: set_set_v,C2: set_set_v,B4: set_set_v,D2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ C2 )
=> ( ( ord_le5216385588623774835_set_v @ B4 @ D2 )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A3 @ B4 ) @ ( sup_sup_set_set_v @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_1044_Un__mono,axiom,
! [A3: set_v,C2: set_v,B4: set_v,D2: set_v] :
( ( ord_less_eq_set_v @ A3 @ C2 )
=> ( ( ord_less_eq_set_v @ B4 @ D2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A3 @ B4 ) @ ( sup_sup_set_v @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_1045_Un__mono,axiom,
! [A3: set_Product_prod_v_v,C2: set_Product_prod_v_v,B4: set_Product_prod_v_v,D2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B4 @ D2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B4 ) @ ( sup_su414716646722978715od_v_v @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_1046_Un__least,axiom,
! [A3: set_set_v,C2: set_set_v,B4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ C2 )
=> ( ( ord_le5216385588623774835_set_v @ B4 @ C2 )
=> ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ A3 @ B4 ) @ C2 ) ) ) ).
% Un_least
thf(fact_1047_Un__least,axiom,
! [A3: set_v,C2: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ A3 @ C2 )
=> ( ( ord_less_eq_set_v @ B4 @ C2 )
=> ( ord_less_eq_set_v @ ( sup_sup_set_v @ A3 @ B4 ) @ C2 ) ) ) ).
% Un_least
thf(fact_1048_Un__least,axiom,
! [A3: set_Product_prod_v_v,C2: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ C2 )
=> ( ( ord_le7336532860387713383od_v_v @ B4 @ C2 )
=> ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B4 ) @ C2 ) ) ) ).
% Un_least
thf(fact_1049_Un__upper1,axiom,
! [A3: set_set_v,B4: set_set_v] : ( ord_le5216385588623774835_set_v @ A3 @ ( sup_sup_set_set_v @ A3 @ B4 ) ) ).
% Un_upper1
thf(fact_1050_Un__upper1,axiom,
! [A3: set_v,B4: set_v] : ( ord_less_eq_set_v @ A3 @ ( sup_sup_set_v @ A3 @ B4 ) ) ).
% Un_upper1
thf(fact_1051_Un__upper1,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ A3 @ B4 ) ) ).
% Un_upper1
thf(fact_1052_Un__upper2,axiom,
! [B4: set_set_v,A3: set_set_v] : ( ord_le5216385588623774835_set_v @ B4 @ ( sup_sup_set_set_v @ A3 @ B4 ) ) ).
% Un_upper2
thf(fact_1053_Un__upper2,axiom,
! [B4: set_v,A3: set_v] : ( ord_less_eq_set_v @ B4 @ ( sup_sup_set_v @ A3 @ B4 ) ) ).
% Un_upper2
thf(fact_1054_Un__upper2,axiom,
! [B4: set_Product_prod_v_v,A3: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ B4 @ ( sup_su414716646722978715od_v_v @ A3 @ B4 ) ) ).
% Un_upper2
thf(fact_1055_Un__absorb1,axiom,
! [A3: set_set_v,B4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ B4 )
=> ( ( sup_sup_set_set_v @ A3 @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_1056_Un__absorb1,axiom,
! [A3: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ A3 @ B4 )
=> ( ( sup_sup_set_v @ A3 @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_1057_Un__absorb1,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B4 )
=> ( ( sup_su414716646722978715od_v_v @ A3 @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_1058_Un__absorb2,axiom,
! [B4: set_set_v,A3: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B4 @ A3 )
=> ( ( sup_sup_set_set_v @ A3 @ B4 )
= A3 ) ) ).
% Un_absorb2
thf(fact_1059_Un__absorb2,axiom,
! [B4: set_v,A3: set_v] :
( ( ord_less_eq_set_v @ B4 @ A3 )
=> ( ( sup_sup_set_v @ A3 @ B4 )
= A3 ) ) ).
% Un_absorb2
thf(fact_1060_Un__absorb2,axiom,
! [B4: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B4 @ A3 )
=> ( ( sup_su414716646722978715od_v_v @ A3 @ B4 )
= A3 ) ) ).
% Un_absorb2
thf(fact_1061_subset__UnE,axiom,
! [C2: set_set_v,A3: set_set_v,B4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ C2 @ ( sup_sup_set_set_v @ A3 @ B4 ) )
=> ~ ! [A8: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A8 @ A3 )
=> ! [B8: set_set_v] :
( ( ord_le5216385588623774835_set_v @ B8 @ B4 )
=> ( C2
!= ( sup_sup_set_set_v @ A8 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_1062_subset__UnE,axiom,
! [C2: set_v,A3: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ C2 @ ( sup_sup_set_v @ A3 @ B4 ) )
=> ~ ! [A8: set_v] :
( ( ord_less_eq_set_v @ A8 @ A3 )
=> ! [B8: set_v] :
( ( ord_less_eq_set_v @ B8 @ B4 )
=> ( C2
!= ( sup_sup_set_v @ A8 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_1063_subset__UnE,axiom,
! [C2: set_Product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ C2 @ ( sup_su414716646722978715od_v_v @ A3 @ B4 ) )
=> ~ ! [A8: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A8 @ A3 )
=> ! [B8: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ B8 @ B4 )
=> ( C2
!= ( sup_su414716646722978715od_v_v @ A8 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_1064_subset__Un__eq,axiom,
( ord_le5216385588623774835_set_v
= ( ^ [A6: set_set_v,B6: set_set_v] :
( ( sup_sup_set_set_v @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_1065_subset__Un__eq,axiom,
( ord_less_eq_set_v
= ( ^ [A6: set_v,B6: set_v] :
( ( sup_sup_set_v @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_1066_subset__Un__eq,axiom,
( ord_le7336532860387713383od_v_v
= ( ^ [A6: set_Product_prod_v_v,B6: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_1067_Un__Int__distrib2,axiom,
! [B4: set_Product_prod_v_v,C2: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B4 @ C2 ) @ A3 )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B4 @ A3 ) @ ( sup_su414716646722978715od_v_v @ C2 @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_1068_Un__Int__distrib2,axiom,
! [B4: set_v,C2: set_v,A3: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ B4 @ C2 ) @ A3 )
= ( inf_inf_set_v @ ( sup_sup_set_v @ B4 @ A3 ) @ ( sup_sup_set_v @ C2 @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_1069_Un__Int__distrib2,axiom,
! [B4: set_set_v,C2: set_set_v,A3: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ B4 @ C2 ) @ A3 )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ B4 @ A3 ) @ ( sup_sup_set_set_v @ C2 @ A3 ) ) ) ).
% Un_Int_distrib2
thf(fact_1070_Int__Un__distrib2,axiom,
! [B4: set_Product_prod_v_v,C2: set_Product_prod_v_v,A3: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ B4 @ C2 ) @ A3 )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ B4 @ A3 ) @ ( inf_in6271465464967711157od_v_v @ C2 @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_1071_Int__Un__distrib2,axiom,
! [B4: set_v,C2: set_v,A3: set_v] :
( ( inf_inf_set_v @ ( sup_sup_set_v @ B4 @ C2 ) @ A3 )
= ( sup_sup_set_v @ ( inf_inf_set_v @ B4 @ A3 ) @ ( inf_inf_set_v @ C2 @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_1072_Int__Un__distrib2,axiom,
! [B4: set_set_v,C2: set_set_v,A3: set_set_v] :
( ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ B4 @ C2 ) @ A3 )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ B4 @ A3 ) @ ( inf_inf_set_set_v @ C2 @ A3 ) ) ) ).
% Int_Un_distrib2
thf(fact_1073_Un__Int__distrib,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ A3 @ ( inf_in6271465464967711157od_v_v @ B4 @ C2 ) )
= ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B4 ) @ ( sup_su414716646722978715od_v_v @ A3 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1074_Un__Int__distrib,axiom,
! [A3: set_v,B4: set_v,C2: set_v] :
( ( sup_sup_set_v @ A3 @ ( inf_inf_set_v @ B4 @ C2 ) )
= ( inf_inf_set_v @ ( sup_sup_set_v @ A3 @ B4 ) @ ( sup_sup_set_v @ A3 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1075_Un__Int__distrib,axiom,
! [A3: set_set_v,B4: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ A3 @ ( inf_inf_set_set_v @ B4 @ C2 ) )
= ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ A3 @ B4 ) @ ( sup_sup_set_set_v @ A3 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_1076_Int__Un__distrib,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( inf_in6271465464967711157od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B4 @ C2 ) )
= ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B4 ) @ ( inf_in6271465464967711157od_v_v @ A3 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1077_Int__Un__distrib,axiom,
! [A3: set_v,B4: set_v,C2: set_v] :
( ( inf_inf_set_v @ A3 @ ( sup_sup_set_v @ B4 @ C2 ) )
= ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B4 ) @ ( inf_inf_set_v @ A3 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1078_Int__Un__distrib,axiom,
! [A3: set_set_v,B4: set_set_v,C2: set_set_v] :
( ( inf_inf_set_set_v @ A3 @ ( sup_sup_set_set_v @ B4 @ C2 ) )
= ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A3 @ B4 ) @ ( inf_inf_set_set_v @ A3 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_1079_Un__Int__crazy,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B4 ) @ ( inf_in6271465464967711157od_v_v @ B4 @ C2 ) ) @ ( inf_in6271465464967711157od_v_v @ C2 @ A3 ) )
= ( inf_in6271465464967711157od_v_v @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B4 ) @ ( sup_su414716646722978715od_v_v @ B4 @ C2 ) ) @ ( sup_su414716646722978715od_v_v @ C2 @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_1080_Un__Int__crazy,axiom,
! [A3: set_v,B4: set_v,C2: set_v] :
( ( sup_sup_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B4 ) @ ( inf_inf_set_v @ B4 @ C2 ) ) @ ( inf_inf_set_v @ C2 @ A3 ) )
= ( inf_inf_set_v @ ( inf_inf_set_v @ ( sup_sup_set_v @ A3 @ B4 ) @ ( sup_sup_set_v @ B4 @ C2 ) ) @ ( sup_sup_set_v @ C2 @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_1081_Un__Int__crazy,axiom,
! [A3: set_set_v,B4: set_set_v,C2: set_set_v] :
( ( sup_sup_set_set_v @ ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A3 @ B4 ) @ ( inf_inf_set_set_v @ B4 @ C2 ) ) @ ( inf_inf_set_set_v @ C2 @ A3 ) )
= ( inf_inf_set_set_v @ ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ A3 @ B4 ) @ ( sup_sup_set_set_v @ B4 @ C2 ) ) @ ( sup_sup_set_set_v @ C2 @ A3 ) ) ) ).
% Un_Int_crazy
thf(fact_1082_Un__Diff,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ ( sup_su414716646722978715od_v_v @ A3 @ B4 ) @ C2 )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ C2 ) @ ( minus_4183494784930505774od_v_v @ B4 @ C2 ) ) ) ).
% Un_Diff
thf(fact_1083_Un__Diff,axiom,
! [A3: set_set_v,B4: set_set_v,C2: set_set_v] :
( ( minus_7228012346218142266_set_v @ ( sup_sup_set_set_v @ A3 @ B4 ) @ C2 )
= ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ A3 @ C2 ) @ ( minus_7228012346218142266_set_v @ B4 @ C2 ) ) ) ).
% Un_Diff
thf(fact_1084_Un__Diff,axiom,
! [A3: set_v,B4: set_v,C2: set_v] :
( ( minus_minus_set_v @ ( sup_sup_set_v @ A3 @ B4 ) @ C2 )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A3 @ C2 ) @ ( minus_minus_set_v @ B4 @ C2 ) ) ) ).
% Un_Diff
thf(fact_1085_distrib__inf__le,axiom,
! [X: set_set_v,Y: set_set_v,Z: 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 @ Z ) ) @ ( inf_inf_set_set_v @ X @ ( sup_sup_set_set_v @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1086_distrib__inf__le,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] : ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ ( inf_inf_Product_unit @ X @ Z ) ) @ ( inf_inf_Product_unit @ X @ ( sup_sup_Product_unit @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1087_distrib__inf__le,axiom,
! [X: set_v,Y: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ ( inf_inf_set_v @ X @ Y ) @ ( inf_inf_set_v @ X @ Z ) ) @ ( inf_inf_set_v @ X @ ( sup_sup_set_v @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1088_distrib__inf__le,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ X @ Y ) @ ( inf_in6271465464967711157od_v_v @ X @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ X @ ( sup_su414716646722978715od_v_v @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_1089_distrib__sup__le,axiom,
! [X: set_set_v,Y: set_set_v,Z: set_set_v] : ( ord_le5216385588623774835_set_v @ ( sup_sup_set_set_v @ X @ ( inf_inf_set_set_v @ Y @ Z ) ) @ ( inf_inf_set_set_v @ ( sup_sup_set_set_v @ X @ Y ) @ ( sup_sup_set_set_v @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1090_distrib__sup__le,axiom,
! [X: product_unit,Y: product_unit,Z: product_unit] : ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) ) @ ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X @ Y ) @ ( sup_sup_Product_unit @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1091_distrib__sup__le,axiom,
! [X: set_v,Y: set_v,Z: set_v] : ( ord_less_eq_set_v @ ( sup_sup_set_v @ X @ ( inf_inf_set_v @ Y @ Z ) ) @ ( inf_inf_set_v @ ( sup_sup_set_v @ X @ Y ) @ ( sup_sup_set_v @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1092_distrib__sup__le,axiom,
! [X: set_Product_prod_v_v,Y: set_Product_prod_v_v,Z: set_Product_prod_v_v] : ( ord_le7336532860387713383od_v_v @ ( sup_su414716646722978715od_v_v @ X @ ( inf_in6271465464967711157od_v_v @ Y @ Z ) ) @ ( inf_in6271465464967711157od_v_v @ ( sup_su414716646722978715od_v_v @ X @ Y ) @ ( sup_su414716646722978715od_v_v @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_1093_singleton__Un__iff,axiom,
! [X: product_prod_v_v,A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v )
= ( sup_su414716646722978715od_v_v @ A3 @ B4 ) )
= ( ( ( A3 = bot_bo723834152578015283od_v_v )
& ( B4
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A3
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B4 = bot_bo723834152578015283od_v_v ) )
| ( ( A3
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B4
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1094_singleton__Un__iff,axiom,
! [X: v,A3: set_v,B4: set_v] :
( ( ( insert_v @ X @ bot_bot_set_v )
= ( sup_sup_set_v @ A3 @ B4 ) )
= ( ( ( A3 = bot_bot_set_v )
& ( B4
= ( insert_v @ X @ bot_bot_set_v ) ) )
| ( ( A3
= ( insert_v @ X @ bot_bot_set_v ) )
& ( B4 = bot_bot_set_v ) )
| ( ( A3
= ( insert_v @ X @ bot_bot_set_v ) )
& ( B4
= ( insert_v @ X @ bot_bot_set_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1095_singleton__Un__iff,axiom,
! [X: set_v,A3: set_set_v,B4: set_set_v] :
( ( ( insert_set_v @ X @ bot_bot_set_set_v )
= ( sup_sup_set_set_v @ A3 @ B4 ) )
= ( ( ( A3 = bot_bot_set_set_v )
& ( B4
= ( insert_set_v @ X @ bot_bot_set_set_v ) ) )
| ( ( A3
= ( insert_set_v @ X @ bot_bot_set_set_v ) )
& ( B4 = bot_bot_set_set_v ) )
| ( ( A3
= ( insert_set_v @ X @ bot_bot_set_set_v ) )
& ( B4
= ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1096_Un__singleton__iff,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v,X: product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ A3 @ B4 )
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
= ( ( ( A3 = bot_bo723834152578015283od_v_v )
& ( B4
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) )
| ( ( A3
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B4 = bot_bo723834152578015283od_v_v ) )
| ( ( A3
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) )
& ( B4
= ( insert1338601472111419319od_v_v @ X @ bot_bo723834152578015283od_v_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1097_Un__singleton__iff,axiom,
! [A3: set_v,B4: set_v,X: v] :
( ( ( sup_sup_set_v @ A3 @ B4 )
= ( insert_v @ X @ bot_bot_set_v ) )
= ( ( ( A3 = bot_bot_set_v )
& ( B4
= ( insert_v @ X @ bot_bot_set_v ) ) )
| ( ( A3
= ( insert_v @ X @ bot_bot_set_v ) )
& ( B4 = bot_bot_set_v ) )
| ( ( A3
= ( insert_v @ X @ bot_bot_set_v ) )
& ( B4
= ( insert_v @ X @ bot_bot_set_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1098_Un__singleton__iff,axiom,
! [A3: set_set_v,B4: set_set_v,X: set_v] :
( ( ( sup_sup_set_set_v @ A3 @ B4 )
= ( insert_set_v @ X @ bot_bot_set_set_v ) )
= ( ( ( A3 = bot_bot_set_set_v )
& ( B4
= ( insert_set_v @ X @ bot_bot_set_set_v ) ) )
| ( ( A3
= ( insert_set_v @ X @ bot_bot_set_set_v ) )
& ( B4 = bot_bot_set_set_v ) )
| ( ( A3
= ( insert_set_v @ X @ bot_bot_set_set_v ) )
& ( B4
= ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1099_insert__is__Un,axiom,
( insert1338601472111419319od_v_v
= ( ^ [A5: product_prod_v_v] : ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A5 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% insert_is_Un
thf(fact_1100_insert__is__Un,axiom,
( insert_v
= ( ^ [A5: v] : ( sup_sup_set_v @ ( insert_v @ A5 @ bot_bot_set_v ) ) ) ) ).
% insert_is_Un
thf(fact_1101_insert__is__Un,axiom,
( insert_set_v
= ( ^ [A5: set_v] : ( sup_sup_set_set_v @ ( insert_set_v @ A5 @ bot_bot_set_set_v ) ) ) ) ).
% insert_is_Un
thf(fact_1102_Un__Int__assoc__eq,axiom,
! [A3: set_set_v,B4: set_set_v,C2: set_set_v] :
( ( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A3 @ B4 ) @ C2 )
= ( inf_inf_set_set_v @ A3 @ ( sup_sup_set_set_v @ B4 @ C2 ) ) )
= ( ord_le5216385588623774835_set_v @ C2 @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_1103_Un__Int__assoc__eq,axiom,
! [A3: set_v,B4: set_v,C2: set_v] :
( ( ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B4 ) @ C2 )
= ( inf_inf_set_v @ A3 @ ( sup_sup_set_v @ B4 @ C2 ) ) )
= ( ord_less_eq_set_v @ C2 @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_1104_Un__Int__assoc__eq,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B4 ) @ C2 )
= ( inf_in6271465464967711157od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B4 @ C2 ) ) )
= ( ord_le7336532860387713383od_v_v @ C2 @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_1105_Diff__partition,axiom,
! [A3: set_set_v,B4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ B4 )
=> ( ( sup_sup_set_set_v @ A3 @ ( minus_7228012346218142266_set_v @ B4 @ A3 ) )
= B4 ) ) ).
% Diff_partition
thf(fact_1106_Diff__partition,axiom,
! [A3: set_v,B4: set_v] :
( ( ord_less_eq_set_v @ A3 @ B4 )
=> ( ( sup_sup_set_v @ A3 @ ( minus_minus_set_v @ B4 @ A3 ) )
= B4 ) ) ).
% Diff_partition
thf(fact_1107_Diff__partition,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ A3 @ B4 )
=> ( ( sup_su414716646722978715od_v_v @ A3 @ ( minus_4183494784930505774od_v_v @ B4 @ A3 ) )
= B4 ) ) ).
% Diff_partition
thf(fact_1108_Diff__subset__conv,axiom,
! [A3: set_set_v,B4: set_set_v,C2: set_set_v] :
( ( ord_le5216385588623774835_set_v @ ( minus_7228012346218142266_set_v @ A3 @ B4 ) @ C2 )
= ( ord_le5216385588623774835_set_v @ A3 @ ( sup_sup_set_set_v @ B4 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1109_Diff__subset__conv,axiom,
! [A3: set_v,B4: set_v,C2: set_v] :
( ( ord_less_eq_set_v @ ( minus_minus_set_v @ A3 @ B4 ) @ C2 )
= ( ord_less_eq_set_v @ A3 @ ( sup_sup_set_v @ B4 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1110_Diff__subset__conv,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B4 ) @ C2 )
= ( ord_le7336532860387713383od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B4 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1111_Un__Diff__Int,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B4 ) @ ( inf_in6271465464967711157od_v_v @ A3 @ B4 ) )
= A3 ) ).
% Un_Diff_Int
thf(fact_1112_Un__Diff__Int,axiom,
! [A3: set_set_v,B4: set_set_v] :
( ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ A3 @ B4 ) @ ( inf_inf_set_set_v @ A3 @ B4 ) )
= A3 ) ).
% Un_Diff_Int
thf(fact_1113_Un__Diff__Int,axiom,
! [A3: set_v,B4: set_v] :
( ( sup_sup_set_v @ ( minus_minus_set_v @ A3 @ B4 ) @ ( inf_inf_set_v @ A3 @ B4 ) )
= A3 ) ).
% Un_Diff_Int
thf(fact_1114_Int__Diff__Un,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v] :
( ( sup_su414716646722978715od_v_v @ ( inf_in6271465464967711157od_v_v @ A3 @ B4 ) @ ( minus_4183494784930505774od_v_v @ A3 @ B4 ) )
= A3 ) ).
% Int_Diff_Un
thf(fact_1115_Int__Diff__Un,axiom,
! [A3: set_set_v,B4: set_set_v] :
( ( sup_sup_set_set_v @ ( inf_inf_set_set_v @ A3 @ B4 ) @ ( minus_7228012346218142266_set_v @ A3 @ B4 ) )
= A3 ) ).
% Int_Diff_Un
thf(fact_1116_Int__Diff__Un,axiom,
! [A3: set_v,B4: set_v] :
( ( sup_sup_set_v @ ( inf_inf_set_v @ A3 @ B4 ) @ ( minus_minus_set_v @ A3 @ B4 ) )
= A3 ) ).
% Int_Diff_Un
thf(fact_1117_Diff__Int,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ ( inf_in6271465464967711157od_v_v @ B4 @ C2 ) )
= ( sup_su414716646722978715od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B4 ) @ ( minus_4183494784930505774od_v_v @ A3 @ C2 ) ) ) ).
% Diff_Int
thf(fact_1118_Diff__Int,axiom,
! [A3: set_set_v,B4: set_set_v,C2: set_set_v] :
( ( minus_7228012346218142266_set_v @ A3 @ ( inf_inf_set_set_v @ B4 @ C2 ) )
= ( sup_sup_set_set_v @ ( minus_7228012346218142266_set_v @ A3 @ B4 ) @ ( minus_7228012346218142266_set_v @ A3 @ C2 ) ) ) ).
% Diff_Int
thf(fact_1119_Diff__Int,axiom,
! [A3: set_v,B4: set_v,C2: set_v] :
( ( minus_minus_set_v @ A3 @ ( inf_inf_set_v @ B4 @ C2 ) )
= ( sup_sup_set_v @ ( minus_minus_set_v @ A3 @ B4 ) @ ( minus_minus_set_v @ A3 @ C2 ) ) ) ).
% Diff_Int
thf(fact_1120_Diff__Un,axiom,
! [A3: set_Product_prod_v_v,B4: set_Product_prod_v_v,C2: set_Product_prod_v_v] :
( ( minus_4183494784930505774od_v_v @ A3 @ ( sup_su414716646722978715od_v_v @ B4 @ C2 ) )
= ( inf_in6271465464967711157od_v_v @ ( minus_4183494784930505774od_v_v @ A3 @ B4 ) @ ( minus_4183494784930505774od_v_v @ A3 @ C2 ) ) ) ).
% Diff_Un
thf(fact_1121_Diff__Un,axiom,
! [A3: set_set_v,B4: set_set_v,C2: set_set_v] :
( ( minus_7228012346218142266_set_v @ A3 @ ( sup_sup_set_set_v @ B4 @ C2 ) )
= ( inf_inf_set_set_v @ ( minus_7228012346218142266_set_v @ A3 @ B4 ) @ ( minus_7228012346218142266_set_v @ A3 @ C2 ) ) ) ).
% Diff_Un
thf(fact_1122_Diff__Un,axiom,
! [A3: set_v,B4: set_v,C2: set_v] :
( ( minus_minus_set_v @ A3 @ ( sup_sup_set_v @ B4 @ C2 ) )
= ( inf_inf_set_v @ ( minus_minus_set_v @ A3 @ B4 ) @ ( minus_minus_set_v @ A3 @ C2 ) ) ) ).
% Diff_Un
thf(fact_1123_is__singleton__the__elem,axiom,
( is_sin9198872032823709915od_v_v
= ( ^ [A6: set_Product_prod_v_v] :
( A6
= ( insert1338601472111419319od_v_v @ ( the_el5392834299063928540od_v_v @ A6 ) @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1124_is__singleton__the__elem,axiom,
( is_singleton_v
= ( ^ [A6: set_v] :
( A6
= ( insert_v @ ( the_elem_v @ A6 ) @ bot_bot_set_v ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1125_is__singleton__the__elem,axiom,
( is_singleton_set_v
= ( ^ [A6: set_set_v] :
( A6
= ( insert_set_v @ ( the_elem_set_v @ A6 ) @ bot_bot_set_set_v ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1126_is__singletonI_H,axiom,
! [A3: set_Product_prod_v_v] :
( ( A3 != bot_bo723834152578015283od_v_v )
=> ( ! [X4: product_prod_v_v,Y4: product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X4 @ A3 )
=> ( ( member7453568604450474000od_v_v @ Y4 @ A3 )
=> ( X4 = Y4 ) ) )
=> ( is_sin9198872032823709915od_v_v @ A3 ) ) ) ).
% is_singletonI'
thf(fact_1127_is__singletonI_H,axiom,
! [A3: set_v] :
( ( A3 != bot_bot_set_v )
=> ( ! [X4: v,Y4: v] :
( ( member_v @ X4 @ A3 )
=> ( ( member_v @ Y4 @ A3 )
=> ( X4 = Y4 ) ) )
=> ( is_singleton_v @ A3 ) ) ) ).
% is_singletonI'
thf(fact_1128_is__singletonI_H,axiom,
! [A3: set_set_v] :
( ( A3 != bot_bot_set_set_v )
=> ( ! [X4: set_v,Y4: set_v] :
( ( member_set_v @ X4 @ A3 )
=> ( ( member_set_v @ Y4 @ A3 )
=> ( X4 = Y4 ) ) )
=> ( is_singleton_set_v @ A3 ) ) ) ).
% is_singletonI'
thf(fact_1129_graph_Ora__add__edge,axiom,
! [Vertices: set_v,Successors: v > set_v,X: v,Y: v,E5: set_Product_prod_v_v,V: v,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E5 )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ ( sup_su414716646722978715od_v_v @ E5 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
| ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ V @ ( sup_su414716646722978715od_v_v @ E5 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) )
& ( sCC_Bl4291963740693775144ding_v @ Successors @ W @ Y @ ( sup_su414716646722978715od_v_v @ E5 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).
% graph.ra_add_edge
thf(fact_1130_is__singletonE,axiom,
! [A3: set_Product_prod_v_v] :
( ( is_sin9198872032823709915od_v_v @ A3 )
=> ~ ! [X4: product_prod_v_v] :
( A3
!= ( insert1338601472111419319od_v_v @ X4 @ bot_bo723834152578015283od_v_v ) ) ) ).
% is_singletonE
thf(fact_1131_is__singletonE,axiom,
! [A3: set_v] :
( ( is_singleton_v @ A3 )
=> ~ ! [X4: v] :
( A3
!= ( insert_v @ X4 @ bot_bot_set_v ) ) ) ).
% is_singletonE
thf(fact_1132_is__singletonE,axiom,
! [A3: set_set_v] :
( ( is_singleton_set_v @ A3 )
=> ~ ! [X4: set_v] :
( A3
!= ( insert_set_v @ X4 @ bot_bot_set_set_v ) ) ) ).
% is_singletonE
thf(fact_1133_is__singleton__def,axiom,
( is_sin9198872032823709915od_v_v
= ( ^ [A6: set_Product_prod_v_v] :
? [X3: product_prod_v_v] :
( A6
= ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% is_singleton_def
thf(fact_1134_is__singleton__def,axiom,
( is_singleton_v
= ( ^ [A6: set_v] :
? [X3: v] :
( A6
= ( insert_v @ X3 @ bot_bot_set_v ) ) ) ) ).
% is_singleton_def
thf(fact_1135_is__singleton__def,axiom,
( is_singleton_set_v
= ( ^ [A6: set_set_v] :
? [X3: set_v] :
( A6
= ( insert_set_v @ X3 @ bot_bot_set_set_v ) ) ) ) ).
% is_singleton_def
thf(fact_1136_avoiding__explored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,X: v,Y: v,E5: set_Product_prod_v_v,W: v,V: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ E5 )
=> ( ~ ( member_v @ Y @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl4291963740693775144ding_v @ successors @ X @ Y @ ( sup_su414716646722978715od_v_v @ E5 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ).
% avoiding_explored
thf(fact_1137_graph_Oavoiding__explored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,X: v,Y: v,E5: set_Product_prod_v_v,W: v,V: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ E5 )
=> ( ~ ( member_v @ Y @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl4291963740693775144ding_v @ Successors @ X @ Y @ ( sup_su414716646722978715od_v_v @ E5 @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ V @ W ) @ bot_bo723834152578015283od_v_v ) ) ) ) ) ) ) ) ).
% graph.avoiding_explored
thf(fact_1138_Field__insert,axiom,
! [A: product_prod_v_v,B: product_prod_v_v,R: set_Pr2149350503807050951od_v_v] :
( ( field_7153129647634986036od_v_v @ ( insert5641704497130386615od_v_v @ ( produc4031800376763917143od_v_v @ A @ B ) @ R ) )
= ( sup_su414716646722978715od_v_v @ ( insert1338601472111419319od_v_v @ A @ ( insert1338601472111419319od_v_v @ B @ bot_bo723834152578015283od_v_v ) ) @ ( field_7153129647634986036od_v_v @ R ) ) ) ).
% Field_insert
thf(fact_1139_Field__insert,axiom,
! [A: v,B: v,R: set_Product_prod_v_v] :
( ( field_v @ ( insert1338601472111419319od_v_v @ ( product_Pair_v_v @ A @ B ) @ R ) )
= ( sup_sup_set_v @ ( insert_v @ A @ ( insert_v @ B @ bot_bot_set_v ) ) @ ( field_v @ R ) ) ) ).
% Field_insert
thf(fact_1140_Field__insert,axiom,
! [A: set_v,B: set_v,R: set_Pr8199228935972127175_set_v] :
( ( field_set_v @ ( insert1457770702614273975_set_v @ ( produc3441907479644599895_set_v @ A @ B ) @ R ) )
= ( sup_sup_set_set_v @ ( insert_set_v @ A @ ( insert_set_v @ B @ bot_bot_set_set_v ) ) @ ( field_set_v @ R ) ) ) ).
% Field_insert
thf(fact_1141_remove__def,axiom,
( remove5001965847480235980od_v_v
= ( ^ [X3: product_prod_v_v,A6: set_Product_prod_v_v] : ( minus_4183494784930505774od_v_v @ A6 @ ( insert1338601472111419319od_v_v @ X3 @ bot_bo723834152578015283od_v_v ) ) ) ) ).
% remove_def
thf(fact_1142_remove__def,axiom,
( remove_set_v
= ( ^ [X3: set_v,A6: set_set_v] : ( minus_7228012346218142266_set_v @ A6 @ ( insert_set_v @ X3 @ bot_bot_set_set_v ) ) ) ) ).
% remove_def
thf(fact_1143_remove__def,axiom,
( remove_v
= ( ^ [X3: v,A6: set_v] : ( minus_minus_set_v @ A6 @ ( insert_v @ X3 @ bot_bot_set_v ) ) ) ) ).
% remove_def
thf(fact_1144_Sup__fin_Oremove,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( member8406446414694345712od_v_v @ X @ A3 )
=> ( ( ( ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ A3 )
= X ) )
& ( ( ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
!= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ A3 )
= ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1145_Sup__fin_Oremove,axiom,
! [A3: set_set_set_v,X: set_set_v] :
( ( finite8701002811114149628_set_v @ A3 )
=> ( ( member_set_set_v @ X @ A3 )
=> ( ( ( ( minus_5754555218759951642_set_v @ A3 @ ( insert_set_set_v @ X @ bot_bo5775917114081396255_set_v ) )
= bot_bo5775917114081396255_set_v )
=> ( ( lattic1829858174534819978_set_v @ A3 )
= X ) )
& ( ( ( minus_5754555218759951642_set_v @ A3 @ ( insert_set_set_v @ X @ bot_bo5775917114081396255_set_v ) )
!= bot_bo5775917114081396255_set_v )
=> ( ( lattic1829858174534819978_set_v @ A3 )
= ( sup_sup_set_set_v @ X @ ( lattic1829858174534819978_set_v @ ( minus_5754555218759951642_set_v @ A3 @ ( insert_set_set_v @ X @ bot_bo5775917114081396255_set_v ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1146_Sup__fin_Oremove,axiom,
! [A3: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A3 )
=> ( ( member_Product_unit @ X @ A3 )
=> ( ( ( ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ A3 )
= X ) )
& ( ( ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
!= bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ A3 )
= ( sup_sup_Product_unit @ X @ ( lattic5294303975357428420t_unit @ ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1147_Sup__fin_Oremove,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ X @ A3 )
=> ( ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ A3 )
= X ) )
& ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
!= bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ A3 )
= ( sup_sup_set_v @ X @ ( lattic2918178447194608042_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1148_member__remove,axiom,
! [X: v,Y: v,A3: set_v] :
( ( member_v @ X @ ( remove_v @ Y @ A3 ) )
= ( ( member_v @ X @ A3 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_1149_member__remove,axiom,
! [X: product_prod_v_v,Y: product_prod_v_v,A3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ X @ ( remove5001965847480235980od_v_v @ Y @ A3 ) )
= ( ( member7453568604450474000od_v_v @ X @ A3 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_1150_Field__empty,axiom,
( ( field_7153129647634986036od_v_v @ bot_bo3282589961317712691od_v_v )
= bot_bo723834152578015283od_v_v ) ).
% Field_empty
thf(fact_1151_Field__empty,axiom,
( ( field_set_v @ bot_bo8153096493302634547_set_v )
= bot_bot_set_set_v ) ).
% Field_empty
thf(fact_1152_Field__empty,axiom,
( ( field_v @ bot_bo723834152578015283od_v_v )
= bot_bot_set_v ) ).
% Field_empty
thf(fact_1153_Field__Un,axiom,
! [R: set_Pr2149350503807050951od_v_v,S7: set_Pr2149350503807050951od_v_v] :
( ( field_7153129647634986036od_v_v @ ( sup_su1742609618068805275od_v_v @ R @ S7 ) )
= ( sup_su414716646722978715od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ ( field_7153129647634986036od_v_v @ S7 ) ) ) ).
% Field_Un
thf(fact_1154_Field__Un,axiom,
! [R: set_Pr8199228935972127175_set_v,S7: set_Pr8199228935972127175_set_v] :
( ( field_set_v @ ( sup_su7977902838240902043_set_v @ R @ S7 ) )
= ( sup_sup_set_set_v @ ( field_set_v @ R ) @ ( field_set_v @ S7 ) ) ) ).
% Field_Un
thf(fact_1155_Field__Un,axiom,
! [R: set_Product_prod_v_v,S7: set_Product_prod_v_v] :
( ( field_v @ ( sup_su414716646722978715od_v_v @ R @ S7 ) )
= ( sup_sup_set_v @ ( field_v @ R ) @ ( field_v @ S7 ) ) ) ).
% Field_Un
thf(fact_1156_Sup__fin_Osingleton,axiom,
! [X: set_v] :
( ( lattic2918178447194608042_set_v @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= X ) ).
% Sup_fin.singleton
thf(fact_1157_inf__Sup__absorb,axiom,
! [A3: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ A @ A3 )
=> ( ( inf_inf_set_v @ A @ ( lattic2918178447194608042_set_v @ A3 ) )
= A ) ) ) ).
% inf_Sup_absorb
thf(fact_1158_inf__Sup__absorb,axiom,
! [A3: set_Product_unit,A: product_unit] :
( ( finite4290736615968046902t_unit @ A3 )
=> ( ( member_Product_unit @ A @ A3 )
=> ( ( inf_inf_Product_unit @ A @ ( lattic5294303975357428420t_unit @ A3 ) )
= A ) ) ) ).
% inf_Sup_absorb
thf(fact_1159_Sup__fin_Oinsert,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A3 ) )
= ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1160_Sup__fin_Oinsert,axiom,
! [A3: set_set_set_v,X: set_set_v] :
( ( finite8701002811114149628_set_v @ A3 )
=> ( ( A3 != bot_bo5775917114081396255_set_v )
=> ( ( lattic1829858174534819978_set_v @ ( insert_set_set_v @ X @ A3 ) )
= ( sup_sup_set_set_v @ X @ ( lattic1829858174534819978_set_v @ A3 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1161_Sup__fin_Oinsert,axiom,
! [A3: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A3 )
=> ( ( A3 != bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X @ A3 ) )
= ( sup_sup_Product_unit @ X @ ( lattic5294303975357428420t_unit @ A3 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1162_Sup__fin_Oinsert,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ ( insert_set_v @ X @ A3 ) )
= ( sup_sup_set_v @ X @ ( lattic2918178447194608042_set_v @ A3 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1163_FieldI2,axiom,
! [I: product_prod_v_v,J: product_prod_v_v,R3: set_Pr2149350503807050951od_v_v] :
( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I @ J ) @ R3 )
=> ( member7453568604450474000od_v_v @ J @ ( field_7153129647634986036od_v_v @ R3 ) ) ) ).
% FieldI2
thf(fact_1164_FieldI2,axiom,
! [I: v,J: v,R3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I @ J ) @ R3 )
=> ( member_v @ J @ ( field_v @ R3 ) ) ) ).
% FieldI2
thf(fact_1165_FieldI1,axiom,
! [I: product_prod_v_v,J: product_prod_v_v,R3: set_Pr2149350503807050951od_v_v] :
( ( member3038538357316246288od_v_v @ ( produc4031800376763917143od_v_v @ I @ J ) @ R3 )
=> ( member7453568604450474000od_v_v @ I @ ( field_7153129647634986036od_v_v @ R3 ) ) ) ).
% FieldI1
thf(fact_1166_FieldI1,axiom,
! [I: v,J: v,R3: set_Product_prod_v_v] :
( ( member7453568604450474000od_v_v @ ( product_Pair_v_v @ I @ J ) @ R3 )
=> ( member_v @ I @ ( field_v @ R3 ) ) ) ).
% FieldI1
thf(fact_1167_finite__Field,axiom,
! [R: set_Product_prod_v_v] :
( ( finite3348123685078250256od_v_v @ R )
=> ( finite_finite_v @ ( field_v @ R ) ) ) ).
% finite_Field
thf(fact_1168_select__convs_I3_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl157864678168468314t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Explored ) ).
% select_convs(3)
thf(fact_1169_mono__Field,axiom,
! [R: set_Pr2149350503807050951od_v_v,S7: set_Pr2149350503807050951od_v_v] :
( ( ord_le6241436655786843239od_v_v @ R @ S7 )
=> ( ord_le7336532860387713383od_v_v @ ( field_7153129647634986036od_v_v @ R ) @ ( field_7153129647634986036od_v_v @ S7 ) ) ) ).
% mono_Field
thf(fact_1170_mono__Field,axiom,
! [R: set_Product_prod_v_v,S7: set_Product_prod_v_v] :
( ( ord_le7336532860387713383od_v_v @ R @ S7 )
=> ( ord_less_eq_set_v @ ( field_v @ R ) @ ( field_v @ S7 ) ) ) ).
% mono_Field
thf(fact_1171_Sup__fin_OcoboundedI,axiom,
! [A3: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ A @ A3 )
=> ( ord_less_eq_set_v @ A @ ( lattic2918178447194608042_set_v @ A3 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1172_Sup__fin_OcoboundedI,axiom,
! [A3: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( member8406446414694345712od_v_v @ A @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A @ ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_1173_Sup__fin_Oin__idem,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( member8406446414694345712od_v_v @ X @ A3 )
=> ( ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ A3 ) )
= ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1174_Sup__fin_Oin__idem,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ X @ A3 )
=> ( ( sup_sup_set_v @ X @ ( lattic2918178447194608042_set_v @ A3 ) )
= ( lattic2918178447194608042_set_v @ A3 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1175_Sup__fin_Oin__idem,axiom,
! [A3: set_set_set_v,X: set_set_v] :
( ( finite8701002811114149628_set_v @ A3 )
=> ( ( member_set_set_v @ X @ A3 )
=> ( ( sup_sup_set_set_v @ X @ ( lattic1829858174534819978_set_v @ A3 ) )
= ( lattic1829858174534819978_set_v @ A3 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1176_Sup__fin_Oin__idem,axiom,
! [A3: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A3 )
=> ( ( member_Product_unit @ X @ A3 )
=> ( ( sup_sup_Product_unit @ X @ ( lattic5294303975357428420t_unit @ A3 ) )
= ( lattic5294303975357428420t_unit @ A3 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1177_Sup__fin_Obounded__iff,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A3 ) @ X )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
=> ( ord_less_eq_set_v @ X3 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1178_Sup__fin_Obounded__iff,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ X )
= ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ X3 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1179_Sup__fin_OboundedI,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ! [A4: set_v] :
( ( member_set_v @ A4 @ A3 )
=> ( ord_less_eq_set_v @ A4 @ X ) )
=> ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A3 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1180_Sup__fin_OboundedI,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ! [A4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A4 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A4 @ X ) )
=> ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1181_Sup__fin_OboundedE,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A3 ) @ X )
=> ! [A9: set_v] :
( ( member_set_v @ A9 @ A3 )
=> ( ord_less_eq_set_v @ A9 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1182_Sup__fin_OboundedE,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ X )
=> ! [A9: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A9 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ A9 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1183_Sup__fin_Osubset__imp,axiom,
! [A3: set_set_v,B4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ B4 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( finite_finite_set_v @ B4 )
=> ( ord_less_eq_set_v @ ( lattic2918178447194608042_set_v @ A3 ) @ ( lattic2918178447194608042_set_v @ B4 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1184_Sup__fin_Osubset__imp,axiom,
! [A3: set_se8455005133513928103od_v_v,B4: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A3 @ B4 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( finite6084192165098772208od_v_v @ B4 )
=> ( ord_le7336532860387713383od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ ( lattic5151207300795964030od_v_v @ B4 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1185_Sup__fin_Osubset,axiom,
! [A3: set_se8455005133513928103od_v_v,B4: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( B4 != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le4714265922333009223od_v_v @ B4 @ A3 )
=> ( ( sup_su414716646722978715od_v_v @ ( lattic5151207300795964030od_v_v @ B4 ) @ ( lattic5151207300795964030od_v_v @ A3 ) )
= ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1186_Sup__fin_Osubset,axiom,
! [A3: set_set_set_v,B4: set_set_set_v] :
( ( finite8701002811114149628_set_v @ A3 )
=> ( ( B4 != bot_bo5775917114081396255_set_v )
=> ( ( ord_le8117609702905084755_set_v @ B4 @ A3 )
=> ( ( sup_sup_set_set_v @ ( lattic1829858174534819978_set_v @ B4 ) @ ( lattic1829858174534819978_set_v @ A3 ) )
= ( lattic1829858174534819978_set_v @ A3 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1187_Sup__fin_Osubset,axiom,
! [A3: set_Product_unit,B4: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A3 )
=> ( ( B4 != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3507040750410214029t_unit @ B4 @ A3 )
=> ( ( sup_sup_Product_unit @ ( lattic5294303975357428420t_unit @ B4 ) @ ( lattic5294303975357428420t_unit @ A3 ) )
= ( lattic5294303975357428420t_unit @ A3 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1188_Sup__fin_Osubset,axiom,
! [A3: set_set_v,B4: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( B4 != bot_bot_set_set_v )
=> ( ( ord_le5216385588623774835_set_v @ B4 @ A3 )
=> ( ( sup_sup_set_v @ ( lattic2918178447194608042_set_v @ B4 ) @ ( lattic2918178447194608042_set_v @ A3 ) )
= ( lattic2918178447194608042_set_v @ A3 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1189_Sup__fin_Oinsert__not__elem,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ~ ( member8406446414694345712od_v_v @ X @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A3 ) )
= ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1190_Sup__fin_Oinsert__not__elem,axiom,
! [A3: set_set_set_v,X: set_set_v] :
( ( finite8701002811114149628_set_v @ A3 )
=> ( ~ ( member_set_set_v @ X @ A3 )
=> ( ( A3 != bot_bo5775917114081396255_set_v )
=> ( ( lattic1829858174534819978_set_v @ ( insert_set_set_v @ X @ A3 ) )
= ( sup_sup_set_set_v @ X @ ( lattic1829858174534819978_set_v @ A3 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1191_Sup__fin_Oinsert__not__elem,axiom,
! [A3: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A3 )
=> ( ~ ( member_Product_unit @ X @ A3 )
=> ( ( A3 != bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X @ A3 ) )
= ( sup_sup_Product_unit @ X @ ( lattic5294303975357428420t_unit @ A3 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1192_Sup__fin_Oinsert__not__elem,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ~ ( member_set_v @ X @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ ( insert_set_v @ X @ A3 ) )
= ( sup_sup_set_v @ X @ ( lattic2918178447194608042_set_v @ A3 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1193_Sup__fin_Oclosed,axiom,
! [A3: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ! [X4: set_Product_prod_v_v,Y4: set_Product_prod_v_v] : ( member8406446414694345712od_v_v @ ( sup_su414716646722978715od_v_v @ X4 @ Y4 ) @ ( insert7504383016908236695od_v_v @ X4 @ ( insert7504383016908236695od_v_v @ Y4 @ bot_bo3497076220358800403od_v_v ) ) )
=> ( member8406446414694345712od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ A3 ) ) ) ) ).
% Sup_fin.closed
thf(fact_1194_Sup__fin_Oclosed,axiom,
! [A3: set_set_set_v] :
( ( finite8701002811114149628_set_v @ A3 )
=> ( ( A3 != bot_bo5775917114081396255_set_v )
=> ( ! [X4: set_set_v,Y4: set_set_v] : ( member_set_set_v @ ( sup_sup_set_set_v @ X4 @ Y4 ) @ ( insert_set_set_v @ X4 @ ( insert_set_set_v @ Y4 @ bot_bo5775917114081396255_set_v ) ) )
=> ( member_set_set_v @ ( lattic1829858174534819978_set_v @ A3 ) @ A3 ) ) ) ) ).
% Sup_fin.closed
thf(fact_1195_Sup__fin_Oclosed,axiom,
! [A3: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A3 )
=> ( ( A3 != bot_bo3957492148770167129t_unit )
=> ( ! [X4: product_unit,Y4: product_unit] : ( member_Product_unit @ ( sup_sup_Product_unit @ X4 @ Y4 ) @ ( insert_Product_unit @ X4 @ ( insert_Product_unit @ Y4 @ bot_bo3957492148770167129t_unit ) ) )
=> ( member_Product_unit @ ( lattic5294303975357428420t_unit @ A3 ) @ A3 ) ) ) ) ).
% Sup_fin.closed
thf(fact_1196_Sup__fin_Oclosed,axiom,
! [A3: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ! [X4: set_v,Y4: set_v] : ( member_set_v @ ( sup_sup_set_v @ X4 @ Y4 ) @ ( insert_set_v @ X4 @ ( insert_set_v @ Y4 @ bot_bot_set_set_v ) ) )
=> ( member_set_v @ ( lattic2918178447194608042_set_v @ A3 ) @ A3 ) ) ) ) ).
% Sup_fin.closed
thf(fact_1197_Sup__fin_Ounion,axiom,
! [A3: set_se8455005133513928103od_v_v,B4: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( finite6084192165098772208od_v_v @ B4 )
=> ( ( B4 != bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( sup_su335656005089752955od_v_v @ A3 @ B4 ) )
= ( sup_su414716646722978715od_v_v @ ( lattic5151207300795964030od_v_v @ A3 ) @ ( lattic5151207300795964030od_v_v @ B4 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1198_Sup__fin_Ounion,axiom,
! [A3: set_set_set_v,B4: set_set_set_v] :
( ( finite8701002811114149628_set_v @ A3 )
=> ( ( A3 != bot_bo5775917114081396255_set_v )
=> ( ( finite8701002811114149628_set_v @ B4 )
=> ( ( B4 != bot_bo5775917114081396255_set_v )
=> ( ( lattic1829858174534819978_set_v @ ( sup_su4471370308589719943_set_v @ A3 @ B4 ) )
= ( sup_sup_set_set_v @ ( lattic1829858174534819978_set_v @ A3 ) @ ( lattic1829858174534819978_set_v @ B4 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1199_Sup__fin_Ounion,axiom,
! [A3: set_Product_unit,B4: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A3 )
=> ( ( A3 != bot_bo3957492148770167129t_unit )
=> ( ( finite4290736615968046902t_unit @ B4 )
=> ( ( B4 != bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( sup_su793286257634532545t_unit @ A3 @ B4 ) )
= ( sup_sup_Product_unit @ ( lattic5294303975357428420t_unit @ A3 ) @ ( lattic5294303975357428420t_unit @ B4 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1200_Sup__fin_Ounion,axiom,
! [A3: set_set_v,B4: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( finite_finite_set_v @ B4 )
=> ( ( B4 != bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ ( sup_sup_set_set_v @ A3 @ B4 ) )
= ( sup_sup_set_v @ ( lattic2918178447194608042_set_v @ A3 ) @ ( lattic2918178447194608042_set_v @ B4 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1201_Sup__fin_Oinsert__remove,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( ( ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A3 ) )
= X ) )
& ( ( ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) )
!= bot_bo3497076220358800403od_v_v )
=> ( ( lattic5151207300795964030od_v_v @ ( insert7504383016908236695od_v_v @ X @ A3 ) )
= ( sup_su414716646722978715od_v_v @ X @ ( lattic5151207300795964030od_v_v @ ( minus_7679383599658060814od_v_v @ A3 @ ( insert7504383016908236695od_v_v @ X @ bot_bo3497076220358800403od_v_v ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1202_Sup__fin_Oinsert__remove,axiom,
! [A3: set_set_set_v,X: set_set_v] :
( ( finite8701002811114149628_set_v @ A3 )
=> ( ( ( ( minus_5754555218759951642_set_v @ A3 @ ( insert_set_set_v @ X @ bot_bo5775917114081396255_set_v ) )
= bot_bo5775917114081396255_set_v )
=> ( ( lattic1829858174534819978_set_v @ ( insert_set_set_v @ X @ A3 ) )
= X ) )
& ( ( ( minus_5754555218759951642_set_v @ A3 @ ( insert_set_set_v @ X @ bot_bo5775917114081396255_set_v ) )
!= bot_bo5775917114081396255_set_v )
=> ( ( lattic1829858174534819978_set_v @ ( insert_set_set_v @ X @ A3 ) )
= ( sup_sup_set_set_v @ X @ ( lattic1829858174534819978_set_v @ ( minus_5754555218759951642_set_v @ A3 @ ( insert_set_set_v @ X @ bot_bo5775917114081396255_set_v ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1203_Sup__fin_Oinsert__remove,axiom,
! [A3: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A3 )
=> ( ( ( ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X @ A3 ) )
= X ) )
& ( ( ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
!= bot_bo3957492148770167129t_unit )
=> ( ( lattic5294303975357428420t_unit @ ( insert_Product_unit @ X @ A3 ) )
= ( sup_sup_Product_unit @ X @ ( lattic5294303975357428420t_unit @ ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1204_Sup__fin_Oinsert__remove,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ ( insert_set_v @ X @ A3 ) )
= X ) )
& ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
!= bot_bot_set_set_v )
=> ( ( lattic2918178447194608042_set_v @ ( insert_set_v @ X @ A3 ) )
= ( sup_sup_set_v @ X @ ( lattic2918178447194608042_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1205_unite__wf__env,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl9196236973127232072t_unit @ successors @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ).
% unite_wf_env
thf(fact_1206_unite__sub__env,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5768913643336123637t_unit @ E @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ).
% unite_sub_env
thf(fact_1207_graph_Ounite__wf__env,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl7798947040364291444t_unit @ Successors @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_wf_env
thf(fact_1208_graph_Ounite__wf__env,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl9196236973127232072t_unit @ Successors @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_wf_env
thf(fact_1209_graph_Ounite__sub__env,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl7963838319573962697t_unit @ E @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_1210_graph_Ounite__sub__env,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl5768913643336123637t_unit @ E @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% graph.unite_sub_env
thf(fact_1211_unite__subscc,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( 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 @ V @ W @ E ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ) ) ).
% unite_subscc
thf(fact_1212_Inf__fin_Oremove,axiom,
! [A3: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A3 )
=> ( ( member_Product_unit @ X @ A3 )
=> ( ( ( ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= bot_bo3957492148770167129t_unit )
=> ( ( lattic1263872656861969706t_unit @ A3 )
= X ) )
& ( ( ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
!= bot_bo3957492148770167129t_unit )
=> ( ( lattic1263872656861969706t_unit @ A3 )
= ( inf_inf_Product_unit @ X @ ( lattic1263872656861969706t_unit @ ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1213_Inf__fin_Oremove,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ X @ A3 )
=> ( ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ A3 )
= X ) )
& ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
!= bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ A3 )
= ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1214_Inf__fin_Oinsert__remove,axiom,
! [A3: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A3 )
=> ( ( ( ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
= bot_bo3957492148770167129t_unit )
=> ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ A3 ) )
= X ) )
& ( ( ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
!= bot_bo3957492148770167129t_unit )
=> ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ A3 ) )
= ( inf_inf_Product_unit @ X @ ( lattic1263872656861969706t_unit @ ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1215_Inf__fin_Oinsert__remove,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ A3 ) )
= X ) )
& ( ( ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
!= bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ A3 ) )
= ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ ( minus_7228012346218142266_set_v @ A3 @ ( insert_set_v @ X @ bot_bot_set_set_v ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1216_Inf__fin_Oinsert,axiom,
! [A3: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A3 )
=> ( ( A3 != bot_bo3957492148770167129t_unit )
=> ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ A3 ) )
= ( inf_inf_Product_unit @ X @ ( lattic1263872656861969706t_unit @ A3 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1217_Inf__fin_Oinsert,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ A3 ) )
= ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ A3 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1218_Inf__fin_Osingleton,axiom,
! [X: set_v] :
( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ bot_bot_set_set_v ) )
= X ) ).
% Inf_fin.singleton
thf(fact_1219_sup__Inf__absorb,axiom,
! [A3: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( member8406446414694345712od_v_v @ A @ A3 )
=> ( ( sup_su414716646722978715od_v_v @ ( lattic4767070952889939172od_v_v @ A3 ) @ A )
= A ) ) ) ).
% sup_Inf_absorb
thf(fact_1220_sup__Inf__absorb,axiom,
! [A3: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ A @ A3 )
=> ( ( sup_sup_set_v @ ( lattic8209813555532694032_set_v @ A3 ) @ A )
= A ) ) ) ).
% sup_Inf_absorb
thf(fact_1221_sup__Inf__absorb,axiom,
! [A3: set_set_set_v,A: set_set_v] :
( ( finite8701002811114149628_set_v @ A3 )
=> ( ( member_set_set_v @ A @ A3 )
=> ( ( sup_sup_set_set_v @ ( lattic6100936567437900016_set_v @ A3 ) @ A )
= A ) ) ) ).
% sup_Inf_absorb
thf(fact_1222_sup__Inf__absorb,axiom,
! [A3: set_Product_unit,A: product_unit] :
( ( finite4290736615968046902t_unit @ A3 )
=> ( ( member_Product_unit @ A @ A3 )
=> ( ( sup_sup_Product_unit @ ( lattic1263872656861969706t_unit @ A3 ) @ A )
= A ) ) ) ).
% sup_Inf_absorb
thf(fact_1223_select__convs_I7_J,axiom,
! [Root: v,S: v > set_v,Explored: set_v,Visited: set_v,Vsuccs: v > set_v,Sccs: set_set_v,Stack: list_v,Cstack: list_v,More: product_unit] :
( ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bl8064756265740546429t_unit @ Root @ S @ Explored @ Visited @ Vsuccs @ Sccs @ Stack @ Cstack @ More ) )
= Stack ) ).
% select_convs(7)
thf(fact_1224_Inf__fin_OcoboundedI,axiom,
! [A3: set_set_v,A: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ A @ A3 )
=> ( ord_less_eq_set_v @ ( lattic8209813555532694032_set_v @ A3 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1225_Inf__fin_OcoboundedI,axiom,
! [A3: set_se8455005133513928103od_v_v,A: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( member8406446414694345712od_v_v @ A @ A3 )
=> ( ord_le7336532860387713383od_v_v @ ( lattic4767070952889939172od_v_v @ A3 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_1226_Inf__fin_Oin__idem,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( member_set_v @ X @ A3 )
=> ( ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ A3 ) )
= ( lattic8209813555532694032_set_v @ A3 ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_1227_Inf__fin_Oin__idem,axiom,
! [A3: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A3 )
=> ( ( member_Product_unit @ X @ A3 )
=> ( ( inf_inf_Product_unit @ X @ ( lattic1263872656861969706t_unit @ A3 ) )
= ( lattic1263872656861969706t_unit @ A3 ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_1228_Inf__fin_Obounded__iff,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( ord_less_eq_set_v @ X @ ( lattic8209813555532694032_set_v @ A3 ) )
= ( ! [X3: set_v] :
( ( member_set_v @ X3 @ A3 )
=> ( ord_less_eq_set_v @ X @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1229_Inf__fin_Obounded__iff,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ X @ ( lattic4767070952889939172od_v_v @ A3 ) )
= ( ! [X3: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ X3 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ X @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1230_Inf__fin_OboundedI,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ! [A4: set_v] :
( ( member_set_v @ A4 @ A3 )
=> ( ord_less_eq_set_v @ X @ A4 ) )
=> ( ord_less_eq_set_v @ X @ ( lattic8209813555532694032_set_v @ A3 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1231_Inf__fin_OboundedI,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ! [A4: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A4 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ X @ A4 ) )
=> ( ord_le7336532860387713383od_v_v @ X @ ( lattic4767070952889939172od_v_v @ A3 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1232_Inf__fin_OboundedE,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( ord_less_eq_set_v @ X @ ( lattic8209813555532694032_set_v @ A3 ) )
=> ! [A9: set_v] :
( ( member_set_v @ A9 @ A3 )
=> ( ord_less_eq_set_v @ X @ A9 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1233_Inf__fin_OboundedE,axiom,
! [A3: set_se8455005133513928103od_v_v,X: set_Product_prod_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( ord_le7336532860387713383od_v_v @ X @ ( lattic4767070952889939172od_v_v @ A3 ) )
=> ! [A9: set_Product_prod_v_v] :
( ( member8406446414694345712od_v_v @ A9 @ A3 )
=> ( ord_le7336532860387713383od_v_v @ X @ A9 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1234_graph_Ounite__subscc,axiom,
! [Vertices: set_Product_prod_v_v,Successors: product_prod_v_v > set_Product_prod_v_v,V: product_prod_v_v,E: sCC_Bl7326425374436813197t_unit,W: product_prod_v_v] :
( ( sCC_Bl8307124943676871238od_v_v @ Vertices @ Successors )
=> ( ( sCC_Bl3607325323686918683t_unit @ Successors @ V @ E )
=> ( ( member7453568604450474000od_v_v @ W @ ( Successors @ V ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl3878977043676959280t_unit @ E @ V ) )
=> ( ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5498988629518860705t_unit @ E ) )
=> ( ~ ( member7453568604450474000od_v_v @ W @ ( sCC_Bl5094201334446601350t_unit @ E ) )
=> ( sCC_Bl2301996248249672505od_v_v @ Successors @ ( sCC_Bl8440648026628373538t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) @ ( hd_Product_prod_v_v @ ( sCC_Bl2021302119412358655t_unit @ ( sCC_Bl4702006153222411093od_v_v @ V @ W @ E ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_subscc
thf(fact_1235_graph_Ounite__subscc,axiom,
! [Vertices: set_v,Successors: v > set_v,V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl1748261141445803503t_unit @ Successors @ V @ E )
=> ( ( member_v @ W @ ( Successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( 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 @ V @ W @ E ) @ ( hd_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ) ) ) ).
% graph.unite_subscc
thf(fact_1236_Inf__fin_Osubset__imp,axiom,
! [A3: set_set_v,B4: set_set_v] :
( ( ord_le5216385588623774835_set_v @ A3 @ B4 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( finite_finite_set_v @ B4 )
=> ( ord_less_eq_set_v @ ( lattic8209813555532694032_set_v @ B4 ) @ ( lattic8209813555532694032_set_v @ A3 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1237_Inf__fin_Osubset__imp,axiom,
! [A3: set_se8455005133513928103od_v_v,B4: set_se8455005133513928103od_v_v] :
( ( ord_le4714265922333009223od_v_v @ A3 @ B4 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ( finite6084192165098772208od_v_v @ B4 )
=> ( ord_le7336532860387713383od_v_v @ ( lattic4767070952889939172od_v_v @ B4 ) @ ( lattic4767070952889939172od_v_v @ A3 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1238_Inf__fin_Osubset,axiom,
! [A3: set_Product_unit,B4: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A3 )
=> ( ( B4 != bot_bo3957492148770167129t_unit )
=> ( ( ord_le3507040750410214029t_unit @ B4 @ A3 )
=> ( ( inf_inf_Product_unit @ ( lattic1263872656861969706t_unit @ B4 ) @ ( lattic1263872656861969706t_unit @ A3 ) )
= ( lattic1263872656861969706t_unit @ A3 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_1239_Inf__fin_Osubset,axiom,
! [A3: set_set_v,B4: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( B4 != bot_bot_set_set_v )
=> ( ( ord_le5216385588623774835_set_v @ B4 @ A3 )
=> ( ( inf_inf_set_v @ ( lattic8209813555532694032_set_v @ B4 ) @ ( lattic8209813555532694032_set_v @ A3 ) )
= ( lattic8209813555532694032_set_v @ A3 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_1240_Inf__fin_Oinsert__not__elem,axiom,
! [A3: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A3 )
=> ( ~ ( member_Product_unit @ X @ A3 )
=> ( ( A3 != bot_bo3957492148770167129t_unit )
=> ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ A3 ) )
= ( inf_inf_Product_unit @ X @ ( lattic1263872656861969706t_unit @ A3 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1241_Inf__fin_Oinsert__not__elem,axiom,
! [A3: set_set_v,X: set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ~ ( member_set_v @ X @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( insert_set_v @ X @ A3 ) )
= ( inf_inf_set_v @ X @ ( lattic8209813555532694032_set_v @ A3 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1242_Inf__fin_Oclosed,axiom,
! [A3: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A3 )
=> ( ( A3 != bot_bo3957492148770167129t_unit )
=> ( ! [X4: product_unit,Y4: product_unit] : ( member_Product_unit @ ( inf_inf_Product_unit @ X4 @ Y4 ) @ ( insert_Product_unit @ X4 @ ( insert_Product_unit @ Y4 @ bot_bo3957492148770167129t_unit ) ) )
=> ( member_Product_unit @ ( lattic1263872656861969706t_unit @ A3 ) @ A3 ) ) ) ) ).
% Inf_fin.closed
thf(fact_1243_Inf__fin_Oclosed,axiom,
! [A3: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ! [X4: set_v,Y4: set_v] : ( member_set_v @ ( inf_inf_set_v @ X4 @ Y4 ) @ ( insert_set_v @ X4 @ ( insert_set_v @ Y4 @ bot_bot_set_set_v ) ) )
=> ( member_set_v @ ( lattic8209813555532694032_set_v @ A3 ) @ A3 ) ) ) ) ).
% Inf_fin.closed
thf(fact_1244_Inf__fin_Ounion,axiom,
! [A3: set_Product_unit,B4: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A3 )
=> ( ( A3 != bot_bo3957492148770167129t_unit )
=> ( ( finite4290736615968046902t_unit @ B4 )
=> ( ( B4 != bot_bo3957492148770167129t_unit )
=> ( ( lattic1263872656861969706t_unit @ ( sup_su793286257634532545t_unit @ A3 @ B4 ) )
= ( inf_inf_Product_unit @ ( lattic1263872656861969706t_unit @ A3 ) @ ( lattic1263872656861969706t_unit @ B4 ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_1245_Inf__fin_Ounion,axiom,
! [A3: set_set_v,B4: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ( finite_finite_set_v @ B4 )
=> ( ( B4 != bot_bot_set_set_v )
=> ( ( lattic8209813555532694032_set_v @ ( sup_sup_set_set_v @ A3 @ B4 ) )
= ( inf_inf_set_v @ ( lattic8209813555532694032_set_v @ A3 ) @ ( lattic8209813555532694032_set_v @ B4 ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_1246_Inf__fin__le__Sup__fin,axiom,
! [A3: set_set_v] :
( ( finite_finite_set_v @ A3 )
=> ( ( A3 != bot_bot_set_set_v )
=> ( ord_less_eq_set_v @ ( lattic8209813555532694032_set_v @ A3 ) @ ( lattic2918178447194608042_set_v @ A3 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_1247_Inf__fin__le__Sup__fin,axiom,
! [A3: set_se8455005133513928103od_v_v] :
( ( finite6084192165098772208od_v_v @ A3 )
=> ( ( A3 != bot_bo3497076220358800403od_v_v )
=> ( ord_le7336532860387713383od_v_v @ ( lattic4767070952889939172od_v_v @ A3 ) @ ( lattic5151207300795964030od_v_v @ A3 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_1248_stack__class,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v,M: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) )
=> ( member_v @ M @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ).
% stack_class
thf(fact_1249_visited__unexplored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,M: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ M @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ M @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ~ ! [N2: v] :
( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ) ) ).
% visited_unexplored
thf(fact_1250_stack__unexplored,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ N @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ).
% stack_unexplored
thf(fact_1251_stack__visited,axiom,
! [E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ).
% stack_visited
thf(fact_1252_graph_Ostack__visited,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( member_v @ N @ ( sCC_Bl4645233313691564917t_unit @ E ) ) ) ) ) ).
% graph.stack_visited
thf(fact_1253_graph_Ostack__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ N @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ).
% graph.stack_unexplored
thf(fact_1254_graph_Ovisited__unexplored,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,M: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ M @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ M @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ~ ! [N2: v] :
( ( member_v @ N2 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ~ ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N2 ) ) ) ) ) ) ) ).
% graph.visited_unexplored
thf(fact_1255_graph_Ostack__class,axiom,
! [Vertices: set_v,Successors: v > set_v,E: sCC_Bl1394983891496994913t_unit,N: v,M: v] :
( ( sCC_Bloemen_graph_v @ Vertices @ Successors )
=> ( ( sCC_Bl9196236973127232072t_unit @ Successors @ E )
=> ( ( member_v @ N @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( ( member_v @ M @ ( sCC_Bl1280885523602775798t_unit @ E @ N ) )
=> ( member_v @ M @ ( minus_minus_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( sCC_Bl157864678168468314t_unit @ E ) ) ) ) ) ) ) ).
% graph.stack_class
thf(fact_1256_pre__dfs__def,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl36166008131615352t_unit @ successors @ V @ E )
= ( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
& ~ ( member_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
& ( sCC_Bl649662514949026229able_v @ successors @ ( sCC_Bl1090238580953940555t_unit @ E ) @ V )
& ( ( sCC_Bl3795065053823578884t_unit @ E @ V )
= bot_bot_set_v )
& ! [X3: v] :
( ( member_v @ X3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) )
=> ( sCC_Bl649662514949026229able_v @ successors @ X3 @ V ) ) ) ) ).
% pre_dfs_def
thf(fact_1257_unite__S__tl,axiom,
! [E: sCC_Bl1394983891496994913t_unit,W: v,V: v,N: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( member_v @ N @ ( set_v2 @ ( tl_v @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) )
=> ( ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ N )
= ( sCC_Bl1280885523602775798t_unit @ E @ N ) ) ) ) ) ) ) ) ).
% unite_S_tl
thf(fact_1258_List_Ofinite__set,axiom,
! [Xs: list_v] : ( finite_finite_v @ ( set_v2 @ Xs ) ) ).
% List.finite_set
thf(fact_1259_pre__dfss__unite__pre__dfss,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( sCC_Bl1748261141445803503t_unit @ successors @ V
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X3: v] : ( if_set_v @ ( X3 = V ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ V ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ X3 ) )
@ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) ) ) ).
% pre_dfss_unite_pre_dfss
thf(fact_1260_dfs__dfss_Odomintros_I1_J,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors )
@ ( sum_In5289330923152326972t_unit
@ ( produc3862955338007567901t_unit @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( insert_v @ V @ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ E ) ) ) ) ) )
=> ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V @ E ) ) ) ) ).
% dfs_dfss.domintros(1)
thf(fact_1261_pre__dfss__post__dfs__pre__dfss,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit,W: v] :
( ( sCC_Bl1748261141445803503t_unit @ successors @ V @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ( sCC_Bl8953792750115413617t_unit @ successors @ W @ E @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) )
=> ( sCC_Bl1748261141445803503t_unit @ successors @ V
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X3: v] : ( if_set_v @ ( X3 = V ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) @ V ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) @ X3 ) )
@ ( sCC_Bloemen_dfs_v @ successors @ W @ E ) ) ) ) ) ) ) ).
% pre_dfss_post_dfs_pre_dfss
thf(fact_1262_pre__dfs__implies__post__dfs,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( sCC_Bl36166008131615352t_unit @ successors @ V @ E )
=> ( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V @ E ) ) )
=> ( ( sCC_Bl6082031138996704384t_unit @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) )
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) )
=> ( sCC_Bl8953792750115413617t_unit @ successors @ V @ E @ ( sCC_Bloemen_dfs_v @ successors @ V @ E ) ) ) ) ) ).
% pre_dfs_implies_post_dfs
thf(fact_1263_dfs_Opsimps,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In526841707622398774t_unit @ ( produc3862955338007567901t_unit @ V @ E ) ) )
=> ( ( sCC_Bloemen_dfs_v @ successors @ V @ E )
= ( if_SCC4926449794303880475t_unit
@ ( V
= ( hd_v
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) ) ) )
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl8828226123343373779t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) ) )
@ ( sCC_Bl2708505634401380163t_unit
@ ^ [Uu: set_v] :
( sup_sup_set_v
@ ( sCC_Bl157864678168468314t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) )
@ V ) )
@ ( sCC_Bl6816368539212994290t_unit
@ ^ [Uu: set_set_v] :
( sup_sup_set_set_v
@ ( sCC_Bl2536197123907397897t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) )
@ ( insert_set_v
@ ( sCC_Bl1280885523602775798t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) )
@ V )
@ bot_bot_set_set_v ) )
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) ) ) ) )
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] :
( tl_v
@ ( sCC_Bl9201514103433284750t_unit
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uv: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uv: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) ) )
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl7876664385711583351t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl9201514103433284750t_unit @ E ) )
@ ( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] : ( cons_v @ V @ ( sCC_Bl8828226123343373779t_unit @ E ) )
@ ( sCC_Bl7870604408699998558t_unit
@ ^ [Uu: set_v] : ( sup_sup_set_v @ ( sCC_Bl4645233313691564917t_unit @ E ) @ ( insert_v @ V @ bot_bot_set_v ) )
@ E ) ) ) ) ) ) ) ) ).
% dfs.psimps
thf(fact_1264_prepostdfss,axiom,
! [Vs: set_v,W: v,E2: sCC_Bl1394983891496994913t_unit,E3: sCC_Bl1394983891496994913t_unit] :
( ( Vs
= ( minus_minus_set_v @ ( successors @ va ) @ ( sCC_Bl3795065053823578884t_unit @ ea @ va ) ) )
=> ( ( Vs != bot_bot_set_v )
=> ( ( W
= ( fChoice_v
@ ^ [X3: v] : ( member_v @ X3 @ Vs ) ) )
=> ( ( ( ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ ea ) )
=> ( E2 = ea ) )
& ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ ea ) )
=> ( ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ ea ) )
=> ( E2
= ( sCC_Bloemen_dfs_v @ successors @ W @ ea ) ) )
& ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ ea ) )
=> ( E2
= ( sCC_Bloemen_unite_v @ va @ W @ ea ) ) ) ) ) )
=> ( ( E3
= ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X3: v] : ( if_set_v @ ( X3 = va ) @ ( sup_sup_set_v @ ( sCC_Bl3795065053823578884t_unit @ E2 @ va ) @ ( insert_v @ W @ bot_bot_set_v ) ) @ ( sCC_Bl3795065053823578884t_unit @ E2 @ X3 ) )
@ E2 ) )
=> ( ( sCC_Bl1748261141445803503t_unit @ successors @ va @ E3 )
=> ( sCC_Bl6082031138996704384t_unit @ successors @ va @ E3 @ ( sCC_Bloemen_dfss_v @ successors @ va @ E3 ) ) ) ) ) ) ) ) ).
% prepostdfss
thf(fact_1265_dfss_Opsimps,axiom,
! [V: v,E: sCC_Bl1394983891496994913t_unit] :
( ( accp_S2303753412255344476t_unit @ ( sCC_Bl907557413677168252_rel_v @ successors ) @ ( sum_In5289330923152326972t_unit @ ( produc3862955338007567901t_unit @ V @ E ) ) )
=> ( ( sCC_Bloemen_dfss_v @ successors @ V @ E )
= ( if_SCC4926449794303880475t_unit
@ ( ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
= bot_bot_set_v )
@ E
@ ( sCC_Bloemen_dfss_v @ successors @ V
@ ( sCC_Bl48393358579903213t_unit
@ ^ [Uu: v > set_v,X3: v] :
( if_set_v @ ( X3 = V )
@ ( sup_sup_set_v
@ ( sCC_Bl3795065053823578884t_unit
@ ( if_SCC4926449794303880475t_unit
@ ( member_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E ) )
@ E
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ E )
@ ( sCC_Bloemen_unite_v @ V
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ E ) ) )
@ V )
@ ( insert_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ bot_bot_set_v ) )
@ ( sCC_Bl3795065053823578884t_unit
@ ( if_SCC4926449794303880475t_unit
@ ( member_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E ) )
@ E
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ E )
@ ( sCC_Bloemen_unite_v @ V
@ ( fChoice_v
@ ^ [Y3: v] : ( member_v @ Y3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ E ) ) )
@ X3 ) )
@ ( if_SCC4926449794303880475t_unit
@ ( member_v
@ ( fChoice_v
@ ^ [X3: v] : ( member_v @ X3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ ( sCC_Bl157864678168468314t_unit @ E ) )
@ E
@ ( if_SCC4926449794303880475t_unit
@ ~ ( member_v
@ ( fChoice_v
@ ^ [X3: v] : ( member_v @ X3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ ( sCC_Bl4645233313691564917t_unit @ E ) )
@ ( sCC_Bloemen_dfs_v @ successors
@ ( fChoice_v
@ ^ [X3: v] : ( member_v @ X3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ E )
@ ( sCC_Bloemen_unite_v @ V
@ ( fChoice_v
@ ^ [X3: v] : ( member_v @ X3 @ ( minus_minus_set_v @ ( successors @ V ) @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) ) ) )
@ E ) ) ) ) ) ) ) ) ).
% dfss.psimps
thf(fact_1266_prepostdfs,axiom,
! [Vs: set_v,W: v] :
( ( Vs
= ( minus_minus_set_v @ ( successors @ va ) @ ( sCC_Bl3795065053823578884t_unit @ ea @ va ) ) )
=> ( ( Vs != bot_bot_set_v )
=> ( ( W
= ( fChoice_v
@ ^ [X3: v] : ( member_v @ X3 @ Vs ) ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ ea ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ ea ) )
=> ( ( sCC_Bl36166008131615352t_unit @ successors @ W @ ea )
=> ( sCC_Bl8953792750115413617t_unit @ successors @ W @ ea @ ( sCC_Bloemen_dfs_v @ successors @ W @ ea ) ) ) ) ) ) ) ) ).
% prepostdfs
thf(fact_1267_unite__S__equal,axiom,
! [E: sCC_Bl1394983891496994913t_unit,W: v,V: v] :
( ( sCC_Bl9196236973127232072t_unit @ successors @ E )
=> ( ( member_v @ W @ ( successors @ V ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl3795065053823578884t_unit @ E @ V ) )
=> ( ( member_v @ W @ ( sCC_Bl4645233313691564917t_unit @ E ) )
=> ( ~ ( member_v @ W @ ( sCC_Bl157864678168468314t_unit @ E ) )
=> ( ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N3: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) @ N3 ) )
& ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ ( sCC_Bloemen_unite_v @ V @ W @ E ) ) ) ) ) ) )
= ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uu: set_v] :
? [N3: v] :
( ( Uu
= ( sCC_Bl1280885523602775798t_unit @ E @ N3 ) )
& ( member_v @ N3 @ ( set_v2 @ ( sCC_Bl8828226123343373779t_unit @ E ) ) ) ) ) ) ) ) ) ) ) ) ).
% unite_S_equal
thf(fact_1268_unite__def,axiom,
( sCC_Bloemen_unite_v
= ( ^ [V4: v,W2: v,E8: sCC_Bl1394983891496994913t_unit] :
( sCC_Bl349061681862590396t_unit
@ ^ [Uu: list_v] :
( dropWhile_v
@ ^ [X3: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ X3 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) )
@ ( sCC_Bl3155122997657187039t_unit
@ ^ [Uu: v > set_v,X3: v] :
( if_set_v
@ ( member_v @ X3
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uv: set_v] :
? [Y3: v] :
( ( Uv
= ( sCC_Bl1280885523602775798t_unit @ E8 @ Y3 ) )
& ( member_v @ Y3
@ ( sup_sup_set_v
@ ( set_v2
@ ( takeWhile_v
@ ^ [Z2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
@ ( insert_v
@ ( hd_v
@ ( dropWhile_v
@ ^ [Z2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
@ bot_bot_set_v ) ) ) ) ) ) )
@ ( comple2307003700295860064_set_v
@ ( collect_set_v
@ ^ [Uv: set_v] :
? [Y3: v] :
( ( Uv
= ( sCC_Bl1280885523602775798t_unit @ E8 @ Y3 ) )
& ( member_v @ Y3
@ ( sup_sup_set_v
@ ( set_v2
@ ( takeWhile_v
@ ^ [Z2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
@ ( insert_v
@ ( hd_v
@ ( dropWhile_v
@ ^ [Z2: v] :
~ ( member_v @ W2 @ ( sCC_Bl1280885523602775798t_unit @ E8 @ Z2 ) )
@ ( sCC_Bl8828226123343373779t_unit @ E8 ) ) )
@ bot_bot_set_v ) ) ) ) ) )
@ ( sCC_Bl1280885523602775798t_unit @ E8 @ X3 ) )
@ E8 ) ) ) ) ).
% unite_def
thf(fact_1269_Sup__unit__def,axiom,
( comple4687483117567863418t_unit
= ( ^ [Uu2: set_Product_unit] : product_Unity ) ) ).
% Sup_unit_def
thf(fact_1270_old_Ounit_Oexhaust,axiom,
! [Y: product_unit] : ( Y = product_Unity ) ).
% old.unit.exhaust
thf(fact_1271_sup__unit__def,axiom,
( sup_sup_Product_unit
= ( ^ [Uu2: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% sup_unit_def
thf(fact_1272_bot__unit__def,axiom,
bot_bot_Product_unit = product_Unity ).
% bot_unit_def
thf(fact_1273_inf__unit__def,axiom,
( inf_inf_Product_unit
= ( ^ [Uu2: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% inf_unit_def
thf(fact_1274_default__unit__def,axiom,
defaul566961228789861419t_unit = product_Unity ).
% default_unit_def
% Helper facts (6)
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_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,
member_v @ va @ ( sCC_Bl4645233313691564917t_unit @ ea ) ).
%------------------------------------------------------------------------------