TPTP Problem File: SLH0843^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Query_Optimization/0002_Graph_Definitions/prob_00079_002633__14929170_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1483 ( 598 unt; 204 typ; 0 def)
% Number of atoms : 3828 (1485 equ; 0 cnn)
% Maximal formula atoms : 81 ( 2 avg)
% Number of connectives : 11193 ( 507 ~; 65 |; 551 &;8552 @)
% ( 0 <=>;1518 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 6 avg)
% Number of types : 25 ( 24 usr)
% Number of type conns : 1046 (1046 >; 0 *; 0 +; 0 <<)
% Number of symbols : 182 ( 180 usr; 19 con; 0-5 aty)
% Number of variables : 3645 ( 381 ^;2983 !; 281 ?;3645 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:01:15.428
%------------------------------------------------------------------------------
% Could-be-implicit typings (24)
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% Explicit typings (180)
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ord_less_set_b: set_b > set_b > $o ).
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ord_le8458015892825683628unit_o: ( pre_pr7278220950009878019t_unit > $o ) > ( pre_pr7278220950009878019t_unit > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Extended____Real__Oereal_M_Eo_J,type,
ord_le6694447793465728271real_o: ( extended_ereal > $o ) > ( extended_ereal > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
ord_less_eq_set_a_o: ( set_a > $o ) > ( set_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__b_M_Eo_J,type,
ord_less_eq_b_o: ( b > $o ) > ( b > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Real__Oereal,type,
ord_le1083603963089353582_ereal: extended_ereal > extended_ereal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J_J,type,
ord_le8200006823705900825t_unit: set_pr5411798346947241657t_unit > set_pr5411798346947241657t_unit > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
ord_le1644982726543182158_ereal: set_Extended_ereal > set_Extended_ereal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__b_J_J,type,
ord_le8932221534207217157list_b: set_list_b > set_list_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
ord_less_eq_set_b: set_b > set_b > $o ).
thf(sy_c_Set_OCollect_001t__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J,type,
collec8000012497822511960t_unit: ( pre_pr7278220950009878019t_unit > $o ) > set_pr5411798346947241657t_unit ).
thf(sy_c_Set_OCollect_001t__Extended____Real__Oereal,type,
collec5835592288176408249_ereal: ( extended_ereal > $o ) > set_Extended_ereal ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__b_J,type,
collect_list_b: ( list_b > $o ) > set_list_b ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
collec85322473871370393_ereal: ( set_Extended_ereal > $o ) > set_se6634062954251873166_ereal ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
collect_set_list_a: ( set_list_a > $o ) > set_set_list_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__List__Olist_Itf__b_J_J,type,
collect_set_list_b: ( set_list_b > $o ) > set_set_list_b ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
collect_set_set_a: ( set_set_a > $o ) > set_set_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__b_J,type,
collect_set_b: ( set_b > $o ) > set_set_b ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_Set_Oimage_001t__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J_001t__Set__Oset_Itf__a_J,type,
image_7466199892558553556_set_a: ( pre_pr7278220950009878019t_unit > set_a ) > set_pr5411798346947241657t_unit > set_set_a ).
thf(sy_c_Set_Oimage_001t__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J_001tf__a,type,
image_4969699134812999796unit_a: ( pre_pr7278220950009878019t_unit > a ) > set_pr5411798346947241657t_unit > set_a ).
thf(sy_c_Set_Oimage_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
image_6042159593519690757_ereal: ( extended_ereal > extended_ereal ) > set_Extended_ereal > set_Extended_ereal ).
thf(sy_c_Set_Oimage_001t__Extended____Real__Oereal_001t__List__Olist_Itf__b_J,type,
image_3533209195447846460list_b: ( extended_ereal > list_b ) > set_Extended_ereal > set_list_b ).
thf(sy_c_Set_Oimage_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
image_7659842161140344153al_nat: ( extended_ereal > nat ) > set_Extended_ereal > set_nat ).
thf(sy_c_Set_Oimage_001t__Extended____Real__Oereal_001tf__a,type,
image_3724615099042636213real_a: ( extended_ereal > a ) > set_Extended_ereal > set_a ).
thf(sy_c_Set_Oimage_001t__Extended____Real__Oereal_001tf__b,type,
image_3724615099042636214real_b: ( extended_ereal > b ) > set_Extended_ereal > set_b ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__b_J_001t__Extended____Real__Oereal,type,
image_3611896476772571406_ereal: ( list_b > extended_ereal ) > set_list_b > set_Extended_ereal ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__b_J_001t__List__Olist_Itf__b_J,type,
image_list_b_list_b: ( list_b > list_b ) > set_list_b > set_list_b ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
image_4309273772856505399_ereal: ( nat > extended_ereal ) > set_nat > set_Extended_ereal ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__b,type,
image_nat_b: ( nat > b ) > set_nat > set_b ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J,type,
image_6801035452528096924t_unit: ( set_a > pre_pr7278220950009878019t_unit ) > set_set_a > set_pr5411798346947241657t_unit ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001tf__a,type,
image_set_a_a: ( set_a > a ) > set_set_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J,type,
image_5713294457175270716t_unit: ( a > pre_pr7278220950009878019t_unit ) > set_a > set_pr5411798346947241657t_unit ).
thf(sy_c_Set_Oimage_001tf__a_001t__Extended____Real__Oereal,type,
image_8405481351990995413_ereal: ( a > extended_ereal ) > set_a > set_Extended_ereal ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__b,type,
image_a_b: ( a > b ) > set_a > set_b ).
thf(sy_c_Set_Oimage_001tf__b_001t__Extended____Real__Oereal,type,
image_5319725110001000852_ereal: ( b > extended_ereal ) > set_b > set_Extended_ereal ).
thf(sy_c_Set_Oimage_001tf__b_001t__Nat__Onat,type,
image_b_nat: ( b > nat ) > set_b > set_nat ).
thf(sy_c_Set_Oimage_001tf__b_001tf__a,type,
image_b_a: ( b > a ) > set_b > set_a ).
thf(sy_c_Set_Oimage_001tf__b_001tf__b,type,
image_b_b: ( b > b ) > set_b > set_b ).
thf(sy_c_Set_Oinsert_001t__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J,type,
insert6864688055023459379t_unit: pre_pr7278220950009878019t_unit > set_pr5411798346947241657t_unit > set_pr5411798346947241657t_unit ).
thf(sy_c_Set_Oinsert_001t__Extended____Real__Oereal,type,
insert8967887681552722334_ereal: extended_ereal > set_Extended_ereal > set_Extended_ereal ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__b_J,type,
insert_list_b: list_b > set_list_b > set_list_b ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Oinsert_001tf__b,type,
insert_b: b > set_b > set_b ).
thf(sy_c_Shortest__Path_Owf__digraph_O_092_060mu_062_001tf__a_001tf__b,type,
shortest_wf_mu_a_b: pre_pr7278220950009878019t_unit > ( b > real ) > a > a > extended_ereal ).
thf(sy_c_Shortest__Path_Owf__digraph_Omk__cycles__path_001tf__b,type,
shorte6374615165232202367path_b: nat > list_b > list_b ).
thf(sy_c_Vertex__Walk_Ovpath_001tf__a_001tf__b,type,
vertex_vpath_a_b: list_a > pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Weighted__Graph_Owf__digraph_Oawalk__cost_001tf__b,type,
weight7472181610322534790cost_b: ( b > real ) > list_b > real ).
thf(sy_c_member_001t__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J,type,
member6939884229742472986t_unit: pre_pr7278220950009878019t_unit > set_pr5411798346947241657t_unit > $o ).
thf(sy_c_member_001t__Extended____Real__Oereal,type,
member2350847679896131959_ereal: extended_ereal > set_Extended_ereal > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__b_J,type,
member_list_b: list_b > set_list_b > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__b_J,type,
member_set_b: set_b > set_set_b > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_G,type,
g: pre_pr7278220950009878019t_unit ).
thf(sy_v_f,type,
f: b > real ).
% Relevant facts (1278)
thf(fact_0_fin__digraph__axioms,axiom,
fin_digraph_a_b @ g ).
% fin_digraph_axioms
thf(fact_1_source__nmem__k__nh,axiom,
! [V: a,W: b > real,K: real] :
~ ( member_a @ V @ ( graph_3921080825633621230od_a_b @ g @ W @ V @ K ) ) ).
% source_nmem_k_nh
thf(fact_2_sp__costs__finite,axiom,
! [F: b > real] : ( finite7198162374296863863_ereal @ ( graph_1574344591923819902ts_a_b @ g @ F ) ) ).
% sp_costs_finite
thf(fact_3_fin__sp__costs__def,axiom,
! [F: b > real] :
( ( graph_7485366578106294827ts_a_b @ g @ F )
= ( collec5835592288176408249_ereal
@ ^ [Uu: extended_ereal] :
? [U: a,V2: a,C: extended_ereal] :
( ( Uu = C )
& ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
& ( member_a @ V2 @ ( pre_ve642382030648772252t_unit @ g ) )
& ( ( shortest_wf_mu_a_b @ g @ F @ U @ V2 )
= C )
& ( ord_le1188267648640031866_ereal @ C @ extend1530274965995635425_ereal ) ) ) ) ).
% fin_sp_costs_def
thf(fact_4_sp__costs__def,axiom,
! [F: b > real] :
( ( graph_1574344591923819902ts_a_b @ g @ F )
= ( collec5835592288176408249_ereal
@ ^ [Uu: extended_ereal] :
? [U: a,V2: a,C: extended_ereal] :
( ( Uu = C )
& ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
& ( member_a @ V2 @ ( pre_ve642382030648772252t_unit @ g ) )
& ( ( shortest_wf_mu_a_b @ g @ F @ U @ V2 )
= C ) ) ) ) ).
% sp_costs_def
thf(fact_5_finite__Collect__bounded__ex,axiom,
! [P: extended_ereal > $o,Q: extended_ereal > extended_ereal > $o] :
( ( finite7198162374296863863_ereal @ ( collec5835592288176408249_ereal @ P ) )
=> ( ( finite7198162374296863863_ereal
@ ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
? [Y: extended_ereal] :
( ( P @ Y )
& ( Q @ X @ Y ) ) ) )
= ( ! [Y: extended_ereal] :
( ( P @ Y )
=> ( finite7198162374296863863_ereal
@ ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_6_finite__Collect__bounded__ex,axiom,
! [P: extended_ereal > $o,Q: nat > extended_ereal > $o] :
( ( finite7198162374296863863_ereal @ ( collec5835592288176408249_ereal @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
? [Y: extended_ereal] :
( ( P @ Y )
& ( Q @ X @ Y ) ) ) )
= ( ! [Y: extended_ereal] :
( ( P @ Y )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_7_finite__Collect__bounded__ex,axiom,
! [P: extended_ereal > $o,Q: b > extended_ereal > $o] :
( ( finite7198162374296863863_ereal @ ( collec5835592288176408249_ereal @ P ) )
=> ( ( finite_finite_b
@ ( collect_b
@ ^ [X: b] :
? [Y: extended_ereal] :
( ( P @ Y )
& ( Q @ X @ Y ) ) ) )
= ( ! [Y: extended_ereal] :
( ( P @ Y )
=> ( finite_finite_b
@ ( collect_b
@ ^ [X: b] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_8_finite__Collect__bounded__ex,axiom,
! [P: extended_ereal > $o,Q: a > extended_ereal > $o] :
( ( finite7198162374296863863_ereal @ ( collec5835592288176408249_ereal @ P ) )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
? [Y: extended_ereal] :
( ( P @ Y )
& ( Q @ X @ Y ) ) ) )
= ( ! [Y: extended_ereal] :
( ( P @ Y )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X: a] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_9_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q: extended_ereal > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite7198162374296863863_ereal
@ ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
? [Y: nat] :
( ( P @ Y )
& ( Q @ X @ Y ) ) ) )
= ( ! [Y: nat] :
( ( P @ Y )
=> ( finite7198162374296863863_ereal
@ ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_10_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q: nat > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
? [Y: nat] :
( ( P @ Y )
& ( Q @ X @ Y ) ) ) )
= ( ! [Y: nat] :
( ( P @ Y )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_11_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q: b > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_b
@ ( collect_b
@ ^ [X: b] :
? [Y: nat] :
( ( P @ Y )
& ( Q @ X @ Y ) ) ) )
= ( ! [Y: nat] :
( ( P @ Y )
=> ( finite_finite_b
@ ( collect_b
@ ^ [X: b] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_12_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q: a > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
? [Y: nat] :
( ( P @ Y )
& ( Q @ X @ Y ) ) ) )
= ( ! [Y: nat] :
( ( P @ Y )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X: a] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_13_finite__Collect__bounded__ex,axiom,
! [P: b > $o,Q: extended_ereal > b > $o] :
( ( finite_finite_b @ ( collect_b @ P ) )
=> ( ( finite7198162374296863863_ereal
@ ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
? [Y: b] :
( ( P @ Y )
& ( Q @ X @ Y ) ) ) )
= ( ! [Y: b] :
( ( P @ Y )
=> ( finite7198162374296863863_ereal
@ ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_14_finite__Collect__bounded__ex,axiom,
! [P: b > $o,Q: nat > b > $o] :
( ( finite_finite_b @ ( collect_b @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
? [Y: b] :
( ( P @ Y )
& ( Q @ X @ Y ) ) ) )
= ( ! [Y: b] :
( ( P @ Y )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_15_strongly__con__imp__sp__finite,axiom,
! [U2: a,V: a,W: b > real] :
( ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( digrap8691851296217657702ed_a_b @ g )
=> ( ord_le1188267648640031866_ereal @ ( shortest_wf_mu_a_b @ g @ W @ U2 @ V ) @ extend1530274965995635425_ereal ) ) ) ) ).
% strongly_con_imp_sp_finite
thf(fact_16_in__scc__of__self,axiom,
! [U2: a] :
( ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( member_a @ U2 @ ( digrap2937667069914300949of_a_b @ g @ U2 ) ) ) ).
% in_scc_of_self
thf(fact_17_ereal__infty__less_I1_J,axiom,
! [X2: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X2 @ extend1530274965995635425_ereal )
= ( X2 != extend1530274965995635425_ereal ) ) ).
% ereal_infty_less(1)
thf(fact_18_ereal__less__PInfty,axiom,
! [A: extended_ereal] :
( ( A != extend1530274965995635425_ereal )
=> ( ord_le1188267648640031866_ereal @ A @ extend1530274965995635425_ereal ) ) ).
% ereal_less_PInfty
thf(fact_19_scc__of__eq,axiom,
! [U2: a,V: a] :
( ( member_a @ U2 @ ( digrap2937667069914300949of_a_b @ g @ V ) )
=> ( ( digrap2937667069914300949of_a_b @ g @ U2 )
= ( digrap2937667069914300949of_a_b @ g @ V ) ) ) ).
% scc_of_eq
thf(fact_20_finite__Collect__conjI,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( ( finite_finite_list_a @ ( collect_list_a @ P ) )
| ( finite_finite_list_a @ ( collect_list_a @ Q ) ) )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X: list_a] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_21_finite__Collect__conjI,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ( finite_finite_set_a @ ( collect_set_a @ P ) )
| ( finite_finite_set_a @ ( collect_set_a @ Q ) ) )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X: set_a] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_22_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_23_finite__Collect__conjI,axiom,
! [P: b > $o,Q: b > $o] :
( ( ( finite_finite_b @ ( collect_b @ P ) )
| ( finite_finite_b @ ( collect_b @ Q ) ) )
=> ( finite_finite_b
@ ( collect_b
@ ^ [X: b] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_24_finite__Collect__conjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P ) )
| ( finite_finite_a @ ( collect_a @ Q ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_25_finite__Collect__conjI,axiom,
! [P: list_b > $o,Q: list_b > $o] :
( ( ( finite_finite_list_b @ ( collect_list_b @ P ) )
| ( finite_finite_list_b @ ( collect_list_b @ Q ) ) )
=> ( finite_finite_list_b
@ ( collect_list_b
@ ^ [X: list_b] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_26_finite__Collect__conjI,axiom,
! [P: extended_ereal > $o,Q: extended_ereal > $o] :
( ( ( finite7198162374296863863_ereal @ ( collec5835592288176408249_ereal @ P ) )
| ( finite7198162374296863863_ereal @ ( collec5835592288176408249_ereal @ Q ) ) )
=> ( finite7198162374296863863_ereal
@ ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_27_finite__Collect__disjI,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X: list_a] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_list_a @ ( collect_list_a @ P ) )
& ( finite_finite_list_a @ ( collect_list_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_28_finite__Collect__disjI,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X: set_a] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_set_a @ ( collect_set_a @ P ) )
& ( finite_finite_set_a @ ( collect_set_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_29_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_30_finite__Collect__disjI,axiom,
! [P: b > $o,Q: b > $o] :
( ( finite_finite_b
@ ( collect_b
@ ^ [X: b] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_b @ ( collect_b @ P ) )
& ( finite_finite_b @ ( collect_b @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_31_finite__Collect__disjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_32_finite__Collect__disjI,axiom,
! [P: list_b > $o,Q: list_b > $o] :
( ( finite_finite_list_b
@ ( collect_list_b
@ ^ [X: list_b] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_list_b @ ( collect_list_b @ P ) )
& ( finite_finite_list_b @ ( collect_list_b @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_33_finite__Collect__disjI,axiom,
! [P: extended_ereal > $o,Q: extended_ereal > $o] :
( ( finite7198162374296863863_ereal
@ ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite7198162374296863863_ereal @ ( collec5835592288176408249_ereal @ P ) )
& ( finite7198162374296863863_ereal @ ( collec5835592288176408249_ereal @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_34__092_060mu_062__reach__conv,axiom,
! [F: b > real,U2: a,V: a] :
( ( ord_le1188267648640031866_ereal @ ( shortest_wf_mu_a_b @ g @ F @ U2 @ V ) @ extend1530274965995635425_ereal )
= ( reachable_a_b @ g @ U2 @ V ) ) ).
% \<mu>_reach_conv
thf(fact_35_shortest__path__inf,axiom,
! [U2: a,V: a,F: b > real] :
( ~ ( reachable_a_b @ g @ U2 @ V )
=> ( ( shortest_wf_mu_a_b @ g @ F @ U2 @ V )
= extend1530274965995635425_ereal ) ) ).
% shortest_path_inf
thf(fact_36_finite__image__set,axiom,
! [P: nat > $o,F: nat > nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X: nat] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) ) ) ) ).
% finite_image_set
thf(fact_37_finite__image__set,axiom,
! [P: nat > $o,F: nat > b] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite_finite_b
@ ( collect_b
@ ^ [Uu: b] :
? [X: nat] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) ) ) ) ).
% finite_image_set
thf(fact_38_finite__image__set,axiom,
! [P: nat > $o,F: nat > a] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [Uu: a] :
? [X: nat] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) ) ) ) ).
% finite_image_set
thf(fact_39_finite__image__set,axiom,
! [P: nat > $o,F: nat > extended_ereal] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite7198162374296863863_ereal
@ ( collec5835592288176408249_ereal
@ ^ [Uu: extended_ereal] :
? [X: nat] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) ) ) ) ).
% finite_image_set
thf(fact_40_finite__image__set,axiom,
! [P: b > $o,F: b > nat] :
( ( finite_finite_b @ ( collect_b @ P ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X: b] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) ) ) ) ).
% finite_image_set
thf(fact_41_finite__image__set,axiom,
! [P: b > $o,F: b > b] :
( ( finite_finite_b @ ( collect_b @ P ) )
=> ( finite_finite_b
@ ( collect_b
@ ^ [Uu: b] :
? [X: b] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) ) ) ) ).
% finite_image_set
thf(fact_42_finite__image__set,axiom,
! [P: b > $o,F: b > a] :
( ( finite_finite_b @ ( collect_b @ P ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [Uu: a] :
? [X: b] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) ) ) ) ).
% finite_image_set
thf(fact_43_finite__image__set,axiom,
! [P: b > $o,F: b > extended_ereal] :
( ( finite_finite_b @ ( collect_b @ P ) )
=> ( finite7198162374296863863_ereal
@ ( collec5835592288176408249_ereal
@ ^ [Uu: extended_ereal] :
? [X: b] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) ) ) ) ).
% finite_image_set
thf(fact_44_finite__image__set,axiom,
! [P: a > $o,F: a > nat] :
( ( finite_finite_a @ ( collect_a @ P ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X: a] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) ) ) ) ).
% finite_image_set
thf(fact_45_finite__image__set,axiom,
! [P: a > $o,F: a > b] :
( ( finite_finite_a @ ( collect_a @ P ) )
=> ( finite_finite_b
@ ( collect_b
@ ^ [Uu: b] :
? [X: a] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) ) ) ) ).
% finite_image_set
thf(fact_46_finite__verts,axiom,
finite_finite_a @ ( pre_ve642382030648772252t_unit @ g ) ).
% finite_verts
thf(fact_47_reachable__trans,axiom,
! [U2: a,V: a,W: a] :
( ( reachable_a_b @ g @ U2 @ V )
=> ( ( reachable_a_b @ g @ V @ W )
=> ( reachable_a_b @ g @ U2 @ W ) ) ) ).
% reachable_trans
thf(fact_48_reachable__verts__finite,axiom,
! [U2: a] : ( finite_finite_a @ ( collect_a @ ( reachable_a_b @ g @ U2 ) ) ) ).
% reachable_verts_finite
thf(fact_49_reachable__in__verts_I1_J,axiom,
! [U2: a,V: a] :
( ( reachable_a_b @ g @ U2 @ V )
=> ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% reachable_in_verts(1)
thf(fact_50_reachable__in__verts_I2_J,axiom,
! [U2: a,V: a] :
( ( reachable_a_b @ g @ U2 @ V )
=> ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% reachable_in_verts(2)
thf(fact_51_scc__of__def,axiom,
! [U2: a] :
( ( digrap2937667069914300949of_a_b @ g @ U2 )
= ( collect_a
@ ^ [V2: a] :
( ( reachable_a_b @ g @ U2 @ V2 )
& ( reachable_a_b @ g @ V2 @ U2 ) ) ) ) ).
% scc_of_def
thf(fact_52_k__nh__reachable,axiom,
! [U2: a,W: b > real,V: a,K: real] :
( ( member_a @ U2 @ ( graph_3921080825633621230od_a_b @ g @ W @ V @ K ) )
=> ( reachable_a_b @ g @ V @ U2 ) ) ).
% k_nh_reachable
thf(fact_53_strongly__con__imp__reachable__eq__verts,axiom,
! [R: a] :
( ( member_a @ R @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( digrap8691851296217657702ed_a_b @ g )
=> ( ( collect_a @ ( reachable_a_b @ g @ R ) )
= ( pre_ve642382030648772252t_unit @ g ) ) ) ) ).
% strongly_con_imp_reachable_eq_verts
thf(fact_54_reachable__refl,axiom,
! [V: a] :
( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( reachable_a_b @ g @ V @ V ) ) ).
% reachable_refl
thf(fact_55_strongly__connected__spanning__imp__strongly__connected,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digraph_spanning_a_b @ H @ g )
=> ( ( digrap8691851296217657702ed_a_b @ H )
=> ( digrap8691851296217657702ed_a_b @ g ) ) ) ).
% strongly_connected_spanning_imp_strongly_connected
thf(fact_56_wf__digraph_Ok__neighborhood_Ocong,axiom,
graph_3921080825633621230od_a_b = graph_3921080825633621230od_a_b ).
% wf_digraph.k_neighborhood.cong
thf(fact_57_wf__digraph_Osp__costs_Ocong,axiom,
graph_1574344591923819902ts_a_b = graph_1574344591923819902ts_a_b ).
% wf_digraph.sp_costs.cong
thf(fact_58_wf__digraph_Ofin__sp__costs_Ocong,axiom,
graph_7485366578106294827ts_a_b = graph_7485366578106294827ts_a_b ).
% wf_digraph.fin_sp_costs.cong
thf(fact_59_fin__digraph_Osp__costs__finite,axiom,
! [G: pre_pr7278220950009878019t_unit,F: b > real] :
( ( fin_digraph_a_b @ G )
=> ( finite7198162374296863863_ereal @ ( graph_1574344591923819902ts_a_b @ G @ F ) ) ) ).
% fin_digraph.sp_costs_finite
thf(fact_60_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B: set_nat,R2: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_61_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B: set_b,R2: nat > b > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_b @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: b] :
( ( member_b @ Xa @ B )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: b] :
( ( member_b @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_62_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B: set_a,R2: nat > a > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_63_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B: set_Extended_ereal,R2: nat > extended_ereal > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite7198162374296863863_ereal @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa @ B )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_64_pigeonhole__infinite__rel,axiom,
! [A2: set_b,B: set_nat,R2: b > nat > $o] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A3: b] :
( ( member_b @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_65_pigeonhole__infinite__rel,axiom,
! [A2: set_b,B: set_b,R2: b > b > $o] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_b @ B )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ? [Xa: b] :
( ( member_b @ Xa @ B )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: b] :
( ( member_b @ X3 @ B )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A3: b] :
( ( member_b @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_66_pigeonhole__infinite__rel,axiom,
! [A2: set_b,B: set_a,R2: b > a > $o] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A3: b] :
( ( member_b @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_67_pigeonhole__infinite__rel,axiom,
! [A2: set_b,B: set_Extended_ereal,R2: b > extended_ereal > $o] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite7198162374296863863_ereal @ B )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ? [Xa: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa @ B )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ B )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A3: b] :
( ( member_b @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_68_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B: set_nat,R2: a > nat > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A3: a] :
( ( member_a @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_69_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B: set_b,R2: a > b > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_b @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ? [Xa: b] :
( ( member_b @ Xa @ B )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: b] :
( ( member_b @ X3 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A3: a] :
( ( member_a @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_70_not__finite__existsD,axiom,
! [P: list_a > $o] :
( ~ ( finite_finite_list_a @ ( collect_list_a @ P ) )
=> ? [X_1: list_a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_71_not__finite__existsD,axiom,
! [P: set_a > $o] :
( ~ ( finite_finite_set_a @ ( collect_set_a @ P ) )
=> ? [X_1: set_a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_72_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_73_not__finite__existsD,axiom,
! [P: b > $o] :
( ~ ( finite_finite_b @ ( collect_b @ P ) )
=> ? [X_1: b] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_74_not__finite__existsD,axiom,
! [P: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P ) )
=> ? [X_1: a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_75_not__finite__existsD,axiom,
! [P: list_b > $o] :
( ~ ( finite_finite_list_b @ ( collect_list_b @ P ) )
=> ? [X_1: list_b] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_76_not__finite__existsD,axiom,
! [P: extended_ereal > $o] :
( ~ ( finite7198162374296863863_ereal @ ( collec5835592288176408249_ereal @ P ) )
=> ? [X_1: extended_ereal] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_77_less__ereal_Osimps_I2_J,axiom,
! [A: extended_ereal] :
~ ( ord_le1188267648640031866_ereal @ extend1530274965995635425_ereal @ A ) ).
% less_ereal.simps(2)
thf(fact_78_finite__image__set2,axiom,
! [P: nat > $o,Q: nat > $o,F: nat > nat > nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X: nat,Y: nat] :
( ( Uu
= ( F @ X @ Y ) )
& ( P @ X )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_79_finite__image__set2,axiom,
! [P: nat > $o,Q: nat > $o,F: nat > nat > b] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite_finite_b
@ ( collect_b
@ ^ [Uu: b] :
? [X: nat,Y: nat] :
( ( Uu
= ( F @ X @ Y ) )
& ( P @ X )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_80_finite__image__set2,axiom,
! [P: nat > $o,Q: nat > $o,F: nat > nat > a] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [Uu: a] :
? [X: nat,Y: nat] :
( ( Uu
= ( F @ X @ Y ) )
& ( P @ X )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_81_finite__image__set2,axiom,
! [P: nat > $o,Q: nat > $o,F: nat > nat > extended_ereal] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite7198162374296863863_ereal
@ ( collec5835592288176408249_ereal
@ ^ [Uu: extended_ereal] :
? [X: nat,Y: nat] :
( ( Uu
= ( F @ X @ Y ) )
& ( P @ X )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_82_finite__image__set2,axiom,
! [P: nat > $o,Q: b > $o,F: nat > b > nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_b @ ( collect_b @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X: nat,Y: b] :
( ( Uu
= ( F @ X @ Y ) )
& ( P @ X )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_83_finite__image__set2,axiom,
! [P: nat > $o,Q: b > $o,F: nat > b > b] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_b @ ( collect_b @ Q ) )
=> ( finite_finite_b
@ ( collect_b
@ ^ [Uu: b] :
? [X: nat,Y: b] :
( ( Uu
= ( F @ X @ Y ) )
& ( P @ X )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_84_finite__image__set2,axiom,
! [P: nat > $o,Q: b > $o,F: nat > b > a] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_b @ ( collect_b @ Q ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [Uu: a] :
? [X: nat,Y: b] :
( ( Uu
= ( F @ X @ Y ) )
& ( P @ X )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_85_finite__image__set2,axiom,
! [P: nat > $o,Q: b > $o,F: nat > b > extended_ereal] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_b @ ( collect_b @ Q ) )
=> ( finite7198162374296863863_ereal
@ ( collec5835592288176408249_ereal
@ ^ [Uu: extended_ereal] :
? [X: nat,Y: b] :
( ( Uu
= ( F @ X @ Y ) )
& ( P @ X )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_86_finite__image__set2,axiom,
! [P: nat > $o,Q: a > $o,F: nat > a > nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_a @ ( collect_a @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu: nat] :
? [X: nat,Y: a] :
( ( Uu
= ( F @ X @ Y ) )
& ( P @ X )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_87_finite__image__set2,axiom,
! [P: nat > $o,Q: a > $o,F: nat > a > b] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_a @ ( collect_a @ Q ) )
=> ( finite_finite_b
@ ( collect_b
@ ^ [Uu: b] :
? [X: nat,Y: a] :
( ( Uu
= ( F @ X @ Y ) )
& ( P @ X )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_88_vpaths__finite,axiom,
( finite_finite_list_a
@ ( collect_list_a
@ ^ [P2: list_a] : ( vertex_vpath_a_b @ P2 @ g ) ) ) ).
% vpaths_finite
thf(fact_89_fin__digraph__del__vert,axiom,
! [U2: a] : ( fin_digraph_a_b @ ( pre_del_vert_a_b @ g @ U2 ) ) ).
% fin_digraph_del_vert
thf(fact_90_scc__of__in__sccs__verts,axiom,
! [U2: a] :
( ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( member_set_a @ ( digrap2937667069914300949of_a_b @ g @ U2 ) @ ( digrap2871191568752656621ts_a_b @ g ) ) ) ).
% scc_of_in_sccs_verts
thf(fact_91_scc__of__empty__conv,axiom,
! [U2: a] :
( ( ( digrap2937667069914300949of_a_b @ g @ U2 )
= bot_bot_set_a )
= ( ~ ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ g ) ) ) ) ).
% scc_of_empty_conv
thf(fact_92_trails__finite,axiom,
( finite_finite_list_b
@ ( collect_list_b
@ ^ [P2: list_b] :
? [U: a,X4: a] : ( arc_pre_trail_a_b @ g @ U @ P2 @ X4 ) ) ) ).
% trails_finite
thf(fact_93_fin__digraph_Oreachable__verts__finite,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a] :
( ( fin_digraph_a_b @ G )
=> ( finite_finite_a @ ( collect_a @ ( reachable_a_b @ G @ U2 ) ) ) ) ).
% fin_digraph.reachable_verts_finite
thf(fact_94_strongly__connectedE,axiom,
! [G: pre_pr7278220950009878019t_unit] :
( ( digrap8691851296217657702ed_a_b @ G )
=> ! [U3: a,V3: a] :
( ( ( member_a @ U3 @ ( pre_ve642382030648772252t_unit @ G ) )
& ( member_a @ V3 @ ( pre_ve642382030648772252t_unit @ G ) ) )
=> ( reachable_a_b @ G @ U3 @ V3 ) ) ) ).
% strongly_connectedE
thf(fact_95_fin__digraph_Ofinite__verts,axiom,
! [G: pre_pr7278220950009878019t_unit] :
( ( fin_digraph_a_b @ G )
=> ( finite_finite_a @ ( pre_ve642382030648772252t_unit @ G ) ) ) ).
% fin_digraph.finite_verts
thf(fact_96_mem__Collect__eq,axiom,
! [A: pre_pr7278220950009878019t_unit,P: pre_pr7278220950009878019t_unit > $o] :
( ( member6939884229742472986t_unit @ A @ ( collec8000012497822511960t_unit @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_97_mem__Collect__eq,axiom,
! [A: extended_ereal,P: extended_ereal > $o] :
( ( member2350847679896131959_ereal @ A @ ( collec5835592288176408249_ereal @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_98_mem__Collect__eq,axiom,
! [A: list_b,P: list_b > $o] :
( ( member_list_b @ A @ ( collect_list_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_99_mem__Collect__eq,axiom,
! [A: list_a,P: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_100_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_101_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_102_mem__Collect__eq,axiom,
! [A: b,P: b > $o] :
( ( member_b @ A @ ( collect_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_103_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_104_Collect__mem__eq,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ( collec8000012497822511960t_unit
@ ^ [X: pre_pr7278220950009878019t_unit] : ( member6939884229742472986t_unit @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_105_Collect__mem__eq,axiom,
! [A2: set_Extended_ereal] :
( ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] : ( member2350847679896131959_ereal @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_106_Collect__mem__eq,axiom,
! [A2: set_list_b] :
( ( collect_list_b
@ ^ [X: list_b] : ( member_list_b @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_107_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X: list_a] : ( member_list_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_108_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_109_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_110_Collect__mem__eq,axiom,
! [A2: set_b] :
( ( collect_b
@ ^ [X: b] : ( member_b @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_111_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X: set_a] : ( member_set_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_112_Collect__cong,axiom,
! [P: extended_ereal > $o,Q: extended_ereal > $o] :
( ! [X3: extended_ereal] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec5835592288176408249_ereal @ P )
= ( collec5835592288176408249_ereal @ Q ) ) ) ).
% Collect_cong
thf(fact_113_Collect__cong,axiom,
! [P: list_b > $o,Q: list_b > $o] :
( ! [X3: list_b] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_list_b @ P )
= ( collect_list_b @ Q ) ) ) ).
% Collect_cong
thf(fact_114_Collect__cong,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X3: list_a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_list_a @ P )
= ( collect_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_115_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_116_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_117_Collect__cong,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X3: b] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_b @ P )
= ( collect_b @ Q ) ) ) ).
% Collect_cong
thf(fact_118_Collect__cong,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X3: set_a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_set_a @ P )
= ( collect_set_a @ Q ) ) ) ).
% Collect_cong
thf(fact_119_fin__digraph__del__arc,axiom,
! [A: b] : ( fin_digraph_a_b @ ( pre_del_arc_a_b @ g @ A ) ) ).
% fin_digraph_del_arc
thf(fact_120_induce__reachable__preserves__paths,axiom,
! [U2: a,V: a] :
( ( reachable_a_b @ g @ U2 @ V )
=> ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ g @ ( collect_a @ ( reachable_a_b @ g @ U2 ) ) ) @ U2 @ V ) ) ).
% induce_reachable_preserves_paths
thf(fact_121_pre__digraph_Oscc__of__def,axiom,
( digrap2937667069914300949of_a_b
= ( ^ [G2: pre_pr7278220950009878019t_unit,U: a] :
( collect_a
@ ^ [V2: a] :
( ( reachable_a_b @ G2 @ U @ V2 )
& ( reachable_a_b @ G2 @ V2 @ U ) ) ) ) ) ).
% pre_digraph.scc_of_def
thf(fact_122_wf__digraph_Ofin__sp__costs__def,axiom,
! [G: pre_pr7278220950009878019t_unit,F: b > real] :
( ( wf_digraph_a_b @ G )
=> ( ( graph_7485366578106294827ts_a_b @ G @ F )
= ( collec5835592288176408249_ereal
@ ^ [Uu: extended_ereal] :
? [U: a,V2: a,C: extended_ereal] :
( ( Uu = C )
& ( member_a @ U @ ( pre_ve642382030648772252t_unit @ G ) )
& ( member_a @ V2 @ ( pre_ve642382030648772252t_unit @ G ) )
& ( ( shortest_wf_mu_a_b @ G @ F @ U @ V2 )
= C )
& ( ord_le1188267648640031866_ereal @ C @ extend1530274965995635425_ereal ) ) ) ) ) ).
% wf_digraph.fin_sp_costs_def
thf(fact_123_wf__digraph__axioms,axiom,
wf_digraph_a_b @ g ).
% wf_digraph_axioms
thf(fact_124_del__arc__commute,axiom,
! [B2: b,A: b] :
( ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ g @ B2 ) @ A )
= ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ g @ A ) @ B2 ) ) ).
% del_arc_commute
thf(fact_125_finite__sccs__verts,axiom,
finite_finite_set_a @ ( digrap2871191568752656621ts_a_b @ g ) ).
% finite_sccs_verts
thf(fact_126_wf__digraph__del__vert,axiom,
! [U2: a] : ( wf_digraph_a_b @ ( pre_del_vert_a_b @ g @ U2 ) ) ).
% wf_digraph_del_vert
thf(fact_127_in__sccs__verts__conv__reachable,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ g ) )
= ( ( S != bot_bot_set_a )
& ! [X: a] :
( ( member_a @ X @ S )
=> ! [Y: a] :
( ( member_a @ Y @ S )
=> ( reachable_a_b @ g @ X @ Y ) ) )
& ! [X: a] :
( ( member_a @ X @ S )
=> ! [V2: a] :
( ~ ( member_a @ V2 @ S )
=> ( ~ ( reachable_a_b @ g @ X @ V2 )
| ~ ( reachable_a_b @ g @ V2 @ X ) ) ) ) ) ) ).
% in_sccs_verts_conv_reachable
thf(fact_128_induce__subgraph__verts,axiom,
! [G: pre_pr7278220950009878019t_unit,Vs: set_a] :
( ( pre_ve642382030648772252t_unit @ ( digrap7873285959652527175ph_a_b @ G @ Vs ) )
= Vs ) ).
% induce_subgraph_verts
thf(fact_129_wf__digraph__del__arc,axiom,
! [A: b] : ( wf_digraph_a_b @ ( pre_del_arc_a_b @ g @ A ) ) ).
% wf_digraph_del_arc
thf(fact_130_del__del__arc__collapse,axiom,
! [A: b] :
( ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ g @ A ) @ A )
= ( pre_del_arc_a_b @ g @ A ) ) ).
% del_del_arc_collapse
thf(fact_131_wellformed__induce__subgraph,axiom,
! [Vs: set_a] : ( wf_digraph_a_b @ ( digrap7873285959652527175ph_a_b @ g @ Vs ) ) ).
% wellformed_induce_subgraph
thf(fact_132_verts__del__arc,axiom,
! [A: b] :
( ( pre_ve642382030648772252t_unit @ ( pre_del_arc_a_b @ g @ A ) )
= ( pre_ve642382030648772252t_unit @ g ) ) ).
% verts_del_arc
thf(fact_133_strongly__connectedI,axiom,
! [G: pre_pr7278220950009878019t_unit] :
( ( ( pre_ve642382030648772252t_unit @ G )
!= bot_bot_set_a )
=> ( ! [U4: a,V4: a] :
( ( member_a @ U4 @ ( pre_ve642382030648772252t_unit @ G ) )
=> ( ( member_a @ V4 @ ( pre_ve642382030648772252t_unit @ G ) )
=> ( reachable_a_b @ G @ U4 @ V4 ) ) )
=> ( digrap8691851296217657702ed_a_b @ G ) ) ) ).
% strongly_connectedI
thf(fact_134_wf__digraph_Oin__sccs__verts__conv__reachable,axiom,
! [G: pre_pr7278220950009878019t_unit,S: set_a] :
( ( wf_digraph_a_b @ G )
=> ( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ G ) )
= ( ( S != bot_bot_set_a )
& ! [X: a] :
( ( member_a @ X @ S )
=> ! [Y: a] :
( ( member_a @ Y @ S )
=> ( reachable_a_b @ G @ X @ Y ) ) )
& ! [X: a] :
( ( member_a @ X @ S )
=> ! [V2: a] :
( ~ ( member_a @ V2 @ S )
=> ( ~ ( reachable_a_b @ G @ X @ V2 )
| ~ ( reachable_a_b @ G @ V2 @ X ) ) ) ) ) ) ) ).
% wf_digraph.in_sccs_verts_conv_reachable
thf(fact_135_wf__digraph_Owellformed__induce__subgraph,axiom,
! [G: pre_pr7278220950009878019t_unit,Vs: set_a] :
( ( wf_digraph_a_b @ G )
=> ( wf_digraph_a_b @ ( digrap7873285959652527175ph_a_b @ G @ Vs ) ) ) ).
% wf_digraph.wellformed_induce_subgraph
thf(fact_136_pre__digraph_Osccs__verts_Ocong,axiom,
digrap2871191568752656621ts_a_b = digrap2871191568752656621ts_a_b ).
% pre_digraph.sccs_verts.cong
thf(fact_137_pre__digraph_Odel__del__arc__collapse,axiom,
! [G: pre_pr7278220950009878019t_unit,A: b] :
( ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ G @ A ) @ A )
= ( pre_del_arc_a_b @ G @ A ) ) ).
% pre_digraph.del_del_arc_collapse
thf(fact_138_wf__digraph_Owf__digraph__del__vert,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a] :
( ( wf_digraph_a_b @ G )
=> ( wf_digraph_a_b @ ( pre_del_vert_a_b @ G @ U2 ) ) ) ).
% wf_digraph.wf_digraph_del_vert
thf(fact_139_wf__digraph_Owf__digraph__del__arc,axiom,
! [G: pre_pr7278220950009878019t_unit,A: b] :
( ( wf_digraph_a_b @ G )
=> ( wf_digraph_a_b @ ( pre_del_arc_a_b @ G @ A ) ) ) ).
% wf_digraph.wf_digraph_del_arc
thf(fact_140_pre__digraph_Odel__arc__commute,axiom,
! [G: pre_pr7278220950009878019t_unit,B2: b,A: b] :
( ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ G @ B2 ) @ A )
= ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ G @ A ) @ B2 ) ) ).
% pre_digraph.del_arc_commute
thf(fact_141_pre__digraph_Odel__vert_Ocong,axiom,
pre_del_vert_a_b = pre_del_vert_a_b ).
% pre_digraph.del_vert.cong
thf(fact_142_pre__digraph_Odel__arc_Ocong,axiom,
pre_del_arc_a_b = pre_del_arc_a_b ).
% pre_digraph.del_arc.cong
thf(fact_143_wf__digraph_Owf__digraph,axiom,
! [G: pre_pr7278220950009878019t_unit] :
( ( wf_digraph_a_b @ G )
=> ( wf_digraph_a_b @ G ) ) ).
% wf_digraph.wf_digraph
thf(fact_144_wf__digraph_Ostrongly__connected__spanning__imp__strongly__connected,axiom,
! [G: pre_pr7278220950009878019t_unit,H: pre_pr7278220950009878019t_unit] :
( ( wf_digraph_a_b @ G )
=> ( ( digraph_spanning_a_b @ H @ G )
=> ( ( digrap8691851296217657702ed_a_b @ H )
=> ( digrap8691851296217657702ed_a_b @ G ) ) ) ) ).
% wf_digraph.strongly_connected_spanning_imp_strongly_connected
thf(fact_145_wf__digraph_Oinduce__reachable__preserves__paths,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a,V: a] :
( ( wf_digraph_a_b @ G )
=> ( ( reachable_a_b @ G @ U2 @ V )
=> ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ G @ ( collect_a @ ( reachable_a_b @ G @ U2 ) ) ) @ U2 @ V ) ) ) ).
% wf_digraph.induce_reachable_preserves_paths
thf(fact_146_wf__digraph_Oreachable__trans,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a,V: a,W: a] :
( ( wf_digraph_a_b @ G )
=> ( ( reachable_a_b @ G @ U2 @ V )
=> ( ( reachable_a_b @ G @ V @ W )
=> ( reachable_a_b @ G @ U2 @ W ) ) ) ) ).
% wf_digraph.reachable_trans
thf(fact_147_fin__digraph_Oaxioms_I1_J,axiom,
! [G: pre_pr7278220950009878019t_unit] :
( ( fin_digraph_a_b @ G )
=> ( wf_digraph_a_b @ G ) ) ).
% fin_digraph.axioms(1)
thf(fact_148_wf__digraph_Oscc__of__eq,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a,V: a] :
( ( wf_digraph_a_b @ G )
=> ( ( member_a @ U2 @ ( digrap2937667069914300949of_a_b @ G @ V ) )
=> ( ( digrap2937667069914300949of_a_b @ G @ U2 )
= ( digrap2937667069914300949of_a_b @ G @ V ) ) ) ) ).
% wf_digraph.scc_of_eq
thf(fact_149_pre__digraph_Overts__del__arc,axiom,
! [G: pre_pr7278220950009878019t_unit,A: b] :
( ( pre_ve642382030648772252t_unit @ ( pre_del_arc_a_b @ G @ A ) )
= ( pre_ve642382030648772252t_unit @ G ) ) ).
% pre_digraph.verts_del_arc
thf(fact_150_wf__digraph_Oscc__of__empty__conv,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a] :
( ( wf_digraph_a_b @ G )
=> ( ( ( digrap2937667069914300949of_a_b @ G @ U2 )
= bot_bot_set_a )
= ( ~ ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ G ) ) ) ) ) ).
% wf_digraph.scc_of_empty_conv
thf(fact_151_fin__digraph_Ofin__digraph__del__arc,axiom,
! [G: pre_pr7278220950009878019t_unit,A: b] :
( ( fin_digraph_a_b @ G )
=> ( fin_digraph_a_b @ ( pre_del_arc_a_b @ G @ A ) ) ) ).
% fin_digraph.fin_digraph_del_arc
thf(fact_152_wf__digraph_Oscc__of__in__sccs__verts,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a] :
( ( wf_digraph_a_b @ G )
=> ( ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ G ) )
=> ( member_set_a @ ( digrap2937667069914300949of_a_b @ G @ U2 ) @ ( digrap2871191568752656621ts_a_b @ G ) ) ) ) ).
% wf_digraph.scc_of_in_sccs_verts
thf(fact_153_fin__digraph_Ofinite__sccs__verts,axiom,
! [G: pre_pr7278220950009878019t_unit] :
( ( fin_digraph_a_b @ G )
=> ( finite_finite_set_a @ ( digrap2871191568752656621ts_a_b @ G ) ) ) ).
% fin_digraph.finite_sccs_verts
thf(fact_154_fin__digraph_Ofin__digraph__del__vert,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a] :
( ( fin_digraph_a_b @ G )
=> ( fin_digraph_a_b @ ( pre_del_vert_a_b @ G @ U2 ) ) ) ).
% fin_digraph.fin_digraph_del_vert
thf(fact_155_finite_OemptyI,axiom,
finite_finite_list_a @ bot_bot_set_list_a ).
% finite.emptyI
thf(fact_156_finite_OemptyI,axiom,
finite_finite_set_a @ bot_bot_set_set_a ).
% finite.emptyI
thf(fact_157_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_158_finite_OemptyI,axiom,
finite_finite_list_b @ bot_bot_set_list_b ).
% finite.emptyI
thf(fact_159_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_160_finite_OemptyI,axiom,
finite8852549406693098522t_unit @ bot_bo1839476491465656141t_unit ).
% finite.emptyI
thf(fact_161_finite_OemptyI,axiom,
finite_finite_b @ bot_bot_set_b ).
% finite.emptyI
thf(fact_162_finite_OemptyI,axiom,
finite7198162374296863863_ereal @ bot_bo8367695208629047834_ereal ).
% finite.emptyI
thf(fact_163_infinite__imp__nonempty,axiom,
! [S: set_list_a] :
( ~ ( finite_finite_list_a @ S )
=> ( S != bot_bot_set_list_a ) ) ).
% infinite_imp_nonempty
thf(fact_164_infinite__imp__nonempty,axiom,
! [S: set_set_a] :
( ~ ( finite_finite_set_a @ S )
=> ( S != bot_bot_set_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_165_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_166_infinite__imp__nonempty,axiom,
! [S: set_list_b] :
( ~ ( finite_finite_list_b @ S )
=> ( S != bot_bot_set_list_b ) ) ).
% infinite_imp_nonempty
thf(fact_167_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_168_infinite__imp__nonempty,axiom,
! [S: set_pr5411798346947241657t_unit] :
( ~ ( finite8852549406693098522t_unit @ S )
=> ( S != bot_bo1839476491465656141t_unit ) ) ).
% infinite_imp_nonempty
thf(fact_169_infinite__imp__nonempty,axiom,
! [S: set_b] :
( ~ ( finite_finite_b @ S )
=> ( S != bot_bot_set_b ) ) ).
% infinite_imp_nonempty
thf(fact_170_infinite__imp__nonempty,axiom,
! [S: set_Extended_ereal] :
( ~ ( finite7198162374296863863_ereal @ S )
=> ( S != bot_bo8367695208629047834_ereal ) ) ).
% infinite_imp_nonempty
thf(fact_171_wf__digraph_Oreachable__refl,axiom,
! [G: pre_pr7278220950009878019t_unit,V: a] :
( ( wf_digraph_a_b @ G )
=> ( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ G ) )
=> ( reachable_a_b @ G @ V @ V ) ) ) ).
% wf_digraph.reachable_refl
thf(fact_172_wf__digraph_Oreachable__in__verts_I1_J,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a,V: a] :
( ( wf_digraph_a_b @ G )
=> ( ( reachable_a_b @ G @ U2 @ V )
=> ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ G ) ) ) ) ).
% wf_digraph.reachable_in_verts(1)
thf(fact_173_wf__digraph_Oreachable__in__verts_I2_J,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a,V: a] :
( ( wf_digraph_a_b @ G )
=> ( ( reachable_a_b @ G @ U2 @ V )
=> ( member_a @ V @ ( pre_ve642382030648772252t_unit @ G ) ) ) ) ).
% wf_digraph.reachable_in_verts(2)
thf(fact_174_wf__digraph_Oin__scc__of__self,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a] :
( ( wf_digraph_a_b @ G )
=> ( ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ G ) )
=> ( member_a @ U2 @ ( digrap2937667069914300949of_a_b @ G @ U2 ) ) ) ) ).
% wf_digraph.in_scc_of_self
thf(fact_175_wf__digraph_Osource__nmem__k__nh,axiom,
! [G: pre_pr7278220950009878019t_unit,V: a,W: b > real,K: real] :
( ( wf_digraph_a_b @ G )
=> ~ ( member_a @ V @ ( graph_3921080825633621230od_a_b @ G @ W @ V @ K ) ) ) ).
% wf_digraph.source_nmem_k_nh
thf(fact_176_strongly__connected__def,axiom,
( digrap8691851296217657702ed_a_b
= ( ^ [G2: pre_pr7278220950009878019t_unit] :
( ( ( pre_ve642382030648772252t_unit @ G2 )
!= bot_bot_set_a )
& ! [X: a] :
( ( member_a @ X @ ( pre_ve642382030648772252t_unit @ G2 ) )
=> ! [Y: a] :
( ( member_a @ Y @ ( pre_ve642382030648772252t_unit @ G2 ) )
=> ( reachable_a_b @ G2 @ X @ Y ) ) ) ) ) ) ).
% strongly_connected_def
thf(fact_177_wf__digraph_Ostrongly__con__imp__reachable__eq__verts,axiom,
! [G: pre_pr7278220950009878019t_unit,R: a] :
( ( wf_digraph_a_b @ G )
=> ( ( member_a @ R @ ( pre_ve642382030648772252t_unit @ G ) )
=> ( ( digrap8691851296217657702ed_a_b @ G )
=> ( ( collect_a @ ( reachable_a_b @ G @ R ) )
= ( pre_ve642382030648772252t_unit @ G ) ) ) ) ) ).
% wf_digraph.strongly_con_imp_reachable_eq_verts
thf(fact_178_wf__digraph_Ok__nh__reachable,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a,W: b > real,V: a,K: real] :
( ( wf_digraph_a_b @ G )
=> ( ( member_a @ U2 @ ( graph_3921080825633621230od_a_b @ G @ W @ V @ K ) )
=> ( reachable_a_b @ G @ V @ U2 ) ) ) ).
% wf_digraph.k_nh_reachable
thf(fact_179_wf__digraph_Ostrongly__con__imp__sp__finite,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a,V: a,W: b > real] :
( ( wf_digraph_a_b @ G )
=> ( ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ G ) )
=> ( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ G ) )
=> ( ( digrap8691851296217657702ed_a_b @ G )
=> ( ord_le1188267648640031866_ereal @ ( shortest_wf_mu_a_b @ G @ W @ U2 @ V ) @ extend1530274965995635425_ereal ) ) ) ) ) ).
% wf_digraph.strongly_con_imp_sp_finite
thf(fact_180_fin__digraph_Ofin__digraph,axiom,
! [G: pre_pr7278220950009878019t_unit] :
( ( fin_digraph_a_b @ G )
=> ( fin_digraph_a_b @ G ) ) ).
% fin_digraph.fin_digraph
thf(fact_181_pre__digraph_Oscc__of_Ocong,axiom,
digrap2937667069914300949of_a_b = digrap2937667069914300949of_a_b ).
% pre_digraph.scc_of.cong
thf(fact_182_wf__digraph_Osp__costs__def,axiom,
! [G: pre_pr7278220950009878019t_unit,F: b > real] :
( ( wf_digraph_a_b @ G )
=> ( ( graph_1574344591923819902ts_a_b @ G @ F )
= ( collec5835592288176408249_ereal
@ ^ [Uu: extended_ereal] :
? [U: a,V2: a,C: extended_ereal] :
( ( Uu = C )
& ( member_a @ U @ ( pre_ve642382030648772252t_unit @ G ) )
& ( member_a @ V2 @ ( pre_ve642382030648772252t_unit @ G ) )
& ( ( shortest_wf_mu_a_b @ G @ F @ U @ V2 )
= C ) ) ) ) ) ).
% wf_digraph.sp_costs_def
thf(fact_183_reachable__in__vertsE,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a,V: a] :
( ( reachable_a_b @ G @ U2 @ V )
=> ~ ( ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ G ) )
=> ~ ( member_a @ V @ ( pre_ve642382030648772252t_unit @ G ) ) ) ) ).
% reachable_in_vertsE
thf(fact_184_del__vert__add__vert,axiom,
! [U2: a] :
( ~ ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( pre_del_vert_a_b @ ( pre_add_vert_a_b @ g @ U2 ) @ U2 )
= g ) ) ).
% del_vert_add_vert
thf(fact_185_wf__digraph__add__vert,axiom,
! [U2: a] : ( wf_digraph_a_b @ ( pre_add_vert_a_b @ g @ U2 ) ) ).
% wf_digraph_add_vert
thf(fact_186_wf__digraph_O_092_060mu_062__reach__conv,axiom,
! [G: pre_pr7278220950009878019t_unit,F: b > real,U2: a,V: a] :
( ( wf_digraph_a_b @ G )
=> ( ( ord_le1188267648640031866_ereal @ ( shortest_wf_mu_a_b @ G @ F @ U2 @ V ) @ extend1530274965995635425_ereal )
= ( reachable_a_b @ G @ U2 @ V ) ) ) ).
% wf_digraph.\<mu>_reach_conv
thf(fact_187_fin__digraph_Otrails__finite,axiom,
! [G: pre_pr7278220950009878019t_unit] :
( ( fin_digraph_a_b @ G )
=> ( finite_finite_list_b
@ ( collect_list_b
@ ^ [P2: list_b] :
? [U: a,X4: a] : ( arc_pre_trail_a_b @ G @ U @ P2 @ X4 ) ) ) ) ).
% fin_digraph.trails_finite
thf(fact_188_induce__eq__iff__induced,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ H @ g )
=> ( ( digrap7873285959652527175ph_a_b @ g @ ( pre_ve642382030648772252t_unit @ H ) )
= H ) ) ).
% induce_eq_iff_induced
thf(fact_189_wf__digraph_Oshortest__path__inf,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a,V: a,F: b > real] :
( ( wf_digraph_a_b @ G )
=> ( ~ ( reachable_a_b @ G @ U2 @ V )
=> ( ( shortest_wf_mu_a_b @ G @ F @ U2 @ V )
= extend1530274965995635425_ereal ) ) ) ).
% wf_digraph.shortest_path_inf
thf(fact_190_reachable__induce__subgraphD,axiom,
! [S: set_a,U2: a,V: a] :
( ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ g @ S ) @ U2 @ V )
=> ( ( ord_less_eq_set_a @ S @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( reachable_a_b @ g @ U2 @ V ) ) ) ).
% reachable_induce_subgraphD
thf(fact_191_fin__digraph_Ovpaths__finite,axiom,
! [G: pre_pr7278220950009878019t_unit] :
( ( fin_digraph_a_b @ G )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [P2: list_a] : ( vertex_vpath_a_b @ P2 @ G ) ) ) ) ).
% fin_digraph.vpaths_finite
thf(fact_192_in__sccs__verts__conv,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ g ) )
= ( member6939884229742472986t_unit @ ( digrap7873285959652527175ph_a_b @ g @ S ) @ ( digraph_pre_sccs_a_b @ g ) ) ) ).
% in_sccs_verts_conv
thf(fact_193_in__verts__sccsD__sccs,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ g ) )
=> ( member6939884229742472986t_unit @ ( digrap7873285959652527175ph_a_b @ g @ S ) @ ( digraph_pre_sccs_a_b @ g ) ) ) ).
% in_verts_sccsD_sccs
thf(fact_194_sccs__verts__conv__scc__of,axiom,
( ( digrap2871191568752656621ts_a_b @ g )
= ( image_a_set_a @ ( digrap2937667069914300949of_a_b @ g ) @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% sccs_verts_conv_scc_of
thf(fact_195_induced__subgraph__refl,axiom,
digrap5251062021860773499ph_a_b @ g @ g ).
% induced_subgraph_refl
thf(fact_196_in__sccs__imp__induced,axiom,
! [C2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( digraph_pre_sccs_a_b @ g ) )
=> ( digrap5251062021860773499ph_a_b @ C2 @ g ) ) ).
% in_sccs_imp_induced
thf(fact_197_in__sccs__subset__imp__eq,axiom,
! [C2: pre_pr7278220950009878019t_unit,D: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( digraph_pre_sccs_a_b @ g ) )
=> ( ( member6939884229742472986t_unit @ D @ ( digraph_pre_sccs_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( pre_ve642382030648772252t_unit @ C2 ) @ ( pre_ve642382030648772252t_unit @ D ) )
=> ( C2 = D ) ) ) ) ).
% in_sccs_subset_imp_eq
thf(fact_198_sccs__verts__subsets,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ g ) )
=> ( ord_less_eq_set_a @ S @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% sccs_verts_subsets
thf(fact_199_reachable__induce__ss,axiom,
! [S: set_a,U2: a,V: a,T: set_a] :
( ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ g @ S ) @ U2 @ V )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ g @ T ) @ U2 @ V ) ) ) ).
% reachable_induce_ss
thf(fact_200_finite__imageI,axiom,
! [F2: set_nat,H2: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_nat @ ( image_nat_nat @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_201_finite__imageI,axiom,
! [F2: set_nat,H2: nat > b] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_b @ ( image_nat_b @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_202_finite__imageI,axiom,
! [F2: set_nat,H2: nat > a] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_a @ ( image_nat_a @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_203_finite__imageI,axiom,
! [F2: set_nat,H2: nat > extended_ereal] :
( ( finite_finite_nat @ F2 )
=> ( finite7198162374296863863_ereal @ ( image_4309273772856505399_ereal @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_204_finite__imageI,axiom,
! [F2: set_b,H2: b > nat] :
( ( finite_finite_b @ F2 )
=> ( finite_finite_nat @ ( image_b_nat @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_205_finite__imageI,axiom,
! [F2: set_b,H2: b > b] :
( ( finite_finite_b @ F2 )
=> ( finite_finite_b @ ( image_b_b @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_206_finite__imageI,axiom,
! [F2: set_b,H2: b > a] :
( ( finite_finite_b @ F2 )
=> ( finite_finite_a @ ( image_b_a @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_207_finite__imageI,axiom,
! [F2: set_b,H2: b > extended_ereal] :
( ( finite_finite_b @ F2 )
=> ( finite7198162374296863863_ereal @ ( image_5319725110001000852_ereal @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_208_finite__imageI,axiom,
! [F2: set_a,H2: a > nat] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_nat @ ( image_a_nat @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_209_finite__imageI,axiom,
! [F2: set_a,H2: a > b] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_b @ ( image_a_b @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_210_finite__Collect__subsets,axiom,
! [A2: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( finite5282473924520328461list_a
@ ( collect_set_list_a
@ ^ [B3: set_list_a] : ( ord_le8861187494160871172list_a @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_211_finite__Collect__subsets,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( finite7209287970140883943_set_a
@ ( collect_set_set_a
@ ^ [B3: set_set_a] : ( ord_le3724670747650509150_set_a @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_212_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B3: set_nat] : ( ord_less_eq_set_nat @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_213_finite__Collect__subsets,axiom,
! [A2: set_list_b] :
( ( finite_finite_list_b @ A2 )
=> ( finite5353507964566674446list_b
@ ( collect_set_list_b
@ ^ [B3: set_list_b] : ( ord_le8932221534207217157list_b @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_214_finite__Collect__subsets,axiom,
! [A2: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ A2 )
=> ( finite2737741666826350167_ereal
@ ( collec85322473871370393_ereal
@ ^ [B3: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_215_finite__Collect__subsets,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B3: set_a] : ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_216_finite__Collect__subsets,axiom,
! [A2: set_b] :
( ( finite_finite_b @ A2 )
=> ( finite_finite_set_b
@ ( collect_set_b
@ ^ [B3: set_b] : ( ord_less_eq_set_b @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_217_induced__induce,axiom,
! [Vs: set_a] :
( ( ord_less_eq_set_a @ Vs @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( digrap5251062021860773499ph_a_b @ ( digrap7873285959652527175ph_a_b @ g @ Vs ) @ g ) ) ).
% induced_induce
thf(fact_218_pre__digraph_Oin__sccs__imp__induced,axiom,
! [C2: pre_pr7278220950009878019t_unit,G: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( digraph_pre_sccs_a_b @ G ) )
=> ( digrap5251062021860773499ph_a_b @ C2 @ G ) ) ).
% pre_digraph.in_sccs_imp_induced
thf(fact_219_pre__digraph_Osccs_Ocong,axiom,
digraph_pre_sccs_a_b = digraph_pre_sccs_a_b ).
% pre_digraph.sccs.cong
thf(fact_220_all__subset__image,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,P: set_set_a > $o] :
( ( ! [B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_7466199892558553556_set_a @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ B3 @ A2 )
=> ( P @ ( image_7466199892558553556_set_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_221_all__subset__image,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a,P: set_pr5411798346947241657t_unit > $o] :
( ( ! [B3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ B3 @ ( image_6801035452528096924t_unit @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ A2 )
=> ( P @ ( image_6801035452528096924t_unit @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_222_all__subset__image,axiom,
! [F: extended_ereal > extended_ereal,A2: set_Extended_ereal,P: set_Extended_ereal > $o] :
( ( ! [B3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B3 @ ( image_6042159593519690757_ereal @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B3 @ A2 )
=> ( P @ ( image_6042159593519690757_ereal @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_223_all__subset__image,axiom,
! [F: list_b > extended_ereal,A2: set_list_b,P: set_Extended_ereal > $o] :
( ( ! [B3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B3 @ ( image_3611896476772571406_ereal @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_list_b] :
( ( ord_le8932221534207217157list_b @ B3 @ A2 )
=> ( P @ ( image_3611896476772571406_ereal @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_224_all__subset__image,axiom,
! [F: a > set_a,A2: set_a,P: set_set_a > $o] :
( ( ! [B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_a_set_a @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( P @ ( image_a_set_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_225_all__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( P @ ( image_a_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_226_all__subset__image,axiom,
! [F: b > a,A2: set_b,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_b_a @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_b] :
( ( ord_less_eq_set_b @ B3 @ A2 )
=> ( P @ ( image_b_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_227_all__subset__image,axiom,
! [F: a > b,A2: set_a,P: set_b > $o] :
( ( ! [B3: set_b] :
( ( ord_less_eq_set_b @ B3 @ ( image_a_b @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( P @ ( image_a_b @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_228_all__subset__image,axiom,
! [F: b > b,A2: set_b,P: set_b > $o] :
( ( ! [B3: set_b] :
( ( ord_less_eq_set_b @ B3 @ ( image_b_b @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_b] :
( ( ord_less_eq_set_b @ B3 @ A2 )
=> ( P @ ( image_b_b @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_229_finite__surj,axiom,
! [A2: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_230_finite__surj,axiom,
! [A2: set_nat,B: set_Extended_ereal,F: nat > extended_ereal] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ B @ ( image_4309273772856505399_ereal @ F @ A2 ) )
=> ( finite7198162374296863863_ereal @ B ) ) ) ).
% finite_surj
thf(fact_231_finite__surj,axiom,
! [A2: set_b,B: set_nat,F: b > nat] :
( ( finite_finite_b @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_b_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_232_finite__surj,axiom,
! [A2: set_b,B: set_Extended_ereal,F: b > extended_ereal] :
( ( finite_finite_b @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ B @ ( image_5319725110001000852_ereal @ F @ A2 ) )
=> ( finite7198162374296863863_ereal @ B ) ) ) ).
% finite_surj
thf(fact_233_finite__surj,axiom,
! [A2: set_a,B: set_nat,F: a > nat] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_234_finite__surj,axiom,
! [A2: set_a,B: set_Extended_ereal,F: a > extended_ereal] :
( ( finite_finite_a @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ B @ ( image_8405481351990995413_ereal @ F @ A2 ) )
=> ( finite7198162374296863863_ereal @ B ) ) ) ).
% finite_surj
thf(fact_235_finite__surj,axiom,
! [A2: set_Extended_ereal,B: set_nat,F: extended_ereal > nat] :
( ( finite7198162374296863863_ereal @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_7659842161140344153al_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_236_finite__surj,axiom,
! [A2: set_Extended_ereal,B: set_Extended_ereal,F: extended_ereal > extended_ereal] :
( ( finite7198162374296863863_ereal @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ B @ ( image_6042159593519690757_ereal @ F @ A2 ) )
=> ( finite7198162374296863863_ereal @ B ) ) ) ).
% finite_surj
thf(fact_237_finite__surj,axiom,
! [A2: set_nat,B: set_a,F: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A2 ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_surj
thf(fact_238_finite__surj,axiom,
! [A2: set_b,B: set_a,F: b > a] :
( ( finite_finite_b @ A2 )
=> ( ( ord_less_eq_set_a @ B @ ( image_b_a @ F @ A2 ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_surj
thf(fact_239_finite__subset__image,axiom,
! [B: set_nat,F: nat > nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_nat_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_240_finite__subset__image,axiom,
! [B: set_nat,F: extended_ereal > nat,A2: set_Extended_ereal] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_7659842161140344153al_nat @ F @ A2 ) )
=> ? [C3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ C3 @ A2 )
& ( finite7198162374296863863_ereal @ C3 )
& ( B
= ( image_7659842161140344153al_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_241_finite__subset__image,axiom,
! [B: set_Extended_ereal,F: nat > extended_ereal,A2: set_nat] :
( ( finite7198162374296863863_ereal @ B )
=> ( ( ord_le1644982726543182158_ereal @ B @ ( image_4309273772856505399_ereal @ F @ A2 ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_4309273772856505399_ereal @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_242_finite__subset__image,axiom,
! [B: set_Extended_ereal,F: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ B )
=> ( ( ord_le1644982726543182158_ereal @ B @ ( image_6042159593519690757_ereal @ F @ A2 ) )
=> ? [C3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ C3 @ A2 )
& ( finite7198162374296863863_ereal @ C3 )
& ( B
= ( image_6042159593519690757_ereal @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_243_finite__subset__image,axiom,
! [B: set_nat,F: a > nat,A2: set_a] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F @ A2 ) )
=> ? [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
& ( finite_finite_a @ C3 )
& ( B
= ( image_a_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_244_finite__subset__image,axiom,
! [B: set_Extended_ereal,F: a > extended_ereal,A2: set_a] :
( ( finite7198162374296863863_ereal @ B )
=> ( ( ord_le1644982726543182158_ereal @ B @ ( image_8405481351990995413_ereal @ F @ A2 ) )
=> ? [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
& ( finite_finite_a @ C3 )
& ( B
= ( image_8405481351990995413_ereal @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_245_finite__subset__image,axiom,
! [B: set_nat,F: b > nat,A2: set_b] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_b_nat @ F @ A2 ) )
=> ? [C3: set_b] :
( ( ord_less_eq_set_b @ C3 @ A2 )
& ( finite_finite_b @ C3 )
& ( B
= ( image_b_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_246_finite__subset__image,axiom,
! [B: set_Extended_ereal,F: b > extended_ereal,A2: set_b] :
( ( finite7198162374296863863_ereal @ B )
=> ( ( ord_le1644982726543182158_ereal @ B @ ( image_5319725110001000852_ereal @ F @ A2 ) )
=> ? [C3: set_b] :
( ( ord_less_eq_set_b @ C3 @ A2 )
& ( finite_finite_b @ C3 )
& ( B
= ( image_5319725110001000852_ereal @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_247_finite__subset__image,axiom,
! [B: set_a,F: nat > a,A2: set_nat] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A2 ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_nat_a @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_248_finite__subset__image,axiom,
! [B: set_a,F: extended_ereal > a,A2: set_Extended_ereal] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_3724615099042636213real_a @ F @ A2 ) )
=> ? [C3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ C3 @ A2 )
& ( finite7198162374296863863_ereal @ C3 )
& ( B
= ( image_3724615099042636213real_a @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_249_ex__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A2 )
& ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_250_ex__finite__subset__image,axiom,
! [F: extended_ereal > nat,A2: set_Extended_ereal,P: set_nat > $o] :
( ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_7659842161140344153al_nat @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ B3 )
& ( ord_le1644982726543182158_ereal @ B3 @ A2 )
& ( P @ ( image_7659842161140344153al_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_251_ex__finite__subset__image,axiom,
! [F: nat > extended_ereal,A2: set_nat,P: set_Extended_ereal > $o] :
( ( ? [B3: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ B3 )
& ( ord_le1644982726543182158_ereal @ B3 @ ( image_4309273772856505399_ereal @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A2 )
& ( P @ ( image_4309273772856505399_ereal @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_252_ex__finite__subset__image,axiom,
! [F: extended_ereal > extended_ereal,A2: set_Extended_ereal,P: set_Extended_ereal > $o] :
( ( ? [B3: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ B3 )
& ( ord_le1644982726543182158_ereal @ B3 @ ( image_6042159593519690757_ereal @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ B3 )
& ( ord_le1644982726543182158_ereal @ B3 @ A2 )
& ( P @ ( image_6042159593519690757_ereal @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_253_ex__finite__subset__image,axiom,
! [F: a > nat,A2: set_a,P: set_nat > $o] :
( ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_a_nat @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A2 )
& ( P @ ( image_a_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_254_ex__finite__subset__image,axiom,
! [F: a > extended_ereal,A2: set_a,P: set_Extended_ereal > $o] :
( ( ? [B3: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ B3 )
& ( ord_le1644982726543182158_ereal @ B3 @ ( image_8405481351990995413_ereal @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A2 )
& ( P @ ( image_8405481351990995413_ereal @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_255_ex__finite__subset__image,axiom,
! [F: b > nat,A2: set_b,P: set_nat > $o] :
( ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_b_nat @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_b] :
( ( finite_finite_b @ B3 )
& ( ord_less_eq_set_b @ B3 @ A2 )
& ( P @ ( image_b_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_256_ex__finite__subset__image,axiom,
! [F: b > extended_ereal,A2: set_b,P: set_Extended_ereal > $o] :
( ( ? [B3: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ B3 )
& ( ord_le1644982726543182158_ereal @ B3 @ ( image_5319725110001000852_ereal @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_b] :
( ( finite_finite_b @ B3 )
& ( ord_less_eq_set_b @ B3 @ A2 )
& ( P @ ( image_5319725110001000852_ereal @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_257_ex__finite__subset__image,axiom,
! [F: nat > a,A2: set_nat,P: set_a > $o] :
( ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ ( image_nat_a @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A2 )
& ( P @ ( image_nat_a @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_258_ex__finite__subset__image,axiom,
! [F: extended_ereal > a,A2: set_Extended_ereal,P: set_a > $o] :
( ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ ( image_3724615099042636213real_a @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ B3 )
& ( ord_le1644982726543182158_ereal @ B3 @ A2 )
& ( P @ ( image_3724615099042636213real_a @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_259_all__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A2 ) )
=> ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_260_all__finite__subset__image,axiom,
! [F: extended_ereal > nat,A2: set_Extended_ereal,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_7659842161140344153al_nat @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_Extended_ereal] :
( ( ( finite7198162374296863863_ereal @ B3 )
& ( ord_le1644982726543182158_ereal @ B3 @ A2 ) )
=> ( P @ ( image_7659842161140344153al_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_261_all__finite__subset__image,axiom,
! [F: nat > extended_ereal,A2: set_nat,P: set_Extended_ereal > $o] :
( ( ! [B3: set_Extended_ereal] :
( ( ( finite7198162374296863863_ereal @ B3 )
& ( ord_le1644982726543182158_ereal @ B3 @ ( image_4309273772856505399_ereal @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A2 ) )
=> ( P @ ( image_4309273772856505399_ereal @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_262_all__finite__subset__image,axiom,
! [F: extended_ereal > extended_ereal,A2: set_Extended_ereal,P: set_Extended_ereal > $o] :
( ( ! [B3: set_Extended_ereal] :
( ( ( finite7198162374296863863_ereal @ B3 )
& ( ord_le1644982726543182158_ereal @ B3 @ ( image_6042159593519690757_ereal @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_Extended_ereal] :
( ( ( finite7198162374296863863_ereal @ B3 )
& ( ord_le1644982726543182158_ereal @ B3 @ A2 ) )
=> ( P @ ( image_6042159593519690757_ereal @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_263_all__finite__subset__image,axiom,
! [F: a > nat,A2: set_a,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_a_nat @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A2 ) )
=> ( P @ ( image_a_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_264_all__finite__subset__image,axiom,
! [F: a > extended_ereal,A2: set_a,P: set_Extended_ereal > $o] :
( ( ! [B3: set_Extended_ereal] :
( ( ( finite7198162374296863863_ereal @ B3 )
& ( ord_le1644982726543182158_ereal @ B3 @ ( image_8405481351990995413_ereal @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A2 ) )
=> ( P @ ( image_8405481351990995413_ereal @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_265_all__finite__subset__image,axiom,
! [F: b > nat,A2: set_b,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_b_nat @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_b] :
( ( ( finite_finite_b @ B3 )
& ( ord_less_eq_set_b @ B3 @ A2 ) )
=> ( P @ ( image_b_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_266_all__finite__subset__image,axiom,
! [F: b > extended_ereal,A2: set_b,P: set_Extended_ereal > $o] :
( ( ! [B3: set_Extended_ereal] :
( ( ( finite7198162374296863863_ereal @ B3 )
& ( ord_le1644982726543182158_ereal @ B3 @ ( image_5319725110001000852_ereal @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_b] :
( ( ( finite_finite_b @ B3 )
& ( ord_less_eq_set_b @ B3 @ A2 ) )
=> ( P @ ( image_5319725110001000852_ereal @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_267_all__finite__subset__image,axiom,
! [F: nat > a,A2: set_nat,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ ( image_nat_a @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A2 ) )
=> ( P @ ( image_nat_a @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_268_all__finite__subset__image,axiom,
! [F: extended_ereal > a,A2: set_Extended_ereal,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ ( image_3724615099042636213real_a @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_Extended_ereal] :
( ( ( finite7198162374296863863_ereal @ B3 )
& ( ord_le1644982726543182158_ereal @ B3 @ A2 ) )
=> ( P @ ( image_3724615099042636213real_a @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_269_pre__digraph_Oadd__vert_Ocong,axiom,
pre_add_vert_a_b = pre_add_vert_a_b ).
% pre_digraph.add_vert.cong
thf(fact_270_pre__digraph_Oin__sccs__subset__imp__eq,axiom,
! [C2: pre_pr7278220950009878019t_unit,G: pre_pr7278220950009878019t_unit,D: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( digraph_pre_sccs_a_b @ G ) )
=> ( ( member6939884229742472986t_unit @ D @ ( digraph_pre_sccs_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( pre_ve642382030648772252t_unit @ C2 ) @ ( pre_ve642382030648772252t_unit @ D ) )
=> ( C2 = D ) ) ) ) ).
% pre_digraph.in_sccs_subset_imp_eq
thf(fact_271_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_272_finite__has__minimal2,axiom,
! [A2: set_Extended_ereal,A: extended_ereal] :
( ( finite7198162374296863863_ereal @ A2 )
=> ( ( member2350847679896131959_ereal @ A @ A2 )
=> ? [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A2 )
& ( ord_le1083603963089353582_ereal @ X3 @ A )
& ! [Xa: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa @ A2 )
=> ( ( ord_le1083603963089353582_ereal @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_273_finite__has__minimal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ( ord_less_eq_real @ X3 @ A )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_274_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_275_finite__has__minimal2,axiom,
! [A2: set_set_b,A: set_b] :
( ( finite_finite_set_b @ A2 )
=> ( ( member_set_b @ A @ A2 )
=> ? [X3: set_b] :
( ( member_set_b @ X3 @ A2 )
& ( ord_less_eq_set_b @ X3 @ A )
& ! [Xa: set_b] :
( ( member_set_b @ Xa @ A2 )
=> ( ( ord_less_eq_set_b @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_276_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ A @ X3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_277_finite__has__maximal2,axiom,
! [A2: set_Extended_ereal,A: extended_ereal] :
( ( finite7198162374296863863_ereal @ A2 )
=> ( ( member2350847679896131959_ereal @ A @ A2 )
=> ? [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A2 )
& ( ord_le1083603963089353582_ereal @ A @ X3 )
& ! [Xa: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa @ A2 )
=> ( ( ord_le1083603963089353582_ereal @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_278_finite__has__maximal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ( ord_less_eq_real @ A @ X3 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_279_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_280_finite__has__maximal2,axiom,
! [A2: set_set_b,A: set_b] :
( ( finite_finite_set_b @ A2 )
=> ( ( member_set_b @ A @ A2 )
=> ? [X3: set_b] :
( ( member_set_b @ X3 @ A2 )
& ( ord_less_eq_set_b @ A @ X3 )
& ! [Xa: set_b] :
( ( member_set_b @ Xa @ A2 )
=> ( ( ord_less_eq_set_b @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_281_rev__finite__subset,axiom,
! [B: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B )
=> ( finite_finite_list_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_282_rev__finite__subset,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_283_rev__finite__subset,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_284_rev__finite__subset,axiom,
! [B: set_list_b,A2: set_list_b] :
( ( finite_finite_list_b @ B )
=> ( ( ord_le8932221534207217157list_b @ A2 @ B )
=> ( finite_finite_list_b @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_285_rev__finite__subset,axiom,
! [B: set_Extended_ereal,A2: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ B )
=> ( ( ord_le1644982726543182158_ereal @ A2 @ B )
=> ( finite7198162374296863863_ereal @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_286_rev__finite__subset,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_287_rev__finite__subset,axiom,
! [B: set_b,A2: set_b] :
( ( finite_finite_b @ B )
=> ( ( ord_less_eq_set_b @ A2 @ B )
=> ( finite_finite_b @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_288_infinite__super,axiom,
! [S: set_list_a,T: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ T )
=> ( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ T ) ) ) ).
% infinite_super
thf(fact_289_infinite__super,axiom,
! [S: set_set_a,T: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S @ T )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ T ) ) ) ).
% infinite_super
thf(fact_290_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_291_infinite__super,axiom,
! [S: set_list_b,T: set_list_b] :
( ( ord_le8932221534207217157list_b @ S @ T )
=> ( ~ ( finite_finite_list_b @ S )
=> ~ ( finite_finite_list_b @ T ) ) ) ).
% infinite_super
thf(fact_292_infinite__super,axiom,
! [S: set_Extended_ereal,T: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ S @ T )
=> ( ~ ( finite7198162374296863863_ereal @ S )
=> ~ ( finite7198162374296863863_ereal @ T ) ) ) ).
% infinite_super
thf(fact_293_infinite__super,axiom,
! [S: set_a,T: set_a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T ) ) ) ).
% infinite_super
thf(fact_294_infinite__super,axiom,
! [S: set_b,T: set_b] :
( ( ord_less_eq_set_b @ S @ T )
=> ( ~ ( finite_finite_b @ S )
=> ~ ( finite_finite_b @ T ) ) ) ).
% infinite_super
thf(fact_295_finite__subset,axiom,
! [A2: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B )
=> ( ( finite_finite_list_a @ B )
=> ( finite_finite_list_a @ A2 ) ) ) ).
% finite_subset
thf(fact_296_finite__subset,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( finite_finite_set_a @ B )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% finite_subset
thf(fact_297_finite__subset,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_298_finite__subset,axiom,
! [A2: set_list_b,B: set_list_b] :
( ( ord_le8932221534207217157list_b @ A2 @ B )
=> ( ( finite_finite_list_b @ B )
=> ( finite_finite_list_b @ A2 ) ) ) ).
% finite_subset
thf(fact_299_finite__subset,axiom,
! [A2: set_Extended_ereal,B: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A2 @ B )
=> ( ( finite7198162374296863863_ereal @ B )
=> ( finite7198162374296863863_ereal @ A2 ) ) ) ).
% finite_subset
thf(fact_300_finite__subset,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_301_finite__subset,axiom,
! [A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( finite_finite_b @ B )
=> ( finite_finite_b @ A2 ) ) ) ).
% finite_subset
thf(fact_302_pigeonhole__infinite,axiom,
! [A2: set_nat,F: nat > nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A2 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_303_pigeonhole__infinite,axiom,
! [A2: set_nat,F: nat > b] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_b @ ( image_nat_b @ F @ A2 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_304_pigeonhole__infinite,axiom,
! [A2: set_nat,F: nat > a] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_a @ ( image_nat_a @ F @ A2 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_305_pigeonhole__infinite,axiom,
! [A2: set_nat,F: nat > extended_ereal] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite7198162374296863863_ereal @ ( image_4309273772856505399_ereal @ F @ A2 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_306_pigeonhole__infinite,axiom,
! [A2: set_b,F: b > nat] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_nat @ ( image_b_nat @ F @ A2 ) )
=> ? [X3: b] :
( ( member_b @ X3 @ A2 )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A3: b] :
( ( member_b @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_307_pigeonhole__infinite,axiom,
! [A2: set_b,F: b > b] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_b @ ( image_b_b @ F @ A2 ) )
=> ? [X3: b] :
( ( member_b @ X3 @ A2 )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A3: b] :
( ( member_b @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_308_pigeonhole__infinite,axiom,
! [A2: set_b,F: b > a] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_a @ ( image_b_a @ F @ A2 ) )
=> ? [X3: b] :
( ( member_b @ X3 @ A2 )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A3: b] :
( ( member_b @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_309_pigeonhole__infinite,axiom,
! [A2: set_b,F: b > extended_ereal] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite7198162374296863863_ereal @ ( image_5319725110001000852_ereal @ F @ A2 ) )
=> ? [X3: b] :
( ( member_b @ X3 @ A2 )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A3: b] :
( ( member_b @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_310_pigeonhole__infinite,axiom,
! [A2: set_a,F: a > nat] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_nat @ ( image_a_nat @ F @ A2 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A3: a] :
( ( member_a @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_311_pigeonhole__infinite,axiom,
! [A2: set_a,F: a > b] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_b @ ( image_a_b @ F @ A2 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A3: a] :
( ( member_a @ A3 @ A2 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_312_induced__eq__verts__imp__eq,axiom,
! [G: pre_pr7278220950009878019t_unit,H: pre_pr7278220950009878019t_unit,G3: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ G @ H )
=> ( ( digrap5251062021860773499ph_a_b @ G3 @ H )
=> ( ( ( pre_ve642382030648772252t_unit @ G )
= ( pre_ve642382030648772252t_unit @ G3 ) )
=> ( G = G3 ) ) ) ) ).
% induced_eq_verts_imp_eq
thf(fact_313_wf__digraph_Oinduced__subgraph__refl,axiom,
! [G: pre_pr7278220950009878019t_unit] :
( ( wf_digraph_a_b @ G )
=> ( digrap5251062021860773499ph_a_b @ G @ G ) ) ).
% wf_digraph.induced_subgraph_refl
thf(fact_314_wf__digraphI__induced,axiom,
! [H: pre_pr7278220950009878019t_unit,G: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ H @ G )
=> ( wf_digraph_a_b @ H ) ) ).
% wf_digraphI_induced
thf(fact_315_wf__digraph_Oinduced__induce,axiom,
! [G: pre_pr7278220950009878019t_unit,Vs: set_a] :
( ( wf_digraph_a_b @ G )
=> ( ( ord_less_eq_set_a @ Vs @ ( pre_ve642382030648772252t_unit @ G ) )
=> ( digrap5251062021860773499ph_a_b @ ( digrap7873285959652527175ph_a_b @ G @ Vs ) @ G ) ) ) ).
% wf_digraph.induced_induce
thf(fact_316_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_317_finite__has__maximal,axiom,
! [A2: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ A2 )
=> ( ( A2 != bot_bo8367695208629047834_ereal )
=> ? [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A2 )
& ! [Xa: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa @ A2 )
=> ( ( ord_le1083603963089353582_ereal @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_318_finite__has__maximal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_319_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_320_finite__has__maximal,axiom,
! [A2: set_set_b] :
( ( finite_finite_set_b @ A2 )
=> ( ( A2 != bot_bot_set_set_b )
=> ? [X3: set_b] :
( ( member_set_b @ X3 @ A2 )
& ! [Xa: set_b] :
( ( member_set_b @ Xa @ A2 )
=> ( ( ord_less_eq_set_b @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_321_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_322_finite__has__minimal,axiom,
! [A2: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ A2 )
=> ( ( A2 != bot_bo8367695208629047834_ereal )
=> ? [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A2 )
& ! [Xa: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa @ A2 )
=> ( ( ord_le1083603963089353582_ereal @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_323_finite__has__minimal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_324_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_325_finite__has__minimal,axiom,
! [A2: set_set_b] :
( ( finite_finite_set_b @ A2 )
=> ( ( A2 != bot_bot_set_set_b )
=> ? [X3: set_b] :
( ( member_set_b @ X3 @ A2 )
& ! [Xa: set_b] :
( ( member_set_b @ Xa @ A2 )
=> ( ( ord_less_eq_set_b @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_326_wf__digraph_Owf__digraph__add__vert,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a] :
( ( wf_digraph_a_b @ G )
=> ( wf_digraph_a_b @ ( pre_add_vert_a_b @ G @ U2 ) ) ) ).
% wf_digraph.wf_digraph_add_vert
thf(fact_327_wf__digraph_Osccs__verts__subsets,axiom,
! [G: pre_pr7278220950009878019t_unit,S: set_a] :
( ( wf_digraph_a_b @ G )
=> ( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ G ) )
=> ( ord_less_eq_set_a @ S @ ( pre_ve642382030648772252t_unit @ G ) ) ) ) ).
% wf_digraph.sccs_verts_subsets
thf(fact_328_wf__digraph_Oreachable__induce__ss,axiom,
! [G: pre_pr7278220950009878019t_unit,S: set_a,U2: a,V: a,T: set_a] :
( ( wf_digraph_a_b @ G )
=> ( ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ G @ S ) @ U2 @ V )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ G @ T ) @ U2 @ V ) ) ) ) ).
% wf_digraph.reachable_induce_ss
thf(fact_329_wf__digraph_O_092_060mu_062_Ocong,axiom,
shortest_wf_mu_a_b = shortest_wf_mu_a_b ).
% wf_digraph.\<mu>.cong
thf(fact_330_wf__digraph_Oin__sccs__verts__conv,axiom,
! [G: pre_pr7278220950009878019t_unit,S: set_a] :
( ( wf_digraph_a_b @ G )
=> ( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ G ) )
= ( member6939884229742472986t_unit @ ( digrap7873285959652527175ph_a_b @ G @ S ) @ ( digraph_pre_sccs_a_b @ G ) ) ) ) ).
% wf_digraph.in_sccs_verts_conv
thf(fact_331_wf__digraph_Oin__verts__sccsD__sccs,axiom,
! [G: pre_pr7278220950009878019t_unit,S: set_a] :
( ( wf_digraph_a_b @ G )
=> ( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ G ) )
=> ( member6939884229742472986t_unit @ ( digrap7873285959652527175ph_a_b @ G @ S ) @ ( digraph_pre_sccs_a_b @ G ) ) ) ) ).
% wf_digraph.in_verts_sccsD_sccs
thf(fact_332_wf__digraph_Oinduce__eq__iff__induced,axiom,
! [G: pre_pr7278220950009878019t_unit,H: pre_pr7278220950009878019t_unit] :
( ( wf_digraph_a_b @ G )
=> ( ( digrap5251062021860773499ph_a_b @ H @ G )
=> ( ( digrap7873285959652527175ph_a_b @ G @ ( pre_ve642382030648772252t_unit @ H ) )
= H ) ) ) ).
% wf_digraph.induce_eq_iff_induced
thf(fact_333_wf__digraph_Osccs__verts__conv__scc__of,axiom,
! [G: pre_pr7278220950009878019t_unit] :
( ( wf_digraph_a_b @ G )
=> ( ( digrap2871191568752656621ts_a_b @ G )
= ( image_a_set_a @ ( digrap2937667069914300949of_a_b @ G ) @ ( pre_ve642382030648772252t_unit @ G ) ) ) ) ).
% wf_digraph.sccs_verts_conv_scc_of
thf(fact_334_wf__digraph_Oreachable__induce__subgraphD,axiom,
! [G: pre_pr7278220950009878019t_unit,S: set_a,U2: a,V: a] :
( ( wf_digraph_a_b @ G )
=> ( ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ G @ S ) @ U2 @ V )
=> ( ( ord_less_eq_set_a @ S @ ( pre_ve642382030648772252t_unit @ G ) )
=> ( reachable_a_b @ G @ U2 @ V ) ) ) ) ).
% wf_digraph.reachable_induce_subgraphD
thf(fact_335_wf__digraph_Odel__vert__add__vert,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a] :
( ( wf_digraph_a_b @ G )
=> ( ~ ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ G ) )
=> ( ( pre_del_vert_a_b @ ( pre_add_vert_a_b @ G @ U2 ) @ U2 )
= G ) ) ) ).
% wf_digraph.del_vert_add_vert
thf(fact_336_sccs__def,axiom,
( ( digraph_pre_sccs_a_b @ g )
= ( collec8000012497822511960t_unit
@ ^ [H3: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ H3 @ g )
& ( digrap8691851296217657702ed_a_b @ H3 )
& ~ ? [H4: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ H4 @ g )
& ( digrap8691851296217657702ed_a_b @ H4 )
& ( ord_less_set_a @ ( pre_ve642382030648772252t_unit @ H3 ) @ ( pre_ve642382030648772252t_unit @ H4 ) ) ) ) ) ) ).
% sccs_def
thf(fact_337_in__sccsE,axiom,
! [C2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( digraph_pre_sccs_a_b @ g ) )
=> ~ ( ( digrap5251062021860773499ph_a_b @ C2 @ g )
=> ( ( digrap8691851296217657702ed_a_b @ C2 )
=> ? [D2: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ D2 @ g )
& ( digrap8691851296217657702ed_a_b @ D2 )
& ( ord_less_set_a @ ( pre_ve642382030648772252t_unit @ C2 ) @ ( pre_ve642382030648772252t_unit @ D2 ) ) ) ) ) ) ).
% in_sccsE
thf(fact_338_sccs__altdef2,axiom,
( ( digraph_pre_sccs_a_b @ g )
= ( collec8000012497822511960t_unit @ ( digrap4729920478598810909ph_a_b @ g @ digrap8691851296217657702ed_a_b ) ) ) ).
% sccs_altdef2
thf(fact_339_subset__empty,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ bot_bo1839476491465656141t_unit )
= ( A2 = bot_bo1839476491465656141t_unit ) ) ).
% subset_empty
thf(fact_340_subset__empty,axiom,
! [A2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A2 @ bot_bo8367695208629047834_ereal )
= ( A2 = bot_bo8367695208629047834_ereal ) ) ).
% subset_empty
thf(fact_341_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_342_subset__empty,axiom,
! [A2: set_b] :
( ( ord_less_eq_set_b @ A2 @ bot_bot_set_b )
= ( A2 = bot_bot_set_b ) ) ).
% subset_empty
thf(fact_343_empty__subsetI,axiom,
! [A2: set_pr5411798346947241657t_unit] : ( ord_le8200006823705900825t_unit @ bot_bo1839476491465656141t_unit @ A2 ) ).
% empty_subsetI
thf(fact_344_empty__subsetI,axiom,
! [A2: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ bot_bo8367695208629047834_ereal @ A2 ) ).
% empty_subsetI
thf(fact_345_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_346_empty__subsetI,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A2 ) ).
% empty_subsetI
thf(fact_347_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_348_image__empty,axiom,
! [F: a > b] :
( ( image_a_b @ F @ bot_bot_set_a )
= bot_bot_set_b ) ).
% image_empty
thf(fact_349_image__empty,axiom,
! [F: a > extended_ereal] :
( ( image_8405481351990995413_ereal @ F @ bot_bot_set_a )
= bot_bo8367695208629047834_ereal ) ).
% image_empty
thf(fact_350_image__empty,axiom,
! [F: b > a] :
( ( image_b_a @ F @ bot_bot_set_b )
= bot_bot_set_a ) ).
% image_empty
thf(fact_351_image__empty,axiom,
! [F: b > b] :
( ( image_b_b @ F @ bot_bot_set_b )
= bot_bot_set_b ) ).
% image_empty
thf(fact_352_image__empty,axiom,
! [F: b > extended_ereal] :
( ( image_5319725110001000852_ereal @ F @ bot_bot_set_b )
= bot_bo8367695208629047834_ereal ) ).
% image_empty
thf(fact_353_image__empty,axiom,
! [F: extended_ereal > a] :
( ( image_3724615099042636213real_a @ F @ bot_bo8367695208629047834_ereal )
= bot_bot_set_a ) ).
% image_empty
thf(fact_354_image__empty,axiom,
! [F: extended_ereal > b] :
( ( image_3724615099042636214real_b @ F @ bot_bo8367695208629047834_ereal )
= bot_bot_set_b ) ).
% image_empty
thf(fact_355_image__empty,axiom,
! [F: extended_ereal > extended_ereal] :
( ( image_6042159593519690757_ereal @ F @ bot_bo8367695208629047834_ereal )
= bot_bo8367695208629047834_ereal ) ).
% image_empty
thf(fact_356_image__empty,axiom,
! [F: list_b > extended_ereal] :
( ( image_3611896476772571406_ereal @ F @ bot_bot_set_list_b )
= bot_bo8367695208629047834_ereal ) ).
% image_empty
thf(fact_357_empty__is__image,axiom,
! [F: a > a,A2: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_358_empty__is__image,axiom,
! [F: b > a,A2: set_b] :
( ( bot_bot_set_a
= ( image_b_a @ F @ A2 ) )
= ( A2 = bot_bot_set_b ) ) ).
% empty_is_image
thf(fact_359_empty__is__image,axiom,
! [F: extended_ereal > a,A2: set_Extended_ereal] :
( ( bot_bot_set_a
= ( image_3724615099042636213real_a @ F @ A2 ) )
= ( A2 = bot_bo8367695208629047834_ereal ) ) ).
% empty_is_image
thf(fact_360_empty__is__image,axiom,
! [F: a > b,A2: set_a] :
( ( bot_bot_set_b
= ( image_a_b @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_361_empty__is__image,axiom,
! [F: b > b,A2: set_b] :
( ( bot_bot_set_b
= ( image_b_b @ F @ A2 ) )
= ( A2 = bot_bot_set_b ) ) ).
% empty_is_image
thf(fact_362_empty__is__image,axiom,
! [F: extended_ereal > b,A2: set_Extended_ereal] :
( ( bot_bot_set_b
= ( image_3724615099042636214real_b @ F @ A2 ) )
= ( A2 = bot_bo8367695208629047834_ereal ) ) ).
% empty_is_image
thf(fact_363_empty__is__image,axiom,
! [F: a > extended_ereal,A2: set_a] :
( ( bot_bo8367695208629047834_ereal
= ( image_8405481351990995413_ereal @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_364_empty__is__image,axiom,
! [F: b > extended_ereal,A2: set_b] :
( ( bot_bo8367695208629047834_ereal
= ( image_5319725110001000852_ereal @ F @ A2 ) )
= ( A2 = bot_bot_set_b ) ) ).
% empty_is_image
thf(fact_365_empty__is__image,axiom,
! [F: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( bot_bo8367695208629047834_ereal
= ( image_6042159593519690757_ereal @ F @ A2 ) )
= ( A2 = bot_bo8367695208629047834_ereal ) ) ).
% empty_is_image
thf(fact_366_empty__is__image,axiom,
! [F: a > set_a,A2: set_a] :
( ( bot_bot_set_set_a
= ( image_a_set_a @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_367_image__is__empty,axiom,
! [F: a > a,A2: set_a] :
( ( ( image_a_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_368_image__is__empty,axiom,
! [F: b > a,A2: set_b] :
( ( ( image_b_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_b ) ) ).
% image_is_empty
thf(fact_369_image__is__empty,axiom,
! [F: extended_ereal > a,A2: set_Extended_ereal] :
( ( ( image_3724615099042636213real_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bo8367695208629047834_ereal ) ) ).
% image_is_empty
thf(fact_370_image__is__empty,axiom,
! [F: a > b,A2: set_a] :
( ( ( image_a_b @ F @ A2 )
= bot_bot_set_b )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_371_image__is__empty,axiom,
! [F: b > b,A2: set_b] :
( ( ( image_b_b @ F @ A2 )
= bot_bot_set_b )
= ( A2 = bot_bot_set_b ) ) ).
% image_is_empty
thf(fact_372_image__is__empty,axiom,
! [F: extended_ereal > b,A2: set_Extended_ereal] :
( ( ( image_3724615099042636214real_b @ F @ A2 )
= bot_bot_set_b )
= ( A2 = bot_bo8367695208629047834_ereal ) ) ).
% image_is_empty
thf(fact_373_image__is__empty,axiom,
! [F: a > extended_ereal,A2: set_a] :
( ( ( image_8405481351990995413_ereal @ F @ A2 )
= bot_bo8367695208629047834_ereal )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_374_image__is__empty,axiom,
! [F: b > extended_ereal,A2: set_b] :
( ( ( image_5319725110001000852_ereal @ F @ A2 )
= bot_bo8367695208629047834_ereal )
= ( A2 = bot_bot_set_b ) ) ).
% image_is_empty
thf(fact_375_image__is__empty,axiom,
! [F: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( ( image_6042159593519690757_ereal @ F @ A2 )
= bot_bo8367695208629047834_ereal )
= ( A2 = bot_bo8367695208629047834_ereal ) ) ).
% image_is_empty
thf(fact_376_image__is__empty,axiom,
! [F: a > set_a,A2: set_a] :
( ( ( image_a_set_a @ F @ A2 )
= bot_bot_set_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_377_induced__subgraph__altdef2,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ H @ g )
= ( digrap4729920478598810909ph_a_b @ g
@ ^ [H4: pre_pr7278220950009878019t_unit] :
( ( pre_ve642382030648772252t_unit @ H4 )
= ( pre_ve642382030648772252t_unit @ H ) )
@ H ) ) ).
% induced_subgraph_altdef2
thf(fact_378_max__subgraph__mp,axiom,
! [Q: pre_pr7278220950009878019t_unit > $o,X2: pre_pr7278220950009878019t_unit,P: pre_pr7278220950009878019t_unit > $o] :
( ( digrap4729920478598810909ph_a_b @ g @ Q @ X2 )
=> ( ! [X3: pre_pr7278220950009878019t_unit] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ( P @ X2 )
=> ( digrap4729920478598810909ph_a_b @ g @ P @ X2 ) ) ) ) ).
% max_subgraph_mp
thf(fact_379_max__subgraph__prop,axiom,
! [P: pre_pr7278220950009878019t_unit > $o,X2: pre_pr7278220950009878019t_unit] :
( ( digrap4729920478598810909ph_a_b @ g @ P @ X2 )
=> ( P @ X2 ) ) ).
% max_subgraph_prop
thf(fact_380_image__eqI,axiom,
! [B2: a,F: a > a,X2: a,A2: set_a] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A2 )
=> ( member_a @ B2 @ ( image_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_381_image__eqI,axiom,
! [B2: b,F: a > b,X2: a,A2: set_a] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A2 )
=> ( member_b @ B2 @ ( image_a_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_382_image__eqI,axiom,
! [B2: extended_ereal,F: a > extended_ereal,X2: a,A2: set_a] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A2 )
=> ( member2350847679896131959_ereal @ B2 @ ( image_8405481351990995413_ereal @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_383_image__eqI,axiom,
! [B2: a,F: b > a,X2: b,A2: set_b] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_b @ X2 @ A2 )
=> ( member_a @ B2 @ ( image_b_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_384_image__eqI,axiom,
! [B2: b,F: b > b,X2: b,A2: set_b] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_b @ X2 @ A2 )
=> ( member_b @ B2 @ ( image_b_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_385_image__eqI,axiom,
! [B2: extended_ereal,F: b > extended_ereal,X2: b,A2: set_b] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_b @ X2 @ A2 )
=> ( member2350847679896131959_ereal @ B2 @ ( image_5319725110001000852_ereal @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_386_image__eqI,axiom,
! [B2: a,F: extended_ereal > a,X2: extended_ereal,A2: set_Extended_ereal] :
( ( B2
= ( F @ X2 ) )
=> ( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( member_a @ B2 @ ( image_3724615099042636213real_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_387_image__eqI,axiom,
! [B2: b,F: extended_ereal > b,X2: extended_ereal,A2: set_Extended_ereal] :
( ( B2
= ( F @ X2 ) )
=> ( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( member_b @ B2 @ ( image_3724615099042636214real_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_388_image__eqI,axiom,
! [B2: extended_ereal,F: extended_ereal > extended_ereal,X2: extended_ereal,A2: set_Extended_ereal] :
( ( B2
= ( F @ X2 ) )
=> ( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( member2350847679896131959_ereal @ B2 @ ( image_6042159593519690757_ereal @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_389_image__eqI,axiom,
! [B2: extended_ereal,F: list_b > extended_ereal,X2: list_b,A2: set_list_b] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_list_b @ X2 @ A2 )
=> ( member2350847679896131959_ereal @ B2 @ ( image_3611896476772571406_ereal @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_390_empty__Collect__eq,axiom,
! [P: list_b > $o] :
( ( bot_bot_set_list_b
= ( collect_list_b @ P ) )
= ( ! [X: list_b] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_391_empty__Collect__eq,axiom,
! [P: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P ) )
= ( ! [X: list_a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_392_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_393_empty__Collect__eq,axiom,
! [P: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P ) )
= ( ! [X: set_a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_394_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_395_empty__Collect__eq,axiom,
! [P: pre_pr7278220950009878019t_unit > $o] :
( ( bot_bo1839476491465656141t_unit
= ( collec8000012497822511960t_unit @ P ) )
= ( ! [X: pre_pr7278220950009878019t_unit] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_396_empty__Collect__eq,axiom,
! [P: b > $o] :
( ( bot_bot_set_b
= ( collect_b @ P ) )
= ( ! [X: b] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_397_empty__Collect__eq,axiom,
! [P: extended_ereal > $o] :
( ( bot_bo8367695208629047834_ereal
= ( collec5835592288176408249_ereal @ P ) )
= ( ! [X: extended_ereal] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_398_Collect__empty__eq,axiom,
! [P: list_b > $o] :
( ( ( collect_list_b @ P )
= bot_bot_set_list_b )
= ( ! [X: list_b] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_399_Collect__empty__eq,axiom,
! [P: list_a > $o] :
( ( ( collect_list_a @ P )
= bot_bot_set_list_a )
= ( ! [X: list_a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_400_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_401_Collect__empty__eq,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( ! [X: set_a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_402_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_403_Collect__empty__eq,axiom,
! [P: pre_pr7278220950009878019t_unit > $o] :
( ( ( collec8000012497822511960t_unit @ P )
= bot_bo1839476491465656141t_unit )
= ( ! [X: pre_pr7278220950009878019t_unit] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_404_Collect__empty__eq,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( ! [X: b] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_405_Collect__empty__eq,axiom,
! [P: extended_ereal > $o] :
( ( ( collec5835592288176408249_ereal @ P )
= bot_bo8367695208629047834_ereal )
= ( ! [X: extended_ereal] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_406_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X: set_a] :
~ ( member_set_a @ X @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_407_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X: a] :
~ ( member_a @ X @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_408_all__not__in__conv,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ( ! [X: pre_pr7278220950009878019t_unit] :
~ ( member6939884229742472986t_unit @ X @ A2 ) )
= ( A2 = bot_bo1839476491465656141t_unit ) ) ).
% all_not_in_conv
thf(fact_409_all__not__in__conv,axiom,
! [A2: set_b] :
( ( ! [X: b] :
~ ( member_b @ X @ A2 ) )
= ( A2 = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_410_all__not__in__conv,axiom,
! [A2: set_Extended_ereal] :
( ( ! [X: extended_ereal] :
~ ( member2350847679896131959_ereal @ X @ A2 ) )
= ( A2 = bot_bo8367695208629047834_ereal ) ) ).
% all_not_in_conv
thf(fact_411_empty__iff,axiom,
! [C2: set_a] :
~ ( member_set_a @ C2 @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_412_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_413_empty__iff,axiom,
! [C2: pre_pr7278220950009878019t_unit] :
~ ( member6939884229742472986t_unit @ C2 @ bot_bo1839476491465656141t_unit ) ).
% empty_iff
thf(fact_414_empty__iff,axiom,
! [C2: b] :
~ ( member_b @ C2 @ bot_bot_set_b ) ).
% empty_iff
thf(fact_415_empty__iff,axiom,
! [C2: extended_ereal] :
~ ( member2350847679896131959_ereal @ C2 @ bot_bo8367695208629047834_ereal ) ).
% empty_iff
thf(fact_416_subset__antisym,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_417_subset__antisym,axiom,
! [A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( ord_less_eq_set_b @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_418_subsetI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( member_set_a @ X3 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_419_subsetI,axiom,
! [A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ! [X3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ A2 )
=> ( member6939884229742472986t_unit @ X3 @ B ) )
=> ( ord_le8200006823705900825t_unit @ A2 @ B ) ) ).
% subsetI
thf(fact_420_subsetI,axiom,
! [A2: set_Extended_ereal,B: set_Extended_ereal] :
( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A2 )
=> ( member2350847679896131959_ereal @ X3 @ B ) )
=> ( ord_le1644982726543182158_ereal @ A2 @ B ) ) ).
% subsetI
thf(fact_421_subsetI,axiom,
! [A2: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_422_subsetI,axiom,
! [A2: set_b,B: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( member_b @ X3 @ B ) )
=> ( ord_less_eq_set_b @ A2 @ B ) ) ).
% subsetI
thf(fact_423_ereal__infty__less__eq_I1_J,axiom,
! [X2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ extend1530274965995635425_ereal @ X2 )
= ( X2 = extend1530274965995635425_ereal ) ) ).
% ereal_infty_less_eq(1)
thf(fact_424_in__sccs__vertsI__sccs,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( image_7466199892558553556_set_a @ pre_ve642382030648772252t_unit @ ( digraph_pre_sccs_a_b @ g ) ) )
=> ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ g ) ) ) ).
% in_sccs_vertsI_sccs
thf(fact_425_sccs__verts__conv,axiom,
( ( digrap2871191568752656621ts_a_b @ g )
= ( image_7466199892558553556_set_a @ pre_ve642382030648772252t_unit @ ( digraph_pre_sccs_a_b @ g ) ) ) ).
% sccs_verts_conv
thf(fact_426_sccs__conv__sccs__verts,axiom,
( ( digraph_pre_sccs_a_b @ g )
= ( image_6801035452528096924t_unit @ ( digrap7873285959652527175ph_a_b @ g ) @ ( digrap2871191568752656621ts_a_b @ g ) ) ) ).
% sccs_conv_sccs_verts
thf(fact_427_image__ident,axiom,
! [Y2: set_Extended_ereal] :
( ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : X
@ Y2 )
= Y2 ) ).
% image_ident
thf(fact_428_psubsetI,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_a @ A2 @ B ) ) ) ).
% psubsetI
thf(fact_429_psubsetI,axiom,
! [A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_b @ A2 @ B ) ) ) ).
% psubsetI
thf(fact_430_induced__subgraphI__arc__mono,axiom,
! [P: pre_pr7278220950009878019t_unit > $o,H: pre_pr7278220950009878019t_unit] :
( ( digrap4729920478598810909ph_a_b @ g @ P @ H )
=> ( ( digraph_arc_mono_a_b @ P )
=> ( digrap5251062021860773499ph_a_b @ H @ g ) ) ) ).
% induced_subgraphI_arc_mono
thf(fact_431_in__sccsI,axiom,
! [C2: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ C2 @ g )
=> ( ( digrap8691851296217657702ed_a_b @ C2 )
=> ( ~ ? [C4: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ C4 @ g )
& ( digrap8691851296217657702ed_a_b @ C4 )
& ( ord_less_set_a @ ( pre_ve642382030648772252t_unit @ C2 ) @ ( pre_ve642382030648772252t_unit @ C4 ) ) )
=> ( member6939884229742472986t_unit @ C2 @ ( digraph_pre_sccs_a_b @ g ) ) ) ) ) ).
% in_sccsI
thf(fact_432_fin__dia__lowerB,axiom,
! [U2: a,V: a,W: b > real] :
( ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( ord_le1188267648640031866_ereal @ ( shortest_wf_mu_a_b @ g @ W @ U2 @ V ) @ extend1530274965995635425_ereal )
=> ( ord_le1083603963089353582_ereal @ ( shortest_wf_mu_a_b @ g @ W @ U2 @ V ) @ ( graph_1932031826008834157er_a_b @ g @ W ) ) ) ) ) ).
% fin_dia_lowerB
thf(fact_433_psubset__trans,axiom,
! [A2: set_a,B: set_a,C5: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ( ( ord_less_set_a @ B @ C5 )
=> ( ord_less_set_a @ A2 @ C5 ) ) ) ).
% psubset_trans
thf(fact_434_psubsetD,axiom,
! [A2: set_set_a,B: set_set_a,C2: set_a] :
( ( ord_less_set_set_a @ A2 @ B )
=> ( ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_435_psubsetD,axiom,
! [A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit,C2: pre_pr7278220950009878019t_unit] :
( ( ord_le2693654750756130573t_unit @ A2 @ B )
=> ( ( member6939884229742472986t_unit @ C2 @ A2 )
=> ( member6939884229742472986t_unit @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_436_psubsetD,axiom,
! [A2: set_b,B: set_b,C2: b] :
( ( ord_less_set_b @ A2 @ B )
=> ( ( member_b @ C2 @ A2 )
=> ( member_b @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_437_psubsetD,axiom,
! [A2: set_Extended_ereal,B: set_Extended_ereal,C2: extended_ereal] :
( ( ord_le5321083090456148570_ereal @ A2 @ B )
=> ( ( member2350847679896131959_ereal @ C2 @ A2 )
=> ( member2350847679896131959_ereal @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_438_psubsetD,axiom,
! [A2: set_a,B: set_a,C2: a] :
( ( ord_less_set_a @ A2 @ B )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_439_pre__digraph_Omax__subgraph__mp,axiom,
! [G: pre_pr7278220950009878019t_unit,Q: pre_pr7278220950009878019t_unit > $o,X2: pre_pr7278220950009878019t_unit,P: pre_pr7278220950009878019t_unit > $o] :
( ( digrap4729920478598810909ph_a_b @ G @ Q @ X2 )
=> ( ! [X3: pre_pr7278220950009878019t_unit] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ( P @ X2 )
=> ( digrap4729920478598810909ph_a_b @ G @ P @ X2 ) ) ) ) ).
% pre_digraph.max_subgraph_mp
thf(fact_440_pre__digraph_Omax__subgraph_Ocong,axiom,
digrap4729920478598810909ph_a_b = digrap4729920478598810909ph_a_b ).
% pre_digraph.max_subgraph.cong
thf(fact_441_pre__digraph_Omax__subgraph__prop,axiom,
! [G: pre_pr7278220950009878019t_unit,P: pre_pr7278220950009878019t_unit > $o,X2: pre_pr7278220950009878019t_unit] :
( ( digrap4729920478598810909ph_a_b @ G @ P @ X2 )
=> ( P @ X2 ) ) ).
% pre_digraph.max_subgraph_prop
thf(fact_442_not__psubset__empty,axiom,
! [A2: set_pr5411798346947241657t_unit] :
~ ( ord_le2693654750756130573t_unit @ A2 @ bot_bo1839476491465656141t_unit ) ).
% not_psubset_empty
thf(fact_443_not__psubset__empty,axiom,
! [A2: set_b] :
~ ( ord_less_set_b @ A2 @ bot_bot_set_b ) ).
% not_psubset_empty
thf(fact_444_not__psubset__empty,axiom,
! [A2: set_Extended_ereal] :
~ ( ord_le5321083090456148570_ereal @ A2 @ bot_bo8367695208629047834_ereal ) ).
% not_psubset_empty
thf(fact_445_not__psubset__empty,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_446_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_set_a @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_447_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B3: set_b] :
( ( ord_less_set_b @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_448_subset__psubset__trans,axiom,
! [A2: set_a,B: set_a,C5: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_set_a @ B @ C5 )
=> ( ord_less_set_a @ A2 @ C5 ) ) ) ).
% subset_psubset_trans
thf(fact_449_subset__psubset__trans,axiom,
! [A2: set_b,B: set_b,C5: set_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( ord_less_set_b @ B @ C5 )
=> ( ord_less_set_b @ A2 @ C5 ) ) ) ).
% subset_psubset_trans
thf(fact_450_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_451_subset__not__subset__eq,axiom,
( ord_less_set_b
= ( ^ [A4: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A4 @ B3 )
& ~ ( ord_less_eq_set_b @ B3 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_452_psubset__subset__trans,axiom,
! [A2: set_a,B: set_a,C5: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C5 )
=> ( ord_less_set_a @ A2 @ C5 ) ) ) ).
% psubset_subset_trans
thf(fact_453_psubset__subset__trans,axiom,
! [A2: set_b,B: set_b,C5: set_b] :
( ( ord_less_set_b @ A2 @ B )
=> ( ( ord_less_eq_set_b @ B @ C5 )
=> ( ord_less_set_b @ A2 @ C5 ) ) ) ).
% psubset_subset_trans
thf(fact_454_psubset__imp__subset,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% psubset_imp_subset
thf(fact_455_psubset__imp__subset,axiom,
! [A2: set_b,B: set_b] :
( ( ord_less_set_b @ A2 @ B )
=> ( ord_less_eq_set_b @ A2 @ B ) ) ).
% psubset_imp_subset
thf(fact_456_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_457_psubset__eq,axiom,
( ord_less_set_b
= ( ^ [A4: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_458_psubsetE,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_set_a @ A2 @ B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% psubsetE
thf(fact_459_psubsetE,axiom,
! [A2: set_b,B: set_b] :
( ( ord_less_set_b @ A2 @ B )
=> ~ ( ( ord_less_eq_set_b @ A2 @ B )
=> ( ord_less_eq_set_b @ B @ A2 ) ) ) ).
% psubsetE
thf(fact_460_less__eq__set__def,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B3: set_set_a] :
( ord_less_eq_set_a_o
@ ^ [X: set_a] : ( member_set_a @ X @ A4 )
@ ^ [X: set_a] : ( member_set_a @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_461_less__eq__set__def,axiom,
( ord_le8200006823705900825t_unit
= ( ^ [A4: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ord_le8458015892825683628unit_o
@ ^ [X: pre_pr7278220950009878019t_unit] : ( member6939884229742472986t_unit @ X @ A4 )
@ ^ [X: pre_pr7278220950009878019t_unit] : ( member6939884229742472986t_unit @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_462_less__eq__set__def,axiom,
( ord_le1644982726543182158_ereal
= ( ^ [A4: set_Extended_ereal,B3: set_Extended_ereal] :
( ord_le6694447793465728271real_o
@ ^ [X: extended_ereal] : ( member2350847679896131959_ereal @ X @ A4 )
@ ^ [X: extended_ereal] : ( member2350847679896131959_ereal @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_463_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ord_less_eq_a_o
@ ^ [X: a] : ( member_a @ X @ A4 )
@ ^ [X: a] : ( member_a @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_464_less__eq__set__def,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B3: set_b] :
( ord_less_eq_b_o
@ ^ [X: b] : ( member_b @ X @ A4 )
@ ^ [X: b] : ( member_b @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_465_finite__psubset__induct,axiom,
! [A2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ A2 )
=> ( ! [A5: set_list_a] :
( ( finite_finite_list_a @ A5 )
=> ( ! [B4: set_list_a] :
( ( ord_less_set_list_a @ B4 @ A5 )
=> ( P @ B4 ) )
=> ( P @ A5 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_466_finite__psubset__induct,axiom,
! [A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ A2 )
=> ( ! [A5: set_set_a] :
( ( finite_finite_set_a @ A5 )
=> ( ! [B4: set_set_a] :
( ( ord_less_set_set_a @ B4 @ A5 )
=> ( P @ B4 ) )
=> ( P @ A5 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_467_finite__psubset__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ! [B4: set_nat] :
( ( ord_less_set_nat @ B4 @ A5 )
=> ( P @ B4 ) )
=> ( P @ A5 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_468_finite__psubset__induct,axiom,
! [A2: set_b,P: set_b > $o] :
( ( finite_finite_b @ A2 )
=> ( ! [A5: set_b] :
( ( finite_finite_b @ A5 )
=> ( ! [B4: set_b] :
( ( ord_less_set_b @ B4 @ A5 )
=> ( P @ B4 ) )
=> ( P @ A5 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_469_finite__psubset__induct,axiom,
! [A2: set_list_b,P: set_list_b > $o] :
( ( finite_finite_list_b @ A2 )
=> ( ! [A5: set_list_b] :
( ( finite_finite_list_b @ A5 )
=> ( ! [B4: set_list_b] :
( ( ord_less_set_list_b @ B4 @ A5 )
=> ( P @ B4 ) )
=> ( P @ A5 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_470_finite__psubset__induct,axiom,
! [A2: set_Extended_ereal,P: set_Extended_ereal > $o] :
( ( finite7198162374296863863_ereal @ A2 )
=> ( ! [A5: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ A5 )
=> ( ! [B4: set_Extended_ereal] :
( ( ord_le5321083090456148570_ereal @ B4 @ A5 )
=> ( P @ B4 ) )
=> ( P @ A5 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_471_finite__psubset__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ! [B4: set_a] :
( ( ord_less_set_a @ B4 @ A5 )
=> ( P @ B4 ) )
=> ( P @ A5 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_472_ereal__less__eq_I1_J,axiom,
! [X2: extended_ereal] : ( ord_le1083603963089353582_ereal @ X2 @ extend1530274965995635425_ereal ) ).
% ereal_less_eq(1)
thf(fact_473_ereal__infty__less__eq2_I1_J,axiom,
! [A: extended_ereal,B2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B2 )
=> ( ( A = extend1530274965995635425_ereal )
=> ( B2 = extend1530274965995635425_ereal ) ) ) ).
% ereal_infty_less_eq2(1)
thf(fact_474_neq__PInf__trans,axiom,
! [Y3: extended_ereal,X2: extended_ereal] :
( ( Y3 != extend1530274965995635425_ereal )
=> ( ( ord_le1083603963089353582_ereal @ X2 @ Y3 )
=> ( X2 != extend1530274965995635425_ereal ) ) ) ).
% neq_PInf_trans
thf(fact_475_less__eq__ereal__def,axiom,
( ord_le1083603963089353582_ereal
= ( ^ [X: extended_ereal,Y: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X @ Y )
| ( X = Y ) ) ) ) ).
% less_eq_ereal_def
thf(fact_476_rev__image__eqI,axiom,
! [X2: a,A2: set_a,B2: a,F: a > a] :
( ( member_a @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_a @ B2 @ ( image_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_477_rev__image__eqI,axiom,
! [X2: a,A2: set_a,B2: b,F: a > b] :
( ( member_a @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_b @ B2 @ ( image_a_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_478_rev__image__eqI,axiom,
! [X2: a,A2: set_a,B2: extended_ereal,F: a > extended_ereal] :
( ( member_a @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member2350847679896131959_ereal @ B2 @ ( image_8405481351990995413_ereal @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_479_rev__image__eqI,axiom,
! [X2: b,A2: set_b,B2: a,F: b > a] :
( ( member_b @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_a @ B2 @ ( image_b_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_480_rev__image__eqI,axiom,
! [X2: b,A2: set_b,B2: b,F: b > b] :
( ( member_b @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_b @ B2 @ ( image_b_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_481_rev__image__eqI,axiom,
! [X2: b,A2: set_b,B2: extended_ereal,F: b > extended_ereal] :
( ( member_b @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member2350847679896131959_ereal @ B2 @ ( image_5319725110001000852_ereal @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_482_rev__image__eqI,axiom,
! [X2: extended_ereal,A2: set_Extended_ereal,B2: a,F: extended_ereal > a] :
( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_a @ B2 @ ( image_3724615099042636213real_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_483_rev__image__eqI,axiom,
! [X2: extended_ereal,A2: set_Extended_ereal,B2: b,F: extended_ereal > b] :
( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_b @ B2 @ ( image_3724615099042636214real_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_484_rev__image__eqI,axiom,
! [X2: extended_ereal,A2: set_Extended_ereal,B2: extended_ereal,F: extended_ereal > extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member2350847679896131959_ereal @ B2 @ ( image_6042159593519690757_ereal @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_485_rev__image__eqI,axiom,
! [X2: list_b,A2: set_list_b,B2: extended_ereal,F: list_b > extended_ereal] :
( ( member_list_b @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member2350847679896131959_ereal @ B2 @ ( image_3611896476772571406_ereal @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_486_ball__imageD,axiom,
! [F: a > set_a,A2: set_a,P: set_a > $o] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( image_a_set_a @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_487_ball__imageD,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,P: set_a > $o] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( image_7466199892558553556_set_a @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X5: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_488_ball__imageD,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a,P: pre_pr7278220950009878019t_unit > $o] :
( ! [X3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ ( image_6801035452528096924t_unit @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_489_ball__imageD,axiom,
! [F: extended_ereal > extended_ereal,A2: set_Extended_ereal,P: extended_ereal > $o] :
( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ ( image_6042159593519690757_ereal @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X5: extended_ereal] :
( ( member2350847679896131959_ereal @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_490_ball__imageD,axiom,
! [F: list_b > extended_ereal,A2: set_list_b,P: extended_ereal > $o] :
( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ ( image_3611896476772571406_ereal @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X5: list_b] :
( ( member_list_b @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_491_image__cong,axiom,
! [M: set_list_b,N: set_list_b,F: list_b > extended_ereal,G4: list_b > extended_ereal] :
( ( M = N )
=> ( ! [X3: list_b] :
( ( member_list_b @ X3 @ N )
=> ( ( F @ X3 )
= ( G4 @ X3 ) ) )
=> ( ( image_3611896476772571406_ereal @ F @ M )
= ( image_3611896476772571406_ereal @ G4 @ N ) ) ) ) ).
% image_cong
thf(fact_492_image__cong,axiom,
! [M: set_a,N: set_a,F: a > set_a,G4: a > set_a] :
( ( M = N )
=> ( ! [X3: a] :
( ( member_a @ X3 @ N )
=> ( ( F @ X3 )
= ( G4 @ X3 ) ) )
=> ( ( image_a_set_a @ F @ M )
= ( image_a_set_a @ G4 @ N ) ) ) ) ).
% image_cong
thf(fact_493_image__cong,axiom,
! [M: set_set_a,N: set_set_a,F: set_a > pre_pr7278220950009878019t_unit,G4: set_a > pre_pr7278220950009878019t_unit] :
( ( M = N )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ N )
=> ( ( F @ X3 )
= ( G4 @ X3 ) ) )
=> ( ( image_6801035452528096924t_unit @ F @ M )
= ( image_6801035452528096924t_unit @ G4 @ N ) ) ) ) ).
% image_cong
thf(fact_494_image__cong,axiom,
! [M: set_pr5411798346947241657t_unit,N: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > set_a,G4: pre_pr7278220950009878019t_unit > set_a] :
( ( M = N )
=> ( ! [X3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ N )
=> ( ( F @ X3 )
= ( G4 @ X3 ) ) )
=> ( ( image_7466199892558553556_set_a @ F @ M )
= ( image_7466199892558553556_set_a @ G4 @ N ) ) ) ) ).
% image_cong
thf(fact_495_image__cong,axiom,
! [M: set_Extended_ereal,N: set_Extended_ereal,F: extended_ereal > extended_ereal,G4: extended_ereal > extended_ereal] :
( ( M = N )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ N )
=> ( ( F @ X3 )
= ( G4 @ X3 ) ) )
=> ( ( image_6042159593519690757_ereal @ F @ M )
= ( image_6042159593519690757_ereal @ G4 @ N ) ) ) ) ).
% image_cong
thf(fact_496_bex__imageD,axiom,
! [F: a > set_a,A2: set_a,P: set_a > $o] :
( ? [X5: set_a] :
( ( member_set_a @ X5 @ ( image_a_set_a @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_497_bex__imageD,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,P: set_a > $o] :
( ? [X5: set_a] :
( ( member_set_a @ X5 @ ( image_7466199892558553556_set_a @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_498_bex__imageD,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a,P: pre_pr7278220950009878019t_unit > $o] :
( ? [X5: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X5 @ ( image_6801035452528096924t_unit @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_499_bex__imageD,axiom,
! [F: extended_ereal > extended_ereal,A2: set_Extended_ereal,P: extended_ereal > $o] :
( ? [X5: extended_ereal] :
( ( member2350847679896131959_ereal @ X5 @ ( image_6042159593519690757_ereal @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_500_bex__imageD,axiom,
! [F: list_b > extended_ereal,A2: set_list_b,P: extended_ereal > $o] :
( ? [X5: extended_ereal] :
( ( member2350847679896131959_ereal @ X5 @ ( image_3611896476772571406_ereal @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X3: list_b] :
( ( member_list_b @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_501_image__iff,axiom,
! [Z: set_a,F: a > set_a,A2: set_a] :
( ( member_set_a @ Z @ ( image_a_set_a @ F @ A2 ) )
= ( ? [X: a] :
( ( member_a @ X @ A2 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_502_image__iff,axiom,
! [Z: set_a,F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit] :
( ( member_set_a @ Z @ ( image_7466199892558553556_set_a @ F @ A2 ) )
= ( ? [X: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X @ A2 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_503_image__iff,axiom,
! [Z: pre_pr7278220950009878019t_unit,F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a] :
( ( member6939884229742472986t_unit @ Z @ ( image_6801035452528096924t_unit @ F @ A2 ) )
= ( ? [X: set_a] :
( ( member_set_a @ X @ A2 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_504_image__iff,axiom,
! [Z: extended_ereal,F: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ Z @ ( image_6042159593519690757_ereal @ F @ A2 ) )
= ( ? [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A2 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_505_image__iff,axiom,
! [Z: extended_ereal,F: list_b > extended_ereal,A2: set_list_b] :
( ( member2350847679896131959_ereal @ Z @ ( image_3611896476772571406_ereal @ F @ A2 ) )
= ( ? [X: list_b] :
( ( member_list_b @ X @ A2 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_506_imageI,axiom,
! [X2: a,A2: set_a,F: a > a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( image_a_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_507_imageI,axiom,
! [X2: a,A2: set_a,F: a > b] :
( ( member_a @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( image_a_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_508_imageI,axiom,
! [X2: a,A2: set_a,F: a > extended_ereal] :
( ( member_a @ X2 @ A2 )
=> ( member2350847679896131959_ereal @ ( F @ X2 ) @ ( image_8405481351990995413_ereal @ F @ A2 ) ) ) ).
% imageI
thf(fact_509_imageI,axiom,
! [X2: b,A2: set_b,F: b > a] :
( ( member_b @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( image_b_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_510_imageI,axiom,
! [X2: b,A2: set_b,F: b > b] :
( ( member_b @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( image_b_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_511_imageI,axiom,
! [X2: b,A2: set_b,F: b > extended_ereal] :
( ( member_b @ X2 @ A2 )
=> ( member2350847679896131959_ereal @ ( F @ X2 ) @ ( image_5319725110001000852_ereal @ F @ A2 ) ) ) ).
% imageI
thf(fact_512_imageI,axiom,
! [X2: extended_ereal,A2: set_Extended_ereal,F: extended_ereal > a] :
( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( image_3724615099042636213real_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_513_imageI,axiom,
! [X2: extended_ereal,A2: set_Extended_ereal,F: extended_ereal > b] :
( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( image_3724615099042636214real_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_514_imageI,axiom,
! [X2: extended_ereal,A2: set_Extended_ereal,F: extended_ereal > extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( member2350847679896131959_ereal @ ( F @ X2 ) @ ( image_6042159593519690757_ereal @ F @ A2 ) ) ) ).
% imageI
thf(fact_515_imageI,axiom,
! [X2: list_b,A2: set_list_b,F: list_b > extended_ereal] :
( ( member_list_b @ X2 @ A2 )
=> ( member2350847679896131959_ereal @ ( F @ X2 ) @ ( image_3611896476772571406_ereal @ F @ A2 ) ) ) ).
% imageI
thf(fact_516_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X: set_a] : ( member_set_a @ X @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_517_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X: a] : ( member_a @ X @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_518_ex__in__conv,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ( ? [X: pre_pr7278220950009878019t_unit] : ( member6939884229742472986t_unit @ X @ A2 ) )
= ( A2 != bot_bo1839476491465656141t_unit ) ) ).
% ex_in_conv
thf(fact_519_ex__in__conv,axiom,
! [A2: set_b] :
( ( ? [X: b] : ( member_b @ X @ A2 ) )
= ( A2 != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_520_ex__in__conv,axiom,
! [A2: set_Extended_ereal] :
( ( ? [X: extended_ereal] : ( member2350847679896131959_ereal @ X @ A2 ) )
= ( A2 != bot_bo8367695208629047834_ereal ) ) ).
% ex_in_conv
thf(fact_521_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y4: set_a] :
~ ( member_set_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_522_equals0I,axiom,
! [A2: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_523_equals0I,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ! [Y4: pre_pr7278220950009878019t_unit] :
~ ( member6939884229742472986t_unit @ Y4 @ A2 )
=> ( A2 = bot_bo1839476491465656141t_unit ) ) ).
% equals0I
thf(fact_524_equals0I,axiom,
! [A2: set_b] :
( ! [Y4: b] :
~ ( member_b @ Y4 @ A2 )
=> ( A2 = bot_bot_set_b ) ) ).
% equals0I
thf(fact_525_equals0I,axiom,
! [A2: set_Extended_ereal] :
( ! [Y4: extended_ereal] :
~ ( member2350847679896131959_ereal @ Y4 @ A2 )
=> ( A2 = bot_bo8367695208629047834_ereal ) ) ).
% equals0I
thf(fact_526_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_527_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_528_equals0D,axiom,
! [A2: set_pr5411798346947241657t_unit,A: pre_pr7278220950009878019t_unit] :
( ( A2 = bot_bo1839476491465656141t_unit )
=> ~ ( member6939884229742472986t_unit @ A @ A2 ) ) ).
% equals0D
thf(fact_529_equals0D,axiom,
! [A2: set_b,A: b] :
( ( A2 = bot_bot_set_b )
=> ~ ( member_b @ A @ A2 ) ) ).
% equals0D
thf(fact_530_equals0D,axiom,
! [A2: set_Extended_ereal,A: extended_ereal] :
( ( A2 = bot_bo8367695208629047834_ereal )
=> ~ ( member2350847679896131959_ereal @ A @ A2 ) ) ).
% equals0D
thf(fact_531_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_532_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_533_emptyE,axiom,
! [A: pre_pr7278220950009878019t_unit] :
~ ( member6939884229742472986t_unit @ A @ bot_bo1839476491465656141t_unit ) ).
% emptyE
thf(fact_534_emptyE,axiom,
! [A: b] :
~ ( member_b @ A @ bot_bot_set_b ) ).
% emptyE
thf(fact_535_emptyE,axiom,
! [A: extended_ereal] :
~ ( member2350847679896131959_ereal @ A @ bot_bo8367695208629047834_ereal ) ).
% emptyE
thf(fact_536_Collect__mono__iff,axiom,
! [P: extended_ereal > $o,Q: extended_ereal > $o] :
( ( ord_le1644982726543182158_ereal @ ( collec5835592288176408249_ereal @ P ) @ ( collec5835592288176408249_ereal @ Q ) )
= ( ! [X: extended_ereal] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_537_Collect__mono__iff,axiom,
! [P: list_b > $o,Q: list_b > $o] :
( ( ord_le8932221534207217157list_b @ ( collect_list_b @ P ) @ ( collect_list_b @ Q ) )
= ( ! [X: list_b] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_538_Collect__mono__iff,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) )
= ( ! [X: list_a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_539_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_540_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X: set_a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_541_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_542_Collect__mono__iff,axiom,
! [P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) )
= ( ! [X: b] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_543_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_544_set__eq__subset,axiom,
( ( ^ [Y5: set_b,Z2: set_b] : ( Y5 = Z2 ) )
= ( ^ [A4: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A4 @ B3 )
& ( ord_less_eq_set_b @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_545_subset__trans,axiom,
! [A2: set_a,B: set_a,C5: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C5 )
=> ( ord_less_eq_set_a @ A2 @ C5 ) ) ) ).
% subset_trans
thf(fact_546_subset__trans,axiom,
! [A2: set_b,B: set_b,C5: set_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( ord_less_eq_set_b @ B @ C5 )
=> ( ord_less_eq_set_b @ A2 @ C5 ) ) ) ).
% subset_trans
thf(fact_547_Collect__mono,axiom,
! [P: extended_ereal > $o,Q: extended_ereal > $o] :
( ! [X3: extended_ereal] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le1644982726543182158_ereal @ ( collec5835592288176408249_ereal @ P ) @ ( collec5835592288176408249_ereal @ Q ) ) ) ).
% Collect_mono
thf(fact_548_Collect__mono,axiom,
! [P: list_b > $o,Q: list_b > $o] :
( ! [X3: list_b] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le8932221534207217157list_b @ ( collect_list_b @ P ) @ ( collect_list_b @ Q ) ) ) ).
% Collect_mono
thf(fact_549_Collect__mono,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X3: list_a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) ) ) ).
% Collect_mono
thf(fact_550_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_551_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_552_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_553_Collect__mono,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X3: b] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% Collect_mono
thf(fact_554_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_555_subset__refl,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ A2 @ A2 ) ).
% subset_refl
thf(fact_556_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B3: set_set_a] :
! [T2: set_a] :
( ( member_set_a @ T2 @ A4 )
=> ( member_set_a @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_557_subset__iff,axiom,
( ord_le8200006823705900825t_unit
= ( ^ [A4: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
! [T2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ T2 @ A4 )
=> ( member6939884229742472986t_unit @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_558_subset__iff,axiom,
( ord_le1644982726543182158_ereal
= ( ^ [A4: set_Extended_ereal,B3: set_Extended_ereal] :
! [T2: extended_ereal] :
( ( member2350847679896131959_ereal @ T2 @ A4 )
=> ( member2350847679896131959_ereal @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_559_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A4 )
=> ( member_a @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_560_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B3: set_b] :
! [T2: b] :
( ( member_b @ T2 @ A4 )
=> ( member_b @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_561_equalityD2,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_562_equalityD2,axiom,
! [A2: set_b,B: set_b] :
( ( A2 = B )
=> ( ord_less_eq_set_b @ B @ A2 ) ) ).
% equalityD2
thf(fact_563_equalityD1,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_564_equalityD1,axiom,
! [A2: set_b,B: set_b] :
( ( A2 = B )
=> ( ord_less_eq_set_b @ A2 @ B ) ) ).
% equalityD1
thf(fact_565_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B3: set_set_a] :
! [X: set_a] :
( ( member_set_a @ X @ A4 )
=> ( member_set_a @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_566_subset__eq,axiom,
( ord_le8200006823705900825t_unit
= ( ^ [A4: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
! [X: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X @ A4 )
=> ( member6939884229742472986t_unit @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_567_subset__eq,axiom,
( ord_le1644982726543182158_ereal
= ( ^ [A4: set_Extended_ereal,B3: set_Extended_ereal] :
! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A4 )
=> ( member2350847679896131959_ereal @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_568_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
! [X: a] :
( ( member_a @ X @ A4 )
=> ( member_a @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_569_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B3: set_b] :
! [X: b] :
( ( member_b @ X @ A4 )
=> ( member_b @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_570_equalityE,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ~ ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_571_equalityE,axiom,
! [A2: set_b,B: set_b] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_b @ A2 @ B )
=> ~ ( ord_less_eq_set_b @ B @ A2 ) ) ) ).
% equalityE
thf(fact_572_subsetD,axiom,
! [A2: set_set_a,B: set_set_a,C2: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_573_subsetD,axiom,
! [A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit,C2: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ B )
=> ( ( member6939884229742472986t_unit @ C2 @ A2 )
=> ( member6939884229742472986t_unit @ C2 @ B ) ) ) ).
% subsetD
thf(fact_574_subsetD,axiom,
! [A2: set_Extended_ereal,B: set_Extended_ereal,C2: extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A2 @ B )
=> ( ( member2350847679896131959_ereal @ C2 @ A2 )
=> ( member2350847679896131959_ereal @ C2 @ B ) ) ) ).
% subsetD
thf(fact_575_subsetD,axiom,
! [A2: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_576_subsetD,axiom,
! [A2: set_b,B: set_b,C2: b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( member_b @ C2 @ A2 )
=> ( member_b @ C2 @ B ) ) ) ).
% subsetD
thf(fact_577_in__mono,axiom,
! [A2: set_set_a,B: set_set_a,X2: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ X2 @ A2 )
=> ( member_set_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_578_in__mono,axiom,
! [A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit,X2: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ B )
=> ( ( member6939884229742472986t_unit @ X2 @ A2 )
=> ( member6939884229742472986t_unit @ X2 @ B ) ) ) ).
% in_mono
thf(fact_579_in__mono,axiom,
! [A2: set_Extended_ereal,B: set_Extended_ereal,X2: extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A2 @ B )
=> ( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( member2350847679896131959_ereal @ X2 @ B ) ) ) ).
% in_mono
thf(fact_580_in__mono,axiom,
! [A2: set_a,B: set_a,X2: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_581_in__mono,axiom,
! [A2: set_b,B: set_b,X2: b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( member_b @ X2 @ A2 )
=> ( member_b @ X2 @ B ) ) ) ).
% in_mono
thf(fact_582_pre__digraph_Oinduced__subgraph__altdef2,axiom,
( digrap5251062021860773499ph_a_b
= ( ^ [H3: pre_pr7278220950009878019t_unit,G2: pre_pr7278220950009878019t_unit] :
( digrap4729920478598810909ph_a_b @ G2
@ ^ [H4: pre_pr7278220950009878019t_unit] :
( ( pre_ve642382030648772252t_unit @ H4 )
= ( pre_ve642382030648772252t_unit @ H3 ) )
@ H3 ) ) ) ).
% pre_digraph.induced_subgraph_altdef2
thf(fact_583_pre__digraph_Osccs__altdef2,axiom,
( digraph_pre_sccs_a_b
= ( ^ [G2: pre_pr7278220950009878019t_unit] : ( collec8000012497822511960t_unit @ ( digrap4729920478598810909ph_a_b @ G2 @ digrap8691851296217657702ed_a_b ) ) ) ) ).
% pre_digraph.sccs_altdef2
thf(fact_584_wf__digraph_Oin__sccs__vertsI__sccs,axiom,
! [G: pre_pr7278220950009878019t_unit,S: set_a] :
( ( wf_digraph_a_b @ G )
=> ( ( member_set_a @ S @ ( image_7466199892558553556_set_a @ pre_ve642382030648772252t_unit @ ( digraph_pre_sccs_a_b @ G ) ) )
=> ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ G ) ) ) ) ).
% wf_digraph.in_sccs_vertsI_sccs
thf(fact_585_wf__digraph_Osccs__verts__conv,axiom,
! [G: pre_pr7278220950009878019t_unit] :
( ( wf_digraph_a_b @ G )
=> ( ( digrap2871191568752656621ts_a_b @ G )
= ( image_7466199892558553556_set_a @ pre_ve642382030648772252t_unit @ ( digraph_pre_sccs_a_b @ G ) ) ) ) ).
% wf_digraph.sccs_verts_conv
thf(fact_586_wf__digraph_Osccs__conv__sccs__verts,axiom,
! [G: pre_pr7278220950009878019t_unit] :
( ( wf_digraph_a_b @ G )
=> ( ( digraph_pre_sccs_a_b @ G )
= ( image_6801035452528096924t_unit @ ( digrap7873285959652527175ph_a_b @ G ) @ ( digrap2871191568752656621ts_a_b @ G ) ) ) ) ).
% wf_digraph.sccs_conv_sccs_verts
thf(fact_587_Compr__image__eq,axiom,
! [F: extended_ereal > extended_ereal,A2: set_Extended_ereal,P: extended_ereal > $o] :
( ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ ( image_6042159593519690757_ereal @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_6042159593519690757_ereal @ F
@ ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_588_Compr__image__eq,axiom,
! [F: nat > extended_ereal,A2: set_nat,P: extended_ereal > $o] :
( ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ ( image_4309273772856505399_ereal @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_4309273772856505399_ereal @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_589_Compr__image__eq,axiom,
! [F: a > extended_ereal,A2: set_a,P: extended_ereal > $o] :
( ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ ( image_8405481351990995413_ereal @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_8405481351990995413_ereal @ F
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_590_Compr__image__eq,axiom,
! [F: b > extended_ereal,A2: set_b,P: extended_ereal > $o] :
( ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ ( image_5319725110001000852_ereal @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_5319725110001000852_ereal @ F
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_591_Compr__image__eq,axiom,
! [F: extended_ereal > nat,A2: set_Extended_ereal,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_7659842161140344153al_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_7659842161140344153al_nat @ F
@ ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_592_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_593_Compr__image__eq,axiom,
! [F: a > nat,A2: set_a,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_a_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_a_nat @ F
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_594_Compr__image__eq,axiom,
! [F: b > nat,A2: set_b,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_b_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_b_nat @ F
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_595_Compr__image__eq,axiom,
! [F: extended_ereal > a,A2: set_Extended_ereal,P: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( image_3724615099042636213real_a @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_3724615099042636213real_a @ F
@ ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_596_Compr__image__eq,axiom,
! [F: nat > a,A2: set_nat,P: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( image_nat_a @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_nat_a @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_597_image__image,axiom,
! [F: extended_ereal > extended_ereal,G4: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( image_6042159593519690757_ereal @ F @ ( image_6042159593519690757_ereal @ G4 @ A2 ) )
= ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : ( F @ ( G4 @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_598_image__image,axiom,
! [F: a > set_a,G4: a > a,A2: set_a] :
( ( image_a_set_a @ F @ ( image_a_a @ G4 @ A2 ) )
= ( image_a_set_a
@ ^ [X: a] : ( F @ ( G4 @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_599_image__image,axiom,
! [F: extended_ereal > extended_ereal,G4: list_b > extended_ereal,A2: set_list_b] :
( ( image_6042159593519690757_ereal @ F @ ( image_3611896476772571406_ereal @ G4 @ A2 ) )
= ( image_3611896476772571406_ereal
@ ^ [X: list_b] : ( F @ ( G4 @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_600_image__image,axiom,
! [F: list_b > extended_ereal,G4: extended_ereal > list_b,A2: set_Extended_ereal] :
( ( image_3611896476772571406_ereal @ F @ ( image_3533209195447846460list_b @ G4 @ A2 ) )
= ( image_6042159593519690757_ereal
@ ^ [X: extended_ereal] : ( F @ ( G4 @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_601_image__image,axiom,
! [F: set_a > set_a,G4: a > set_a,A2: set_a] :
( ( image_set_a_set_a @ F @ ( image_a_set_a @ G4 @ A2 ) )
= ( image_a_set_a
@ ^ [X: a] : ( F @ ( G4 @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_602_image__image,axiom,
! [F: list_b > extended_ereal,G4: list_b > list_b,A2: set_list_b] :
( ( image_3611896476772571406_ereal @ F @ ( image_list_b_list_b @ G4 @ A2 ) )
= ( image_3611896476772571406_ereal
@ ^ [X: list_b] : ( F @ ( G4 @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_603_image__image,axiom,
! [F: a > set_a,G4: pre_pr7278220950009878019t_unit > a,A2: set_pr5411798346947241657t_unit] :
( ( image_a_set_a @ F @ ( image_4969699134812999796unit_a @ G4 @ A2 ) )
= ( image_7466199892558553556_set_a
@ ^ [X: pre_pr7278220950009878019t_unit] : ( F @ ( G4 @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_604_image__image,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,G4: a > pre_pr7278220950009878019t_unit,A2: set_a] :
( ( image_7466199892558553556_set_a @ F @ ( image_5713294457175270716t_unit @ G4 @ A2 ) )
= ( image_a_set_a
@ ^ [X: a] : ( F @ ( G4 @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_605_image__image,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,G4: a > set_a,A2: set_a] :
( ( image_6801035452528096924t_unit @ F @ ( image_a_set_a @ G4 @ A2 ) )
= ( image_5713294457175270716t_unit
@ ^ [X: a] : ( F @ ( G4 @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_606_image__image,axiom,
! [F: set_a > set_a,G4: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit] :
( ( image_set_a_set_a @ F @ ( image_7466199892558553556_set_a @ G4 @ A2 ) )
= ( image_7466199892558553556_set_a
@ ^ [X: pre_pr7278220950009878019t_unit] : ( F @ ( G4 @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_607_imageE,axiom,
! [B2: a,F: a > a,A2: set_a] :
( ( member_a @ B2 @ ( image_a_a @ F @ A2 ) )
=> ~ ! [X3: a] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_608_imageE,axiom,
! [B2: a,F: b > a,A2: set_b] :
( ( member_a @ B2 @ ( image_b_a @ F @ A2 ) )
=> ~ ! [X3: b] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_b @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_609_imageE,axiom,
! [B2: a,F: extended_ereal > a,A2: set_Extended_ereal] :
( ( member_a @ B2 @ ( image_3724615099042636213real_a @ F @ A2 ) )
=> ~ ! [X3: extended_ereal] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member2350847679896131959_ereal @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_610_imageE,axiom,
! [B2: b,F: a > b,A2: set_a] :
( ( member_b @ B2 @ ( image_a_b @ F @ A2 ) )
=> ~ ! [X3: a] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_611_imageE,axiom,
! [B2: b,F: b > b,A2: set_b] :
( ( member_b @ B2 @ ( image_b_b @ F @ A2 ) )
=> ~ ! [X3: b] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_b @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_612_imageE,axiom,
! [B2: b,F: extended_ereal > b,A2: set_Extended_ereal] :
( ( member_b @ B2 @ ( image_3724615099042636214real_b @ F @ A2 ) )
=> ~ ! [X3: extended_ereal] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member2350847679896131959_ereal @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_613_imageE,axiom,
! [B2: extended_ereal,F: a > extended_ereal,A2: set_a] :
( ( member2350847679896131959_ereal @ B2 @ ( image_8405481351990995413_ereal @ F @ A2 ) )
=> ~ ! [X3: a] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_614_imageE,axiom,
! [B2: extended_ereal,F: b > extended_ereal,A2: set_b] :
( ( member2350847679896131959_ereal @ B2 @ ( image_5319725110001000852_ereal @ F @ A2 ) )
=> ~ ! [X3: b] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_b @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_615_imageE,axiom,
! [B2: extended_ereal,F: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ B2 @ ( image_6042159593519690757_ereal @ F @ A2 ) )
=> ~ ! [X3: extended_ereal] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member2350847679896131959_ereal @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_616_imageE,axiom,
! [B2: a,F: set_a > a,A2: set_set_a] :
( ( member_a @ B2 @ ( image_set_a_a @ F @ A2 ) )
=> ~ ! [X3: set_a] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_set_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_617_empty__def,axiom,
( bot_bot_set_list_b
= ( collect_list_b
@ ^ [X: list_b] : $false ) ) ).
% empty_def
thf(fact_618_empty__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a
@ ^ [X: list_a] : $false ) ) ).
% empty_def
thf(fact_619_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X: nat] : $false ) ) ).
% empty_def
thf(fact_620_empty__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a
@ ^ [X: set_a] : $false ) ) ).
% empty_def
thf(fact_621_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X: a] : $false ) ) ).
% empty_def
thf(fact_622_empty__def,axiom,
( bot_bo1839476491465656141t_unit
= ( collec8000012497822511960t_unit
@ ^ [X: pre_pr7278220950009878019t_unit] : $false ) ) ).
% empty_def
thf(fact_623_empty__def,axiom,
( bot_bot_set_b
= ( collect_b
@ ^ [X: b] : $false ) ) ).
% empty_def
thf(fact_624_empty__def,axiom,
( bot_bo8367695208629047834_ereal
= ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] : $false ) ) ).
% empty_def
thf(fact_625_Collect__subset,axiom,
! [A2: set_pr5411798346947241657t_unit,P: pre_pr7278220950009878019t_unit > $o] :
( ord_le8200006823705900825t_unit
@ ( collec8000012497822511960t_unit
@ ^ [X: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_626_Collect__subset,axiom,
! [A2: set_Extended_ereal,P: extended_ereal > $o] :
( ord_le1644982726543182158_ereal
@ ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_627_Collect__subset,axiom,
! [A2: set_list_b,P: list_b > $o] :
( ord_le8932221534207217157list_b
@ ( collect_list_b
@ ^ [X: list_b] :
( ( member_list_b @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_628_Collect__subset,axiom,
! [A2: set_list_a,P: list_a > $o] :
( ord_le8861187494160871172list_a
@ ( collect_list_a
@ ^ [X: list_a] :
( ( member_list_a @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_629_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_630_Collect__subset,axiom,
! [A2: set_set_a,P: set_a > $o] :
( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X: set_a] :
( ( member_set_a @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_631_Collect__subset,axiom,
! [A2: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_632_Collect__subset,axiom,
! [A2: set_b,P: b > $o] :
( ord_less_eq_set_b
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_633_pre__digraph_Oin__sccsI,axiom,
! [C2: pre_pr7278220950009878019t_unit,G: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ C2 @ G )
=> ( ( digrap8691851296217657702ed_a_b @ C2 )
=> ( ~ ? [C4: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ C4 @ G )
& ( digrap8691851296217657702ed_a_b @ C4 )
& ( ord_less_set_a @ ( pre_ve642382030648772252t_unit @ C2 ) @ ( pre_ve642382030648772252t_unit @ C4 ) ) )
=> ( member6939884229742472986t_unit @ C2 @ ( digraph_pre_sccs_a_b @ G ) ) ) ) ) ).
% pre_digraph.in_sccsI
thf(fact_634_pre__digraph_Oin__sccsE,axiom,
! [C2: pre_pr7278220950009878019t_unit,G: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( digraph_pre_sccs_a_b @ G ) )
=> ~ ( ( digrap5251062021860773499ph_a_b @ C2 @ G )
=> ( ( digrap8691851296217657702ed_a_b @ C2 )
=> ? [D2: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ D2 @ G )
& ( digrap8691851296217657702ed_a_b @ D2 )
& ( ord_less_set_a @ ( pre_ve642382030648772252t_unit @ C2 ) @ ( pre_ve642382030648772252t_unit @ D2 ) ) ) ) ) ) ).
% pre_digraph.in_sccsE
thf(fact_635_pre__digraph_Osccs__def,axiom,
( digraph_pre_sccs_a_b
= ( ^ [G2: pre_pr7278220950009878019t_unit] :
( collec8000012497822511960t_unit
@ ^ [H3: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ H3 @ G2 )
& ( digrap8691851296217657702ed_a_b @ H3 )
& ~ ? [H4: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ H4 @ G2 )
& ( digrap8691851296217657702ed_a_b @ H4 )
& ( ord_less_set_a @ ( pre_ve642382030648772252t_unit @ H3 ) @ ( pre_ve642382030648772252t_unit @ H4 ) ) ) ) ) ) ) ).
% pre_digraph.sccs_def
thf(fact_636_subset__image__iff,axiom,
! [B: set_set_a,F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_7466199892558553556_set_a @ F @ A2 ) )
= ( ? [AA: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ AA @ A2 )
& ( B
= ( image_7466199892558553556_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_637_subset__image__iff,axiom,
! [B: set_pr5411798346947241657t_unit,F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a] :
( ( ord_le8200006823705900825t_unit @ B @ ( image_6801035452528096924t_unit @ F @ A2 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A2 )
& ( B
= ( image_6801035452528096924t_unit @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_638_subset__image__iff,axiom,
! [B: set_Extended_ereal,F: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B @ ( image_6042159593519690757_ereal @ F @ A2 ) )
= ( ? [AA: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ AA @ A2 )
& ( B
= ( image_6042159593519690757_ereal @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_639_subset__image__iff,axiom,
! [B: set_Extended_ereal,F: list_b > extended_ereal,A2: set_list_b] :
( ( ord_le1644982726543182158_ereal @ B @ ( image_3611896476772571406_ereal @ F @ A2 ) )
= ( ? [AA: set_list_b] :
( ( ord_le8932221534207217157list_b @ AA @ A2 )
& ( B
= ( image_3611896476772571406_ereal @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_640_subset__image__iff,axiom,
! [B: set_set_a,F: a > set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B
= ( image_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_641_subset__image__iff,axiom,
! [B: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_642_subset__image__iff,axiom,
! [B: set_a,F: b > a,A2: set_b] :
( ( ord_less_eq_set_a @ B @ ( image_b_a @ F @ A2 ) )
= ( ? [AA: set_b] :
( ( ord_less_eq_set_b @ AA @ A2 )
& ( B
= ( image_b_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_643_subset__image__iff,axiom,
! [B: set_b,F: a > b,A2: set_a] :
( ( ord_less_eq_set_b @ B @ ( image_a_b @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B
= ( image_a_b @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_644_subset__image__iff,axiom,
! [B: set_b,F: b > b,A2: set_b] :
( ( ord_less_eq_set_b @ B @ ( image_b_b @ F @ A2 ) )
= ( ? [AA: set_b] :
( ( ord_less_eq_set_b @ AA @ A2 )
& ( B
= ( image_b_b @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_645_image__subset__iff,axiom,
! [F: a > set_a,A2: set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ B )
= ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( member_set_a @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_646_image__subset__iff,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_7466199892558553556_set_a @ F @ A2 ) @ B )
= ( ! [X: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X @ A2 )
=> ( member_set_a @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_647_image__subset__iff,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a,B: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ ( image_6801035452528096924t_unit @ F @ A2 ) @ B )
= ( ! [X: set_a] :
( ( member_set_a @ X @ A2 )
=> ( member6939884229742472986t_unit @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_648_image__subset__iff,axiom,
! [F: extended_ereal > extended_ereal,A2: set_Extended_ereal,B: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F @ A2 ) @ B )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A2 )
=> ( member2350847679896131959_ereal @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_649_image__subset__iff,axiom,
! [F: list_b > extended_ereal,A2: set_list_b,B: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( image_3611896476772571406_ereal @ F @ A2 ) @ B )
= ( ! [X: list_b] :
( ( member_list_b @ X @ A2 )
=> ( member2350847679896131959_ereal @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_650_subset__imageE,axiom,
! [B: set_set_a,F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_7466199892558553556_set_a @ F @ A2 ) )
=> ~ ! [C3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ C3 @ A2 )
=> ( B
!= ( image_7466199892558553556_set_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_651_subset__imageE,axiom,
! [B: set_pr5411798346947241657t_unit,F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a] :
( ( ord_le8200006823705900825t_unit @ B @ ( image_6801035452528096924t_unit @ F @ A2 ) )
=> ~ ! [C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ A2 )
=> ( B
!= ( image_6801035452528096924t_unit @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_652_subset__imageE,axiom,
! [B: set_Extended_ereal,F: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ B @ ( image_6042159593519690757_ereal @ F @ A2 ) )
=> ~ ! [C3: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ C3 @ A2 )
=> ( B
!= ( image_6042159593519690757_ereal @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_653_subset__imageE,axiom,
! [B: set_Extended_ereal,F: list_b > extended_ereal,A2: set_list_b] :
( ( ord_le1644982726543182158_ereal @ B @ ( image_3611896476772571406_ereal @ F @ A2 ) )
=> ~ ! [C3: set_list_b] :
( ( ord_le8932221534207217157list_b @ C3 @ A2 )
=> ( B
!= ( image_3611896476772571406_ereal @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_654_subset__imageE,axiom,
! [B: set_set_a,F: a > set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ B @ ( image_a_set_a @ F @ A2 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( B
!= ( image_a_set_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_655_subset__imageE,axiom,
! [B: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A2 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( B
!= ( image_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_656_subset__imageE,axiom,
! [B: set_a,F: b > a,A2: set_b] :
( ( ord_less_eq_set_a @ B @ ( image_b_a @ F @ A2 ) )
=> ~ ! [C3: set_b] :
( ( ord_less_eq_set_b @ C3 @ A2 )
=> ( B
!= ( image_b_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_657_subset__imageE,axiom,
! [B: set_b,F: a > b,A2: set_a] :
( ( ord_less_eq_set_b @ B @ ( image_a_b @ F @ A2 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( B
!= ( image_a_b @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_658_subset__imageE,axiom,
! [B: set_b,F: b > b,A2: set_b] :
( ( ord_less_eq_set_b @ B @ ( image_b_b @ F @ A2 ) )
=> ~ ! [C3: set_b] :
( ( ord_less_eq_set_b @ C3 @ A2 )
=> ( B
!= ( image_b_b @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_659_image__subsetI,axiom,
! [A2: set_a,F: a > extended_ereal,B: set_Extended_ereal] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member2350847679896131959_ereal @ ( F @ X3 ) @ B ) )
=> ( ord_le1644982726543182158_ereal @ ( image_8405481351990995413_ereal @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_660_image__subsetI,axiom,
! [A2: set_b,F: b > extended_ereal,B: set_Extended_ereal] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( member2350847679896131959_ereal @ ( F @ X3 ) @ B ) )
=> ( ord_le1644982726543182158_ereal @ ( image_5319725110001000852_ereal @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_661_image__subsetI,axiom,
! [A2: set_Extended_ereal,F: extended_ereal > extended_ereal,B: set_Extended_ereal] :
( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A2 )
=> ( member2350847679896131959_ereal @ ( F @ X3 ) @ B ) )
=> ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_662_image__subsetI,axiom,
! [A2: set_a,F: a > a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_663_image__subsetI,axiom,
! [A2: set_b,F: b > a,B: set_a] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_b_a @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_664_image__subsetI,axiom,
! [A2: set_Extended_ereal,F: extended_ereal > a,B: set_a] :
( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A2 )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_3724615099042636213real_a @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_665_image__subsetI,axiom,
! [A2: set_a,F: a > b,B: set_b] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_b @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_b @ ( image_a_b @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_666_image__subsetI,axiom,
! [A2: set_b,F: b > b,B: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( member_b @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_b @ ( image_b_b @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_667_image__subsetI,axiom,
! [A2: set_Extended_ereal,F: extended_ereal > b,B: set_b] :
( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A2 )
=> ( member_b @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_b @ ( image_3724615099042636214real_b @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_668_image__subsetI,axiom,
! [A2: set_list_b,F: list_b > extended_ereal,B: set_Extended_ereal] :
( ! [X3: list_b] :
( ( member_list_b @ X3 @ A2 )
=> ( member2350847679896131959_ereal @ ( F @ X3 ) @ B ) )
=> ( ord_le1644982726543182158_ereal @ ( image_3611896476772571406_ereal @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_669_image__mono,axiom,
! [A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > set_a] :
( ( ord_le8200006823705900825t_unit @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_7466199892558553556_set_a @ F @ A2 ) @ ( image_7466199892558553556_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_670_image__mono,axiom,
! [A2: set_set_a,B: set_set_a,F: set_a > pre_pr7278220950009878019t_unit] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ord_le8200006823705900825t_unit @ ( image_6801035452528096924t_unit @ F @ A2 ) @ ( image_6801035452528096924t_unit @ F @ B ) ) ) ).
% image_mono
thf(fact_671_image__mono,axiom,
! [A2: set_Extended_ereal,B: set_Extended_ereal,F: extended_ereal > extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A2 @ B )
=> ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F @ A2 ) @ ( image_6042159593519690757_ereal @ F @ B ) ) ) ).
% image_mono
thf(fact_672_image__mono,axiom,
! [A2: set_list_b,B: set_list_b,F: list_b > extended_ereal] :
( ( ord_le8932221534207217157list_b @ A2 @ B )
=> ( ord_le1644982726543182158_ereal @ ( image_3611896476772571406_ereal @ F @ A2 ) @ ( image_3611896476772571406_ereal @ F @ B ) ) ) ).
% image_mono
thf(fact_673_image__mono,axiom,
! [A2: set_a,B: set_a,F: a > set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ ( image_a_set_a @ F @ B ) ) ) ).
% image_mono
thf(fact_674_image__mono,axiom,
! [A2: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_675_image__mono,axiom,
! [A2: set_a,B: set_a,F: a > b] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_b @ ( image_a_b @ F @ A2 ) @ ( image_a_b @ F @ B ) ) ) ).
% image_mono
thf(fact_676_image__mono,axiom,
! [A2: set_b,B: set_b,F: b > a] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ord_less_eq_set_a @ ( image_b_a @ F @ A2 ) @ ( image_b_a @ F @ B ) ) ) ).
% image_mono
thf(fact_677_image__mono,axiom,
! [A2: set_b,B: set_b,F: b > b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ord_less_eq_set_b @ ( image_b_b @ F @ A2 ) @ ( image_b_b @ F @ B ) ) ) ).
% image_mono
thf(fact_678_Setcompr__eq__image,axiom,
! [F: a > extended_ereal,A2: set_a] :
( ( collec5835592288176408249_ereal
@ ^ [Uu: extended_ereal] :
? [X: a] :
( ( Uu
= ( F @ X ) )
& ( member_a @ X @ A2 ) ) )
= ( image_8405481351990995413_ereal @ F @ A2 ) ) ).
% Setcompr_eq_image
thf(fact_679_Setcompr__eq__image,axiom,
! [F: b > extended_ereal,A2: set_b] :
( ( collec5835592288176408249_ereal
@ ^ [Uu: extended_ereal] :
? [X: b] :
( ( Uu
= ( F @ X ) )
& ( member_b @ X @ A2 ) ) )
= ( image_5319725110001000852_ereal @ F @ A2 ) ) ).
% Setcompr_eq_image
thf(fact_680_Setcompr__eq__image,axiom,
! [F: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( collec5835592288176408249_ereal
@ ^ [Uu: extended_ereal] :
? [X: extended_ereal] :
( ( Uu
= ( F @ X ) )
& ( member2350847679896131959_ereal @ X @ A2 ) ) )
= ( image_6042159593519690757_ereal @ F @ A2 ) ) ).
% Setcompr_eq_image
thf(fact_681_Setcompr__eq__image,axiom,
! [F: a > nat,A2: set_a] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X: a] :
( ( Uu
= ( F @ X ) )
& ( member_a @ X @ A2 ) ) )
= ( image_a_nat @ F @ A2 ) ) ).
% Setcompr_eq_image
thf(fact_682_Setcompr__eq__image,axiom,
! [F: b > nat,A2: set_b] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X: b] :
( ( Uu
= ( F @ X ) )
& ( member_b @ X @ A2 ) ) )
= ( image_b_nat @ F @ A2 ) ) ).
% Setcompr_eq_image
thf(fact_683_Setcompr__eq__image,axiom,
! [F: extended_ereal > nat,A2: set_Extended_ereal] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X: extended_ereal] :
( ( Uu
= ( F @ X ) )
& ( member2350847679896131959_ereal @ X @ A2 ) ) )
= ( image_7659842161140344153al_nat @ F @ A2 ) ) ).
% Setcompr_eq_image
thf(fact_684_Setcompr__eq__image,axiom,
! [F: a > a,A2: set_a] :
( ( collect_a
@ ^ [Uu: a] :
? [X: a] :
( ( Uu
= ( F @ X ) )
& ( member_a @ X @ A2 ) ) )
= ( image_a_a @ F @ A2 ) ) ).
% Setcompr_eq_image
thf(fact_685_Setcompr__eq__image,axiom,
! [F: b > a,A2: set_b] :
( ( collect_a
@ ^ [Uu: a] :
? [X: b] :
( ( Uu
= ( F @ X ) )
& ( member_b @ X @ A2 ) ) )
= ( image_b_a @ F @ A2 ) ) ).
% Setcompr_eq_image
thf(fact_686_Setcompr__eq__image,axiom,
! [F: extended_ereal > a,A2: set_Extended_ereal] :
( ( collect_a
@ ^ [Uu: a] :
? [X: extended_ereal] :
( ( Uu
= ( F @ X ) )
& ( member2350847679896131959_ereal @ X @ A2 ) ) )
= ( image_3724615099042636213real_a @ F @ A2 ) ) ).
% Setcompr_eq_image
thf(fact_687_Setcompr__eq__image,axiom,
! [F: a > b,A2: set_a] :
( ( collect_b
@ ^ [Uu: b] :
? [X: a] :
( ( Uu
= ( F @ X ) )
& ( member_a @ X @ A2 ) ) )
= ( image_a_b @ F @ A2 ) ) ).
% Setcompr_eq_image
thf(fact_688_setcompr__eq__image,axiom,
! [F: extended_ereal > extended_ereal,P: extended_ereal > $o] :
( ( collec5835592288176408249_ereal
@ ^ [Uu: extended_ereal] :
? [X: extended_ereal] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) )
= ( image_6042159593519690757_ereal @ F @ ( collec5835592288176408249_ereal @ P ) ) ) ).
% setcompr_eq_image
thf(fact_689_setcompr__eq__image,axiom,
! [F: nat > extended_ereal,P: nat > $o] :
( ( collec5835592288176408249_ereal
@ ^ [Uu: extended_ereal] :
? [X: nat] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) )
= ( image_4309273772856505399_ereal @ F @ ( collect_nat @ P ) ) ) ).
% setcompr_eq_image
thf(fact_690_setcompr__eq__image,axiom,
! [F: a > extended_ereal,P: a > $o] :
( ( collec5835592288176408249_ereal
@ ^ [Uu: extended_ereal] :
? [X: a] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) )
= ( image_8405481351990995413_ereal @ F @ ( collect_a @ P ) ) ) ).
% setcompr_eq_image
thf(fact_691_setcompr__eq__image,axiom,
! [F: b > extended_ereal,P: b > $o] :
( ( collec5835592288176408249_ereal
@ ^ [Uu: extended_ereal] :
? [X: b] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) )
= ( image_5319725110001000852_ereal @ F @ ( collect_b @ P ) ) ) ).
% setcompr_eq_image
thf(fact_692_setcompr__eq__image,axiom,
! [F: extended_ereal > nat,P: extended_ereal > $o] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X: extended_ereal] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) )
= ( image_7659842161140344153al_nat @ F @ ( collec5835592288176408249_ereal @ P ) ) ) ).
% setcompr_eq_image
thf(fact_693_setcompr__eq__image,axiom,
! [F: nat > nat,P: nat > $o] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X: nat] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) )
= ( image_nat_nat @ F @ ( collect_nat @ P ) ) ) ).
% setcompr_eq_image
thf(fact_694_setcompr__eq__image,axiom,
! [F: a > nat,P: a > $o] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X: a] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) )
= ( image_a_nat @ F @ ( collect_a @ P ) ) ) ).
% setcompr_eq_image
thf(fact_695_setcompr__eq__image,axiom,
! [F: b > nat,P: b > $o] :
( ( collect_nat
@ ^ [Uu: nat] :
? [X: b] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) )
= ( image_b_nat @ F @ ( collect_b @ P ) ) ) ).
% setcompr_eq_image
thf(fact_696_setcompr__eq__image,axiom,
! [F: extended_ereal > a,P: extended_ereal > $o] :
( ( collect_a
@ ^ [Uu: a] :
? [X: extended_ereal] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) )
= ( image_3724615099042636213real_a @ F @ ( collec5835592288176408249_ereal @ P ) ) ) ).
% setcompr_eq_image
thf(fact_697_setcompr__eq__image,axiom,
! [F: nat > a,P: nat > $o] :
( ( collect_a
@ ^ [Uu: a] :
? [X: nat] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) )
= ( image_nat_a @ F @ ( collect_nat @ P ) ) ) ).
% setcompr_eq_image
thf(fact_698_dia__lowerB,axiom,
! [U2: a,V: a,W: b > real] :
( ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ord_le1083603963089353582_ereal @ ( shortest_wf_mu_a_b @ g @ W @ U2 @ V ) @ ( graph_926353876199057498er_a_b @ g @ W ) ) ) ) ).
% dia_lowerB
thf(fact_699_scc__disj,axiom,
! [C2: pre_pr7278220950009878019t_unit,D: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( digraph_pre_sccs_a_b @ g ) )
=> ( ( member6939884229742472986t_unit @ D @ ( digraph_pre_sccs_a_b @ g ) )
=> ( ( C2 != D )
=> ( ( inf_inf_set_a @ ( pre_ve642382030648772252t_unit @ C2 ) @ ( pre_ve642382030648772252t_unit @ D ) )
= bot_bot_set_a ) ) ) ) ).
% scc_disj
thf(fact_700_connected__spanning__imp__connected,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digraph_spanning_a_b @ H @ g )
=> ( ( digrap8783888973171253482ed_a_b @ H )
=> ( digrap8783888973171253482ed_a_b @ g ) ) ) ).
% connected_spanning_imp_connected
thf(fact_701_ends__del__vert,axiom,
! [U2: a] :
( ( arc_to_ends_a_b @ ( pre_del_vert_a_b @ g @ U2 ) )
= ( arc_to_ends_a_b @ g ) ) ).
% ends_del_vert
thf(fact_702_trail__Nil__iff,axiom,
! [U2: a,V: a] :
( ( arc_pre_trail_a_b @ g @ U2 @ nil_b @ V )
= ( ( U2 = V )
& ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ g ) ) ) ) ).
% trail_Nil_iff
thf(fact_703_apaths__finite,axiom,
! [U2: a,V: a] :
( finite_finite_list_b
@ ( collect_list_b
@ ^ [P2: list_b] : ( arc_pre_apath_a_b @ g @ U2 @ P2 @ V ) ) ) ).
% apaths_finite
thf(fact_704_wf__digraph_Ofin__dia__lowerB,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a,V: a,W: b > real] :
( ( wf_digraph_a_b @ G )
=> ( ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ G ) )
=> ( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ G ) )
=> ( ( ord_le1188267648640031866_ereal @ ( shortest_wf_mu_a_b @ G @ W @ U2 @ V ) @ extend1530274965995635425_ereal )
=> ( ord_le1083603963089353582_ereal @ ( shortest_wf_mu_a_b @ G @ W @ U2 @ V ) @ ( graph_1932031826008834157er_a_b @ G @ W ) ) ) ) ) ) ).
% wf_digraph.fin_dia_lowerB
thf(fact_705_strongly__connected__eq__iff,axiom,
( ( digrap8691851296217657702ed_a_b @ g )
= ( ( digraph_pre_sccs_a_b @ g )
= ( insert6864688055023459379t_unit @ g @ bot_bo1839476491465656141t_unit ) ) ) ).
% strongly_connected_eq_iff
thf(fact_706_verts__add__vert,axiom,
! [U2: a] :
( ( pre_ve642382030648772252t_unit @ ( pre_add_vert_a_b @ g @ U2 ) )
= ( insert_a @ U2 @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% verts_add_vert
thf(fact_707_reachable__apath,axiom,
! [U2: a,V: a] :
( ( reachable_a_b @ g @ U2 @ V )
= ( ? [P2: list_b] : ( arc_pre_apath_a_b @ g @ U2 @ P2 @ V ) ) ) ).
% reachable_apath
thf(fact_708_apath__nonempty__ends,axiom,
! [U2: a,P3: list_b,V: a] :
( ( arc_pre_apath_a_b @ g @ U2 @ P3 @ V )
=> ( ( P3 != nil_b )
=> ( U2 != V ) ) ) ).
% apath_nonempty_ends
thf(fact_709_apath__ends,axiom,
! [U2: a,P3: list_b,V: a,U5: a,V5: a] :
( ( arc_pre_apath_a_b @ g @ U2 @ P3 @ V )
=> ( ( arc_pre_apath_a_b @ g @ U5 @ P3 @ V5 )
=> ( ( ( P3 != nil_b )
& ( U2 != V )
& ( U2 = U5 )
& ( V = V5 ) )
| ( ( P3 = nil_b )
& ( U2 = V )
& ( U5 = V5 ) ) ) ) ) ).
% apath_ends
thf(fact_710_insert__absorb2,axiom,
! [X2: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( insert6864688055023459379t_unit @ X2 @ ( insert6864688055023459379t_unit @ X2 @ A2 ) )
= ( insert6864688055023459379t_unit @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_711_insert__absorb2,axiom,
! [X2: a,A2: set_a] :
( ( insert_a @ X2 @ ( insert_a @ X2 @ A2 ) )
= ( insert_a @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_712_insert__absorb2,axiom,
! [X2: b,A2: set_b] :
( ( insert_b @ X2 @ ( insert_b @ X2 @ A2 ) )
= ( insert_b @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_713_insert__absorb2,axiom,
! [X2: extended_ereal,A2: set_Extended_ereal] :
( ( insert8967887681552722334_ereal @ X2 @ ( insert8967887681552722334_ereal @ X2 @ A2 ) )
= ( insert8967887681552722334_ereal @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_714_insert__iff,axiom,
! [A: a,B2: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_715_insert__iff,axiom,
! [A: set_a,B2: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_set_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_716_insert__iff,axiom,
! [A: pre_pr7278220950009878019t_unit,B2: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ A @ ( insert6864688055023459379t_unit @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member6939884229742472986t_unit @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_717_insert__iff,axiom,
! [A: b,B2: b,A2: set_b] :
( ( member_b @ A @ ( insert_b @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_b @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_718_insert__iff,axiom,
! [A: extended_ereal,B2: extended_ereal,A2: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ A @ ( insert8967887681552722334_ereal @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member2350847679896131959_ereal @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_719_insertCI,axiom,
! [A: a,B: set_a,B2: a] :
( ( ~ ( member_a @ A @ B )
=> ( A = B2 ) )
=> ( member_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_720_insertCI,axiom,
! [A: set_a,B: set_set_a,B2: set_a] :
( ( ~ ( member_set_a @ A @ B )
=> ( A = B2 ) )
=> ( member_set_a @ A @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_721_insertCI,axiom,
! [A: pre_pr7278220950009878019t_unit,B: set_pr5411798346947241657t_unit,B2: pre_pr7278220950009878019t_unit] :
( ( ~ ( member6939884229742472986t_unit @ A @ B )
=> ( A = B2 ) )
=> ( member6939884229742472986t_unit @ A @ ( insert6864688055023459379t_unit @ B2 @ B ) ) ) ).
% insertCI
thf(fact_722_insertCI,axiom,
! [A: b,B: set_b,B2: b] :
( ( ~ ( member_b @ A @ B )
=> ( A = B2 ) )
=> ( member_b @ A @ ( insert_b @ B2 @ B ) ) ) ).
% insertCI
thf(fact_723_insertCI,axiom,
! [A: extended_ereal,B: set_Extended_ereal,B2: extended_ereal] :
( ( ~ ( member2350847679896131959_ereal @ A @ B )
=> ( A = B2 ) )
=> ( member2350847679896131959_ereal @ A @ ( insert8967887681552722334_ereal @ B2 @ B ) ) ) ).
% insertCI
thf(fact_724_Int__iff,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
= ( ( member_set_a @ C2 @ A2 )
& ( member_set_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_725_Int__iff,axiom,
! [C2: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( inf_in1092213268631476299t_unit @ A2 @ B ) )
= ( ( member6939884229742472986t_unit @ C2 @ A2 )
& ( member6939884229742472986t_unit @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_726_Int__iff,axiom,
! [C2: b,A2: set_b,B: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A2 @ B ) )
= ( ( member_b @ C2 @ A2 )
& ( member_b @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_727_Int__iff,axiom,
! [C2: extended_ereal,A2: set_Extended_ereal,B: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ C2 @ ( inf_in2779415704524776092_ereal @ A2 @ B ) )
= ( ( member2350847679896131959_ereal @ C2 @ A2 )
& ( member2350847679896131959_ereal @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_728_Int__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
& ( member_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_729_IntI,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ A2 )
=> ( ( member_set_a @ C2 @ B )
=> ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_730_IntI,axiom,
! [C2: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C2 @ A2 )
=> ( ( member6939884229742472986t_unit @ C2 @ B )
=> ( member6939884229742472986t_unit @ C2 @ ( inf_in1092213268631476299t_unit @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_731_IntI,axiom,
! [C2: b,A2: set_b,B: set_b] :
( ( member_b @ C2 @ A2 )
=> ( ( member_b @ C2 @ B )
=> ( member_b @ C2 @ ( inf_inf_set_b @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_732_IntI,axiom,
! [C2: extended_ereal,A2: set_Extended_ereal,B: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ C2 @ A2 )
=> ( ( member2350847679896131959_ereal @ C2 @ B )
=> ( member2350847679896131959_ereal @ C2 @ ( inf_in2779415704524776092_ereal @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_733_IntI,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_734_apath__Nil__iff,axiom,
! [U2: a,V: a] :
( ( arc_pre_apath_a_b @ g @ U2 @ nil_b @ V )
= ( ( U2 = V )
& ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ g ) ) ) ) ).
% apath_Nil_iff
thf(fact_735_sccs__verts__disjoint,axiom,
! [S: set_a,T: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ g ) )
=> ( ( member_set_a @ T @ ( digrap2871191568752656621ts_a_b @ g ) )
=> ( ( S != T )
=> ( ( inf_inf_set_a @ S @ T )
= bot_bot_set_a ) ) ) ) ).
% sccs_verts_disjoint
thf(fact_736_dia__eq__fin__dia__if__finite,axiom,
! [F: b > real] :
( ( ord_le1188267648640031866_ereal @ ( graph_926353876199057498er_a_b @ g @ F ) @ extend1530274965995635425_ereal )
=> ( ( graph_926353876199057498er_a_b @ g @ F )
= ( graph_1932031826008834157er_a_b @ g @ F ) ) ) ).
% dia_eq_fin_dia_if_finite
thf(fact_737_image__insert,axiom,
! [F: a > a,A: a,B: set_a] :
( ( image_a_a @ F @ ( insert_a @ A @ B ) )
= ( insert_a @ ( F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_insert
thf(fact_738_image__insert,axiom,
! [F: a > b,A: a,B: set_a] :
( ( image_a_b @ F @ ( insert_a @ A @ B ) )
= ( insert_b @ ( F @ A ) @ ( image_a_b @ F @ B ) ) ) ).
% image_insert
thf(fact_739_image__insert,axiom,
! [F: a > extended_ereal,A: a,B: set_a] :
( ( image_8405481351990995413_ereal @ F @ ( insert_a @ A @ B ) )
= ( insert8967887681552722334_ereal @ ( F @ A ) @ ( image_8405481351990995413_ereal @ F @ B ) ) ) ).
% image_insert
thf(fact_740_image__insert,axiom,
! [F: b > a,A: b,B: set_b] :
( ( image_b_a @ F @ ( insert_b @ A @ B ) )
= ( insert_a @ ( F @ A ) @ ( image_b_a @ F @ B ) ) ) ).
% image_insert
thf(fact_741_image__insert,axiom,
! [F: b > b,A: b,B: set_b] :
( ( image_b_b @ F @ ( insert_b @ A @ B ) )
= ( insert_b @ ( F @ A ) @ ( image_b_b @ F @ B ) ) ) ).
% image_insert
thf(fact_742_image__insert,axiom,
! [F: b > extended_ereal,A: b,B: set_b] :
( ( image_5319725110001000852_ereal @ F @ ( insert_b @ A @ B ) )
= ( insert8967887681552722334_ereal @ ( F @ A ) @ ( image_5319725110001000852_ereal @ F @ B ) ) ) ).
% image_insert
thf(fact_743_image__insert,axiom,
! [F: extended_ereal > a,A: extended_ereal,B: set_Extended_ereal] :
( ( image_3724615099042636213real_a @ F @ ( insert8967887681552722334_ereal @ A @ B ) )
= ( insert_a @ ( F @ A ) @ ( image_3724615099042636213real_a @ F @ B ) ) ) ).
% image_insert
thf(fact_744_image__insert,axiom,
! [F: extended_ereal > b,A: extended_ereal,B: set_Extended_ereal] :
( ( image_3724615099042636214real_b @ F @ ( insert8967887681552722334_ereal @ A @ B ) )
= ( insert_b @ ( F @ A ) @ ( image_3724615099042636214real_b @ F @ B ) ) ) ).
% image_insert
thf(fact_745_image__insert,axiom,
! [F: extended_ereal > extended_ereal,A: extended_ereal,B: set_Extended_ereal] :
( ( image_6042159593519690757_ereal @ F @ ( insert8967887681552722334_ereal @ A @ B ) )
= ( insert8967887681552722334_ereal @ ( F @ A ) @ ( image_6042159593519690757_ereal @ F @ B ) ) ) ).
% image_insert
thf(fact_746_image__insert,axiom,
! [F: list_b > extended_ereal,A: list_b,B: set_list_b] :
( ( image_3611896476772571406_ereal @ F @ ( insert_list_b @ A @ B ) )
= ( insert8967887681552722334_ereal @ ( F @ A ) @ ( image_3611896476772571406_ereal @ F @ B ) ) ) ).
% image_insert
thf(fact_747_insert__image,axiom,
! [X2: a,A2: set_a,F: a > a] :
( ( member_a @ X2 @ A2 )
=> ( ( insert_a @ ( F @ X2 ) @ ( image_a_a @ F @ A2 ) )
= ( image_a_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_748_insert__image,axiom,
! [X2: a,A2: set_a,F: a > b] :
( ( member_a @ X2 @ A2 )
=> ( ( insert_b @ ( F @ X2 ) @ ( image_a_b @ F @ A2 ) )
= ( image_a_b @ F @ A2 ) ) ) ).
% insert_image
thf(fact_749_insert__image,axiom,
! [X2: a,A2: set_a,F: a > extended_ereal] :
( ( member_a @ X2 @ A2 )
=> ( ( insert8967887681552722334_ereal @ ( F @ X2 ) @ ( image_8405481351990995413_ereal @ F @ A2 ) )
= ( image_8405481351990995413_ereal @ F @ A2 ) ) ) ).
% insert_image
thf(fact_750_insert__image,axiom,
! [X2: b,A2: set_b,F: b > a] :
( ( member_b @ X2 @ A2 )
=> ( ( insert_a @ ( F @ X2 ) @ ( image_b_a @ F @ A2 ) )
= ( image_b_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_751_insert__image,axiom,
! [X2: b,A2: set_b,F: b > b] :
( ( member_b @ X2 @ A2 )
=> ( ( insert_b @ ( F @ X2 ) @ ( image_b_b @ F @ A2 ) )
= ( image_b_b @ F @ A2 ) ) ) ).
% insert_image
thf(fact_752_insert__image,axiom,
! [X2: b,A2: set_b,F: b > extended_ereal] :
( ( member_b @ X2 @ A2 )
=> ( ( insert8967887681552722334_ereal @ ( F @ X2 ) @ ( image_5319725110001000852_ereal @ F @ A2 ) )
= ( image_5319725110001000852_ereal @ F @ A2 ) ) ) ).
% insert_image
thf(fact_753_insert__image,axiom,
! [X2: extended_ereal,A2: set_Extended_ereal,F: extended_ereal > a] :
( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( ( insert_a @ ( F @ X2 ) @ ( image_3724615099042636213real_a @ F @ A2 ) )
= ( image_3724615099042636213real_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_754_insert__image,axiom,
! [X2: extended_ereal,A2: set_Extended_ereal,F: extended_ereal > b] :
( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( ( insert_b @ ( F @ X2 ) @ ( image_3724615099042636214real_b @ F @ A2 ) )
= ( image_3724615099042636214real_b @ F @ A2 ) ) ) ).
% insert_image
thf(fact_755_insert__image,axiom,
! [X2: extended_ereal,A2: set_Extended_ereal,F: extended_ereal > extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( ( insert8967887681552722334_ereal @ ( F @ X2 ) @ ( image_6042159593519690757_ereal @ F @ A2 ) )
= ( image_6042159593519690757_ereal @ F @ A2 ) ) ) ).
% insert_image
thf(fact_756_insert__image,axiom,
! [X2: list_b,A2: set_list_b,F: list_b > extended_ereal] :
( ( member_list_b @ X2 @ A2 )
=> ( ( insert8967887681552722334_ereal @ ( F @ X2 ) @ ( image_3611896476772571406_ereal @ F @ A2 ) )
= ( image_3611896476772571406_ereal @ F @ A2 ) ) ) ).
% insert_image
thf(fact_757_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_758_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_759_singletonI,axiom,
! [A: pre_pr7278220950009878019t_unit] : ( member6939884229742472986t_unit @ A @ ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) ) ).
% singletonI
thf(fact_760_singletonI,axiom,
! [A: b] : ( member_b @ A @ ( insert_b @ A @ bot_bot_set_b ) ) ).
% singletonI
thf(fact_761_singletonI,axiom,
! [A: extended_ereal] : ( member2350847679896131959_ereal @ A @ ( insert8967887681552722334_ereal @ A @ bot_bo8367695208629047834_ereal ) ) ).
% singletonI
thf(fact_762_finite__insert,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( finite8852549406693098522t_unit @ ( insert6864688055023459379t_unit @ A @ A2 ) )
= ( finite8852549406693098522t_unit @ A2 ) ) ).
% finite_insert
thf(fact_763_finite__insert,axiom,
! [A: list_a,A2: set_list_a] :
( ( finite_finite_list_a @ ( insert_list_a @ A @ A2 ) )
= ( finite_finite_list_a @ A2 ) ) ).
% finite_insert
thf(fact_764_finite__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( finite_finite_set_a @ ( insert_set_a @ A @ A2 ) )
= ( finite_finite_set_a @ A2 ) ) ).
% finite_insert
thf(fact_765_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_766_finite__insert,axiom,
! [A: b,A2: set_b] :
( ( finite_finite_b @ ( insert_b @ A @ A2 ) )
= ( finite_finite_b @ A2 ) ) ).
% finite_insert
thf(fact_767_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_768_finite__insert,axiom,
! [A: list_b,A2: set_list_b] :
( ( finite_finite_list_b @ ( insert_list_b @ A @ A2 ) )
= ( finite_finite_list_b @ A2 ) ) ).
% finite_insert
thf(fact_769_finite__insert,axiom,
! [A: extended_ereal,A2: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ ( insert8967887681552722334_ereal @ A @ A2 ) )
= ( finite7198162374296863863_ereal @ A2 ) ) ).
% finite_insert
thf(fact_770_insert__subset,axiom,
! [X2: set_a,A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X2 @ A2 ) @ B )
= ( ( member_set_a @ X2 @ B )
& ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_771_insert__subset,axiom,
! [X2: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ ( insert6864688055023459379t_unit @ X2 @ A2 ) @ B )
= ( ( member6939884229742472986t_unit @ X2 @ B )
& ( ord_le8200006823705900825t_unit @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_772_insert__subset,axiom,
! [X2: extended_ereal,A2: set_Extended_ereal,B: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ ( insert8967887681552722334_ereal @ X2 @ A2 ) @ B )
= ( ( member2350847679896131959_ereal @ X2 @ B )
& ( ord_le1644982726543182158_ereal @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_773_insert__subset,axiom,
! [X2: a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X2 @ A2 ) @ B )
= ( ( member_a @ X2 @ B )
& ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_774_insert__subset,axiom,
! [X2: b,A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ ( insert_b @ X2 @ A2 ) @ B )
= ( ( member_b @ X2 @ B )
& ( ord_less_eq_set_b @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_775_finite__Int,axiom,
! [F2: set_list_a,G: set_list_a] :
( ( ( finite_finite_list_a @ F2 )
| ( finite_finite_list_a @ G ) )
=> ( finite_finite_list_a @ ( inf_inf_set_list_a @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_776_finite__Int,axiom,
! [F2: set_set_a,G: set_set_a] :
( ( ( finite_finite_set_a @ F2 )
| ( finite_finite_set_a @ G ) )
=> ( finite_finite_set_a @ ( inf_inf_set_set_a @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_777_finite__Int,axiom,
! [F2: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_778_finite__Int,axiom,
! [F2: set_b,G: set_b] :
( ( ( finite_finite_b @ F2 )
| ( finite_finite_b @ G ) )
=> ( finite_finite_b @ ( inf_inf_set_b @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_779_finite__Int,axiom,
! [F2: set_list_b,G: set_list_b] :
( ( ( finite_finite_list_b @ F2 )
| ( finite_finite_list_b @ G ) )
=> ( finite_finite_list_b @ ( inf_inf_set_list_b @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_780_finite__Int,axiom,
! [F2: set_Extended_ereal,G: set_Extended_ereal] :
( ( ( finite7198162374296863863_ereal @ F2 )
| ( finite7198162374296863863_ereal @ G ) )
=> ( finite7198162374296863863_ereal @ ( inf_in2779415704524776092_ereal @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_781_finite__Int,axiom,
! [F2: set_a,G: set_a] :
( ( ( finite_finite_a @ F2 )
| ( finite_finite_a @ G ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_782_Int__subset__iff,axiom,
! [C5: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C5 @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( ord_less_eq_set_a @ C5 @ A2 )
& ( ord_less_eq_set_a @ C5 @ B ) ) ) ).
% Int_subset_iff
thf(fact_783_Int__subset__iff,axiom,
! [C5: set_b,A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ C5 @ ( inf_inf_set_b @ A2 @ B ) )
= ( ( ord_less_eq_set_b @ C5 @ A2 )
& ( ord_less_eq_set_b @ C5 @ B ) ) ) ).
% Int_subset_iff
thf(fact_784_Int__insert__right__if1,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_785_Int__insert__right__if1,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ A @ A2 )
=> ( ( inf_in1092213268631476299t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ B ) )
= ( insert6864688055023459379t_unit @ A @ ( inf_in1092213268631476299t_unit @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_786_Int__insert__right__if1,axiom,
! [A: b,A2: set_b,B: set_b] :
( ( member_b @ A @ A2 )
=> ( ( inf_inf_set_b @ A2 @ ( insert_b @ A @ B ) )
= ( insert_b @ A @ ( inf_inf_set_b @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_787_Int__insert__right__if1,axiom,
! [A: extended_ereal,A2: set_Extended_ereal,B: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ A @ A2 )
=> ( ( inf_in2779415704524776092_ereal @ A2 @ ( insert8967887681552722334_ereal @ A @ B ) )
= ( insert8967887681552722334_ereal @ A @ ( inf_in2779415704524776092_ereal @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_788_Int__insert__right__if1,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_789_Int__insert__right__if0,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_790_Int__insert__right__if0,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ A @ A2 )
=> ( ( inf_in1092213268631476299t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ B ) )
= ( inf_in1092213268631476299t_unit @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_791_Int__insert__right__if0,axiom,
! [A: b,A2: set_b,B: set_b] :
( ~ ( member_b @ A @ A2 )
=> ( ( inf_inf_set_b @ A2 @ ( insert_b @ A @ B ) )
= ( inf_inf_set_b @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_792_Int__insert__right__if0,axiom,
! [A: extended_ereal,A2: set_Extended_ereal,B: set_Extended_ereal] :
( ~ ( member2350847679896131959_ereal @ A @ A2 )
=> ( ( inf_in2779415704524776092_ereal @ A2 @ ( insert8967887681552722334_ereal @ A @ B ) )
= ( inf_in2779415704524776092_ereal @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_793_Int__insert__right__if0,axiom,
! [A: a,A2: set_a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_794_insert__inter__insert,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A @ A2 ) @ ( insert6864688055023459379t_unit @ A @ B ) )
= ( insert6864688055023459379t_unit @ A @ ( inf_in1092213268631476299t_unit @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_795_insert__inter__insert,axiom,
! [A: b,A2: set_b,B: set_b] :
( ( inf_inf_set_b @ ( insert_b @ A @ A2 ) @ ( insert_b @ A @ B ) )
= ( insert_b @ A @ ( inf_inf_set_b @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_796_insert__inter__insert,axiom,
! [A: extended_ereal,A2: set_Extended_ereal,B: set_Extended_ereal] :
( ( inf_in2779415704524776092_ereal @ ( insert8967887681552722334_ereal @ A @ A2 ) @ ( insert8967887681552722334_ereal @ A @ B ) )
= ( insert8967887681552722334_ereal @ A @ ( inf_in2779415704524776092_ereal @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_797_insert__inter__insert,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_798_Int__insert__left__if1,axiom,
! [A: set_a,C5: set_set_a,B: set_set_a] :
( ( member_set_a @ A @ C5 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C5 )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B @ C5 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_799_Int__insert__left__if1,axiom,
! [A: pre_pr7278220950009878019t_unit,C5: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ A @ C5 )
=> ( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A @ B ) @ C5 )
= ( insert6864688055023459379t_unit @ A @ ( inf_in1092213268631476299t_unit @ B @ C5 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_800_Int__insert__left__if1,axiom,
! [A: b,C5: set_b,B: set_b] :
( ( member_b @ A @ C5 )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B ) @ C5 )
= ( insert_b @ A @ ( inf_inf_set_b @ B @ C5 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_801_Int__insert__left__if1,axiom,
! [A: extended_ereal,C5: set_Extended_ereal,B: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ A @ C5 )
=> ( ( inf_in2779415704524776092_ereal @ ( insert8967887681552722334_ereal @ A @ B ) @ C5 )
= ( insert8967887681552722334_ereal @ A @ ( inf_in2779415704524776092_ereal @ B @ C5 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_802_Int__insert__left__if1,axiom,
! [A: a,C5: set_a,B: set_a] :
( ( member_a @ A @ C5 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C5 )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C5 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_803_Int__insert__left__if0,axiom,
! [A: set_a,C5: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ A @ C5 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C5 )
= ( inf_inf_set_set_a @ B @ C5 ) ) ) ).
% Int_insert_left_if0
thf(fact_804_Int__insert__left__if0,axiom,
! [A: pre_pr7278220950009878019t_unit,C5: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ A @ C5 )
=> ( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A @ B ) @ C5 )
= ( inf_in1092213268631476299t_unit @ B @ C5 ) ) ) ).
% Int_insert_left_if0
thf(fact_805_Int__insert__left__if0,axiom,
! [A: b,C5: set_b,B: set_b] :
( ~ ( member_b @ A @ C5 )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B ) @ C5 )
= ( inf_inf_set_b @ B @ C5 ) ) ) ).
% Int_insert_left_if0
thf(fact_806_Int__insert__left__if0,axiom,
! [A: extended_ereal,C5: set_Extended_ereal,B: set_Extended_ereal] :
( ~ ( member2350847679896131959_ereal @ A @ C5 )
=> ( ( inf_in2779415704524776092_ereal @ ( insert8967887681552722334_ereal @ A @ B ) @ C5 )
= ( inf_in2779415704524776092_ereal @ B @ C5 ) ) ) ).
% Int_insert_left_if0
thf(fact_807_Int__insert__left__if0,axiom,
! [A: a,C5: set_a,B: set_a] :
( ~ ( member_a @ A @ C5 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C5 )
= ( inf_inf_set_a @ B @ C5 ) ) ) ).
% Int_insert_left_if0
thf(fact_808_singleton__conv,axiom,
! [A: list_b] :
( ( collect_list_b
@ ^ [X: list_b] : ( X = A ) )
= ( insert_list_b @ A @ bot_bot_set_list_b ) ) ).
% singleton_conv
thf(fact_809_singleton__conv,axiom,
! [A: list_a] :
( ( collect_list_a
@ ^ [X: list_a] : ( X = A ) )
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).
% singleton_conv
thf(fact_810_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X: nat] : ( X = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_811_singleton__conv,axiom,
! [A: set_a] :
( ( collect_set_a
@ ^ [X: set_a] : ( X = A ) )
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singleton_conv
thf(fact_812_singleton__conv,axiom,
! [A: a] :
( ( collect_a
@ ^ [X: a] : ( X = A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_813_singleton__conv,axiom,
! [A: pre_pr7278220950009878019t_unit] :
( ( collec8000012497822511960t_unit
@ ^ [X: pre_pr7278220950009878019t_unit] : ( X = A ) )
= ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) ) ).
% singleton_conv
thf(fact_814_singleton__conv,axiom,
! [A: b] :
( ( collect_b
@ ^ [X: b] : ( X = A ) )
= ( insert_b @ A @ bot_bot_set_b ) ) ).
% singleton_conv
thf(fact_815_singleton__conv,axiom,
! [A: extended_ereal] :
( ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] : ( X = A ) )
= ( insert8967887681552722334_ereal @ A @ bot_bo8367695208629047834_ereal ) ) ).
% singleton_conv
thf(fact_816_singleton__conv2,axiom,
! [A: list_b] :
( ( collect_list_b
@ ( ^ [Y5: list_b,Z2: list_b] : ( Y5 = Z2 )
@ A ) )
= ( insert_list_b @ A @ bot_bot_set_list_b ) ) ).
% singleton_conv2
thf(fact_817_singleton__conv2,axiom,
! [A: list_a] :
( ( collect_list_a
@ ( ^ [Y5: list_a,Z2: list_a] : ( Y5 = Z2 )
@ A ) )
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).
% singleton_conv2
thf(fact_818_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_819_singleton__conv2,axiom,
! [A: set_a] :
( ( collect_set_a
@ ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 )
@ A ) )
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singleton_conv2
thf(fact_820_singleton__conv2,axiom,
! [A: a] :
( ( collect_a
@ ( ^ [Y5: a,Z2: a] : ( Y5 = Z2 )
@ A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_821_singleton__conv2,axiom,
! [A: pre_pr7278220950009878019t_unit] :
( ( collec8000012497822511960t_unit
@ ( ^ [Y5: pre_pr7278220950009878019t_unit,Z2: pre_pr7278220950009878019t_unit] : ( Y5 = Z2 )
@ A ) )
= ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) ) ).
% singleton_conv2
thf(fact_822_singleton__conv2,axiom,
! [A: b] :
( ( collect_b
@ ( ^ [Y5: b,Z2: b] : ( Y5 = Z2 )
@ A ) )
= ( insert_b @ A @ bot_bot_set_b ) ) ).
% singleton_conv2
thf(fact_823_singleton__conv2,axiom,
! [A: extended_ereal] :
( ( collec5835592288176408249_ereal
@ ( ^ [Y5: extended_ereal,Z2: extended_ereal] : ( Y5 = Z2 )
@ A ) )
= ( insert8967887681552722334_ereal @ A @ bot_bo8367695208629047834_ereal ) ) ).
% singleton_conv2
thf(fact_824_induce__subgraph__ends,axiom,
! [G: pre_pr7278220950009878019t_unit,S: set_a] :
( ( arc_to_ends_a_b @ ( digrap7873285959652527175ph_a_b @ G @ S ) )
= ( arc_to_ends_a_b @ G ) ) ).
% induce_subgraph_ends
thf(fact_825_arc__mono__strongly__connected,axiom,
digraph_arc_mono_a_b @ digrap8691851296217657702ed_a_b ).
% arc_mono_strongly_connected
thf(fact_826_singleton__insert__inj__eq_H,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B2: pre_pr7278220950009878019t_unit] :
( ( ( insert6864688055023459379t_unit @ A @ A2 )
= ( insert6864688055023459379t_unit @ B2 @ bot_bo1839476491465656141t_unit ) )
= ( ( A = B2 )
& ( ord_le8200006823705900825t_unit @ A2 @ ( insert6864688055023459379t_unit @ B2 @ bot_bo1839476491465656141t_unit ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_827_singleton__insert__inj__eq_H,axiom,
! [A: extended_ereal,A2: set_Extended_ereal,B2: extended_ereal] :
( ( ( insert8967887681552722334_ereal @ A @ A2 )
= ( insert8967887681552722334_ereal @ B2 @ bot_bo8367695208629047834_ereal ) )
= ( ( A = B2 )
& ( ord_le1644982726543182158_ereal @ A2 @ ( insert8967887681552722334_ereal @ B2 @ bot_bo8367695208629047834_ereal ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_828_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B2: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B2 @ bot_bot_set_a ) )
= ( ( A = B2 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_829_singleton__insert__inj__eq_H,axiom,
! [A: b,A2: set_b,B2: b] :
( ( ( insert_b @ A @ A2 )
= ( insert_b @ B2 @ bot_bot_set_b ) )
= ( ( A = B2 )
& ( ord_less_eq_set_b @ A2 @ ( insert_b @ B2 @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_830_singleton__insert__inj__eq,axiom,
! [B2: pre_pr7278220950009878019t_unit,A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( ( insert6864688055023459379t_unit @ B2 @ bot_bo1839476491465656141t_unit )
= ( insert6864688055023459379t_unit @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le8200006823705900825t_unit @ A2 @ ( insert6864688055023459379t_unit @ B2 @ bot_bo1839476491465656141t_unit ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_831_singleton__insert__inj__eq,axiom,
! [B2: extended_ereal,A: extended_ereal,A2: set_Extended_ereal] :
( ( ( insert8967887681552722334_ereal @ B2 @ bot_bo8367695208629047834_ereal )
= ( insert8967887681552722334_ereal @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le1644982726543182158_ereal @ A2 @ ( insert8967887681552722334_ereal @ B2 @ bot_bo8367695208629047834_ereal ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_832_singleton__insert__inj__eq,axiom,
! [B2: a,A: a,A2: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_833_singleton__insert__inj__eq,axiom,
! [B2: b,A: b,A2: set_b] :
( ( ( insert_b @ B2 @ bot_bot_set_b )
= ( insert_b @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_b @ A2 @ ( insert_b @ B2 @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_834_insert__disjoint_I1_J,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ B )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B )
& ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_835_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_836_insert__disjoint_I1_J,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A @ A2 ) @ B )
= bot_bo1839476491465656141t_unit )
= ( ~ ( member6939884229742472986t_unit @ A @ B )
& ( ( inf_in1092213268631476299t_unit @ A2 @ B )
= bot_bo1839476491465656141t_unit ) ) ) ).
% insert_disjoint(1)
thf(fact_837_insert__disjoint_I1_J,axiom,
! [A: b,A2: set_b,B: set_b] :
( ( ( inf_inf_set_b @ ( insert_b @ A @ A2 ) @ B )
= bot_bot_set_b )
= ( ~ ( member_b @ A @ B )
& ( ( inf_inf_set_b @ A2 @ B )
= bot_bot_set_b ) ) ) ).
% insert_disjoint(1)
thf(fact_838_insert__disjoint_I1_J,axiom,
! [A: extended_ereal,A2: set_Extended_ereal,B: set_Extended_ereal] :
( ( ( inf_in2779415704524776092_ereal @ ( insert8967887681552722334_ereal @ A @ A2 ) @ B )
= bot_bo8367695208629047834_ereal )
= ( ~ ( member2350847679896131959_ereal @ A @ B )
& ( ( inf_in2779415704524776092_ereal @ A2 @ B )
= bot_bo8367695208629047834_ereal ) ) ) ).
% insert_disjoint(1)
thf(fact_839_insert__disjoint_I2_J,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ B ) )
= ( ~ ( member_set_a @ A @ B )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_840_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B ) )
= ( ~ ( member_a @ A @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_841_insert__disjoint_I2_J,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( bot_bo1839476491465656141t_unit
= ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A @ A2 ) @ B ) )
= ( ~ ( member6939884229742472986t_unit @ A @ B )
& ( bot_bo1839476491465656141t_unit
= ( inf_in1092213268631476299t_unit @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_842_insert__disjoint_I2_J,axiom,
! [A: b,A2: set_b,B: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ ( insert_b @ A @ A2 ) @ B ) )
= ( ~ ( member_b @ A @ B )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_843_insert__disjoint_I2_J,axiom,
! [A: extended_ereal,A2: set_Extended_ereal,B: set_Extended_ereal] :
( ( bot_bo8367695208629047834_ereal
= ( inf_in2779415704524776092_ereal @ ( insert8967887681552722334_ereal @ A @ A2 ) @ B ) )
= ( ~ ( member2350847679896131959_ereal @ A @ B )
& ( bot_bo8367695208629047834_ereal
= ( inf_in2779415704524776092_ereal @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_844_disjoint__insert_I1_J,axiom,
! [B: set_set_a,A: set_a,A2: set_set_a] :
( ( ( inf_inf_set_set_a @ B @ ( insert_set_a @ A @ A2 ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B )
& ( ( inf_inf_set_set_a @ B @ A2 )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_845_disjoint__insert_I1_J,axiom,
! [B: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B )
& ( ( inf_inf_set_a @ B @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_846_disjoint__insert_I1_J,axiom,
! [B: set_pr5411798346947241657t_unit,A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( ( inf_in1092213268631476299t_unit @ B @ ( insert6864688055023459379t_unit @ A @ A2 ) )
= bot_bo1839476491465656141t_unit )
= ( ~ ( member6939884229742472986t_unit @ A @ B )
& ( ( inf_in1092213268631476299t_unit @ B @ A2 )
= bot_bo1839476491465656141t_unit ) ) ) ).
% disjoint_insert(1)
thf(fact_847_disjoint__insert_I1_J,axiom,
! [B: set_b,A: b,A2: set_b] :
( ( ( inf_inf_set_b @ B @ ( insert_b @ A @ A2 ) )
= bot_bot_set_b )
= ( ~ ( member_b @ A @ B )
& ( ( inf_inf_set_b @ B @ A2 )
= bot_bot_set_b ) ) ) ).
% disjoint_insert(1)
thf(fact_848_disjoint__insert_I1_J,axiom,
! [B: set_Extended_ereal,A: extended_ereal,A2: set_Extended_ereal] :
( ( ( inf_in2779415704524776092_ereal @ B @ ( insert8967887681552722334_ereal @ A @ A2 ) )
= bot_bo8367695208629047834_ereal )
= ( ~ ( member2350847679896131959_ereal @ A @ B )
& ( ( inf_in2779415704524776092_ereal @ B @ A2 )
= bot_bo8367695208629047834_ereal ) ) ) ).
% disjoint_insert(1)
thf(fact_849_disjoint__insert_I2_J,axiom,
! [A2: set_set_a,B2: set_a,B: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ B2 @ B ) ) )
= ( ~ ( member_set_a @ B2 @ A2 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_850_disjoint__insert_I2_J,axiom,
! [A2: set_a,B2: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B2 @ B ) ) )
= ( ~ ( member_a @ B2 @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_851_disjoint__insert_I2_J,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: pre_pr7278220950009878019t_unit,B: set_pr5411798346947241657t_unit] :
( ( bot_bo1839476491465656141t_unit
= ( inf_in1092213268631476299t_unit @ A2 @ ( insert6864688055023459379t_unit @ B2 @ B ) ) )
= ( ~ ( member6939884229742472986t_unit @ B2 @ A2 )
& ( bot_bo1839476491465656141t_unit
= ( inf_in1092213268631476299t_unit @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_852_disjoint__insert_I2_J,axiom,
! [A2: set_b,B2: b,B: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ A2 @ ( insert_b @ B2 @ B ) ) )
= ( ~ ( member_b @ B2 @ A2 )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_853_disjoint__insert_I2_J,axiom,
! [A2: set_Extended_ereal,B2: extended_ereal,B: set_Extended_ereal] :
( ( bot_bo8367695208629047834_ereal
= ( inf_in2779415704524776092_ereal @ A2 @ ( insert8967887681552722334_ereal @ B2 @ B ) ) )
= ( ~ ( member2350847679896131959_ereal @ B2 @ A2 )
& ( bot_bo8367695208629047834_ereal
= ( inf_in2779415704524776092_ereal @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_854_spanning__tree__imp__connected,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digrap5718416180170401981ee_a_b @ H @ g )
=> ( digrap8783888973171253482ed_a_b @ g ) ) ).
% spanning_tree_imp_connected
thf(fact_855_ereal__complete__Sup,axiom,
! [S: set_Extended_ereal] :
? [X3: extended_ereal] :
( ! [Xa: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa @ S )
=> ( ord_le1083603963089353582_ereal @ Xa @ X3 ) )
& ! [Z3: extended_ereal] :
( ! [Xa2: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa2 @ S )
=> ( ord_le1083603963089353582_ereal @ Xa2 @ Z3 ) )
=> ( ord_le1083603963089353582_ereal @ X3 @ Z3 ) ) ) ).
% ereal_complete_Sup
thf(fact_856_ereal__complete__Inf,axiom,
! [S: set_Extended_ereal] :
? [X3: extended_ereal] :
( ! [Xa: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa @ S )
=> ( ord_le1083603963089353582_ereal @ X3 @ Xa ) )
& ! [Z3: extended_ereal] :
( ! [Xa2: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa2 @ S )
=> ( ord_le1083603963089353582_ereal @ Z3 @ Xa2 ) )
=> ( ord_le1083603963089353582_ereal @ Z3 @ X3 ) ) ) ).
% ereal_complete_Inf
thf(fact_857_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B5: set_a] :
( ( A2
= ( insert_a @ A @ B5 ) )
& ~ ( member_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_858_mk__disjoint__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ? [B5: set_set_a] :
( ( A2
= ( insert_set_a @ A @ B5 ) )
& ~ ( member_set_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_859_mk__disjoint__insert,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ A @ A2 )
=> ? [B5: set_pr5411798346947241657t_unit] :
( ( A2
= ( insert6864688055023459379t_unit @ A @ B5 ) )
& ~ ( member6939884229742472986t_unit @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_860_mk__disjoint__insert,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ? [B5: set_b] :
( ( A2
= ( insert_b @ A @ B5 ) )
& ~ ( member_b @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_861_mk__disjoint__insert,axiom,
! [A: extended_ereal,A2: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ A @ A2 )
=> ? [B5: set_Extended_ereal] :
( ( A2
= ( insert8967887681552722334_ereal @ A @ B5 ) )
& ~ ( member2350847679896131959_ereal @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_862_Int__left__commute,axiom,
! [A2: set_a,B: set_a,C5: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C5 ) )
= ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A2 @ C5 ) ) ) ).
% Int_left_commute
thf(fact_863_Int__insert__right,axiom,
! [A: set_a,A2: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B ) ) ) )
& ( ~ ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( inf_inf_set_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_864_Int__insert__right,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( ( member6939884229742472986t_unit @ A @ A2 )
=> ( ( inf_in1092213268631476299t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ B ) )
= ( insert6864688055023459379t_unit @ A @ ( inf_in1092213268631476299t_unit @ A2 @ B ) ) ) )
& ( ~ ( member6939884229742472986t_unit @ A @ A2 )
=> ( ( inf_in1092213268631476299t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ B ) )
= ( inf_in1092213268631476299t_unit @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_865_Int__insert__right,axiom,
! [A: b,A2: set_b,B: set_b] :
( ( ( member_b @ A @ A2 )
=> ( ( inf_inf_set_b @ A2 @ ( insert_b @ A @ B ) )
= ( insert_b @ A @ ( inf_inf_set_b @ A2 @ B ) ) ) )
& ( ~ ( member_b @ A @ A2 )
=> ( ( inf_inf_set_b @ A2 @ ( insert_b @ A @ B ) )
= ( inf_inf_set_b @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_866_Int__insert__right,axiom,
! [A: extended_ereal,A2: set_Extended_ereal,B: set_Extended_ereal] :
( ( ( member2350847679896131959_ereal @ A @ A2 )
=> ( ( inf_in2779415704524776092_ereal @ A2 @ ( insert8967887681552722334_ereal @ A @ B ) )
= ( insert8967887681552722334_ereal @ A @ ( inf_in2779415704524776092_ereal @ A2 @ B ) ) ) )
& ( ~ ( member2350847679896131959_ereal @ A @ A2 )
=> ( ( inf_in2779415704524776092_ereal @ A2 @ ( insert8967887681552722334_ereal @ A @ B ) )
= ( inf_in2779415704524776092_ereal @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_867_Int__insert__right,axiom,
! [A: a,A2: set_a,B: set_a] :
( ( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_868_Int__left__absorb,axiom,
! [A2: set_a,B: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( inf_inf_set_a @ A2 @ B ) ) ).
% Int_left_absorb
thf(fact_869_Int__insert__left,axiom,
! [A: set_a,C5: set_set_a,B: set_set_a] :
( ( ( member_set_a @ A @ C5 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C5 )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B @ C5 ) ) ) )
& ( ~ ( member_set_a @ A @ C5 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B ) @ C5 )
= ( inf_inf_set_set_a @ B @ C5 ) ) ) ) ).
% Int_insert_left
thf(fact_870_Int__insert__left,axiom,
! [A: pre_pr7278220950009878019t_unit,C5: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( ( member6939884229742472986t_unit @ A @ C5 )
=> ( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A @ B ) @ C5 )
= ( insert6864688055023459379t_unit @ A @ ( inf_in1092213268631476299t_unit @ B @ C5 ) ) ) )
& ( ~ ( member6939884229742472986t_unit @ A @ C5 )
=> ( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A @ B ) @ C5 )
= ( inf_in1092213268631476299t_unit @ B @ C5 ) ) ) ) ).
% Int_insert_left
thf(fact_871_Int__insert__left,axiom,
! [A: b,C5: set_b,B: set_b] :
( ( ( member_b @ A @ C5 )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B ) @ C5 )
= ( insert_b @ A @ ( inf_inf_set_b @ B @ C5 ) ) ) )
& ( ~ ( member_b @ A @ C5 )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B ) @ C5 )
= ( inf_inf_set_b @ B @ C5 ) ) ) ) ).
% Int_insert_left
thf(fact_872_Int__insert__left,axiom,
! [A: extended_ereal,C5: set_Extended_ereal,B: set_Extended_ereal] :
( ( ( member2350847679896131959_ereal @ A @ C5 )
=> ( ( inf_in2779415704524776092_ereal @ ( insert8967887681552722334_ereal @ A @ B ) @ C5 )
= ( insert8967887681552722334_ereal @ A @ ( inf_in2779415704524776092_ereal @ B @ C5 ) ) ) )
& ( ~ ( member2350847679896131959_ereal @ A @ C5 )
=> ( ( inf_in2779415704524776092_ereal @ ( insert8967887681552722334_ereal @ A @ B ) @ C5 )
= ( inf_in2779415704524776092_ereal @ B @ C5 ) ) ) ) ).
% Int_insert_left
thf(fact_873_Int__insert__left,axiom,
! [A: a,C5: set_a,B: set_a] :
( ( ( member_a @ A @ C5 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C5 )
= ( insert_a @ A @ ( inf_inf_set_a @ B @ C5 ) ) ) )
& ( ~ ( member_a @ A @ C5 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B ) @ C5 )
= ( inf_inf_set_a @ B @ C5 ) ) ) ) ).
% Int_insert_left
thf(fact_874_Collect__conj__eq,axiom,
! [P: extended_ereal > $o,Q: extended_ereal > $o] :
( ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_in2779415704524776092_ereal @ ( collec5835592288176408249_ereal @ P ) @ ( collec5835592288176408249_ereal @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_875_Collect__conj__eq,axiom,
! [P: list_b > $o,Q: list_b > $o] :
( ( collect_list_b
@ ^ [X: list_b] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_list_b @ ( collect_list_b @ P ) @ ( collect_list_b @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_876_Collect__conj__eq,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( collect_list_a
@ ^ [X: list_a] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_877_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_878_Collect__conj__eq,axiom,
! [P: b > $o,Q: b > $o] :
( ( collect_b
@ ^ [X: b] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_879_Collect__conj__eq,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( collect_set_a
@ ^ [X: set_a] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_880_Collect__conj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_881_insert__commute,axiom,
! [X2: pre_pr7278220950009878019t_unit,Y3: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( insert6864688055023459379t_unit @ X2 @ ( insert6864688055023459379t_unit @ Y3 @ A2 ) )
= ( insert6864688055023459379t_unit @ Y3 @ ( insert6864688055023459379t_unit @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_882_insert__commute,axiom,
! [X2: a,Y3: a,A2: set_a] :
( ( insert_a @ X2 @ ( insert_a @ Y3 @ A2 ) )
= ( insert_a @ Y3 @ ( insert_a @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_883_insert__commute,axiom,
! [X2: b,Y3: b,A2: set_b] :
( ( insert_b @ X2 @ ( insert_b @ Y3 @ A2 ) )
= ( insert_b @ Y3 @ ( insert_b @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_884_insert__commute,axiom,
! [X2: extended_ereal,Y3: extended_ereal,A2: set_Extended_ereal] :
( ( insert8967887681552722334_ereal @ X2 @ ( insert8967887681552722334_ereal @ Y3 @ A2 ) )
= ( insert8967887681552722334_ereal @ Y3 @ ( insert8967887681552722334_ereal @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_885_insert__Collect,axiom,
! [A: pre_pr7278220950009878019t_unit,P: pre_pr7278220950009878019t_unit > $o] :
( ( insert6864688055023459379t_unit @ A @ ( collec8000012497822511960t_unit @ P ) )
= ( collec8000012497822511960t_unit
@ ^ [U: pre_pr7278220950009878019t_unit] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_886_insert__Collect,axiom,
! [A: extended_ereal,P: extended_ereal > $o] :
( ( insert8967887681552722334_ereal @ A @ ( collec5835592288176408249_ereal @ P ) )
= ( collec5835592288176408249_ereal
@ ^ [U: extended_ereal] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_887_insert__Collect,axiom,
! [A: list_b,P: list_b > $o] :
( ( insert_list_b @ A @ ( collect_list_b @ P ) )
= ( collect_list_b
@ ^ [U: list_b] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_888_insert__Collect,axiom,
! [A: list_a,P: list_a > $o] :
( ( insert_list_a @ A @ ( collect_list_a @ P ) )
= ( collect_list_a
@ ^ [U: list_a] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_889_insert__Collect,axiom,
! [A: nat,P: nat > $o] :
( ( insert_nat @ A @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U: nat] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_890_insert__Collect,axiom,
! [A: a,P: a > $o] :
( ( insert_a @ A @ ( collect_a @ P ) )
= ( collect_a
@ ^ [U: a] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_891_insert__Collect,axiom,
! [A: b,P: b > $o] :
( ( insert_b @ A @ ( collect_b @ P ) )
= ( collect_b
@ ^ [U: b] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_892_insert__Collect,axiom,
! [A: set_a,P: set_a > $o] :
( ( insert_set_a @ A @ ( collect_set_a @ P ) )
= ( collect_set_a
@ ^ [U: set_a] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_893_insert__eq__iff,axiom,
! [A: a,A2: set_a,B2: a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B2 @ B )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C6: set_a] :
( ( A2
= ( insert_a @ B2 @ C6 ) )
& ~ ( member_a @ B2 @ C6 )
& ( B
= ( insert_a @ A @ C6 ) )
& ~ ( member_a @ A @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_894_insert__eq__iff,axiom,
! [A: set_a,A2: set_set_a,B2: set_a,B: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ~ ( member_set_a @ B2 @ B )
=> ( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C6: set_set_a] :
( ( A2
= ( insert_set_a @ B2 @ C6 ) )
& ~ ( member_set_a @ B2 @ C6 )
& ( B
= ( insert_set_a @ A @ C6 ) )
& ~ ( member_set_a @ A @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_895_insert__eq__iff,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B2: pre_pr7278220950009878019t_unit,B: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ A @ A2 )
=> ( ~ ( member6939884229742472986t_unit @ B2 @ B )
=> ( ( ( insert6864688055023459379t_unit @ A @ A2 )
= ( insert6864688055023459379t_unit @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C6: set_pr5411798346947241657t_unit] :
( ( A2
= ( insert6864688055023459379t_unit @ B2 @ C6 ) )
& ~ ( member6939884229742472986t_unit @ B2 @ C6 )
& ( B
= ( insert6864688055023459379t_unit @ A @ C6 ) )
& ~ ( member6939884229742472986t_unit @ A @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_896_insert__eq__iff,axiom,
! [A: b,A2: set_b,B2: b,B: set_b] :
( ~ ( member_b @ A @ A2 )
=> ( ~ ( member_b @ B2 @ B )
=> ( ( ( insert_b @ A @ A2 )
= ( insert_b @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C6: set_b] :
( ( A2
= ( insert_b @ B2 @ C6 ) )
& ~ ( member_b @ B2 @ C6 )
& ( B
= ( insert_b @ A @ C6 ) )
& ~ ( member_b @ A @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_897_insert__eq__iff,axiom,
! [A: extended_ereal,A2: set_Extended_ereal,B2: extended_ereal,B: set_Extended_ereal] :
( ~ ( member2350847679896131959_ereal @ A @ A2 )
=> ( ~ ( member2350847679896131959_ereal @ B2 @ B )
=> ( ( ( insert8967887681552722334_ereal @ A @ A2 )
= ( insert8967887681552722334_ereal @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C6: set_Extended_ereal] :
( ( A2
= ( insert8967887681552722334_ereal @ B2 @ C6 ) )
& ~ ( member2350847679896131959_ereal @ B2 @ C6 )
& ( B
= ( insert8967887681552722334_ereal @ A @ C6 ) )
& ~ ( member2350847679896131959_ereal @ A @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_898_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_899_insert__absorb,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( insert_set_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_900_insert__absorb,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ A @ A2 )
=> ( ( insert6864688055023459379t_unit @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_901_insert__absorb,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ( ( insert_b @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_902_insert__absorb,axiom,
! [A: extended_ereal,A2: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ A @ A2 )
=> ( ( insert8967887681552722334_ereal @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_903_insert__ident,axiom,
! [X2: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X2 @ A2 )
=> ( ~ ( member_a @ X2 @ B )
=> ( ( ( insert_a @ X2 @ A2 )
= ( insert_a @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_904_insert__ident,axiom,
! [X2: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X2 @ A2 )
=> ( ~ ( member_set_a @ X2 @ B )
=> ( ( ( insert_set_a @ X2 @ A2 )
= ( insert_set_a @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_905_insert__ident,axiom,
! [X2: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ X2 @ A2 )
=> ( ~ ( member6939884229742472986t_unit @ X2 @ B )
=> ( ( ( insert6864688055023459379t_unit @ X2 @ A2 )
= ( insert6864688055023459379t_unit @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_906_insert__ident,axiom,
! [X2: b,A2: set_b,B: set_b] :
( ~ ( member_b @ X2 @ A2 )
=> ( ~ ( member_b @ X2 @ B )
=> ( ( ( insert_b @ X2 @ A2 )
= ( insert_b @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_907_insert__ident,axiom,
! [X2: extended_ereal,A2: set_Extended_ereal,B: set_Extended_ereal] :
( ~ ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( ~ ( member2350847679896131959_ereal @ X2 @ B )
=> ( ( ( insert8967887681552722334_ereal @ X2 @ A2 )
= ( insert8967887681552722334_ereal @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_908_insert__compr,axiom,
( insert6864688055023459379t_unit
= ( ^ [A3: pre_pr7278220950009878019t_unit,B3: set_pr5411798346947241657t_unit] :
( collec8000012497822511960t_unit
@ ^ [X: pre_pr7278220950009878019t_unit] :
( ( X = A3 )
| ( member6939884229742472986t_unit @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_909_insert__compr,axiom,
( insert8967887681552722334_ereal
= ( ^ [A3: extended_ereal,B3: set_Extended_ereal] :
( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( X = A3 )
| ( member2350847679896131959_ereal @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_910_insert__compr,axiom,
( insert_list_b
= ( ^ [A3: list_b,B3: set_list_b] :
( collect_list_b
@ ^ [X: list_b] :
( ( X = A3 )
| ( member_list_b @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_911_insert__compr,axiom,
( insert_list_a
= ( ^ [A3: list_a,B3: set_list_a] :
( collect_list_a
@ ^ [X: list_a] :
( ( X = A3 )
| ( member_list_a @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_912_insert__compr,axiom,
( insert_nat
= ( ^ [A3: nat,B3: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( X = A3 )
| ( member_nat @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_913_insert__compr,axiom,
( insert_a
= ( ^ [A3: a,B3: set_a] :
( collect_a
@ ^ [X: a] :
( ( X = A3 )
| ( member_a @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_914_insert__compr,axiom,
( insert_b
= ( ^ [A3: b,B3: set_b] :
( collect_b
@ ^ [X: b] :
( ( X = A3 )
| ( member_b @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_915_insert__compr,axiom,
( insert_set_a
= ( ^ [A3: set_a,B3: set_set_a] :
( collect_set_a
@ ^ [X: set_a] :
( ( X = A3 )
| ( member_set_a @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_916_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A4 ) ) ) ).
% Int_commute
thf(fact_917_Int__Collect,axiom,
! [X2: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,P: pre_pr7278220950009878019t_unit > $o] :
( ( member6939884229742472986t_unit @ X2 @ ( inf_in1092213268631476299t_unit @ A2 @ ( collec8000012497822511960t_unit @ P ) ) )
= ( ( member6939884229742472986t_unit @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_918_Int__Collect,axiom,
! [X2: extended_ereal,A2: set_Extended_ereal,P: extended_ereal > $o] :
( ( member2350847679896131959_ereal @ X2 @ ( inf_in2779415704524776092_ereal @ A2 @ ( collec5835592288176408249_ereal @ P ) ) )
= ( ( member2350847679896131959_ereal @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_919_Int__Collect,axiom,
! [X2: list_b,A2: set_list_b,P: list_b > $o] :
( ( member_list_b @ X2 @ ( inf_inf_set_list_b @ A2 @ ( collect_list_b @ P ) ) )
= ( ( member_list_b @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_920_Int__Collect,axiom,
! [X2: list_a,A2: set_list_a,P: list_a > $o] :
( ( member_list_a @ X2 @ ( inf_inf_set_list_a @ A2 @ ( collect_list_a @ P ) ) )
= ( ( member_list_a @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_921_Int__Collect,axiom,
! [X2: nat,A2: set_nat,P: nat > $o] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_922_Int__Collect,axiom,
! [X2: b,A2: set_b,P: b > $o] :
( ( member_b @ X2 @ ( inf_inf_set_b @ A2 @ ( collect_b @ P ) ) )
= ( ( member_b @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_923_Int__Collect,axiom,
! [X2: set_a,A2: set_set_a,P: set_a > $o] :
( ( member_set_a @ X2 @ ( inf_inf_set_set_a @ A2 @ ( collect_set_a @ P ) ) )
= ( ( member_set_a @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_924_Int__Collect,axiom,
! [X2: a,A2: set_a,P: a > $o] :
( ( member_a @ X2 @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) )
= ( ( member_a @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_925_Set_Oset__insert,axiom,
! [X2: a,A2: set_a] :
( ( member_a @ X2 @ A2 )
=> ~ ! [B5: set_a] :
( ( A2
= ( insert_a @ X2 @ B5 ) )
=> ( member_a @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_926_Set_Oset__insert,axiom,
! [X2: set_a,A2: set_set_a] :
( ( member_set_a @ X2 @ A2 )
=> ~ ! [B5: set_set_a] :
( ( A2
= ( insert_set_a @ X2 @ B5 ) )
=> ( member_set_a @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_927_Set_Oset__insert,axiom,
! [X2: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ X2 @ A2 )
=> ~ ! [B5: set_pr5411798346947241657t_unit] :
( ( A2
= ( insert6864688055023459379t_unit @ X2 @ B5 ) )
=> ( member6939884229742472986t_unit @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_928_Set_Oset__insert,axiom,
! [X2: b,A2: set_b] :
( ( member_b @ X2 @ A2 )
=> ~ ! [B5: set_b] :
( ( A2
= ( insert_b @ X2 @ B5 ) )
=> ( member_b @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_929_Set_Oset__insert,axiom,
! [X2: extended_ereal,A2: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ X2 @ A2 )
=> ~ ! [B5: set_Extended_ereal] :
( ( A2
= ( insert8967887681552722334_ereal @ X2 @ B5 ) )
=> ( member2350847679896131959_ereal @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_930_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_931_Int__assoc,axiom,
! [A2: set_a,B: set_a,C5: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B ) @ C5 )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B @ C5 ) ) ) ).
% Int_assoc
thf(fact_932_insertI2,axiom,
! [A: a,B: set_a,B2: a] :
( ( member_a @ A @ B )
=> ( member_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_933_insertI2,axiom,
! [A: set_a,B: set_set_a,B2: set_a] :
( ( member_set_a @ A @ B )
=> ( member_set_a @ A @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_934_insertI2,axiom,
! [A: pre_pr7278220950009878019t_unit,B: set_pr5411798346947241657t_unit,B2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ A @ B )
=> ( member6939884229742472986t_unit @ A @ ( insert6864688055023459379t_unit @ B2 @ B ) ) ) ).
% insertI2
thf(fact_935_insertI2,axiom,
! [A: b,B: set_b,B2: b] :
( ( member_b @ A @ B )
=> ( member_b @ A @ ( insert_b @ B2 @ B ) ) ) ).
% insertI2
thf(fact_936_insertI2,axiom,
! [A: extended_ereal,B: set_Extended_ereal,B2: extended_ereal] :
( ( member2350847679896131959_ereal @ A @ B )
=> ( member2350847679896131959_ereal @ A @ ( insert8967887681552722334_ereal @ B2 @ B ) ) ) ).
% insertI2
thf(fact_937_insertI1,axiom,
! [A: a,B: set_a] : ( member_a @ A @ ( insert_a @ A @ B ) ) ).
% insertI1
thf(fact_938_insertI1,axiom,
! [A: set_a,B: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B ) ) ).
% insertI1
thf(fact_939_insertI1,axiom,
! [A: pre_pr7278220950009878019t_unit,B: set_pr5411798346947241657t_unit] : ( member6939884229742472986t_unit @ A @ ( insert6864688055023459379t_unit @ A @ B ) ) ).
% insertI1
thf(fact_940_insertI1,axiom,
! [A: b,B: set_b] : ( member_b @ A @ ( insert_b @ A @ B ) ) ).
% insertI1
thf(fact_941_insertI1,axiom,
! [A: extended_ereal,B: set_Extended_ereal] : ( member2350847679896131959_ereal @ A @ ( insert8967887681552722334_ereal @ A @ B ) ) ).
% insertI1
thf(fact_942_insertE,axiom,
! [A: a,B2: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_943_insertE,axiom,
! [A: set_a,B2: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_set_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_944_insertE,axiom,
! [A: pre_pr7278220950009878019t_unit,B2: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ A @ ( insert6864688055023459379t_unit @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member6939884229742472986t_unit @ A @ A2 ) ) ) ).
% insertE
thf(fact_945_insertE,axiom,
! [A: b,B2: b,A2: set_b] :
( ( member_b @ A @ ( insert_b @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_b @ A @ A2 ) ) ) ).
% insertE
thf(fact_946_insertE,axiom,
! [A: extended_ereal,B2: extended_ereal,A2: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ A @ ( insert8967887681552722334_ereal @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member2350847679896131959_ereal @ A @ A2 ) ) ) ).
% insertE
thf(fact_947_Int__def,axiom,
( inf_in1092213268631476299t_unit
= ( ^ [A4: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( collec8000012497822511960t_unit
@ ^ [X: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X @ A4 )
& ( member6939884229742472986t_unit @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_948_Int__def,axiom,
( inf_in2779415704524776092_ereal
= ( ^ [A4: set_Extended_ereal,B3: set_Extended_ereal] :
( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A4 )
& ( member2350847679896131959_ereal @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_949_Int__def,axiom,
( inf_inf_set_list_b
= ( ^ [A4: set_list_b,B3: set_list_b] :
( collect_list_b
@ ^ [X: list_b] :
( ( member_list_b @ X @ A4 )
& ( member_list_b @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_950_Int__def,axiom,
( inf_inf_set_list_a
= ( ^ [A4: set_list_a,B3: set_list_a] :
( collect_list_a
@ ^ [X: list_a] :
( ( member_list_a @ X @ A4 )
& ( member_list_a @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_951_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A4 )
& ( member_nat @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_952_Int__def,axiom,
( inf_inf_set_b
= ( ^ [A4: set_b,B3: set_b] :
( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A4 )
& ( member_b @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_953_Int__def,axiom,
( inf_inf_set_set_a
= ( ^ [A4: set_set_a,B3: set_set_a] :
( collect_set_a
@ ^ [X: set_a] :
( ( member_set_a @ X @ A4 )
& ( member_set_a @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_954_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B3: set_a] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A4 )
& ( member_a @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_955_IntD2,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ( member_set_a @ C2 @ B ) ) ).
% IntD2
thf(fact_956_IntD2,axiom,
! [C2: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( inf_in1092213268631476299t_unit @ A2 @ B ) )
=> ( member6939884229742472986t_unit @ C2 @ B ) ) ).
% IntD2
thf(fact_957_IntD2,axiom,
! [C2: b,A2: set_b,B: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A2 @ B ) )
=> ( member_b @ C2 @ B ) ) ).
% IntD2
thf(fact_958_IntD2,axiom,
! [C2: extended_ereal,A2: set_Extended_ereal,B: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ C2 @ ( inf_in2779415704524776092_ereal @ A2 @ B ) )
=> ( member2350847679896131959_ereal @ C2 @ B ) ) ).
% IntD2
thf(fact_959_IntD2,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C2 @ B ) ) ).
% IntD2
thf(fact_960_IntD1,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ( member_set_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_961_IntD1,axiom,
! [C2: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( inf_in1092213268631476299t_unit @ A2 @ B ) )
=> ( member6939884229742472986t_unit @ C2 @ A2 ) ) ).
% IntD1
thf(fact_962_IntD1,axiom,
! [C2: b,A2: set_b,B: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A2 @ B ) )
=> ( member_b @ C2 @ A2 ) ) ).
% IntD1
thf(fact_963_IntD1,axiom,
! [C2: extended_ereal,A2: set_Extended_ereal,B: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ C2 @ ( inf_in2779415704524776092_ereal @ A2 @ B ) )
=> ( member2350847679896131959_ereal @ C2 @ A2 ) ) ).
% IntD1
thf(fact_964_IntD1,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ( member_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_965_IntE,axiom,
! [C2: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B ) )
=> ~ ( ( member_set_a @ C2 @ A2 )
=> ~ ( member_set_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_966_IntE,axiom,
! [C2: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( inf_in1092213268631476299t_unit @ A2 @ B ) )
=> ~ ( ( member6939884229742472986t_unit @ C2 @ A2 )
=> ~ ( member6939884229742472986t_unit @ C2 @ B ) ) ) ).
% IntE
thf(fact_967_IntE,axiom,
! [C2: b,A2: set_b,B: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A2 @ B ) )
=> ~ ( ( member_b @ C2 @ A2 )
=> ~ ( member_b @ C2 @ B ) ) ) ).
% IntE
thf(fact_968_IntE,axiom,
! [C2: extended_ereal,A2: set_Extended_ereal,B: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ C2 @ ( inf_in2779415704524776092_ereal @ A2 @ B ) )
=> ~ ( ( member2350847679896131959_ereal @ C2 @ A2 )
=> ~ ( member2350847679896131959_ereal @ C2 @ B ) ) ) ).
% IntE
thf(fact_969_IntE,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C2 @ A2 )
=> ~ ( member_a @ C2 @ B ) ) ) ).
% IntE
thf(fact_970_wf__digraph_Ofin__diameter_Ocong,axiom,
graph_1932031826008834157er_a_b = graph_1932031826008834157er_a_b ).
% wf_digraph.fin_diameter.cong
thf(fact_971_wf__digraph_Odiameter_Ocong,axiom,
graph_926353876199057498er_a_b = graph_926353876199057498er_a_b ).
% wf_digraph.diameter.cong
thf(fact_972_Int__emptyI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ~ ( member_set_a @ X3 @ B ) )
=> ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_973_Int__emptyI,axiom,
! [A2: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_974_Int__emptyI,axiom,
! [A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ! [X3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ A2 )
=> ~ ( member6939884229742472986t_unit @ X3 @ B ) )
=> ( ( inf_in1092213268631476299t_unit @ A2 @ B )
= bot_bo1839476491465656141t_unit ) ) ).
% Int_emptyI
thf(fact_975_Int__emptyI,axiom,
! [A2: set_b,B: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ~ ( member_b @ X3 @ B ) )
=> ( ( inf_inf_set_b @ A2 @ B )
= bot_bot_set_b ) ) ).
% Int_emptyI
thf(fact_976_Int__emptyI,axiom,
! [A2: set_Extended_ereal,B: set_Extended_ereal] :
( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A2 )
=> ~ ( member2350847679896131959_ereal @ X3 @ B ) )
=> ( ( inf_in2779415704524776092_ereal @ A2 @ B )
= bot_bo8367695208629047834_ereal ) ) ).
% Int_emptyI
thf(fact_977_disjoint__iff,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B )
= bot_bot_set_set_a )
= ( ! [X: set_a] :
( ( member_set_a @ X @ A2 )
=> ~ ( member_set_a @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_978_disjoint__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ~ ( member_a @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_979_disjoint__iff,axiom,
! [A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( ( inf_in1092213268631476299t_unit @ A2 @ B )
= bot_bo1839476491465656141t_unit )
= ( ! [X: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X @ A2 )
=> ~ ( member6939884229742472986t_unit @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_980_disjoint__iff,axiom,
! [A2: set_b,B: set_b] :
( ( ( inf_inf_set_b @ A2 @ B )
= bot_bot_set_b )
= ( ! [X: b] :
( ( member_b @ X @ A2 )
=> ~ ( member_b @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_981_disjoint__iff,axiom,
! [A2: set_Extended_ereal,B: set_Extended_ereal] :
( ( ( inf_in2779415704524776092_ereal @ A2 @ B )
= bot_bo8367695208629047834_ereal )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A2 )
=> ~ ( member2350847679896131959_ereal @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_982_Int__empty__left,axiom,
! [B: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_983_Int__empty__left,axiom,
! [B: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ bot_bo1839476491465656141t_unit @ B )
= bot_bo1839476491465656141t_unit ) ).
% Int_empty_left
thf(fact_984_Int__empty__left,axiom,
! [B: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ B )
= bot_bot_set_b ) ).
% Int_empty_left
thf(fact_985_Int__empty__left,axiom,
! [B: set_Extended_ereal] :
( ( inf_in2779415704524776092_ereal @ bot_bo8367695208629047834_ereal @ B )
= bot_bo8367695208629047834_ereal ) ).
% Int_empty_left
thf(fact_986_Int__empty__right,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_987_Int__empty__right,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ A2 @ bot_bo1839476491465656141t_unit )
= bot_bo1839476491465656141t_unit ) ).
% Int_empty_right
thf(fact_988_Int__empty__right,axiom,
! [A2: set_b] :
( ( inf_inf_set_b @ A2 @ bot_bot_set_b )
= bot_bot_set_b ) ).
% Int_empty_right
thf(fact_989_Int__empty__right,axiom,
! [A2: set_Extended_ereal] :
( ( inf_in2779415704524776092_ereal @ A2 @ bot_bo8367695208629047834_ereal )
= bot_bo8367695208629047834_ereal ) ).
% Int_empty_right
thf(fact_990_disjoint__iff__not__equal,axiom,
! [A2: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ! [Y: a] :
( ( member_a @ Y @ B )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_991_disjoint__iff__not__equal,axiom,
! [A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( ( inf_in1092213268631476299t_unit @ A2 @ B )
= bot_bo1839476491465656141t_unit )
= ( ! [X: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X @ A2 )
=> ! [Y: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ Y @ B )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_992_disjoint__iff__not__equal,axiom,
! [A2: set_b,B: set_b] :
( ( ( inf_inf_set_b @ A2 @ B )
= bot_bot_set_b )
= ( ! [X: b] :
( ( member_b @ X @ A2 )
=> ! [Y: b] :
( ( member_b @ Y @ B )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_993_disjoint__iff__not__equal,axiom,
! [A2: set_Extended_ereal,B: set_Extended_ereal] :
( ( ( inf_in2779415704524776092_ereal @ A2 @ B )
= bot_bo8367695208629047834_ereal )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ A2 )
=> ! [Y: extended_ereal] :
( ( member2350847679896131959_ereal @ Y @ B )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_994_singletonD,axiom,
! [B2: set_a,A: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_995_singletonD,axiom,
! [B2: a,A: a] :
( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_996_singletonD,axiom,
! [B2: pre_pr7278220950009878019t_unit,A: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ B2 @ ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_997_singletonD,axiom,
! [B2: b,A: b] :
( ( member_b @ B2 @ ( insert_b @ A @ bot_bot_set_b ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_998_singletonD,axiom,
! [B2: extended_ereal,A: extended_ereal] :
( ( member2350847679896131959_ereal @ B2 @ ( insert8967887681552722334_ereal @ A @ bot_bo8367695208629047834_ereal ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_999_singleton__iff,axiom,
! [B2: set_a,A: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_1000_singleton__iff,axiom,
! [B2: a,A: a] :
( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_1001_singleton__iff,axiom,
! [B2: pre_pr7278220950009878019t_unit,A: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ B2 @ ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_1002_singleton__iff,axiom,
! [B2: b,A: b] :
( ( member_b @ B2 @ ( insert_b @ A @ bot_bot_set_b ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_1003_singleton__iff,axiom,
! [B2: extended_ereal,A: extended_ereal] :
( ( member2350847679896131959_ereal @ B2 @ ( insert8967887681552722334_ereal @ A @ bot_bo8367695208629047834_ereal ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_1004_doubleton__eq__iff,axiom,
! [A: a,B2: a,C2: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) )
= ( insert_a @ C2 @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C2 )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1005_doubleton__eq__iff,axiom,
! [A: pre_pr7278220950009878019t_unit,B2: pre_pr7278220950009878019t_unit,C2: pre_pr7278220950009878019t_unit,D: pre_pr7278220950009878019t_unit] :
( ( ( insert6864688055023459379t_unit @ A @ ( insert6864688055023459379t_unit @ B2 @ bot_bo1839476491465656141t_unit ) )
= ( insert6864688055023459379t_unit @ C2 @ ( insert6864688055023459379t_unit @ D @ bot_bo1839476491465656141t_unit ) ) )
= ( ( ( A = C2 )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1006_doubleton__eq__iff,axiom,
! [A: b,B2: b,C2: b,D: b] :
( ( ( insert_b @ A @ ( insert_b @ B2 @ bot_bot_set_b ) )
= ( insert_b @ C2 @ ( insert_b @ D @ bot_bot_set_b ) ) )
= ( ( ( A = C2 )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1007_doubleton__eq__iff,axiom,
! [A: extended_ereal,B2: extended_ereal,C2: extended_ereal,D: extended_ereal] :
( ( ( insert8967887681552722334_ereal @ A @ ( insert8967887681552722334_ereal @ B2 @ bot_bo8367695208629047834_ereal ) )
= ( insert8967887681552722334_ereal @ C2 @ ( insert8967887681552722334_ereal @ D @ bot_bo8367695208629047834_ereal ) ) )
= ( ( ( A = C2 )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1008_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_1009_insert__not__empty,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( insert6864688055023459379t_unit @ A @ A2 )
!= bot_bo1839476491465656141t_unit ) ).
% insert_not_empty
thf(fact_1010_insert__not__empty,axiom,
! [A: b,A2: set_b] :
( ( insert_b @ A @ A2 )
!= bot_bot_set_b ) ).
% insert_not_empty
thf(fact_1011_insert__not__empty,axiom,
! [A: extended_ereal,A2: set_Extended_ereal] :
( ( insert8967887681552722334_ereal @ A @ A2 )
!= bot_bo8367695208629047834_ereal ) ).
% insert_not_empty
thf(fact_1012_singleton__inject,axiom,
! [A: a,B2: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B2 @ bot_bot_set_a ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_1013_singleton__inject,axiom,
! [A: pre_pr7278220950009878019t_unit,B2: pre_pr7278220950009878019t_unit] :
( ( ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit )
= ( insert6864688055023459379t_unit @ B2 @ bot_bo1839476491465656141t_unit ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_1014_singleton__inject,axiom,
! [A: b,B2: b] :
( ( ( insert_b @ A @ bot_bot_set_b )
= ( insert_b @ B2 @ bot_bot_set_b ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_1015_singleton__inject,axiom,
! [A: extended_ereal,B2: extended_ereal] :
( ( ( insert8967887681552722334_ereal @ A @ bot_bo8367695208629047834_ereal )
= ( insert8967887681552722334_ereal @ B2 @ bot_bo8367695208629047834_ereal ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_1016_Int__mono,axiom,
! [A2: set_a,C5: set_a,B: set_a,D3: set_a] :
( ( ord_less_eq_set_a @ A2 @ C5 )
=> ( ( ord_less_eq_set_a @ B @ D3 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ ( inf_inf_set_a @ C5 @ D3 ) ) ) ) ).
% Int_mono
thf(fact_1017_Int__mono,axiom,
! [A2: set_b,C5: set_b,B: set_b,D3: set_b] :
( ( ord_less_eq_set_b @ A2 @ C5 )
=> ( ( ord_less_eq_set_b @ B @ D3 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B ) @ ( inf_inf_set_b @ C5 @ D3 ) ) ) ) ).
% Int_mono
thf(fact_1018_Int__lower1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_1019_Int__lower1,axiom,
! [A2: set_b,B: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_1020_Int__lower2,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_1021_Int__lower2,axiom,
! [A2: set_b,B: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_1022_Int__absorb1,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_1023_Int__absorb1,axiom,
! [B: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B @ A2 )
=> ( ( inf_inf_set_b @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_1024_Int__absorb2,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( inf_inf_set_a @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_1025_Int__absorb2,axiom,
! [A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( inf_inf_set_b @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_1026_Int__greatest,axiom,
! [C5: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C5 @ A2 )
=> ( ( ord_less_eq_set_a @ C5 @ B )
=> ( ord_less_eq_set_a @ C5 @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_1027_Int__greatest,axiom,
! [C5: set_b,A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ C5 @ A2 )
=> ( ( ord_less_eq_set_b @ C5 @ B )
=> ( ord_less_eq_set_b @ C5 @ ( inf_inf_set_b @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_1028_Int__Collect__mono,axiom,
! [A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit,P: pre_pr7278220950009878019t_unit > $o,Q: pre_pr7278220950009878019t_unit > $o] :
( ( ord_le8200006823705900825t_unit @ A2 @ B )
=> ( ! [X3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le8200006823705900825t_unit @ ( inf_in1092213268631476299t_unit @ A2 @ ( collec8000012497822511960t_unit @ P ) ) @ ( inf_in1092213268631476299t_unit @ B @ ( collec8000012497822511960t_unit @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1029_Int__Collect__mono,axiom,
! [A2: set_Extended_ereal,B: set_Extended_ereal,P: extended_ereal > $o,Q: extended_ereal > $o] :
( ( ord_le1644982726543182158_ereal @ A2 @ B )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le1644982726543182158_ereal @ ( inf_in2779415704524776092_ereal @ A2 @ ( collec5835592288176408249_ereal @ P ) ) @ ( inf_in2779415704524776092_ereal @ B @ ( collec5835592288176408249_ereal @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1030_Int__Collect__mono,axiom,
! [A2: set_list_b,B: set_list_b,P: list_b > $o,Q: list_b > $o] :
( ( ord_le8932221534207217157list_b @ A2 @ B )
=> ( ! [X3: list_b] :
( ( member_list_b @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le8932221534207217157list_b @ ( inf_inf_set_list_b @ A2 @ ( collect_list_b @ P ) ) @ ( inf_inf_set_list_b @ B @ ( collect_list_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1031_Int__Collect__mono,axiom,
! [A2: set_list_a,B: set_list_a,P: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ A2 @ B )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ ( collect_list_a @ P ) ) @ ( inf_inf_set_list_a @ B @ ( collect_list_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1032_Int__Collect__mono,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1033_Int__Collect__mono,axiom,
! [A2: set_set_a,B: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1034_Int__Collect__mono,axiom,
! [A2: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1035_Int__Collect__mono,axiom,
! [A2: set_b,B: set_b,P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ ( collect_b @ P ) ) @ ( inf_inf_set_b @ B @ ( collect_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1036_finite_OinsertI,axiom,
! [A2: set_pr5411798346947241657t_unit,A: pre_pr7278220950009878019t_unit] :
( ( finite8852549406693098522t_unit @ A2 )
=> ( finite8852549406693098522t_unit @ ( insert6864688055023459379t_unit @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_1037_finite_OinsertI,axiom,
! [A2: set_list_a,A: list_a] :
( ( finite_finite_list_a @ A2 )
=> ( finite_finite_list_a @ ( insert_list_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_1038_finite_OinsertI,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( finite_finite_set_a @ ( insert_set_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_1039_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_1040_finite_OinsertI,axiom,
! [A2: set_b,A: b] :
( ( finite_finite_b @ A2 )
=> ( finite_finite_b @ ( insert_b @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_1041_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_1042_finite_OinsertI,axiom,
! [A2: set_list_b,A: list_b] :
( ( finite_finite_list_b @ A2 )
=> ( finite_finite_list_b @ ( insert_list_b @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_1043_finite_OinsertI,axiom,
! [A2: set_Extended_ereal,A: extended_ereal] :
( ( finite7198162374296863863_ereal @ A2 )
=> ( finite7198162374296863863_ereal @ ( insert8967887681552722334_ereal @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_1044_insert__mono,axiom,
! [C5: set_pr5411798346947241657t_unit,D3: set_pr5411798346947241657t_unit,A: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ C5 @ D3 )
=> ( ord_le8200006823705900825t_unit @ ( insert6864688055023459379t_unit @ A @ C5 ) @ ( insert6864688055023459379t_unit @ A @ D3 ) ) ) ).
% insert_mono
thf(fact_1045_insert__mono,axiom,
! [C5: set_Extended_ereal,D3: set_Extended_ereal,A: extended_ereal] :
( ( ord_le1644982726543182158_ereal @ C5 @ D3 )
=> ( ord_le1644982726543182158_ereal @ ( insert8967887681552722334_ereal @ A @ C5 ) @ ( insert8967887681552722334_ereal @ A @ D3 ) ) ) ).
% insert_mono
thf(fact_1046_insert__mono,axiom,
! [C5: set_a,D3: set_a,A: a] :
( ( ord_less_eq_set_a @ C5 @ D3 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C5 ) @ ( insert_a @ A @ D3 ) ) ) ).
% insert_mono
thf(fact_1047_insert__mono,axiom,
! [C5: set_b,D3: set_b,A: b] :
( ( ord_less_eq_set_b @ C5 @ D3 )
=> ( ord_less_eq_set_b @ ( insert_b @ A @ C5 ) @ ( insert_b @ A @ D3 ) ) ) ).
% insert_mono
thf(fact_1048_subset__insert,axiom,
! [X2: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X2 @ B ) )
= ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_1049_subset__insert,axiom,
! [X2: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ X2 @ A2 )
=> ( ( ord_le8200006823705900825t_unit @ A2 @ ( insert6864688055023459379t_unit @ X2 @ B ) )
= ( ord_le8200006823705900825t_unit @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_1050_subset__insert,axiom,
! [X2: extended_ereal,A2: set_Extended_ereal,B: set_Extended_ereal] :
( ~ ( member2350847679896131959_ereal @ X2 @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ A2 @ ( insert8967887681552722334_ereal @ X2 @ B ) )
= ( ord_le1644982726543182158_ereal @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_1051_subset__insert,axiom,
! [X2: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X2 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X2 @ B ) )
= ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_1052_subset__insert,axiom,
! [X2: b,A2: set_b,B: set_b] :
( ~ ( member_b @ X2 @ A2 )
=> ( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X2 @ B ) )
= ( ord_less_eq_set_b @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_1053_subset__insertI,axiom,
! [B: set_pr5411798346947241657t_unit,A: pre_pr7278220950009878019t_unit] : ( ord_le8200006823705900825t_unit @ B @ ( insert6864688055023459379t_unit @ A @ B ) ) ).
% subset_insertI
thf(fact_1054_subset__insertI,axiom,
! [B: set_Extended_ereal,A: extended_ereal] : ( ord_le1644982726543182158_ereal @ B @ ( insert8967887681552722334_ereal @ A @ B ) ) ).
% subset_insertI
thf(fact_1055_subset__insertI,axiom,
! [B: set_a,A: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A @ B ) ) ).
% subset_insertI
thf(fact_1056_subset__insertI,axiom,
! [B: set_b,A: b] : ( ord_less_eq_set_b @ B @ ( insert_b @ A @ B ) ) ).
% subset_insertI
thf(fact_1057_subset__insertI2,axiom,
! [A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit,B2: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ B )
=> ( ord_le8200006823705900825t_unit @ A2 @ ( insert6864688055023459379t_unit @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_1058_subset__insertI2,axiom,
! [A2: set_Extended_ereal,B: set_Extended_ereal,B2: extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A2 @ B )
=> ( ord_le1644982726543182158_ereal @ A2 @ ( insert8967887681552722334_ereal @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_1059_subset__insertI2,axiom,
! [A2: set_a,B: set_a,B2: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_1060_subset__insertI2,axiom,
! [A2: set_b,B: set_b,B2: b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ord_less_eq_set_b @ A2 @ ( insert_b @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_1061_Collect__conv__if,axiom,
! [P: list_b > $o,A: list_b] :
( ( ( P @ A )
=> ( ( collect_list_b
@ ^ [X: list_b] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_list_b @ A @ bot_bot_set_list_b ) ) )
& ( ~ ( P @ A )
=> ( ( collect_list_b
@ ^ [X: list_b] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_list_b ) ) ) ).
% Collect_conv_if
thf(fact_1062_Collect__conv__if,axiom,
! [P: list_a > $o,A: list_a] :
( ( ( P @ A )
=> ( ( collect_list_a
@ ^ [X: list_a] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_list_a
@ ^ [X: list_a] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_list_a ) ) ) ).
% Collect_conv_if
thf(fact_1063_Collect__conv__if,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_1064_Collect__conv__if,axiom,
! [P: set_a > $o,A: set_a] :
( ( ( P @ A )
=> ( ( collect_set_a
@ ^ [X: set_a] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_a
@ ^ [X: set_a] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_set_a ) ) ) ).
% Collect_conv_if
thf(fact_1065_Collect__conv__if,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X: a] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X: a] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_1066_Collect__conv__if,axiom,
! [P: pre_pr7278220950009878019t_unit > $o,A: pre_pr7278220950009878019t_unit] :
( ( ( P @ A )
=> ( ( collec8000012497822511960t_unit
@ ^ [X: pre_pr7278220950009878019t_unit] :
( ( X = A )
& ( P @ X ) ) )
= ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) ) )
& ( ~ ( P @ A )
=> ( ( collec8000012497822511960t_unit
@ ^ [X: pre_pr7278220950009878019t_unit] :
( ( X = A )
& ( P @ X ) ) )
= bot_bo1839476491465656141t_unit ) ) ) ).
% Collect_conv_if
thf(fact_1067_Collect__conv__if,axiom,
! [P: b > $o,A: b] :
( ( ( P @ A )
=> ( ( collect_b
@ ^ [X: b] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_b @ A @ bot_bot_set_b ) ) )
& ( ~ ( P @ A )
=> ( ( collect_b
@ ^ [X: b] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_b ) ) ) ).
% Collect_conv_if
thf(fact_1068_Collect__conv__if,axiom,
! [P: extended_ereal > $o,A: extended_ereal] :
( ( ( P @ A )
=> ( ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( X = A )
& ( P @ X ) ) )
= ( insert8967887681552722334_ereal @ A @ bot_bo8367695208629047834_ereal ) ) )
& ( ~ ( P @ A )
=> ( ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( X = A )
& ( P @ X ) ) )
= bot_bo8367695208629047834_ereal ) ) ) ).
% Collect_conv_if
thf(fact_1069_Collect__conv__if2,axiom,
! [P: list_b > $o,A: list_b] :
( ( ( P @ A )
=> ( ( collect_list_b
@ ^ [X: list_b] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_list_b @ A @ bot_bot_set_list_b ) ) )
& ( ~ ( P @ A )
=> ( ( collect_list_b
@ ^ [X: list_b] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_list_b ) ) ) ).
% Collect_conv_if2
thf(fact_1070_Collect__conv__if2,axiom,
! [P: list_a > $o,A: list_a] :
( ( ( P @ A )
=> ( ( collect_list_a
@ ^ [X: list_a] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_list_a
@ ^ [X: list_a] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_list_a ) ) ) ).
% Collect_conv_if2
thf(fact_1071_Collect__conv__if2,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_1072_Collect__conv__if2,axiom,
! [P: set_a > $o,A: set_a] :
( ( ( P @ A )
=> ( ( collect_set_a
@ ^ [X: set_a] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_a
@ ^ [X: set_a] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_1073_Collect__conv__if2,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X: a] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X: a] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_1074_Collect__conv__if2,axiom,
! [P: pre_pr7278220950009878019t_unit > $o,A: pre_pr7278220950009878019t_unit] :
( ( ( P @ A )
=> ( ( collec8000012497822511960t_unit
@ ^ [X: pre_pr7278220950009878019t_unit] :
( ( A = X )
& ( P @ X ) ) )
= ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) ) )
& ( ~ ( P @ A )
=> ( ( collec8000012497822511960t_unit
@ ^ [X: pre_pr7278220950009878019t_unit] :
( ( A = X )
& ( P @ X ) ) )
= bot_bo1839476491465656141t_unit ) ) ) ).
% Collect_conv_if2
thf(fact_1075_Collect__conv__if2,axiom,
! [P: b > $o,A: b] :
( ( ( P @ A )
=> ( ( collect_b
@ ^ [X: b] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_b @ A @ bot_bot_set_b ) ) )
& ( ~ ( P @ A )
=> ( ( collect_b
@ ^ [X: b] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_b ) ) ) ).
% Collect_conv_if2
thf(fact_1076_Collect__conv__if2,axiom,
! [P: extended_ereal > $o,A: extended_ereal] :
( ( ( P @ A )
=> ( ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( A = X )
& ( P @ X ) ) )
= ( insert8967887681552722334_ereal @ A @ bot_bo8367695208629047834_ereal ) ) )
& ( ~ ( P @ A )
=> ( ( collec5835592288176408249_ereal
@ ^ [X: extended_ereal] :
( ( A = X )
& ( P @ X ) ) )
= bot_bo8367695208629047834_ereal ) ) ) ).
% Collect_conv_if2
thf(fact_1077_pre__digraph_Oends__del__vert,axiom,
! [G: pre_pr7278220950009878019t_unit,U2: a] :
( ( arc_to_ends_a_b @ ( pre_del_vert_a_b @ G @ U2 ) )
= ( arc_to_ends_a_b @ G ) ) ).
% pre_digraph.ends_del_vert
thf(fact_1078_image__Int__subset,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] : ( ord_le3724670747650509150_set_a @ ( image_7466199892558553556_set_a @ F @ ( inf_in1092213268631476299t_unit @ A2 @ B ) ) @ ( inf_inf_set_set_a @ ( image_7466199892558553556_set_a @ F @ A2 ) @ ( image_7466199892558553556_set_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_1079_image__Int__subset,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a,B: set_set_a] : ( ord_le8200006823705900825t_unit @ ( image_6801035452528096924t_unit @ F @ ( inf_inf_set_set_a @ A2 @ B ) ) @ ( inf_in1092213268631476299t_unit @ ( image_6801035452528096924t_unit @ F @ A2 ) @ ( image_6801035452528096924t_unit @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_1080_image__Int__subset,axiom,
! [F: extended_ereal > extended_ereal,A2: set_Extended_ereal,B: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F @ ( inf_in2779415704524776092_ereal @ A2 @ B ) ) @ ( inf_in2779415704524776092_ereal @ ( image_6042159593519690757_ereal @ F @ A2 ) @ ( image_6042159593519690757_ereal @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_1081_image__Int__subset,axiom,
! [F: list_b > extended_ereal,A2: set_list_b,B: set_list_b] : ( ord_le1644982726543182158_ereal @ ( image_3611896476772571406_ereal @ F @ ( inf_inf_set_list_b @ A2 @ B ) ) @ ( inf_in2779415704524776092_ereal @ ( image_3611896476772571406_ereal @ F @ A2 ) @ ( image_3611896476772571406_ereal @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_1082_image__Int__subset,axiom,
! [F: a > set_a,A2: set_a,B: set_a] : ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ ( inf_inf_set_a @ A2 @ B ) ) @ ( inf_inf_set_set_a @ ( image_a_set_a @ F @ A2 ) @ ( image_a_set_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_1083_image__Int__subset,axiom,
! [F: a > a,A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F @ ( inf_inf_set_a @ A2 @ B ) ) @ ( inf_inf_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_1084_image__Int__subset,axiom,
! [F: a > b,A2: set_a,B: set_a] : ( ord_less_eq_set_b @ ( image_a_b @ F @ ( inf_inf_set_a @ A2 @ B ) ) @ ( inf_inf_set_b @ ( image_a_b @ F @ A2 ) @ ( image_a_b @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_1085_finite_Ocases,axiom,
! [A: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( ( A != bot_bot_set_list_a )
=> ~ ! [A5: set_list_a] :
( ? [A6: list_a] :
( A
= ( insert_list_a @ A6 @ A5 ) )
=> ~ ( finite_finite_list_a @ A5 ) ) ) ) ).
% finite.cases
thf(fact_1086_finite_Ocases,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ~ ! [A5: set_set_a] :
( ? [A6: set_a] :
( A
= ( insert_set_a @ A6 @ A5 ) )
=> ~ ( finite_finite_set_a @ A5 ) ) ) ) ).
% finite.cases
thf(fact_1087_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A5: set_nat] :
( ? [A6: nat] :
( A
= ( insert_nat @ A6 @ A5 ) )
=> ~ ( finite_finite_nat @ A5 ) ) ) ) ).
% finite.cases
thf(fact_1088_finite_Ocases,axiom,
! [A: set_list_b] :
( ( finite_finite_list_b @ A )
=> ( ( A != bot_bot_set_list_b )
=> ~ ! [A5: set_list_b] :
( ? [A6: list_b] :
( A
= ( insert_list_b @ A6 @ A5 ) )
=> ~ ( finite_finite_list_b @ A5 ) ) ) ) ).
% finite.cases
thf(fact_1089_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A5: set_a] :
( ? [A6: a] :
( A
= ( insert_a @ A6 @ A5 ) )
=> ~ ( finite_finite_a @ A5 ) ) ) ) ).
% finite.cases
thf(fact_1090_finite_Ocases,axiom,
! [A: set_pr5411798346947241657t_unit] :
( ( finite8852549406693098522t_unit @ A )
=> ( ( A != bot_bo1839476491465656141t_unit )
=> ~ ! [A5: set_pr5411798346947241657t_unit] :
( ? [A6: pre_pr7278220950009878019t_unit] :
( A
= ( insert6864688055023459379t_unit @ A6 @ A5 ) )
=> ~ ( finite8852549406693098522t_unit @ A5 ) ) ) ) ).
% finite.cases
thf(fact_1091_finite_Ocases,axiom,
! [A: set_b] :
( ( finite_finite_b @ A )
=> ( ( A != bot_bot_set_b )
=> ~ ! [A5: set_b] :
( ? [A6: b] :
( A
= ( insert_b @ A6 @ A5 ) )
=> ~ ( finite_finite_b @ A5 ) ) ) ) ).
% finite.cases
thf(fact_1092_finite_Ocases,axiom,
! [A: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ A )
=> ( ( A != bot_bo8367695208629047834_ereal )
=> ~ ! [A5: set_Extended_ereal] :
( ? [A6: extended_ereal] :
( A
= ( insert8967887681552722334_ereal @ A6 @ A5 ) )
=> ~ ( finite7198162374296863863_ereal @ A5 ) ) ) ) ).
% finite.cases
thf(fact_1093_finite_Osimps,axiom,
( finite_finite_list_a
= ( ^ [A3: set_list_a] :
( ( A3 = bot_bot_set_list_a )
| ? [A4: set_list_a,B6: list_a] :
( ( A3
= ( insert_list_a @ B6 @ A4 ) )
& ( finite_finite_list_a @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_1094_finite_Osimps,axiom,
( finite_finite_set_a
= ( ^ [A3: set_set_a] :
( ( A3 = bot_bot_set_set_a )
| ? [A4: set_set_a,B6: set_a] :
( ( A3
= ( insert_set_a @ B6 @ A4 ) )
& ( finite_finite_set_a @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_1095_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A3: set_nat] :
( ( A3 = bot_bot_set_nat )
| ? [A4: set_nat,B6: nat] :
( ( A3
= ( insert_nat @ B6 @ A4 ) )
& ( finite_finite_nat @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_1096_finite_Osimps,axiom,
( finite_finite_list_b
= ( ^ [A3: set_list_b] :
( ( A3 = bot_bot_set_list_b )
| ? [A4: set_list_b,B6: list_b] :
( ( A3
= ( insert_list_b @ B6 @ A4 ) )
& ( finite_finite_list_b @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_1097_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A3: set_a] :
( ( A3 = bot_bot_set_a )
| ? [A4: set_a,B6: a] :
( ( A3
= ( insert_a @ B6 @ A4 ) )
& ( finite_finite_a @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_1098_finite_Osimps,axiom,
( finite8852549406693098522t_unit
= ( ^ [A3: set_pr5411798346947241657t_unit] :
( ( A3 = bot_bo1839476491465656141t_unit )
| ? [A4: set_pr5411798346947241657t_unit,B6: pre_pr7278220950009878019t_unit] :
( ( A3
= ( insert6864688055023459379t_unit @ B6 @ A4 ) )
& ( finite8852549406693098522t_unit @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_1099_finite_Osimps,axiom,
( finite_finite_b
= ( ^ [A3: set_b] :
( ( A3 = bot_bot_set_b )
| ? [A4: set_b,B6: b] :
( ( A3
= ( insert_b @ B6 @ A4 ) )
& ( finite_finite_b @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_1100_finite_Osimps,axiom,
( finite7198162374296863863_ereal
= ( ^ [A3: set_Extended_ereal] :
( ( A3 = bot_bo8367695208629047834_ereal )
| ? [A4: set_Extended_ereal,B6: extended_ereal] :
( ( A3
= ( insert8967887681552722334_ereal @ B6 @ A4 ) )
& ( finite7198162374296863863_ereal @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_1101_finite__induct,axiom,
! [F2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F2 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X3: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ~ ( member_list_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_a @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1102_finite__induct,axiom,
! [F2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X3: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1103_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1104_finite__induct,axiom,
! [F2: set_list_b,P: set_list_b > $o] :
( ( finite_finite_list_b @ F2 )
=> ( ( P @ bot_bot_set_list_b )
=> ( ! [X3: list_b,F3: set_list_b] :
( ( finite_finite_list_b @ F3 )
=> ( ~ ( member_list_b @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_b @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1105_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1106_finite__induct,axiom,
! [F2: set_pr5411798346947241657t_unit,P: set_pr5411798346947241657t_unit > $o] :
( ( finite8852549406693098522t_unit @ F2 )
=> ( ( P @ bot_bo1839476491465656141t_unit )
=> ( ! [X3: pre_pr7278220950009878019t_unit,F3: set_pr5411798346947241657t_unit] :
( ( finite8852549406693098522t_unit @ F3 )
=> ( ~ ( member6939884229742472986t_unit @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert6864688055023459379t_unit @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1107_finite__induct,axiom,
! [F2: set_b,P: set_b > $o] :
( ( finite_finite_b @ F2 )
=> ( ( P @ bot_bot_set_b )
=> ( ! [X3: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ~ ( member_b @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1108_finite__induct,axiom,
! [F2: set_Extended_ereal,P: set_Extended_ereal > $o] :
( ( finite7198162374296863863_ereal @ F2 )
=> ( ( P @ bot_bo8367695208629047834_ereal )
=> ( ! [X3: extended_ereal,F3: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ F3 )
=> ( ~ ( member2350847679896131959_ereal @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert8967887681552722334_ereal @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1109_finite__ne__induct,axiom,
! [F2: set_b,P: set_b > $o] :
( ( finite_finite_b @ F2 )
=> ( ( F2 != bot_bot_set_b )
=> ( ! [X3: b] : ( P @ ( insert_b @ X3 @ bot_bot_set_b ) )
=> ( ! [X3: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ( F3 != bot_bot_set_b )
=> ( ~ ( member_b @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1110_finite__ne__induct,axiom,
! [F2: set_Extended_ereal,P: set_Extended_ereal > $o] :
( ( finite7198162374296863863_ereal @ F2 )
=> ( ( F2 != bot_bo8367695208629047834_ereal )
=> ( ! [X3: extended_ereal] : ( P @ ( insert8967887681552722334_ereal @ X3 @ bot_bo8367695208629047834_ereal ) )
=> ( ! [X3: extended_ereal,F3: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ F3 )
=> ( ( F3 != bot_bo8367695208629047834_ereal )
=> ( ~ ( member2350847679896131959_ereal @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert8967887681552722334_ereal @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1111_symmetric__connected__imp__strongly__connected,axiom,
( ( symmetric_a_b @ g )
=> ( ( digrap8783888973171253482ed_a_b @ g )
=> ( digrap8691851296217657702ed_a_b @ g ) ) ) ).
% symmetric_connected_imp_strongly_connected
thf(fact_1112_empty__imp__fin__dia__minf,axiom,
! [W: b > real] :
( ( ( pre_ve642382030648772252t_unit @ g )
= bot_bot_set_a )
=> ( ( graph_1932031826008834157er_a_b @ g @ W )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ).
% empty_imp_fin_dia_minf
thf(fact_1113_empty__imp__dia__minf,axiom,
! [W: b > real] :
( ( ( pre_ve642382030648772252t_unit @ g )
= bot_bot_set_a )
=> ( ( graph_926353876199057498er_a_b @ g @ W )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ).
% empty_imp_dia_minf
thf(fact_1114_verts__del__vert,axiom,
! [U2: a] :
( ( pre_ve642382030648772252t_unit @ ( pre_del_vert_a_b @ g @ U2 ) )
= ( minus_minus_set_a @ ( pre_ve642382030648772252t_unit @ g ) @ ( insert_a @ U2 @ bot_bot_set_a ) ) ) ).
% verts_del_vert
thf(fact_1115_ereal__minus__minus__image,axiom,
! [S: set_Extended_ereal] :
( ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S ) )
= S ) ).
% ereal_minus_minus_image
thf(fact_1116_ereal__uminus__eq__iff,axiom,
! [A: extended_ereal,B2: extended_ereal] :
( ( ( uminus27091377158695749_ereal @ A )
= ( uminus27091377158695749_ereal @ B2 ) )
= ( A = B2 ) ) ).
% ereal_uminus_eq_iff
thf(fact_1117_ereal__uminus__uminus,axiom,
! [A: extended_ereal] :
( ( uminus27091377158695749_ereal @ ( uminus27091377158695749_ereal @ A ) )
= A ) ).
% ereal_uminus_uminus
thf(fact_1118_ereal__minus__le__minus,axiom,
! [A: extended_ereal,B2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ A ) @ ( uminus27091377158695749_ereal @ B2 ) )
= ( ord_le1083603963089353582_ereal @ B2 @ A ) ) ).
% ereal_minus_le_minus
thf(fact_1119_ereal__minus__less__minus,axiom,
! [A: extended_ereal,B2: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ A ) @ ( uminus27091377158695749_ereal @ B2 ) )
= ( ord_le1188267648640031866_ereal @ B2 @ A ) ) ).
% ereal_minus_less_minus
thf(fact_1120_ereal__infty__less__eq_I2_J,axiom,
! [X2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X2 @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ).
% ereal_infty_less_eq(2)
thf(fact_1121_ereal__infty__less_I2_J,axiom,
! [X2: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ X2 )
= ( X2
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ).
% ereal_infty_less(2)
thf(fact_1122_ereal__MInfty__lessI,axiom,
! [A: extended_ereal] :
( ( A
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ A ) ) ).
% ereal_MInfty_lessI
thf(fact_1123_ereal__image__uminus__shift,axiom,
! [X6: set_Extended_ereal,Y2: set_Extended_ereal] :
( ( ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ X6 )
= Y2 )
= ( X6
= ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ Y2 ) ) ) ).
% ereal_image_uminus_shift
thf(fact_1124_bot__ereal__def,axiom,
( bot_bo2710585358178759738_ereal
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% bot_ereal_def
thf(fact_1125_ereal__uminus__eq__reorder,axiom,
! [A: extended_ereal,B2: extended_ereal] :
( ( ( uminus27091377158695749_ereal @ A )
= B2 )
= ( A
= ( uminus27091377158695749_ereal @ B2 ) ) ) ).
% ereal_uminus_eq_reorder
thf(fact_1126_MInfty__neq__PInfty_I1_J,axiom,
( extend1530274965995635425_ereal
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% MInfty_neq_PInfty(1)
thf(fact_1127_ereal__uminus__le__reorder,axiom,
! [A: extended_ereal,B2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ A ) @ B2 )
= ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ B2 ) @ A ) ) ).
% ereal_uminus_le_reorder
thf(fact_1128_ereal__complete__uminus__eq,axiom,
! [S: set_Extended_ereal,X2: extended_ereal] :
( ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S ) )
=> ( ord_le1083603963089353582_ereal @ X @ X2 ) )
& ! [Z4: extended_ereal] :
( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S ) )
=> ( ord_le1083603963089353582_ereal @ X @ Z4 ) )
=> ( ord_le1083603963089353582_ereal @ X2 @ Z4 ) ) )
= ( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ S )
=> ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ X2 ) @ X ) )
& ! [Z4: extended_ereal] :
( ! [X: extended_ereal] :
( ( member2350847679896131959_ereal @ X @ S )
=> ( ord_le1083603963089353582_ereal @ Z4 @ X ) )
=> ( ord_le1083603963089353582_ereal @ Z4 @ ( uminus27091377158695749_ereal @ X2 ) ) ) ) ) ).
% ereal_complete_uminus_eq
thf(fact_1129_ereal__less__uminus__reorder,axiom,
! [A: extended_ereal,B2: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ ( uminus27091377158695749_ereal @ B2 ) )
= ( ord_le1188267648640031866_ereal @ B2 @ ( uminus27091377158695749_ereal @ A ) ) ) ).
% ereal_less_uminus_reorder
thf(fact_1130_ereal__uminus__less__reorder,axiom,
! [A: extended_ereal,B2: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ A ) @ B2 )
= ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ B2 ) @ A ) ) ).
% ereal_uminus_less_reorder
thf(fact_1131_ereal__less__eq_I2_J,axiom,
! [X2: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ X2 ) ).
% ereal_less_eq(2)
thf(fact_1132_ereal__infty__less__eq2_I2_J,axiom,
! [A: extended_ereal,B2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B2 )
=> ( ( B2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( A
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ).
% ereal_infty_less_eq2(2)
thf(fact_1133_less__ereal_Osimps_I6_J,axiom,
ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ extend1530274965995635425_ereal ).
% less_ereal.simps(6)
thf(fact_1134_less__ereal_Osimps_I3_J,axiom,
! [A: extended_ereal] :
~ ( ord_le1188267648640031866_ereal @ A @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% less_ereal.simps(3)
thf(fact_1135_k__neighborhood__def,axiom,
! [W: b > real,V: a,K: real] :
( ( graph_3921080825633621230od_a_b @ g @ W @ V @ K )
= ( minus_minus_set_a
@ ( collect_a
@ ^ [U: a] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
& ( ord_le1083603963089353582_ereal @ ( shortest_wf_mu_a_b @ g @ W @ V @ U ) @ ( extended_ereal2 @ K ) ) ) )
@ ( insert_a @ V @ bot_bot_set_a ) ) ) ).
% k_neighborhood_def
thf(fact_1136_closed__euler2_I1_J,axiom,
! [U2: a,P3: list_b] :
( ( pre_euler_trail_a_b @ g @ U2 @ P3 @ U2 )
=> ( digrap8783888973171253482ed_a_b @ g ) ) ).
% closed_euler2(1)
thf(fact_1137_euler__imp__connected,axiom,
! [U2: a,P3: list_b,V: a] :
( ( pre_euler_trail_a_b @ g @ U2 @ P3 @ V )
=> ( digrap8783888973171253482ed_a_b @ g ) ) ).
% euler_imp_connected
thf(fact_1138_ereal__minus_I4_J,axiom,
! [X2: extended_ereal] :
( ( minus_2816186181549245109_ereal @ extend1530274965995635425_ereal @ X2 )
= extend1530274965995635425_ereal ) ).
% ereal_minus(4)
thf(fact_1139_ereal__minus_I1_J,axiom,
! [R: real,P3: real] :
( ( minus_2816186181549245109_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P3 ) )
= ( extended_ereal2 @ ( minus_minus_real @ R @ P3 ) ) ) ).
% ereal_minus(1)
thf(fact_1140_ereal__cong,axiom,
! [X2: real,Y3: real] :
( ( X2 = Y3 )
=> ( ( extended_ereal2 @ X2 )
= ( extended_ereal2 @ Y3 ) ) ) ).
% ereal_cong
thf(fact_1141_ereal_Oinject,axiom,
! [X1: real,Y1: real] :
( ( ( extended_ereal2 @ X1 )
= ( extended_ereal2 @ Y1 ) )
= ( X1 = Y1 ) ) ).
% ereal.inject
thf(fact_1142_ereal__minus_I5_J,axiom,
( ( minus_2816186181549245109_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ extend1530274965995635425_ereal )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% ereal_minus(5)
thf(fact_1143_ereal__less__eq_I3_J,axiom,
! [R: real,P3: real] :
( ( ord_le1083603963089353582_ereal @ ( extended_ereal2 @ R ) @ ( extended_ereal2 @ P3 ) )
= ( ord_less_eq_real @ R @ P3 ) ) ).
% ereal_less_eq(3)
thf(fact_1144_ereal__minus_I2_J,axiom,
! [R: real] :
( ( minus_2816186181549245109_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ ( extended_ereal2 @ R ) )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% ereal_minus(2)
thf(fact_1145_ereal__minus_I3_J,axiom,
! [R: real] :
( ( minus_2816186181549245109_ereal @ ( extended_ereal2 @ R ) @ extend1530274965995635425_ereal )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% ereal_minus(3)
thf(fact_1146_uminus__ereal_Osimps_I1_J,axiom,
! [R: real] :
( ( uminus27091377158695749_ereal @ ( extended_ereal2 @ R ) )
= ( extended_ereal2 @ ( uminus_uminus_real @ R ) ) ) ).
% uminus_ereal.simps(1)
thf(fact_1147_ereal__le__less,axiom,
! [Y3: real,A: extended_ereal,X2: real] :
( ( ord_le1083603963089353582_ereal @ ( extended_ereal2 @ Y3 ) @ A )
=> ( ( ord_less_real @ X2 @ Y3 )
=> ( ord_le1188267648640031866_ereal @ ( extended_ereal2 @ X2 ) @ A ) ) ) ).
% ereal_le_less
thf(fact_1148_le__ereal__less,axiom,
! [A: extended_ereal,X2: real,Y3: real] :
( ( ord_le1083603963089353582_ereal @ A @ ( extended_ereal2 @ X2 ) )
=> ( ( ord_less_real @ X2 @ Y3 )
=> ( ord_le1188267648640031866_ereal @ A @ ( extended_ereal2 @ Y3 ) ) ) ) ).
% le_ereal_less
thf(fact_1149_less__ereal_Osimps_I1_J,axiom,
! [X2: real,Y3: real] :
( ( ord_le1188267648640031866_ereal @ ( extended_ereal2 @ X2 ) @ ( extended_ereal2 @ Y3 ) )
= ( ord_less_real @ X2 @ Y3 ) ) ).
% less_ereal.simps(1)
thf(fact_1150_PInfty__neq__ereal_I1_J,axiom,
! [R: real] :
( ( extended_ereal2 @ R )
!= extend1530274965995635425_ereal ) ).
% PInfty_neq_ereal(1)
thf(fact_1151_ereal__le__real,axiom,
! [X2: extended_ereal,Y3: extended_ereal] :
( ! [Z5: real] :
( ( ord_le1083603963089353582_ereal @ X2 @ ( extended_ereal2 @ Z5 ) )
=> ( ord_le1083603963089353582_ereal @ Y3 @ ( extended_ereal2 @ Z5 ) ) )
=> ( ord_le1083603963089353582_ereal @ Y3 @ X2 ) ) ).
% ereal_le_real
thf(fact_1152_le__ereal__le,axiom,
! [A: extended_ereal,X2: real,Y3: real] :
( ( ord_le1083603963089353582_ereal @ A @ ( extended_ereal2 @ X2 ) )
=> ( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_le1083603963089353582_ereal @ A @ ( extended_ereal2 @ Y3 ) ) ) ) ).
% le_ereal_le
thf(fact_1153_ereal__le__le,axiom,
! [Y3: real,A: extended_ereal,X2: real] :
( ( ord_le1083603963089353582_ereal @ ( extended_ereal2 @ Y3 ) @ A )
=> ( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_le1083603963089353582_ereal @ ( extended_ereal2 @ X2 ) @ A ) ) ) ).
% ereal_le_le
thf(fact_1154_ereal__uminus__complement,axiom,
! [S: set_Extended_ereal] :
( ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ ( uminus5895154729394068773_ereal @ S ) )
= ( uminus5895154729394068773_ereal @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S ) ) ) ).
% ereal_uminus_complement
thf(fact_1155_ereal__dense2,axiom,
! [X2: extended_ereal,Y3: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X2 @ Y3 )
=> ? [Z5: real] :
( ( ord_le1188267648640031866_ereal @ X2 @ ( extended_ereal2 @ Z5 ) )
& ( ord_le1188267648640031866_ereal @ ( extended_ereal2 @ Z5 ) @ Y3 ) ) ) ).
% ereal_dense2
thf(fact_1156_less__ereal__le,axiom,
! [A: extended_ereal,X2: real,Y3: real] :
( ( ord_le1188267648640031866_ereal @ A @ ( extended_ereal2 @ X2 ) )
=> ( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_le1188267648640031866_ereal @ A @ ( extended_ereal2 @ Y3 ) ) ) ) ).
% less_ereal_le
thf(fact_1157_ereal__less__le,axiom,
! [Y3: real,A: extended_ereal,X2: real] :
( ( ord_le1188267648640031866_ereal @ ( extended_ereal2 @ Y3 ) @ A )
=> ( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_le1188267648640031866_ereal @ ( extended_ereal2 @ X2 ) @ A ) ) ) ).
% ereal_less_le
thf(fact_1158_ereal__minus__mono,axiom,
! [A2: extended_ereal,B: extended_ereal,D3: extended_ereal,C5: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A2 @ B )
=> ( ( ord_le1083603963089353582_ereal @ D3 @ C5 )
=> ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ A2 @ C5 ) @ ( minus_2816186181549245109_ereal @ B @ D3 ) ) ) ) ).
% ereal_minus_mono
thf(fact_1159_ereal__less__ereal__Ex,axiom,
! [X2: extended_ereal,R: real] :
( ( ord_le1188267648640031866_ereal @ X2 @ ( extended_ereal2 @ R ) )
= ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
| ? [P2: real] :
( ( ord_less_real @ P2 @ R )
& ( X2
= ( extended_ereal2 @ P2 ) ) ) ) ) ).
% ereal_less_ereal_Ex
thf(fact_1160_less__ereal_Oelims_I1_J,axiom,
! [X2: extended_ereal,Xa3: extended_ereal,Y3: $o] :
( ( ( ord_le1188267648640031866_ereal @ X2 @ Xa3 )
= Y3 )
=> ( ! [X3: real] :
( ( X2
= ( extended_ereal2 @ X3 ) )
=> ! [Y4: real] :
( ( Xa3
= ( extended_ereal2 @ Y4 ) )
=> ( Y3
= ( ~ ( ord_less_real @ X3 @ Y4 ) ) ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> Y3 )
=> ( ( ( Xa3
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> Y3 )
=> ( ( ? [X3: real] :
( X2
= ( extended_ereal2 @ X3 ) )
=> ( ( Xa3 = extend1530274965995635425_ereal )
=> ~ Y3 ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ? [R3: real] :
( Xa3
= ( extended_ereal2 @ R3 ) )
=> ~ Y3 ) )
=> ~ ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( Xa3 = extend1530274965995635425_ereal )
=> ~ Y3 ) ) ) ) ) ) ) ) ).
% less_ereal.elims(1)
thf(fact_1161_less__ereal_Oelims_I2_J,axiom,
! [X2: extended_ereal,Xa3: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X2 @ Xa3 )
=> ( ! [X3: real] :
( ( X2
= ( extended_ereal2 @ X3 ) )
=> ! [Y4: real] :
( ( Xa3
= ( extended_ereal2 @ Y4 ) )
=> ~ ( ord_less_real @ X3 @ Y4 ) ) )
=> ( ( ? [X3: real] :
( X2
= ( extended_ereal2 @ X3 ) )
=> ( Xa3 != extend1530274965995635425_ereal ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ! [R3: real] :
( Xa3
!= ( extended_ereal2 @ R3 ) ) )
=> ~ ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Xa3 != extend1530274965995635425_ereal ) ) ) ) ) ) ).
% less_ereal.elims(2)
thf(fact_1162_less__ereal_Oelims_I3_J,axiom,
! [X2: extended_ereal,Xa3: extended_ereal] :
( ~ ( ord_le1188267648640031866_ereal @ X2 @ Xa3 )
=> ( ! [X3: real] :
( ( X2
= ( extended_ereal2 @ X3 ) )
=> ! [Y4: real] :
( ( Xa3
= ( extended_ereal2 @ Y4 ) )
=> ( ord_less_real @ X3 @ Y4 ) ) )
=> ( ( X2 != extend1530274965995635425_ereal )
=> ( Xa3
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ).
% less_ereal.elims(3)
thf(fact_1163_ereal__all__split,axiom,
( ( ^ [P4: extended_ereal > $o] :
! [X7: extended_ereal] : ( P4 @ X7 ) )
= ( ^ [P5: extended_ereal > $o] :
( ( P5 @ extend1530274965995635425_ereal )
& ! [X: real] : ( P5 @ ( extended_ereal2 @ X ) )
& ( P5 @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ).
% ereal_all_split
thf(fact_1164_ereal__ex__split,axiom,
( ( ^ [P4: extended_ereal > $o] :
? [X7: extended_ereal] : ( P4 @ X7 ) )
= ( ^ [P5: extended_ereal > $o] :
( ( P5 @ extend1530274965995635425_ereal )
| ? [X: real] : ( P5 @ ( extended_ereal2 @ X ) )
| ( P5 @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ).
% ereal_ex_split
thf(fact_1165_ereal3__cases,axiom,
! [X2: extended_ereal,Xa3: extended_ereal,Xb: extended_ereal] :
( ( ? [R3: real] :
( X2
= ( extended_ereal2 @ R3 ) )
=> ( ? [Ra: real] :
( Xa3
= ( extended_ereal2 @ Ra ) )
=> ! [Rb: real] :
( Xb
!= ( extended_ereal2 @ Rb ) ) ) )
=> ( ( ? [R3: real] :
( X2
= ( extended_ereal2 @ R3 ) )
=> ( ? [Ra: real] :
( Xa3
= ( extended_ereal2 @ Ra ) )
=> ( Xb != extend1530274965995635425_ereal ) ) )
=> ( ( ? [R3: real] :
( X2
= ( extended_ereal2 @ R3 ) )
=> ( ? [Ra: real] :
( Xa3
= ( extended_ereal2 @ Ra ) )
=> ( Xb
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) )
=> ( ( ? [R3: real] :
( X2
= ( extended_ereal2 @ R3 ) )
=> ( ( Xa3 = extend1530274965995635425_ereal )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ? [R3: real] :
( X2
= ( extended_ereal2 @ R3 ) )
=> ( ( Xa3 = extend1530274965995635425_ereal )
=> ( Xb != extend1530274965995635425_ereal ) ) )
=> ( ( ? [R3: real] :
( X2
= ( extended_ereal2 @ R3 ) )
=> ( ( Xa3 = extend1530274965995635425_ereal )
=> ( Xb
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) )
=> ( ( ? [R3: real] :
( X2
= ( extended_ereal2 @ R3 ) )
=> ( ( Xa3
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ? [R3: real] :
( X2
= ( extended_ereal2 @ R3 ) )
=> ( ( Xa3
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Xb != extend1530274965995635425_ereal ) ) )
=> ( ( ? [R3: real] :
( X2
= ( extended_ereal2 @ R3 ) )
=> ( ( Xa3
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Xb
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( ? [R3: real] :
( Xa3
= ( extended_ereal2 @ R3 ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( ? [R3: real] :
( Xa3
= ( extended_ereal2 @ R3 ) )
=> ( Xb != extend1530274965995635425_ereal ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( ? [R3: real] :
( Xa3
= ( extended_ereal2 @ R3 ) )
=> ( Xb
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( ( Xa3 = extend1530274965995635425_ereal )
=> ! [R3: real] :
( Xb
!= ( extended_ereal2 @ R3 ) ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( ( Xa3 = extend1530274965995635425_ereal )
=> ( Xb != extend1530274965995635425_ereal ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( ( Xa3 = extend1530274965995635425_ereal )
=> ( Xb
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( ( Xa3
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ! [R3: real] :
( Xb
!= ( extended_ereal2 @ R3 ) ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( ( Xa3
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Xb != extend1530274965995635425_ereal ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( ( Xa3
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Xb
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ? [R3: real] :
( Xa3
= ( extended_ereal2 @ R3 ) )
=> ! [Ra: real] :
( Xb
!= ( extended_ereal2 @ Ra ) ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ? [R3: real] :
( Xa3
= ( extended_ereal2 @ R3 ) )
=> ( Xb != extend1530274965995635425_ereal ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ? [R3: real] :
( Xa3
= ( extended_ereal2 @ R3 ) )
=> ( Xb
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( Xa3 = extend1530274965995635425_ereal )
=> ! [R3: real] :
( Xb
!= ( extended_ereal2 @ R3 ) ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( Xa3 = extend1530274965995635425_ereal )
=> ( Xb != extend1530274965995635425_ereal ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( Xa3 = extend1530274965995635425_ereal )
=> ( Xb
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( Xa3
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ! [R3: real] :
( Xb
!= ( extended_ereal2 @ R3 ) ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( Xa3
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Xb != extend1530274965995635425_ereal ) ) )
=> ~ ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ( Xa3
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Xb
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% ereal3_cases
thf(fact_1166_ereal2__cases,axiom,
! [X2: extended_ereal,Xa3: extended_ereal] :
( ( ? [R3: real] :
( X2
= ( extended_ereal2 @ R3 ) )
=> ! [Ra: real] :
( Xa3
!= ( extended_ereal2 @ Ra ) ) )
=> ( ( ? [R3: real] :
( X2
= ( extended_ereal2 @ R3 ) )
=> ( Xa3 != extend1530274965995635425_ereal ) )
=> ( ( ? [R3: real] :
( X2
= ( extended_ereal2 @ R3 ) )
=> ( Xa3
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ! [R3: real] :
( Xa3
!= ( extended_ereal2 @ R3 ) ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( Xa3 != extend1530274965995635425_ereal ) )
=> ( ( ( X2 = extend1530274965995635425_ereal )
=> ( Xa3
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ! [R3: real] :
( Xa3
!= ( extended_ereal2 @ R3 ) ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Xa3 != extend1530274965995635425_ereal ) )
=> ~ ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Xa3
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ) ) ) ) ) ) ).
% ereal2_cases
thf(fact_1167_ereal__cases,axiom,
! [X2: extended_ereal] :
( ! [R3: real] :
( X2
!= ( extended_ereal2 @ R3 ) )
=> ( ( X2 != extend1530274965995635425_ereal )
=> ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ).
% ereal_cases
thf(fact_1168_real__of__ereal_Ocases,axiom,
! [X2: extended_ereal] :
( ! [R3: real] :
( X2
!= ( extended_ereal2 @ R3 ) )
=> ( ( X2 != extend1530274965995635425_ereal )
=> ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ).
% real_of_ereal.cases
thf(fact_1169_abs__ereal_Ocases,axiom,
! [X2: extended_ereal] :
( ! [R3: real] :
( X2
!= ( extended_ereal2 @ R3 ) )
=> ( ( X2
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( X2 = extend1530274965995635425_ereal ) ) ) ).
% abs_ereal.cases
thf(fact_1170_MInfty__neq__ereal_I1_J,axiom,
! [R: real] :
( ( extended_ereal2 @ R )
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% MInfty_neq_ereal(1)
thf(fact_1171_ereal__top,axiom,
! [X2: extended_ereal] :
( ! [B5: real] : ( ord_le1083603963089353582_ereal @ ( extended_ereal2 @ B5 ) @ X2 )
=> ( X2 = extend1530274965995635425_ereal ) ) ).
% ereal_top
thf(fact_1172_ereal__minus__eq__PInfty__iff,axiom,
! [X2: extended_ereal,Y3: extended_ereal] :
( ( ( minus_2816186181549245109_ereal @ X2 @ Y3 )
= extend1530274965995635425_ereal )
= ( ( Y3
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
| ( X2 = extend1530274965995635425_ereal ) ) ) ).
% ereal_minus_eq_PInfty_iff
thf(fact_1173_ereal__minus__eq__minus__iff,axiom,
! [A: extended_ereal,B2: extended_ereal,C2: extended_ereal] :
( ( ( minus_2816186181549245109_ereal @ A @ B2 )
= ( minus_2816186181549245109_ereal @ A @ C2 ) )
= ( ( B2 = C2 )
| ( A = extend1530274965995635425_ereal )
| ( ( A
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( B2
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( C2
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ).
% ereal_minus_eq_minus_iff
thf(fact_1174_ereal__minus__diff__eq,axiom,
! [X2: extended_ereal,Y3: extended_ereal] :
( ( ( X2 = extend1530274965995635425_ereal )
=> ( Y3 != extend1530274965995635425_ereal ) )
=> ( ( ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Y3
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
=> ( ( uminus27091377158695749_ereal @ ( minus_2816186181549245109_ereal @ X2 @ Y3 ) )
= ( minus_2816186181549245109_ereal @ Y3 @ X2 ) ) ) ) ).
% ereal_minus_diff_eq
thf(fact_1175_less__ereal_Osimps_I4_J,axiom,
! [X2: real] : ( ord_le1188267648640031866_ereal @ ( extended_ereal2 @ X2 ) @ extend1530274965995635425_ereal ) ).
% less_ereal.simps(4)
thf(fact_1176_ereal__bot,axiom,
! [X2: extended_ereal] :
( ! [B5: real] : ( ord_le1083603963089353582_ereal @ X2 @ ( extended_ereal2 @ B5 ) )
=> ( X2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ).
% ereal_bot
thf(fact_1177_less__ereal_Osimps_I5_J,axiom,
! [R: real] : ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ ( extended_ereal2 @ R ) ) ).
% less_ereal.simps(5)
thf(fact_1178_min__cost__awalk2,axiom,
! [W: b > real,A: a,B2: a] :
( ( ( shortest_wf_mu_a_b @ g @ W @ A @ B2 )
!= extend1530274965995635425_ereal )
=> ( ( ( shortest_wf_mu_a_b @ g @ W @ A @ B2 )
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ? [P6: list_b] :
( ( arc_pre_apath_a_b @ g @ A @ P6 @ B2 )
& ( ( shortest_wf_mu_a_b @ g @ W @ A @ B2 )
= ( extended_ereal2 @ ( weight7472181610322534790cost_b @ W @ P6 ) ) ) ) ) ) ).
% min_cost_awalk2
thf(fact_1179_connected__arcs__empty,axiom,
( ( digrap8783888973171253482ed_a_b @ g )
=> ( ( ( pre_ar1395965042833527383t_unit @ g )
= bot_bot_set_b )
=> ( ( ( pre_ve642382030648772252t_unit @ g )
!= bot_bot_set_a )
=> ~ ! [V4: a] :
( ( pre_ve642382030648772252t_unit @ g )
!= ( insert_a @ V4 @ bot_bot_set_a ) ) ) ) ) ).
% connected_arcs_empty
thf(fact_1180_finite__arcs,axiom,
finite_finite_b @ ( pre_ar1395965042833527383t_unit @ g ) ).
% finite_arcs
thf(fact_1181_del__arc__in,axiom,
! [A: b] :
( ~ ( member_b @ A @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( pre_del_arc_a_b @ g @ A )
= g ) ) ).
% del_arc_in
thf(fact_1182_arcs__add__vert,axiom,
! [U2: a] :
( ( pre_ar1395965042833527383t_unit @ ( pre_add_vert_a_b @ g @ U2 ) )
= ( pre_ar1395965042833527383t_unit @ g ) ) ).
% arcs_add_vert
thf(fact_1183_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_1184_arcs__del__arc,axiom,
! [A: b] :
( ( pre_ar1395965042833527383t_unit @ ( pre_del_arc_a_b @ g @ A ) )
= ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ g ) @ ( insert_b @ A @ bot_bot_set_b ) ) ) ).
% arcs_del_arc
thf(fact_1185_fin__digraphI,axiom,
( ( finite_finite_a @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( finite_finite_b @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( fin_digraph_a_b @ g ) ) ) ).
% fin_digraphI
thf(fact_1186_diff__diff__commute__ereal,axiom,
! [X2: extended_ereal,Y3: extended_ereal,Z: extended_ereal] :
( ( minus_2816186181549245109_ereal @ ( minus_2816186181549245109_ereal @ X2 @ Y3 ) @ Z )
= ( minus_2816186181549245109_ereal @ ( minus_2816186181549245109_ereal @ X2 @ Z ) @ Y3 ) ) ).
% diff_diff_commute_ereal
thf(fact_1187_min__cost__awalk,axiom,
! [U2: a,V: a,C2: b > real] :
( ( reachable_a_b @ g @ U2 @ V )
=> ( ! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( C2 @ E ) ) )
=> ? [P6: list_b] :
( ( arc_pre_apath_a_b @ g @ U2 @ P6 @ V )
& ( ( shortest_wf_mu_a_b @ g @ C2 @ U2 @ V )
= ( extended_ereal2 @ ( weight7472181610322534790cost_b @ C2 @ P6 ) ) ) ) ) ) ).
% min_cost_awalk
thf(fact_1188_euler__trail__conv__connected,axiom,
! [U2: a,P3: list_b,V: a] :
( ( digrap8783888973171253482ed_a_b @ g )
=> ( ( pre_euler_trail_a_b @ g @ U2 @ P3 @ V )
= ( ( arc_pre_trail_a_b @ g @ U2 @ P3 @ V )
& ( ( set_b2 @ P3 )
= ( pre_ar1395965042833527383t_unit @ g ) ) ) ) ) ).
% euler_trail_conv_connected
thf(fact_1189_min__cost__le__walk__cost,axiom,
! [U2: a,P3: list_b,V: a,C2: b > real] :
( ( arc_pre_awalk_a_b @ g @ U2 @ P3 @ V )
=> ( ord_le1083603963089353582_ereal @ ( shortest_wf_mu_a_b @ g @ C2 @ U2 @ V ) @ ( extended_ereal2 @ ( weight7472181610322534790cost_b @ C2 @ P3 ) ) ) ) ).
% min_cost_le_walk_cost
thf(fact_1190_awalk__ends__eqD,axiom,
! [U2: a,P3: list_b,V: a,W: a] :
( ( arc_pre_awalk_a_b @ g @ U2 @ P3 @ U2 )
=> ( ( arc_pre_awalk_a_b @ g @ V @ P3 @ W )
=> ( V = W ) ) ) ).
% awalk_ends_eqD
thf(fact_1191_awalk__last__in__verts,axiom,
! [U2: a,P3: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U2 @ P3 @ V )
=> ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% awalk_last_in_verts
thf(fact_1192_awalk__hd__in__verts,axiom,
! [U2: a,P3: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U2 @ P3 @ V )
=> ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% awalk_hd_in_verts
thf(fact_1193_reachable__awalkI,axiom,
! [U2: a,P3: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U2 @ P3 @ V )
=> ( reachable_a_b @ g @ U2 @ V ) ) ).
% reachable_awalkI
thf(fact_1194_reachable__awalk,axiom,
! [U2: a,V: a] :
( ( reachable_a_b @ g @ U2 @ V )
= ( ? [P2: list_b] : ( arc_pre_awalk_a_b @ g @ U2 @ P2 @ V ) ) ) ).
% reachable_awalk
thf(fact_1195_awalk__ends,axiom,
! [U2: a,P3: list_b,V: a,U5: a,V5: a] :
( ( arc_pre_awalk_a_b @ g @ U2 @ P3 @ V )
=> ( ( arc_pre_awalk_a_b @ g @ U5 @ P3 @ V5 )
=> ( ( ( P3 != nil_b )
& ( U2 = U5 )
& ( V = V5 ) )
| ( ( P3 = nil_b )
& ( U2 = V )
& ( U5 = V5 ) ) ) ) ) ).
% awalk_ends
thf(fact_1196_awalk__empty__ends,axiom,
! [U2: a,V: a] :
( ( arc_pre_awalk_a_b @ g @ U2 @ nil_b @ V )
=> ( U2 = V ) ) ).
% awalk_empty_ends
thf(fact_1197_awalkI__apath,axiom,
! [U2: a,P3: list_b,V: a] :
( ( arc_pre_apath_a_b @ g @ U2 @ P3 @ V )
=> ( arc_pre_awalk_a_b @ g @ U2 @ P3 @ V ) ) ).
% awalkI_apath
thf(fact_1198_awalk__Nil__iff,axiom,
! [U2: a,V: a] :
( ( arc_pre_awalk_a_b @ g @ U2 @ nil_b @ V )
= ( ( U2 = V )
& ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ g ) ) ) ) ).
% awalk_Nil_iff
thf(fact_1199_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1200_pos__cost__pos__awalk__cost,axiom,
! [U2: a,P3: list_b,V: a,C2: b > real] :
( ( arc_pre_awalk_a_b @ g @ U2 @ P3 @ V )
=> ( ! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( C2 @ E ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( weight7472181610322534790cost_b @ C2 @ P3 ) ) ) ) ).
% pos_cost_pos_awalk_cost
thf(fact_1201_minus__ereal__0,axiom,
! [X2: extended_ereal] :
( ( minus_2816186181549245109_ereal @ X2 @ ( extended_ereal2 @ zero_zero_real ) )
= X2 ) ).
% minus_ereal_0
thf(fact_1202_awalk__cost__Nil,axiom,
! [F: b > real] :
( ( weight7472181610322534790cost_b @ F @ nil_b )
= zero_zero_real ) ).
% awalk_cost_Nil
thf(fact_1203_arc__balancedI__trail,axiom,
! [U2: a,P3: list_b,V: a] :
( ( arc_pre_trail_a_b @ g @ U2 @ P3 @ V )
=> ( pre_ar5931435604406180204ed_a_b @ g @ U2 @ ( set_b2 @ P3 ) @ V ) ) ).
% arc_balancedI_trail
thf(fact_1204_mk__cycles__path__awalk,axiom,
! [U2: a,C2: list_b,N3: nat] :
( ( arc_pre_awalk_a_b @ g @ U2 @ C2 @ U2 )
=> ( arc_pre_awalk_a_b @ g @ U2 @ ( shorte6374615165232202367path_b @ N3 @ C2 ) @ U2 ) ) ).
% mk_cycles_path_awalk
thf(fact_1205_mk__cycles__path_Osimps_I1_J,axiom,
! [C2: list_b] :
( ( shorte6374615165232202367path_b @ zero_zero_nat @ C2 )
= nil_b ) ).
% mk_cycles_path.simps(1)
thf(fact_1206_ereal__uminus__zero,axiom,
( ( uminus27091377158695749_ereal @ zero_z2744965634713055877_ereal )
= zero_z2744965634713055877_ereal ) ).
% ereal_uminus_zero
thf(fact_1207_ereal__uminus__zero__iff,axiom,
! [A: extended_ereal] :
( ( ( uminus27091377158695749_ereal @ A )
= zero_z2744965634713055877_ereal )
= ( A = zero_z2744965634713055877_ereal ) ) ).
% ereal_uminus_zero_iff
thf(fact_1208_ereal__minus_I7_J,axiom,
! [X2: extended_ereal] :
( ( minus_2816186181549245109_ereal @ X2 @ zero_z2744965634713055877_ereal )
= X2 ) ).
% ereal_minus(7)
thf(fact_1209_sp__non__neg__if__w__non__neg,axiom,
! [W: b > real,U2: a,V: a] :
( ! [X3: b] :
( ( member_b @ X3 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( W @ X3 ) ) )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( shortest_wf_mu_a_b @ g @ W @ U2 @ V ) ) ) ).
% sp_non_neg_if_w_non_neg
thf(fact_1210_sp__to__self__if__w__non__neg,axiom,
! [W: b > real,U2: a] :
( ! [X3: b] :
( ( member_b @ X3 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( W @ X3 ) ) )
=> ( ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( shortest_wf_mu_a_b @ g @ W @ U2 @ U2 )
= zero_z2744965634713055877_ereal ) ) ) ).
% sp_to_self_if_w_non_neg
thf(fact_1211_ereal__eq__0_I1_J,axiom,
! [R: real] :
( ( ( extended_ereal2 @ R )
= zero_z2744965634713055877_ereal )
= ( R = zero_zero_real ) ) ).
% ereal_eq_0(1)
thf(fact_1212_ereal__eq__0_I2_J,axiom,
! [R: real] :
( ( zero_z2744965634713055877_ereal
= ( extended_ereal2 @ R ) )
= ( R = zero_zero_real ) ) ).
% ereal_eq_0(2)
thf(fact_1213_ereal__uminus__le__0__iff,axiom,
! [A: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ A ) @ zero_z2744965634713055877_ereal )
= ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A ) ) ).
% ereal_uminus_le_0_iff
thf(fact_1214_ereal__0__le__uminus__iff,axiom,
! [A: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( uminus27091377158695749_ereal @ A ) )
= ( ord_le1083603963089353582_ereal @ A @ zero_z2744965634713055877_ereal ) ) ).
% ereal_0_le_uminus_iff
thf(fact_1215_neg__0__less__iff__less__erea,axiom,
! [A: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ ( uminus27091377158695749_ereal @ A ) )
= ( ord_le1188267648640031866_ereal @ A @ zero_z2744965634713055877_ereal ) ) ).
% neg_0_less_iff_less_erea
thf(fact_1216_ereal__minus_I8_J,axiom,
! [X2: extended_ereal] :
( ( minus_2816186181549245109_ereal @ zero_z2744965634713055877_ereal @ X2 )
= ( uminus27091377158695749_ereal @ X2 ) ) ).
% ereal_minus(8)
thf(fact_1217_ereal__less__eq_I5_J,axiom,
! [R: real] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( extended_ereal2 @ R ) )
= ( ord_less_eq_real @ zero_zero_real @ R ) ) ).
% ereal_less_eq(5)
thf(fact_1218_ereal__less__eq_I4_J,axiom,
! [R: real] :
( ( ord_le1083603963089353582_ereal @ ( extended_ereal2 @ R ) @ zero_z2744965634713055877_ereal )
= ( ord_less_eq_real @ R @ zero_zero_real ) ) ).
% ereal_less_eq(4)
thf(fact_1219_ereal__less_I1_J,axiom,
! [R: real] :
( ( ord_le1188267648640031866_ereal @ ( extended_ereal2 @ R ) @ zero_z2744965634713055877_ereal )
= ( ord_less_real @ R @ zero_zero_real ) ) ).
% ereal_less(1)
thf(fact_1220_ereal__less_I2_J,axiom,
! [R: real] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ ( extended_ereal2 @ R ) )
= ( ord_less_real @ zero_zero_real @ R ) ) ).
% ereal_less(2)
thf(fact_1221_Infty__neq__0_I1_J,axiom,
extend1530274965995635425_ereal != zero_z2744965634713055877_ereal ).
% Infty_neq_0(1)
thf(fact_1222_zero__ereal__def,axiom,
( zero_z2744965634713055877_ereal
= ( extended_ereal2 @ zero_zero_real ) ) ).
% zero_ereal_def
thf(fact_1223_Infty__neq__0_I3_J,axiom,
( ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal )
!= zero_z2744965634713055877_ereal ) ).
% Infty_neq_0(3)
thf(fact_1224_ereal__less_I5_J,axiom,
ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ extend1530274965995635425_ereal ).
% ereal_less(5)
thf(fact_1225_ereal__diff__le__mono__left,axiom,
! [X2: extended_ereal,Z: extended_ereal,Y3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X2 @ Z )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Y3 )
=> ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ X2 @ Y3 ) @ Z ) ) ) ).
% ereal_diff_le_mono_left
thf(fact_1226_ereal__diff__positive,axiom,
! [A: extended_ereal,B2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B2 )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( minus_2816186181549245109_ereal @ B2 @ A ) ) ) ).
% ereal_diff_positive
thf(fact_1227_ereal__diff__le__self,axiom,
! [Y3: extended_ereal,X2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Y3 )
=> ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ X2 @ Y3 ) @ X2 ) ) ).
% ereal_diff_le_self
thf(fact_1228_ereal__diff__gr0,axiom,
! [A: extended_ereal,B2: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ B2 )
=> ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ ( minus_2816186181549245109_ereal @ B2 @ A ) ) ) ).
% ereal_diff_gr0
thf(fact_1229_not__MInfty__nonneg,axiom,
! [X2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ X2 )
=> ( X2
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ).
% not_MInfty_nonneg
thf(fact_1230_ereal__less_I6_J,axiom,
ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ zero_z2744965634713055877_ereal ).
% ereal_less(6)
thf(fact_1231_ereal__diff__nonpos,axiom,
! [A: extended_ereal,B2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B2 )
=> ( ( ( A = extend1530274965995635425_ereal )
=> ( B2 != extend1530274965995635425_ereal ) )
=> ( ( ( A
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( B2
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
=> ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ A @ B2 ) @ zero_z2744965634713055877_ereal ) ) ) ) ).
% ereal_diff_nonpos
thf(fact_1232_ereal__mono__minus__cancel,axiom,
! [C2: extended_ereal,A: extended_ereal,B2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ C2 @ A ) @ ( minus_2816186181549245109_ereal @ C2 @ B2 ) )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ C2 )
=> ( ( ord_le1188267648640031866_ereal @ C2 @ extend1530274965995635425_ereal )
=> ( ord_le1083603963089353582_ereal @ B2 @ A ) ) ) ) ).
% ereal_mono_minus_cancel
thf(fact_1233_no__neg__cyc__reach__imp__path,axiom,
! [U2: a,V: a,F: b > real] :
( ( reachable_a_b @ g @ U2 @ V )
=> ( ! [P6: list_b] :
( ( arc_pre_awalk_a_b @ g @ U2 @ P6 @ V )
=> ~ ? [W2: a,C7: list_b] :
( ( arc_pre_awalk_a_b @ g @ W2 @ C7 @ W2 )
& ( member_a @ W2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U2 @ P6 ) ) )
& ( ord_less_real @ ( weight7472181610322534790cost_b @ F @ C7 ) @ zero_zero_real ) ) )
=> ? [P6: list_b] :
( ( arc_pre_apath_a_b @ g @ U2 @ P6 @ V )
& ( ( shortest_wf_mu_a_b @ g @ F @ U2 @ V )
= ( extended_ereal2 @ ( weight7472181610322534790cost_b @ F @ P6 ) ) ) ) ) ) ).
% no_neg_cyc_reach_imp_path
thf(fact_1234_no__neg__cyc__imp__no__neg__inf,axiom,
! [U2: a,V: a,F: b > real] :
( ! [P6: list_b] :
( ( arc_pre_awalk_a_b @ g @ U2 @ P6 @ V )
=> ~ ? [W2: a,C7: list_b] :
( ( arc_pre_awalk_a_b @ g @ W2 @ C7 @ W2 )
& ( member_a @ W2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U2 @ P6 ) ) )
& ( ord_less_real @ ( weight7472181610322534790cost_b @ F @ C7 ) @ zero_zero_real ) ) )
=> ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ ( shortest_wf_mu_a_b @ g @ F @ U2 @ V ) ) ) ).
% no_neg_cyc_imp_no_neg_inf
thf(fact_1235_awalk__verts__non__Nil,axiom,
! [U2: a,P3: list_b] :
( ( arc_pr7493981781705774526ts_a_b @ g @ U2 @ P3 )
!= nil_a ) ).
% awalk_verts_non_Nil
thf(fact_1236_awalk__verts__ne__eq,axiom,
! [P3: list_b,U2: a,V: a] :
( ( P3 != nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ g @ U2 @ P3 )
= ( arc_pr7493981781705774526ts_a_b @ g @ V @ P3 ) ) ) ).
% awalk_verts_ne_eq
thf(fact_1237_awalk__verts__induce,axiom,
! [S: set_a] :
( ( arc_pr7493981781705774526ts_a_b @ ( digrap7873285959652527175ph_a_b @ g @ S ) )
= ( arc_pr7493981781705774526ts_a_b @ g ) ) ).
% awalk_verts_induce
thf(fact_1238_hd__in__awalk__verts_I1_J,axiom,
! [U2: a,P3: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U2 @ P3 @ V )
=> ( member_a @ U2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U2 @ P3 ) ) ) ) ).
% hd_in_awalk_verts(1)
thf(fact_1239_hd__in__awalk__verts_I2_J,axiom,
! [U2: a,P3: list_b,V: a] :
( ( arc_pre_apath_a_b @ g @ U2 @ P3 @ V )
=> ( member_a @ U2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U2 @ P3 ) ) ) ) ).
% hd_in_awalk_verts(2)
thf(fact_1240_awalk__verts__reachable__to,axiom,
! [U2: a,P3: list_b,V: a,W: a] :
( ( arc_pre_awalk_a_b @ g @ U2 @ P3 @ V )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U2 @ P3 ) ) )
=> ( reachable_a_b @ g @ W @ V ) ) ) ).
% awalk_verts_reachable_to
thf(fact_1241_awalk__verts__reachable__from,axiom,
! [U2: a,P3: list_b,V: a,W: a] :
( ( arc_pre_awalk_a_b @ g @ U2 @ P3 @ V )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U2 @ P3 ) ) )
=> ( reachable_a_b @ g @ U2 @ W ) ) ) ).
% awalk_verts_reachable_from
thf(fact_1242_awalk__del__vert,axiom,
! [U2: a,P3: list_b,V: a,X2: a] :
( ( arc_pre_awalk_a_b @ g @ U2 @ P3 @ V )
=> ( ~ ( member_a @ X2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U2 @ P3 ) ) )
=> ( arc_pre_awalk_a_b @ ( pre_del_vert_a_b @ g @ X2 ) @ U2 @ P3 @ V ) ) ) ).
% awalk_del_vert
thf(fact_1243_rotate__trailE_H,axiom,
! [U2: a,P3: list_b,W: a] :
( ( arc_pre_trail_a_b @ g @ U2 @ P3 @ U2 )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U2 @ P3 ) ) )
=> ~ ! [Q2: list_b] :
( ( arc_pre_trail_a_b @ g @ W @ Q2 @ W )
=> ( ( ( set_b2 @ Q2 )
= ( set_b2 @ P3 ) )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ W @ Q2 ) )
!= ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U2 @ P3 ) ) ) ) ) ) ) ).
% rotate_trailE'
thf(fact_1244_awalk__verts__in__verts,axiom,
! [U2: a,P3: list_b,V: a] :
( ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P3 ) @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( member_a @ V @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U2 @ P3 ) ) )
=> ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) ) ) ) ) ).
% awalk_verts_in_verts
thf(fact_1245_awalk__induce,axiom,
! [U2: a,P3: list_b,V: a,S: set_a] :
( ( arc_pre_awalk_a_b @ g @ U2 @ P3 @ V )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U2 @ P3 ) ) @ S )
=> ( arc_pre_awalk_a_b @ ( digrap7873285959652527175ph_a_b @ g @ S ) @ U2 @ P3 @ V ) ) ) ).
% awalk_induce
thf(fact_1246_euler__trail__def,axiom,
! [U2: a,P3: list_b,V: a] :
( ( pre_euler_trail_a_b @ g @ U2 @ P3 @ V )
= ( ( arc_pre_trail_a_b @ g @ U2 @ P3 @ V )
& ( ( set_b2 @ P3 )
= ( pre_ar1395965042833527383t_unit @ g ) )
& ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U2 @ P3 ) )
= ( pre_ve642382030648772252t_unit @ g ) ) ) ) ).
% euler_trail_def
thf(fact_1247_neg__cycle__imp__inf___092_060mu_062,axiom,
! [U2: a,P3: list_b,V: a,W: a,C2: list_b,F: b > real] :
( ( arc_pre_awalk_a_b @ g @ U2 @ P3 @ V )
=> ( ( arc_pre_awalk_a_b @ g @ W @ C2 @ W )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U2 @ P3 ) ) )
=> ( ( ord_less_real @ ( weight7472181610322534790cost_b @ F @ C2 ) @ zero_zero_real )
=> ( ( shortest_wf_mu_a_b @ g @ F @ U2 @ V )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ) ).
% neg_cycle_imp_inf_\<mu>
thf(fact_1248_neg__inf__imp__neg__cyc,axiom,
! [F: b > real,U2: a,V: a] :
( ( ( shortest_wf_mu_a_b @ g @ F @ U2 @ V )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ? [P6: list_b] :
( ( arc_pre_awalk_a_b @ g @ U2 @ P6 @ V )
& ? [W2: a,C7: list_b] :
( ( arc_pre_awalk_a_b @ g @ W2 @ C7 @ W2 )
& ( member_a @ W2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U2 @ P6 ) ) )
& ( ord_less_real @ ( weight7472181610322534790cost_b @ F @ C7 ) @ zero_zero_real ) ) ) ) ).
% neg_inf_imp_neg_cyc
thf(fact_1249_set__inner__verts,axiom,
! [U2: a,P3: list_b,V: a] :
( ( arc_pre_apath_a_b @ g @ U2 @ P3 @ V )
=> ( ( set_a2 @ ( pre_inner_verts_a_b @ g @ P3 ) )
= ( minus_minus_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U2 @ P3 ) ) @ ( insert_a @ U2 @ ( insert_a @ V @ bot_bot_set_a ) ) ) ) ) ).
% set_inner_verts
thf(fact_1250_awalk__to__path__no__neg__cyc__cost,axiom,
! [U2: a,P3: list_b,V: a,F: b > real] :
( ( arc_pre_awalk_a_b @ g @ U2 @ P3 @ V )
=> ( ~ ? [W2: a,C7: list_b] :
( ( arc_pre_awalk_a_b @ g @ W2 @ C7 @ W2 )
& ( member_a @ W2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U2 @ P3 ) ) )
& ( ord_less_real @ ( weight7472181610322534790cost_b @ F @ C7 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( weight7472181610322534790cost_b @ F @ ( arc_wf446166946845163101th_a_b @ g @ P3 ) ) @ ( weight7472181610322534790cost_b @ F @ P3 ) ) ) ) ).
% awalk_to_path_no_neg_cyc_cost
thf(fact_1251_apath__awalk__to__apath,axiom,
! [U2: a,P3: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U2 @ P3 @ V )
=> ( arc_pre_apath_a_b @ g @ U2 @ ( arc_wf446166946845163101th_a_b @ g @ P3 ) @ V ) ) ).
% apath_awalk_to_apath
thf(fact_1252_awalk__to__apath__subset,axiom,
! [U2: a,P3: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U2 @ P3 @ V )
=> ( ord_less_eq_set_b @ ( set_b2 @ ( arc_wf446166946845163101th_a_b @ g @ P3 ) ) @ ( set_b2 @ P3 ) ) ) ).
% awalk_to_apath_subset
thf(fact_1253_inner__verts__Nil,axiom,
( ( pre_inner_verts_a_b @ g @ nil_b )
= nil_a ) ).
% inner_verts_Nil
thf(fact_1254_gen__iapath__def,axiom,
! [V6: set_a,U2: a,P3: list_b,V: a] :
( ( pre_gen_iapath_a_b @ g @ V6 @ U2 @ P3 @ V )
= ( ( member_a @ U2 @ V6 )
& ( member_a @ V @ V6 )
& ( arc_pre_apath_a_b @ g @ U2 @ P3 @ V )
& ( ( inf_inf_set_a @ ( set_a2 @ ( pre_inner_verts_a_b @ g @ P3 ) ) @ V6 )
= bot_bot_set_a )
& ( P3 != nil_b ) ) ) ).
% gen_iapath_def
thf(fact_1255_walk__cheaper__path__imp__neg__cyc,axiom,
! [U2: a,P3: list_b,V: a,F: b > real] :
( ( arc_pre_awalk_a_b @ g @ U2 @ P3 @ V )
=> ( ( ord_le1188267648640031866_ereal @ ( extended_ereal2 @ ( weight7472181610322534790cost_b @ F @ P3 ) )
@ ( comple3556804143462414037_ereal
@ ( image_3611896476772571406_ereal
@ ^ [P2: list_b] : ( extended_ereal2 @ ( weight7472181610322534790cost_b @ F @ P2 ) )
@ ( collect_list_b
@ ^ [P2: list_b] : ( arc_pre_apath_a_b @ g @ U2 @ P2 @ V ) ) ) ) )
=> ? [W2: a,C7: list_b] :
( ( arc_pre_awalk_a_b @ g @ W2 @ C7 @ W2 )
& ( member_a @ W2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U2 @ P3 ) ) )
& ( ord_less_real @ ( weight7472181610322534790cost_b @ F @ C7 ) @ zero_zero_real ) ) ) ) ).
% walk_cheaper_path_imp_neg_cyc
thf(fact_1256_iapath__dist__ends,axiom,
! [U2: a,P3: list_b,V: a] :
( ( pre_gen_iapath_a_b @ g @ ( verts3_a_b @ g ) @ U2 @ P3 @ V )
=> ( U2 != V ) ) ).
% iapath_dist_ends
thf(fact_1257__092_060mu_062__def,axiom,
! [F: b > real,U2: a,V: a] :
( ( shortest_wf_mu_a_b @ g @ F @ U2 @ V )
= ( comple3556804143462414037_ereal
@ ( image_3611896476772571406_ereal
@ ^ [P2: list_b] : ( extended_ereal2 @ ( weight7472181610322534790cost_b @ F @ P2 ) )
@ ( collect_list_b
@ ^ [P2: list_b] : ( arc_pre_awalk_a_b @ g @ U2 @ P2 @ V ) ) ) ) ) ).
% \<mu>_def
thf(fact_1258_Inf__eq__MInfty,axiom,
! [S: set_Extended_ereal] :
( ( member2350847679896131959_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ S )
=> ( ( comple3556804143462414037_ereal @ S )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ).
% Inf_eq_MInfty
thf(fact_1259_Inf__eq__PInfty,axiom,
! [S: set_Extended_ereal] :
( ( ( comple3556804143462414037_ereal @ S )
= extend1530274965995635425_ereal )
= ( ( S = bot_bo8367695208629047834_ereal )
| ( S
= ( insert8967887681552722334_ereal @ extend1530274965995635425_ereal @ bot_bo8367695208629047834_ereal ) ) ) ) ).
% Inf_eq_PInfty
thf(fact_1260_awalkI,axiom,
! [U2: a,P3: list_b,V: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U2 @ P3 ) ) @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P3 ) @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( arc_pre_cas_a_b @ g @ U2 @ P3 @ V )
=> ( arc_pre_awalk_a_b @ g @ U2 @ P3 @ V ) ) ) ) ).
% awalkI
thf(fact_1261_sp__triangle,axiom,
! [A: a,B2: a,C2: a,W: b > real] :
( ( member_a @ A @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( member_a @ B2 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( member_a @ C2 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( W @ X3 ) ) )
=> ( ord_le1083603963089353582_ereal @ ( shortest_wf_mu_a_b @ g @ W @ A @ C2 ) @ ( plus_p7876563987511257093_ereal @ ( shortest_wf_mu_a_b @ g @ W @ A @ B2 ) @ ( shortest_wf_mu_a_b @ g @ W @ B2 @ C2 ) ) ) ) ) ) ) ).
% sp_triangle
thf(fact_1262_cas_Osimps_I1_J,axiom,
! [U2: a,V: a] :
( ( arc_pre_cas_a_b @ g @ U2 @ nil_b @ V )
= ( U2 = V ) ) ).
% cas.simps(1)
thf(fact_1263_cas__ends,axiom,
! [U2: a,P3: list_b,V: a,U5: a,V5: a] :
( ( arc_pre_cas_a_b @ g @ U2 @ P3 @ V )
=> ( ( arc_pre_cas_a_b @ g @ U5 @ P3 @ V5 )
=> ( ( ( P3 != nil_b )
& ( U2 = U5 )
& ( V = V5 ) )
| ( ( P3 = nil_b )
& ( U2 = V )
& ( U5 = V5 ) ) ) ) ) ).
% cas_ends
thf(fact_1264_cas__induce,axiom,
! [U2: a,P3: list_b,V: a,S: set_a] :
( ( arc_pre_cas_a_b @ g @ U2 @ P3 @ V )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ g @ U2 @ P3 ) ) @ S )
=> ( arc_pre_cas_a_b @ ( digrap7873285959652527175ph_a_b @ g @ S ) @ U2 @ P3 @ V ) ) ) ).
% cas_induce
thf(fact_1265_awalk__def,axiom,
! [U2: a,P3: list_b,V: a] :
( ( arc_pre_awalk_a_b @ g @ U2 @ P3 @ V )
= ( ( member_a @ U2 @ ( pre_ve642382030648772252t_unit @ g ) )
& ( ord_less_eq_set_b @ ( set_b2 @ P3 ) @ ( pre_ar1395965042833527383t_unit @ g ) )
& ( arc_pre_cas_a_b @ g @ U2 @ P3 @ V ) ) ) ).
% awalk_def
thf(fact_1266_ereal__plus__eq__PInfty,axiom,
! [A: extended_ereal,B2: extended_ereal] :
( ( ( plus_p7876563987511257093_ereal @ A @ B2 )
= extend1530274965995635425_ereal )
= ( ( A = extend1530274965995635425_ereal )
| ( B2 = extend1530274965995635425_ereal ) ) ) ).
% ereal_plus_eq_PInfty
thf(fact_1267_ereal__PInfty__eq__plus,axiom,
! [A: extended_ereal,B2: extended_ereal] :
( ( extend1530274965995635425_ereal
= ( plus_p7876563987511257093_ereal @ A @ B2 ) )
= ( ( A = extend1530274965995635425_ereal )
| ( B2 = extend1530274965995635425_ereal ) ) ) ).
% ereal_PInfty_eq_plus
thf(fact_1268_ereal__0__plus,axiom,
! [X2: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ ( extended_ereal2 @ zero_zero_real ) @ X2 )
= X2 ) ).
% ereal_0_plus
thf(fact_1269_plus__ereal__0,axiom,
! [X2: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ X2 @ ( extended_ereal2 @ zero_zero_real ) )
= X2 ) ).
% plus_ereal_0
thf(fact_1270_ereal__plus__eq__MInfty,axiom,
! [A: extended_ereal,B2: extended_ereal] :
( ( ( plus_p7876563987511257093_ereal @ A @ B2 )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
= ( ( ( A
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
| ( B2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
& ( A != extend1530274965995635425_ereal )
& ( B2 != extend1530274965995635425_ereal ) ) ) ).
% ereal_plus_eq_MInfty
thf(fact_1271_ereal__MInfty__eq__plus,axiom,
! [A: extended_ereal,B2: extended_ereal] :
( ( ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal )
= ( plus_p7876563987511257093_ereal @ A @ B2 ) )
= ( ( ( A
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( B2 != extend1530274965995635425_ereal ) )
| ( ( B2
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( A != extend1530274965995635425_ereal ) ) ) ) ).
% ereal_MInfty_eq_plus
thf(fact_1272_ereal__minus_I6_J,axiom,
! [X2: extended_ereal,Y3: extended_ereal] :
( ( minus_2816186181549245109_ereal @ X2 @ ( uminus27091377158695749_ereal @ Y3 ) )
= ( plus_p7876563987511257093_ereal @ X2 @ Y3 ) ) ).
% ereal_minus(6)
thf(fact_1273_ereal__add__nonneg__eq__0__iff,axiom,
! [A: extended_ereal,B2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ B2 )
=> ( ( ( plus_p7876563987511257093_ereal @ A @ B2 )
= zero_z2744965634713055877_ereal )
= ( ( A = zero_z2744965634713055877_ereal )
& ( B2 = zero_z2744965634713055877_ereal ) ) ) ) ) ).
% ereal_add_nonneg_eq_0_iff
thf(fact_1274_ereal__le__add__self2,axiom,
! [Y3: extended_ereal,X2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Y3 )
=> ( ord_le1083603963089353582_ereal @ X2 @ ( plus_p7876563987511257093_ereal @ Y3 @ X2 ) ) ) ).
% ereal_le_add_self2
thf(fact_1275_ereal__le__add__mono2,axiom,
! [X2: extended_ereal,Z: extended_ereal,Y3: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X2 @ Z )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Y3 )
=> ( ord_le1083603963089353582_ereal @ X2 @ ( plus_p7876563987511257093_ereal @ Y3 @ Z ) ) ) ) ).
% ereal_le_add_mono2
thf(fact_1276_ereal__le__add__mono1,axiom,
! [X2: extended_ereal,Y3: extended_ereal,Z: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X2 @ Y3 )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Z )
=> ( ord_le1083603963089353582_ereal @ X2 @ ( plus_p7876563987511257093_ereal @ Y3 @ Z ) ) ) ) ).
% ereal_le_add_mono1
thf(fact_1277_ereal__le__add__self,axiom,
! [Y3: extended_ereal,X2: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Y3 )
=> ( ord_le1083603963089353582_ereal @ X2 @ ( plus_p7876563987511257093_ereal @ X2 @ Y3 ) ) ) ).
% ereal_le_add_self
% Conjectures (1)
thf(conj_0,conjecture,
( finite7198162374296863863_ereal
@ ( collec5835592288176408249_ereal
@ ^ [Uu: extended_ereal] :
? [U: a,V2: a,C: extended_ereal] :
( ( Uu = C )
& ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
& ( member_a @ V2 @ ( pre_ve642382030648772252t_unit @ g ) )
& ( ( shortest_wf_mu_a_b @ g @ f @ U @ V2 )
= C )
& ( ord_le1188267648640031866_ereal @ C @ extend1530274965995635425_ereal ) ) ) ) ).
%------------------------------------------------------------------------------