TPTP Problem File: SLH0842^1.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/0011_QueryGraph/prob_00303_012082__15362834_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1474 ( 509 unt; 202 typ; 0 def)
% Number of atoms : 3800 (1255 equ; 0 cnn)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 12222 ( 493 ~; 40 |; 351 &;9441 @)
% ( 0 <=>;1897 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 19 ( 18 usr)
% Number of type conns : 954 ( 954 >; 0 *; 0 +; 0 <<)
% Number of symbols : 185 ( 184 usr; 16 con; 0-4 aty)
% Number of variables : 3435 ( 68 ^;3213 !; 154 ?;3435 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:06:56.548
%------------------------------------------------------------------------------
% Could-be-implicit typings (18)
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thf(ty_n_tf__b,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (184)
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thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
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thf(sy_c_Fun_Oinj__on_001tf__a_001tf__b,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J_M_Eo_J,type,
bot_bo8537066411596906360unit_o: pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__b_M_Eo_J,type,
bot_bot_b_o: b > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J_J,type,
bot_bo1839476491465656141t_unit: set_pr5411798346947241657t_unit ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
bot_bo8367695208629047834_ereal: set_Extended_ereal ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J_J_J,type,
bot_bo1540698489285124355t_unit: set_se2139339572462915695t_unit ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
bot_bot_set_b: set_b ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Real__Oereal,type,
ord_le1188267648640031866_ereal: extended_ereal > extended_ereal > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J_J,type,
ord_le2693654750756130573t_unit: set_pr5411798346947241657t_unit > set_pr5411798346947241657t_unit > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
ord_le5321083090456148570_ereal: set_Extended_ereal > set_Extended_ereal > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
ord_le6819997720685908915od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_less_set_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__b_J,type,
ord_less_set_b: set_b > set_b > $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__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_QueryGraph_Oquery__graph_001tf__a_001tf__b,type,
query_graph_a_b: pre_pr7278220950009878019t_unit > ( b > real ) > ( a > real ) > $o ).
thf(sy_c_QueryGraph_Oquery__graph_Omatch__sel_001tf__a_001tf__b,type,
query_match_sel_a_b: pre_pr7278220950009878019t_unit > ( b > real ) > a > a > real ).
thf(sy_c_QueryGraph_Oquery__graph_Omatching__sel_001tf__a_001tf__b,type,
query_5351540782772103094el_a_b: pre_pr7278220950009878019t_unit > ( b > real ) > ( a > a > real ) > $o ).
thf(sy_c_QueryGraph_Oquery__graph_Oremove__sel_001tf__a_001tf__b,type,
query_remove_sel_a_b: pre_pr7278220950009878019t_unit > ( b > real ) > a > b > real ).
thf(sy_c_Selectivities_Osel__reasonable_001tf__a,type,
sel_reasonable_a: ( a > a > real ) > $o ).
thf(sy_c_Selectivities_Osel__symm_001tf__a,type,
sel_symm_a: ( a > a > real ) > $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__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
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__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J,type,
image_7933780498232994317t_unit: ( pre_pr7278220950009878019t_unit > pre_pr7278220950009878019t_unit ) > set_pr5411798346947241657t_unit > set_pr5411798346947241657t_unit ).
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__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J_001tf__b,type,
image_4969699134812999797unit_b: ( pre_pr7278220950009878019t_unit > b ) > set_pr5411798346947241657t_unit > set_b ).
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_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__Set__Oset_It__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J_J_001t__Set__Oset_It__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J_J,type,
image_4554393186639800441t_unit: ( set_pr5411798346947241657t_unit > set_pr5411798346947241657t_unit ) > set_se2139339572462915695t_unit > set_se2139339572462915695t_unit ).
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_001t__Set__Oset_Itf__a_J_001tf__b,type,
image_set_a_b: ( set_a > b ) > set_set_a > set_b ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__b_J_001t__Set__Oset_Itf__b_J,type,
image_set_b_set_b: ( set_b > set_b ) > set_set_b > set_set_b ).
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__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__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J,type,
image_4434118323594779837t_unit: ( b > pre_pr7278220950009878019t_unit ) > set_b > set_pr5411798346947241657t_unit ).
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__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
image_6761185482258179565od_a_a: ( b > product_prod_a_a ) > set_b > set_Product_prod_a_a ).
thf(sy_c_Set_Oimage_001tf__b_001t__Set__Oset_Itf__a_J,type,
image_b_set_a: ( b > set_a ) > set_b > set_set_a ).
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__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
insert4534936382041156343od_a_a: product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
insert_real: real > set_real > set_real ).
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__Tree_Opre__digraph_Oleaf_001tf__a_001tf__b,type,
shorte1213025427933718126af_a_b: pre_pr7278220950009878019t_unit > a > $o ).
thf(sy_c_Shortest__Path__Tree_Osubgraph_001tf__a_001tf__b,type,
shorte3657265928840388360ph_a_b: pre_pr7278220950009878019t_unit > pre_pr7278220950009878019t_unit > $o ).
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__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J_J,type,
member5449360183034373072t_unit: set_pr5411798346947241657t_unit > set_se2139339572462915695t_unit > $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_cf,type,
cf: a > real ).
thf(sy_v_e,type,
e: b ).
thf(sy_v_sel,type,
sel: b > real ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (1271)
thf(fact_0_sel__leq__1,axiom,
! [E: b] : ( ord_less_eq_real @ ( sel @ E ) @ one_one_real ) ).
% sel_leq_1
thf(fact_1_pseudo__graph__axioms,axiom,
pseudo_graph_a_b @ g ).
% pseudo_graph_axioms
thf(fact_2_query__graph_Oremove__sel_Ocong,axiom,
query_remove_sel_a_b = query_remove_sel_a_b ).
% query_graph.remove_sel.cong
thf(fact_3_loopfree__digraph__axioms,axiom,
loopfree_digraph_a_b @ g ).
% loopfree_digraph_axioms
thf(fact_4_nomulti__digraph__axioms,axiom,
nomulti_digraph_a_b @ g ).
% nomulti_digraph_axioms
thf(fact_5_digraph__axioms,axiom,
digraph_a_b @ g ).
% digraph_axioms
thf(fact_6_graph__axioms,axiom,
graph_a_b @ g ).
% graph_axioms
thf(fact_7_remove__sel__1,axiom,
! [E: b,X: a] :
( ~ ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( query_remove_sel_a_b @ g @ sel @ X @ E )
= one_one_real ) ) ).
% remove_sel_1
thf(fact_8_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_9_matching__sel__unique,axiom,
! [F: a > a > real,G: a > a > real] :
( ( query_5351540782772103094el_a_b @ g @ sel @ F )
=> ( ( query_5351540782772103094el_a_b @ g @ sel @ G )
=> ( F = G ) ) ) ).
% matching_sel_unique
thf(fact_10_matching__sel__unique__aux,axiom,
! [F: a > a > real,G: a > a > real,X: a,Y: a] :
( ( query_5351540782772103094el_a_b @ g @ sel @ F )
=> ( ( query_5351540782772103094el_a_b @ g @ sel @ G )
=> ( ( F @ X @ Y )
= ( G @ X @ Y ) ) ) ) ).
% matching_sel_unique_aux
thf(fact_11_not__arc__sel__1,axiom,
! [E: b] :
( ~ ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( sel @ E )
= one_one_real ) ) ).
% not_arc_sel_1
thf(fact_12_sym__digraph__axioms,axiom,
sym_digraph_a_b @ g ).
% sym_digraph_axioms
thf(fact_13_scc__of__eq,axiom,
! [U: a,V: a] :
( ( member_a @ U @ ( digrap2937667069914300949of_a_b @ g @ V ) )
=> ( ( digrap2937667069914300949of_a_b @ g @ U )
= ( digrap2937667069914300949of_a_b @ g @ V ) ) ) ).
% scc_of_eq
thf(fact_14_query__graph_Omatching__sel_Ocong,axiom,
query_5351540782772103094el_a_b = query_5351540782772103094el_a_b ).
% query_graph.matching_sel.cong
thf(fact_15_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_16_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_17_matching__sel__symm,axiom,
! [F: a > a > real] :
( ( query_5351540782772103094el_a_b @ g @ sel @ F )
=> ( sel_symm_a @ F ) ) ).
% matching_sel_symm
thf(fact_18_match__sel__unique,axiom,
! [F: a > a > real] :
( ( query_5351540782772103094el_a_b @ g @ sel @ F )
=> ( F
= ( query_match_sel_a_b @ g @ sel ) ) ) ).
% match_sel_unique
thf(fact_19_arcs__add__vert,axiom,
! [U: a] :
( ( pre_ar1395965042833527383t_unit @ ( pre_add_vert_a_b @ g @ U ) )
= ( pre_ar1395965042833527383t_unit @ g ) ) ).
% arcs_add_vert
thf(fact_20_del__vert__remove__sel__1,axiom,
! [E: b,X: a] :
( ~ ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ ( pre_del_vert_a_b @ g @ X ) ) )
=> ( ( query_remove_sel_a_b @ g @ sel @ X @ E )
= one_one_real ) ) ).
% del_vert_remove_sel_1
thf(fact_21_matching__sel__reasonable,axiom,
! [F: a > a > real] :
( ( query_5351540782772103094el_a_b @ g @ sel @ F )
=> ( sel_reasonable_a @ F ) ) ).
% matching_sel_reasonable
thf(fact_22_sel__less__arc,axiom,
! [X: b] :
( ( ord_less_real @ ( sel @ X ) @ one_one_real )
=> ( member_b @ X @ ( pre_ar1395965042833527383t_unit @ g ) ) ) ).
% sel_less_arc
thf(fact_23_graph__del__vert,axiom,
! [X: a] : ( graph_a_b @ ( pre_del_vert_a_b @ g @ X ) ) ).
% graph_del_vert
thf(fact_24_graph_Ointro,axiom,
! [G2: pre_pr7278220950009878019t_unit] :
( ( digraph_a_b @ G2 )
=> ( ( pseudo_graph_a_b @ G2 )
=> ( graph_a_b @ G2 ) ) ) ).
% graph.intro
thf(fact_25_Digraph_Ograph__def,axiom,
( graph_a_b
= ( ^ [G3: pre_pr7278220950009878019t_unit] :
( ( digraph_a_b @ G3 )
& ( pseudo_graph_a_b @ G3 ) ) ) ) ).
% Digraph.graph_def
thf(fact_26_induced__graph__imp__graph,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ H @ g )
=> ( sym_digraph_a_b @ H ) ) ).
% induced_graph_imp_graph
thf(fact_27_digraphI__induced,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ H @ g )
=> ( digraph_a_b @ H ) ) ).
% digraphI_induced
thf(fact_28_no__multi__arcs,axiom,
! [E1: b,E2: b] :
( ( member_b @ E1 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( ( arc_to_ends_a_b @ g @ E1 )
= ( arc_to_ends_a_b @ g @ E2 ) )
=> ( E1 = E2 ) ) ) ) ).
% no_multi_arcs
thf(fact_29_induced__subgraph__refl,axiom,
digrap5251062021860773499ph_a_b @ g @ g ).
% induced_subgraph_refl
thf(fact_30_ends__del__vert,axiom,
! [U: a] :
( ( arc_to_ends_a_b @ ( pre_del_vert_a_b @ g @ U ) )
= ( arc_to_ends_a_b @ g ) ) ).
% ends_del_vert
thf(fact_31_match__sel__matching,axiom,
query_5351540782772103094el_a_b @ g @ sel @ ( query_match_sel_a_b @ g @ sel ) ).
% match_sel_matching
thf(fact_32_match__sel__reasonable,axiom,
sel_reasonable_a @ ( query_match_sel_a_b @ g @ sel ) ).
% match_sel_reasonable
thf(fact_33_match__sel__symm,axiom,
sel_symm_a @ ( query_match_sel_a_b @ g @ sel ) ).
% match_sel_symm
thf(fact_34_pre__digraph_Oends__del__vert,axiom,
! [G2: pre_pr7278220950009878019t_unit,U: a] :
( ( arc_to_ends_a_b @ ( pre_del_vert_a_b @ G2 @ U ) )
= ( arc_to_ends_a_b @ G2 ) ) ).
% pre_digraph.ends_del_vert
thf(fact_35_pre__digraph_Odel__vert_Ocong,axiom,
pre_del_vert_a_b = pre_del_vert_a_b ).
% pre_digraph.del_vert.cong
thf(fact_36_pre__digraph_Oadd__vert_Ocong,axiom,
pre_add_vert_a_b = pre_add_vert_a_b ).
% pre_digraph.add_vert.cong
thf(fact_37_query__graph_Omatch__sel_Ocong,axiom,
query_match_sel_a_b = query_match_sel_a_b ).
% query_graph.match_sel.cong
thf(fact_38_pre__digraph_Oarcs__add__vert,axiom,
! [G2: pre_pr7278220950009878019t_unit,U: a] :
( ( pre_ar1395965042833527383t_unit @ ( pre_add_vert_a_b @ G2 @ U ) )
= ( pre_ar1395965042833527383t_unit @ G2 ) ) ).
% pre_digraph.arcs_add_vert
thf(fact_39_nomulti__digraph_Ono__multi__arcs,axiom,
! [G2: pre_pr7278220950009878019t_unit,E1: b,E2: b] :
( ( nomulti_digraph_a_b @ G2 )
=> ( ( member_b @ E1 @ ( pre_ar1395965042833527383t_unit @ G2 ) )
=> ( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ G2 ) )
=> ( ( ( arc_to_ends_a_b @ G2 @ E1 )
= ( arc_to_ends_a_b @ G2 @ E2 ) )
=> ( E1 = E2 ) ) ) ) ) ).
% nomulti_digraph.no_multi_arcs
thf(fact_40_nomulti__digraph_Onomulti__digraph,axiom,
! [G2: pre_pr7278220950009878019t_unit] :
( ( nomulti_digraph_a_b @ G2 )
=> ( nomulti_digraph_a_b @ G2 ) ) ).
% nomulti_digraph.nomulti_digraph
thf(fact_41_graph_Oaxioms_I1_J,axiom,
! [G2: pre_pr7278220950009878019t_unit] :
( ( graph_a_b @ G2 )
=> ( digraph_a_b @ G2 ) ) ).
% graph.axioms(1)
thf(fact_42_digraph_Oaxioms_I2_J,axiom,
! [G2: pre_pr7278220950009878019t_unit] :
( ( digraph_a_b @ G2 )
=> ( loopfree_digraph_a_b @ G2 ) ) ).
% digraph.axioms(2)
thf(fact_43_digraph_Oaxioms_I3_J,axiom,
! [G2: pre_pr7278220950009878019t_unit] :
( ( digraph_a_b @ G2 )
=> ( nomulti_digraph_a_b @ G2 ) ) ).
% digraph.axioms(3)
thf(fact_44_mem__Collect__eq,axiom,
! [A: b,P: b > $o] :
( ( member_b @ A @ ( collect_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_45_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_46_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_47_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_48_Collect__mem__eq,axiom,
! [A2: set_b] :
( ( collect_b
@ ^ [X2: b] : ( member_b @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ( collec8000012497822511960t_unit
@ ^ [X2: pre_pr7278220950009878019t_unit] : ( member6939884229742472986t_unit @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_52_pseudo__graph_Oaxioms_I2_J,axiom,
! [G2: pre_pr7278220950009878019t_unit] :
( ( pseudo_graph_a_b @ G2 )
=> ( sym_digraph_a_b @ G2 ) ) ).
% pseudo_graph.axioms(2)
thf(fact_53_graph_Oaxioms_I2_J,axiom,
! [G2: pre_pr7278220950009878019t_unit] :
( ( graph_a_b @ G2 )
=> ( pseudo_graph_a_b @ G2 ) ) ).
% graph.axioms(2)
thf(fact_54_remove__sel__pos,axiom,
! [X: a,E: b] : ( ord_less_real @ zero_zero_real @ ( query_remove_sel_a_b @ g @ sel @ X @ E ) ) ).
% remove_sel_pos
thf(fact_55_del__vert__add__vert,axiom,
! [U: a] :
( ~ ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( pre_del_vert_a_b @ ( pre_add_vert_a_b @ g @ U ) @ U )
= g ) ) ).
% del_vert_add_vert
thf(fact_56_sel__pos,axiom,
! [E: b] : ( ord_less_real @ zero_zero_real @ ( sel @ E ) ) ).
% sel_pos
thf(fact_57_add__arc__in,axiom,
! [A: b] :
( ( member_b @ A @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( pre_add_arc_a_b @ g @ A )
= g ) ) ).
% add_arc_in
thf(fact_58_inj__on__arc__to__ends,axiom,
inj_on8302219782909765977od_a_a @ ( arc_to_ends_a_b @ g ) @ ( pre_ar1395965042833527383t_unit @ g ) ).
% inj_on_arc_to_ends
thf(fact_59_in__sccs__imp__induced,axiom,
! [C: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ g ) )
=> ( digrap5251062021860773499ph_a_b @ C @ g ) ) ).
% in_sccs_imp_induced
thf(fact_60_digraph_OdigraphI__induced,axiom,
! [G2: pre_pr7278220950009878019t_unit,H: pre_pr7278220950009878019t_unit] :
( ( digraph_a_b @ G2 )
=> ( ( digrap5251062021860773499ph_a_b @ H @ G2 )
=> ( digraph_a_b @ H ) ) ) ).
% digraph.digraphI_induced
thf(fact_61_graph_Ograph__del__vert,axiom,
! [G2: pre_pr7278220950009878019t_unit,X: a] :
( ( graph_a_b @ G2 )
=> ( graph_a_b @ ( pre_del_vert_a_b @ G2 @ X ) ) ) ).
% graph.graph_del_vert
thf(fact_62_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_63_sym__digraph_Oinduced__graph__imp__graph,axiom,
! [G2: pre_pr7278220950009878019t_unit,H: pre_pr7278220950009878019t_unit] :
( ( sym_digraph_a_b @ G2 )
=> ( ( digrap5251062021860773499ph_a_b @ H @ G2 )
=> ( sym_digraph_a_b @ H ) ) ) ).
% sym_digraph.induced_graph_imp_graph
thf(fact_64_k__nh__finite,axiom,
! [W: b > real,V: a,K: real] : ( finite_finite_a @ ( graph_3921080825633621230od_a_b @ g @ W @ V @ K ) ) ).
% k_nh_finite
thf(fact_65_del__arc__commute,axiom,
! [B: b,A: b] :
( ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ g @ B ) @ A )
= ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ g @ A ) @ B ) ) ).
% del_arc_commute
thf(fact_66_add__arc__commute,axiom,
! [B: b,A: b] :
( ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ g @ B ) @ A )
= ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ g @ A ) @ B ) ) ).
% add_arc_commute
thf(fact_67_scc__for__vert__ex,axiom,
! [U: a] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
=> ? [C2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( digraph_pre_sccs_a_b @ g ) )
& ( member_a @ U @ ( pre_ve642382030648772252t_unit @ C2 ) ) ) ) ).
% scc_for_vert_ex
thf(fact_68_in__scc__of__self,axiom,
! [U: a] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( member_a @ U @ ( digrap2937667069914300949of_a_b @ g @ U ) ) ) ).
% in_scc_of_self
thf(fact_69_finite__verts,axiom,
finite_finite_a @ ( pre_ve642382030648772252t_unit @ g ) ).
% finite_verts
thf(fact_70_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_71_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_72_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_73_add__add__arc__collapse,axiom,
! [A: b] :
( ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ g @ A ) @ A )
= ( pre_add_arc_a_b @ g @ A ) ) ).
% add_add_arc_collapse
thf(fact_74_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_75_add__del__arc__collapse,axiom,
! [A: b] :
( ( pre_add_arc_a_b @ ( pre_del_arc_a_b @ g @ A ) @ A )
= ( pre_add_arc_a_b @ g @ A ) ) ).
% add_del_arc_collapse
thf(fact_76_graph__is__union__sccs,axiom,
( ( digrap6752193522309670266on_a_b @ g @ ( digraph_pre_sccs_a_b @ g ) )
= g ) ).
% graph_is_union_sccs
thf(fact_77_pre__digraph_Odel__del__arc__collapse,axiom,
! [G2: pre_pr7278220950009878019t_unit,A: b] :
( ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ G2 @ A ) @ A )
= ( pre_del_arc_a_b @ G2 @ A ) ) ).
% pre_digraph.del_del_arc_collapse
thf(fact_78_pre__digraph_Oadd__del__arc__collapse,axiom,
! [G2: pre_pr7278220950009878019t_unit,A: b] :
( ( pre_add_arc_a_b @ ( pre_del_arc_a_b @ G2 @ A ) @ A )
= ( pre_add_arc_a_b @ G2 @ A ) ) ).
% pre_digraph.add_del_arc_collapse
thf(fact_79_pre__digraph_Oadd__add__arc__collapse,axiom,
! [G2: pre_pr7278220950009878019t_unit,A: b] :
( ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ G2 @ A ) @ A )
= ( pre_add_arc_a_b @ G2 @ A ) ) ).
% pre_digraph.add_add_arc_collapse
thf(fact_80_pre__digraph_Odel__arc__commute,axiom,
! [G2: pre_pr7278220950009878019t_unit,B: b,A: b] :
( ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ G2 @ B ) @ A )
= ( pre_del_arc_a_b @ ( pre_del_arc_a_b @ G2 @ A ) @ B ) ) ).
% pre_digraph.del_arc_commute
thf(fact_81_pre__digraph_Oadd__arc__commute,axiom,
! [G2: pre_pr7278220950009878019t_unit,B: b,A: b] :
( ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ G2 @ B ) @ A )
= ( pre_add_arc_a_b @ ( pre_add_arc_a_b @ G2 @ A ) @ B ) ) ).
% pre_digraph.add_arc_commute
thf(fact_82_pre__digraph_Overts__del__arc,axiom,
! [G2: pre_pr7278220950009878019t_unit,A: b] :
( ( pre_ve642382030648772252t_unit @ ( pre_del_arc_a_b @ G2 @ A ) )
= ( pre_ve642382030648772252t_unit @ G2 ) ) ).
% pre_digraph.verts_del_arc
thf(fact_83_pre__digraph_Odel__arc_Ocong,axiom,
pre_del_arc_a_b = pre_del_arc_a_b ).
% pre_digraph.del_arc.cong
thf(fact_84_pre__digraph_Oadd__arc_Ocong,axiom,
pre_add_arc_a_b = pre_add_arc_a_b ).
% pre_digraph.add_arc.cong
thf(fact_85_pre__digraph_Osccs_Ocong,axiom,
digraph_pre_sccs_a_b = digraph_pre_sccs_a_b ).
% pre_digraph.sccs.cong
thf(fact_86_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_87_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_88_sym__digraph_Oscc__for__vert__ex,axiom,
! [G2: pre_pr7278220950009878019t_unit,U: a] :
( ( sym_digraph_a_b @ G2 )
=> ( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ G2 ) )
=> ? [C2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( digraph_pre_sccs_a_b @ G2 ) )
& ( member_a @ U @ ( pre_ve642382030648772252t_unit @ C2 ) ) ) ) ) ).
% sym_digraph.scc_for_vert_ex
thf(fact_89_induced__eq__verts__imp__eq,axiom,
! [G2: pre_pr7278220950009878019t_unit,H: pre_pr7278220950009878019t_unit,G4: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ G2 @ H )
=> ( ( digrap5251062021860773499ph_a_b @ G4 @ H )
=> ( ( ( pre_ve642382030648772252t_unit @ G2 )
= ( pre_ve642382030648772252t_unit @ G4 ) )
=> ( G2 = G4 ) ) ) ) ).
% induced_eq_verts_imp_eq
thf(fact_90_pre__digraph_Oin__sccs__imp__induced,axiom,
! [C: pre_pr7278220950009878019t_unit,G2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ G2 ) )
=> ( digrap5251062021860773499ph_a_b @ C @ G2 ) ) ).
% pre_digraph.in_sccs_imp_induced
thf(fact_91_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_92_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_93_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_94_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_95_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_96_pre__digraph_Odel__arc__in,axiom,
! [A: b,G2: pre_pr7278220950009878019t_unit] :
( ~ ( member_b @ A @ ( pre_ar1395965042833527383t_unit @ G2 ) )
=> ( ( pre_del_arc_a_b @ G2 @ A )
= G2 ) ) ).
% pre_digraph.del_arc_in
thf(fact_97_nomulti__digraph_Oinj__on__arc__to__ends,axiom,
! [G2: pre_pr7278220950009878019t_unit] :
( ( nomulti_digraph_a_b @ G2 )
=> ( inj_on8302219782909765977od_a_a @ ( arc_to_ends_a_b @ G2 ) @ ( pre_ar1395965042833527383t_unit @ G2 ) ) ) ).
% nomulti_digraph.inj_on_arc_to_ends
thf(fact_98_pre__digraph_Oscc__of_Ocong,axiom,
digrap2937667069914300949of_a_b = digrap2937667069914300949of_a_b ).
% pre_digraph.scc_of.cong
thf(fact_99_pos__cards,axiom,
! [X: a] :
( ( member_a @ X @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ord_less_real @ zero_zero_real @ ( cf @ X ) ) ) ).
% pos_cards
thf(fact_100_merge__in__verts,axiom,
! [X: a] :
( ( member_a @ X @ ( graph_2957805489637798020ts_a_b @ g ) )
=> ( member_a @ X @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% merge_in_verts
thf(fact_101_sel__reasonable__def,axiom,
( sel_reasonable_a
= ( ^ [Sel: a > a > real] :
! [X2: a,Y2: a] :
( ( ord_less_eq_real @ ( Sel @ X2 @ Y2 ) @ one_one_real )
& ( ord_less_real @ zero_zero_real @ ( Sel @ X2 @ Y2 ) ) ) ) ) ).
% sel_reasonable_def
thf(fact_102_scc__of__in__sccs__verts,axiom,
! [U: a] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( member_set_a @ ( digrap2937667069914300949of_a_b @ g @ U ) @ ( digrap2871191568752656621ts_a_b @ g ) ) ) ).
% scc_of_in_sccs_verts
thf(fact_103_scc__of__empty__conv,axiom,
! [U: a] :
( ( ( digrap2937667069914300949of_a_b @ g @ U )
= bot_bot_set_a )
= ( ~ ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) ) ) ) ).
% scc_of_empty_conv
thf(fact_104_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_105_exists__scc,axiom,
( ( ( pre_ve642382030648772252t_unit @ g )
!= bot_bot_set_a )
=> ? [C2: pre_pr7278220950009878019t_unit] : ( member6939884229742472986t_unit @ C2 @ ( digraph_pre_sccs_a_b @ g ) ) ) ).
% exists_scc
thf(fact_106_in__sccs__subset__imp__eq,axiom,
! [C: pre_pr7278220950009878019t_unit,D: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ g ) )
=> ( ( member6939884229742472986t_unit @ D @ ( digraph_pre_sccs_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ D ) )
=> ( C = D ) ) ) ) ).
% in_sccs_subset_imp_eq
thf(fact_107_verts__add__vert,axiom,
! [U: a] :
( ( pre_ve642382030648772252t_unit @ ( pre_add_vert_a_b @ g @ U ) )
= ( insert_a @ U @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% verts_add_vert
thf(fact_108_branch__in__verts,axiom,
! [X: a] :
( ( member_a @ X @ ( graph_4596510882073158607ts_a_b @ g ) )
=> ( member_a @ X @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% branch_in_verts
thf(fact_109_inj__on__verts__sccs,axiom,
inj_on6672326154762249576_set_a @ pre_ve642382030648772252t_unit @ ( digraph_pre_sccs_a_b @ g ) ).
% inj_on_verts_sccs
thf(fact_110_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_111_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_112_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_113_scc__decomp__unique,axiom,
! [S: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ S @ ( digraph_pre_sccs_a_b @ g ) )
=> ( ( ( pre_ve642382030648772252t_unit @ ( digrap6752193522309670266on_a_b @ g @ S ) )
= ( pre_ve642382030648772252t_unit @ g ) )
=> ( S
= ( digraph_pre_sccs_a_b @ g ) ) ) ) ).
% scc_decomp_unique
thf(fact_114_inj__on__empty,axiom,
! [F: b > product_prod_a_a] : ( inj_on8302219782909765977od_a_a @ F @ bot_bot_set_b ) ).
% inj_on_empty
thf(fact_115_inj__on__empty,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a] : ( inj_on6672326154762249576_set_a @ F @ bot_bo1839476491465656141t_unit ) ).
% inj_on_empty
thf(fact_116_induce__subgraph__verts,axiom,
! [G2: pre_pr7278220950009878019t_unit,Vs: set_a] :
( ( pre_ve642382030648772252t_unit @ ( digrap7873285959652527175ph_a_b @ G2 @ Vs ) )
= Vs ) ).
% induce_subgraph_verts
thf(fact_117_induce__subgraph__ends,axiom,
! [G2: pre_pr7278220950009878019t_unit,S: set_a] :
( ( arc_to_ends_a_b @ ( digrap7873285959652527175ph_a_b @ G2 @ S ) )
= ( arc_to_ends_a_b @ G2 ) ) ).
% induce_subgraph_ends
thf(fact_118_is__chain__def,axiom,
( ( graph_3890552050688490787in_a_b @ g )
= ( ( graph_4596510882073158607ts_a_b @ g )
= bot_bot_set_a ) ) ).
% is_chain_def
thf(fact_119_is__chain_H__def,axiom,
( ( graph_8150681439568091980in_a_b @ g )
= ( ( graph_2957805489637798020ts_a_b @ g )
= bot_bot_set_a ) ) ).
% is_chain'_def
thf(fact_120_last__merge__is__merge,axiom,
! [Y: a] :
( ( member_a @ Y @ ( graph_2659413520663303054ts_a_b @ g ) )
=> ( member_a @ Y @ ( graph_2957805489637798020ts_a_b @ g ) ) ) ).
% last_merge_is_merge
thf(fact_121_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_122_query__graph__axioms,axiom,
query_graph_a_b @ g @ sel @ cf ).
% query_graph_axioms
thf(fact_123_pre__digraph_Osccs__verts_Ocong,axiom,
digrap2871191568752656621ts_a_b = digrap2871191568752656621ts_a_b ).
% pre_digraph.sccs_verts.cong
thf(fact_124_wf__digraph_Omerging__points_Ocong,axiom,
graph_2957805489637798020ts_a_b = graph_2957805489637798020ts_a_b ).
% wf_digraph.merging_points.cong
thf(fact_125_wf__digraph_Obranching__points_Ocong,axiom,
graph_4596510882073158607ts_a_b = graph_4596510882073158607ts_a_b ).
% wf_digraph.branching_points.cong
thf(fact_126_subset__inj__on,axiom,
! [F: b > product_prod_a_a,B2: set_b,A2: set_b] :
( ( inj_on8302219782909765977od_a_a @ F @ B2 )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( inj_on8302219782909765977od_a_a @ F @ A2 ) ) ) ).
% subset_inj_on
thf(fact_127_subset__inj__on,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,B2: set_pr5411798346947241657t_unit,A2: set_pr5411798346947241657t_unit] :
( ( inj_on6672326154762249576_set_a @ F @ B2 )
=> ( ( ord_le8200006823705900825t_unit @ A2 @ B2 )
=> ( inj_on6672326154762249576_set_a @ F @ A2 ) ) ) ).
% subset_inj_on
thf(fact_128_inj__on__subset,axiom,
! [F: b > product_prod_a_a,A2: set_b,B2: set_b] :
( ( inj_on8302219782909765977od_a_a @ F @ A2 )
=> ( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( inj_on8302219782909765977od_a_a @ F @ B2 ) ) ) ).
% inj_on_subset
thf(fact_129_inj__on__subset,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( inj_on6672326154762249576_set_a @ F @ A2 )
=> ( ( ord_le8200006823705900825t_unit @ B2 @ A2 )
=> ( inj_on6672326154762249576_set_a @ F @ B2 ) ) ) ).
% inj_on_subset
thf(fact_130_pre__digraph_OUnion_Ocong,axiom,
digrap6752193522309670266on_a_b = digrap6752193522309670266on_a_b ).
% pre_digraph.Union.cong
thf(fact_131_pre__digraph_Oarcs__add__arc,axiom,
! [G2: pre_pr7278220950009878019t_unit,A: b] :
( ( pre_ar1395965042833527383t_unit @ ( pre_add_arc_a_b @ G2 @ A ) )
= ( insert_b @ A @ ( pre_ar1395965042833527383t_unit @ G2 ) ) ) ).
% pre_digraph.arcs_add_arc
thf(fact_132_pre__digraph_Oin__sccs__subset__imp__eq,axiom,
! [C: pre_pr7278220950009878019t_unit,G2: pre_pr7278220950009878019t_unit,D: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ G2 ) )
=> ( ( member6939884229742472986t_unit @ D @ ( digraph_pre_sccs_a_b @ G2 ) )
=> ( ( ord_less_eq_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ D ) )
=> ( C = D ) ) ) ) ).
% pre_digraph.in_sccs_subset_imp_eq
thf(fact_133_pre__digraph_Overts__add__vert,axiom,
! [G2: pre_pr7278220950009878019t_unit,U: a] :
( ( pre_ve642382030648772252t_unit @ ( pre_add_vert_a_b @ G2 @ U ) )
= ( insert_a @ U @ ( pre_ve642382030648772252t_unit @ G2 ) ) ) ).
% pre_digraph.verts_add_vert
thf(fact_134_sym__digraph_Oscc__decomp__unique,axiom,
! [G2: pre_pr7278220950009878019t_unit,S: set_pr5411798346947241657t_unit] :
( ( sym_digraph_a_b @ G2 )
=> ( ( ord_le8200006823705900825t_unit @ S @ ( digraph_pre_sccs_a_b @ G2 ) )
=> ( ( ( pre_ve642382030648772252t_unit @ ( digrap6752193522309670266on_a_b @ G2 @ S ) )
= ( pre_ve642382030648772252t_unit @ G2 ) )
=> ( S
= ( digraph_pre_sccs_a_b @ G2 ) ) ) ) ) ).
% sym_digraph.scc_decomp_unique
thf(fact_135_inj__on__inverseI,axiom,
! [A2: set_b,G: product_prod_a_a > b,F: b > product_prod_a_a] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on8302219782909765977od_a_a @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_136_inj__on__inverseI,axiom,
! [A2: set_pr5411798346947241657t_unit,G: set_a > pre_pr7278220950009878019t_unit,F: pre_pr7278220950009878019t_unit > set_a] :
( ! [X3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ A2 )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on6672326154762249576_set_a @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_137_inj__on__contraD,axiom,
! [F: b > product_prod_a_a,A2: set_b,X: b,Y: b] :
( ( inj_on8302219782909765977od_a_a @ F @ A2 )
=> ( ( X != Y )
=> ( ( member_b @ X @ A2 )
=> ( ( member_b @ Y @ A2 )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_138_inj__on__contraD,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,X: pre_pr7278220950009878019t_unit,Y: pre_pr7278220950009878019t_unit] :
( ( inj_on6672326154762249576_set_a @ F @ A2 )
=> ( ( X != Y )
=> ( ( member6939884229742472986t_unit @ X @ A2 )
=> ( ( member6939884229742472986t_unit @ Y @ A2 )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_139_inj__on__eq__iff,axiom,
! [F: b > product_prod_a_a,A2: set_b,X: b,Y: b] :
( ( inj_on8302219782909765977od_a_a @ F @ A2 )
=> ( ( member_b @ X @ A2 )
=> ( ( member_b @ Y @ A2 )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_140_inj__on__eq__iff,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,X: pre_pr7278220950009878019t_unit,Y: pre_pr7278220950009878019t_unit] :
( ( inj_on6672326154762249576_set_a @ F @ A2 )
=> ( ( member6939884229742472986t_unit @ X @ A2 )
=> ( ( member6939884229742472986t_unit @ Y @ A2 )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_141_inj__on__cong,axiom,
! [A2: set_b,F: b > product_prod_a_a,G: b > product_prod_a_a] :
( ! [A3: b] :
( ( member_b @ A3 @ A2 )
=> ( ( F @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on8302219782909765977od_a_a @ F @ A2 )
= ( inj_on8302219782909765977od_a_a @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_142_inj__on__cong,axiom,
! [A2: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > set_a,G: pre_pr7278220950009878019t_unit > set_a] :
( ! [A3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ A3 @ A2 )
=> ( ( F @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on6672326154762249576_set_a @ F @ A2 )
= ( inj_on6672326154762249576_set_a @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_143_inj__on__def,axiom,
( inj_on8302219782909765977od_a_a
= ( ^ [F2: b > product_prod_a_a,A4: set_b] :
! [X2: b] :
( ( member_b @ X2 @ A4 )
=> ! [Y2: b] :
( ( member_b @ Y2 @ A4 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
=> ( X2 = Y2 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_144_inj__on__def,axiom,
( inj_on6672326154762249576_set_a
= ( ^ [F2: pre_pr7278220950009878019t_unit > set_a,A4: set_pr5411798346947241657t_unit] :
! [X2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X2 @ A4 )
=> ! [Y2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ Y2 @ A4 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
=> ( X2 = Y2 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_145_inj__onI,axiom,
! [A2: set_b,F: b > product_prod_a_a] :
( ! [X3: b,Y3: b] :
( ( member_b @ X3 @ A2 )
=> ( ( member_b @ Y3 @ A2 )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) )
=> ( inj_on8302219782909765977od_a_a @ F @ A2 ) ) ).
% inj_onI
thf(fact_146_inj__onI,axiom,
! [A2: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > set_a] :
( ! [X3: pre_pr7278220950009878019t_unit,Y3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ A2 )
=> ( ( member6939884229742472986t_unit @ Y3 @ A2 )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) )
=> ( inj_on6672326154762249576_set_a @ F @ A2 ) ) ).
% inj_onI
thf(fact_147_inj__onD,axiom,
! [F: b > product_prod_a_a,A2: set_b,X: b,Y: b] :
( ( inj_on8302219782909765977od_a_a @ F @ A2 )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_b @ X @ A2 )
=> ( ( member_b @ Y @ A2 )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_148_inj__onD,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,X: pre_pr7278220950009878019t_unit,Y: pre_pr7278220950009878019t_unit] :
( ( inj_on6672326154762249576_set_a @ F @ A2 )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member6939884229742472986t_unit @ X @ A2 )
=> ( ( member6939884229742472986t_unit @ Y @ A2 )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_149_sym__digraph_Oexists__scc,axiom,
! [G2: pre_pr7278220950009878019t_unit] :
( ( sym_digraph_a_b @ G2 )
=> ( ( ( pre_ve642382030648772252t_unit @ G2 )
!= bot_bot_set_a )
=> ? [C2: pre_pr7278220950009878019t_unit] : ( member6939884229742472986t_unit @ C2 @ ( digraph_pre_sccs_a_b @ G2 ) ) ) ) ).
% sym_digraph.exists_scc
thf(fact_150_sel__symm__def,axiom,
( sel_symm_a
= ( ^ [Sel: a > a > real] :
! [X2: a,Y2: a] :
( ( Sel @ X2 @ Y2 )
= ( Sel @ Y2 @ X2 ) ) ) ) ).
% sel_symm_def
thf(fact_151_sym__digraph_Ograph__is__union__sccs,axiom,
! [G2: pre_pr7278220950009878019t_unit] :
( ( sym_digraph_a_b @ G2 )
=> ( ( digrap6752193522309670266on_a_b @ G2 @ ( digraph_pre_sccs_a_b @ G2 ) )
= G2 ) ) ).
% sym_digraph.graph_is_union_sccs
thf(fact_152_last__branch__is__branch,axiom,
! [Y: a] :
( ( member_a @ Y @ ( graph_1747835947655717337ts_a_b @ g ) )
=> ( member_a @ Y @ ( graph_4596510882073158607ts_a_b @ g ) ) ) ).
% last_branch_is_branch
thf(fact_153_merge__in__supergraph,axiom,
! [C3: pre_pr7278220950009878019t_unit,X: a] :
( ( shorte3657265928840388360ph_a_b @ C3 @ g )
=> ( ( member_a @ X @ ( graph_2957805489637798020ts_a_b @ C3 ) )
=> ( member_a @ X @ ( graph_2957805489637798020ts_a_b @ g ) ) ) ) ).
% merge_in_supergraph
thf(fact_154_branch__in__supergraph,axiom,
! [C3: pre_pr7278220950009878019t_unit,X: a] :
( ( shorte3657265928840388360ph_a_b @ C3 @ g )
=> ( ( member_a @ X @ ( graph_4596510882073158607ts_a_b @ C3 ) )
=> ( member_a @ X @ ( graph_4596510882073158607ts_a_b @ g ) ) ) ) ).
% branch_in_supergraph
thf(fact_155_singleton__insert__inj__eq_H,axiom,
! [A: b,A2: set_b,B: b] :
( ( ( insert_b @ A @ A2 )
= ( insert_b @ B @ bot_bot_set_b ) )
= ( ( A = B )
& ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_156_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_157_singleton__insert__inj__eq_H,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B: pre_pr7278220950009878019t_unit] :
( ( ( insert6864688055023459379t_unit @ A @ A2 )
= ( insert6864688055023459379t_unit @ B @ bot_bo1839476491465656141t_unit ) )
= ( ( A = B )
& ( ord_le8200006823705900825t_unit @ A2 @ ( insert6864688055023459379t_unit @ B @ bot_bo1839476491465656141t_unit ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_158_singleton__insert__inj__eq,axiom,
! [B: b,A: b,A2: set_b] :
( ( ( insert_b @ B @ bot_bot_set_b )
= ( insert_b @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_159_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A2: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_160_singleton__insert__inj__eq,axiom,
! [B: pre_pr7278220950009878019t_unit,A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( ( insert6864688055023459379t_unit @ B @ bot_bo1839476491465656141t_unit )
= ( insert6864688055023459379t_unit @ A @ A2 ) )
= ( ( A = B )
& ( ord_le8200006823705900825t_unit @ A2 @ ( insert6864688055023459379t_unit @ B @ bot_bo1839476491465656141t_unit ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_161_insert__subset,axiom,
! [X: b,A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ ( insert_b @ X @ A2 ) @ B2 )
= ( ( member_b @ X @ B2 )
& ( ord_less_eq_set_b @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_162_insert__subset,axiom,
! [X: set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A2 ) @ B2 )
= ( ( member_set_a @ X @ B2 )
& ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_163_insert__subset,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( ( member_a @ X @ B2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_164_insert__subset,axiom,
! [X: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ ( insert6864688055023459379t_unit @ X @ A2 ) @ B2 )
= ( ( member6939884229742472986t_unit @ X @ B2 )
& ( ord_le8200006823705900825t_unit @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_165_finite__insert,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( finite8852549406693098522t_unit @ ( insert6864688055023459379t_unit @ A @ A2 ) )
= ( finite8852549406693098522t_unit @ A2 ) ) ).
% finite_insert
thf(fact_166_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_167_finite__insert,axiom,
! [A: b,A2: set_b] :
( ( finite_finite_b @ ( insert_b @ A @ A2 ) )
= ( finite_finite_b @ A2 ) ) ).
% finite_insert
thf(fact_168_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_169_finite__insert,axiom,
! [A: extended_ereal,A2: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ ( insert8967887681552722334_ereal @ A @ A2 ) )
= ( finite7198162374296863863_ereal @ A2 ) ) ).
% finite_insert
thf(fact_170_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_171_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_172_singletonI,axiom,
! [A: b] : ( member_b @ A @ ( insert_b @ A @ bot_bot_set_b ) ) ).
% singletonI
thf(fact_173_singletonI,axiom,
! [A: pre_pr7278220950009878019t_unit] : ( member6939884229742472986t_unit @ A @ ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) ) ).
% singletonI
thf(fact_174_empty__subsetI,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A2 ) ).
% empty_subsetI
thf(fact_175_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_176_empty__subsetI,axiom,
! [A2: set_pr5411798346947241657t_unit] : ( ord_le8200006823705900825t_unit @ bot_bo1839476491465656141t_unit @ A2 ) ).
% empty_subsetI
thf(fact_177_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_178_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_179_subset__empty,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ bot_bo1839476491465656141t_unit )
= ( A2 = bot_bo1839476491465656141t_unit ) ) ).
% subset_empty
thf(fact_180_finite__arcs,axiom,
finite_finite_b @ ( pre_ar1395965042833527383t_unit @ g ) ).
% finite_arcs
thf(fact_181_finite__sccs__verts,axiom,
finite_finite_set_a @ ( digrap2871191568752656621ts_a_b @ g ) ).
% finite_sccs_verts
thf(fact_182_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_183_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_184_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_185_empty__iff,axiom,
! [C: pre_pr7278220950009878019t_unit] :
~ ( member6939884229742472986t_unit @ C @ bot_bo1839476491465656141t_unit ) ).
% empty_iff
thf(fact_186_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X2: set_a] :
~ ( member_set_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_187_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_188_all__not__in__conv,axiom,
! [A2: set_b] :
( ( ! [X2: b] :
~ ( member_b @ X2 @ A2 ) )
= ( A2 = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_189_all__not__in__conv,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ( ! [X2: pre_pr7278220950009878019t_unit] :
~ ( member6939884229742472986t_unit @ X2 @ A2 ) )
= ( A2 = bot_bo1839476491465656141t_unit ) ) ).
% all_not_in_conv
thf(fact_190_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_191_Collect__empty__eq,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( ! [X2: b] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_192_Collect__empty__eq,axiom,
! [P: pre_pr7278220950009878019t_unit > $o] :
( ( ( collec8000012497822511960t_unit @ P )
= bot_bo1839476491465656141t_unit )
= ( ! [X2: pre_pr7278220950009878019t_unit] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_193_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_194_empty__Collect__eq,axiom,
! [P: b > $o] :
( ( bot_bot_set_b
= ( collect_b @ P ) )
= ( ! [X2: b] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_195_empty__Collect__eq,axiom,
! [P: pre_pr7278220950009878019t_unit > $o] :
( ( bot_bo1839476491465656141t_unit
= ( collec8000012497822511960t_unit @ P ) )
= ( ! [X2: pre_pr7278220950009878019t_unit] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_196_psubsetI,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_197_psubsetI,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_le2693654750756130573t_unit @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_198_subsetI,axiom,
! [A2: set_b,B2: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( member_b @ X3 @ B2 ) )
=> ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% subsetI
thf(fact_199_subsetI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( member_set_a @ X3 @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_200_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_201_subsetI,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ! [X3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ A2 )
=> ( member6939884229742472986t_unit @ X3 @ B2 ) )
=> ( ord_le8200006823705900825t_unit @ A2 @ B2 ) ) ).
% subsetI
thf(fact_202_subset__antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_203_subset__antisym,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ B2 )
=> ( ( ord_le8200006823705900825t_unit @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_204_insertCI,axiom,
! [A: b,B2: set_b,B: b] :
( ( ~ ( member_b @ A @ B2 )
=> ( A = B ) )
=> ( member_b @ A @ ( insert_b @ B @ B2 ) ) ) ).
% insertCI
thf(fact_205_insertCI,axiom,
! [A: a,B2: set_a,B: a] :
( ( ~ ( member_a @ A @ B2 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_206_insertCI,axiom,
! [A: pre_pr7278220950009878019t_unit,B2: set_pr5411798346947241657t_unit,B: pre_pr7278220950009878019t_unit] :
( ( ~ ( member6939884229742472986t_unit @ A @ B2 )
=> ( A = B ) )
=> ( member6939884229742472986t_unit @ A @ ( insert6864688055023459379t_unit @ B @ B2 ) ) ) ).
% insertCI
thf(fact_207_insertCI,axiom,
! [A: set_a,B2: set_set_a,B: set_a] :
( ( ~ ( member_set_a @ A @ B2 )
=> ( A = B ) )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_208_insert__iff,axiom,
! [A: b,B: b,A2: set_b] :
( ( member_b @ A @ ( insert_b @ B @ A2 ) )
= ( ( A = B )
| ( member_b @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_209_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_210_insert__iff,axiom,
! [A: pre_pr7278220950009878019t_unit,B: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ A @ ( insert6864688055023459379t_unit @ B @ A2 ) )
= ( ( A = B )
| ( member6939884229742472986t_unit @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_211_insert__iff,axiom,
! [A: set_a,B: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A2 ) )
= ( ( A = B )
| ( member_set_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_212_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_213_insert__absorb2,axiom,
! [X: b,A2: set_b] :
( ( insert_b @ X @ ( insert_b @ X @ A2 ) )
= ( insert_b @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_214_insert__absorb2,axiom,
! [X: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( insert6864688055023459379t_unit @ X @ ( insert6864688055023459379t_unit @ X @ A2 ) )
= ( insert6864688055023459379t_unit @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_215_arcs__add__arc,axiom,
! [A: b] :
( ( pre_ar1395965042833527383t_unit @ ( pre_add_arc_a_b @ g @ A ) )
= ( insert_b @ A @ ( pre_ar1395965042833527383t_unit @ g ) ) ) ).
% arcs_add_arc
thf(fact_216_sp__costs__finite,axiom,
! [F: b > real] : ( finite7198162374296863863_ereal @ ( graph_1574344591923819902ts_a_b @ g @ F ) ) ).
% sp_costs_finite
thf(fact_217_fin__sp__costs__finite,axiom,
! [F: b > real] : ( finite7198162374296863863_ereal @ ( graph_7485366578106294827ts_a_b @ g @ F ) ) ).
% fin_sp_costs_finite
thf(fact_218_wf__digraph_Olast__branching__points_Ocong,axiom,
graph_1747835947655717337ts_a_b = graph_1747835947655717337ts_a_b ).
% wf_digraph.last_branching_points.cong
thf(fact_219_wf__digraph_Olast__merging__points_Ocong,axiom,
graph_2659413520663303054ts_a_b = graph_2659413520663303054ts_a_b ).
% wf_digraph.last_merging_points.cong
thf(fact_220_wf__digraph_Ois__chain_Ocong,axiom,
graph_3890552050688490787in_a_b = graph_3890552050688490787in_a_b ).
% wf_digraph.is_chain.cong
thf(fact_221_wf__digraph_Ois__chain_H_Ocong,axiom,
graph_8150681439568091980in_a_b = graph_8150681439568091980in_a_b ).
% wf_digraph.is_chain'.cong
thf(fact_222_query__graph_Omatching__sel__unique,axiom,
! [G2: pre_pr7278220950009878019t_unit,Sel2: b > real,Cf: a > real,F: a > a > real,G: a > a > real] :
( ( query_graph_a_b @ G2 @ Sel2 @ Cf )
=> ( ( query_5351540782772103094el_a_b @ G2 @ Sel2 @ F )
=> ( ( query_5351540782772103094el_a_b @ G2 @ Sel2 @ G )
=> ( F = G ) ) ) ) ).
% query_graph.matching_sel_unique
thf(fact_223_query__graph_Omatching__sel__unique__aux,axiom,
! [G2: pre_pr7278220950009878019t_unit,Sel2: b > real,Cf: a > real,F: a > a > real,G: a > a > real,X: a,Y: a] :
( ( query_graph_a_b @ G2 @ Sel2 @ Cf )
=> ( ( query_5351540782772103094el_a_b @ G2 @ Sel2 @ F )
=> ( ( query_5351540782772103094el_a_b @ G2 @ Sel2 @ G )
=> ( ( F @ X @ Y )
= ( G @ X @ Y ) ) ) ) ) ).
% query_graph.matching_sel_unique_aux
thf(fact_224_query__graph_Oaxioms_I1_J,axiom,
! [G2: pre_pr7278220950009878019t_unit,Sel2: b > real,Cf: a > real] :
( ( query_graph_a_b @ G2 @ Sel2 @ Cf )
=> ( graph_a_b @ G2 ) ) ).
% query_graph.axioms(1)
thf(fact_225_query__graph_Osel__pos,axiom,
! [G2: pre_pr7278220950009878019t_unit,Sel2: b > real,Cf: a > real,E: b] :
( ( query_graph_a_b @ G2 @ Sel2 @ Cf )
=> ( ord_less_real @ zero_zero_real @ ( Sel2 @ E ) ) ) ).
% query_graph.sel_pos
thf(fact_226_query__graph_Osel__leq__1,axiom,
! [G2: pre_pr7278220950009878019t_unit,Sel2: b > real,Cf: a > real,E: b] :
( ( query_graph_a_b @ G2 @ Sel2 @ Cf )
=> ( ord_less_eq_real @ ( Sel2 @ E ) @ one_one_real ) ) ).
% query_graph.sel_leq_1
thf(fact_227_query__graph_Onot__arc__sel__1,axiom,
! [G2: pre_pr7278220950009878019t_unit,Sel2: b > real,Cf: a > real,E: b] :
( ( query_graph_a_b @ G2 @ Sel2 @ Cf )
=> ( ~ ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ G2 ) )
=> ( ( Sel2 @ E )
= one_one_real ) ) ) ).
% query_graph.not_arc_sel_1
thf(fact_228_query__graph_Omatching__sel__reasonable,axiom,
! [G2: pre_pr7278220950009878019t_unit,Sel2: b > real,Cf: a > real,F: a > a > real] :
( ( query_graph_a_b @ G2 @ Sel2 @ Cf )
=> ( ( query_5351540782772103094el_a_b @ G2 @ Sel2 @ F )
=> ( sel_reasonable_a @ F ) ) ) ).
% query_graph.matching_sel_reasonable
thf(fact_229_query__graph_Omatch__sel__unique,axiom,
! [G2: pre_pr7278220950009878019t_unit,Sel2: b > real,Cf: a > real,F: a > a > real] :
( ( query_graph_a_b @ G2 @ Sel2 @ Cf )
=> ( ( query_5351540782772103094el_a_b @ G2 @ Sel2 @ F )
=> ( F
= ( query_match_sel_a_b @ G2 @ Sel2 ) ) ) ) ).
% query_graph.match_sel_unique
thf(fact_230_query__graph_Omatch__sel__matching,axiom,
! [G2: pre_pr7278220950009878019t_unit,Sel2: b > real,Cf: a > real] :
( ( query_graph_a_b @ G2 @ Sel2 @ Cf )
=> ( query_5351540782772103094el_a_b @ G2 @ Sel2 @ ( query_match_sel_a_b @ G2 @ Sel2 ) ) ) ).
% query_graph.match_sel_matching
thf(fact_231_query__graph_Omatching__sel__symm,axiom,
! [G2: pre_pr7278220950009878019t_unit,Sel2: b > real,Cf: a > real,F: a > a > real] :
( ( query_graph_a_b @ G2 @ Sel2 @ Cf )
=> ( ( query_5351540782772103094el_a_b @ G2 @ Sel2 @ F )
=> ( sel_symm_a @ F ) ) ) ).
% query_graph.matching_sel_symm
thf(fact_232_query__graph_Omatch__sel__reasonable,axiom,
! [G2: pre_pr7278220950009878019t_unit,Sel2: b > real,Cf: a > real] :
( ( query_graph_a_b @ G2 @ Sel2 @ Cf )
=> ( sel_reasonable_a @ ( query_match_sel_a_b @ G2 @ Sel2 ) ) ) ).
% query_graph.match_sel_reasonable
thf(fact_233_query__graph_Omatch__sel__symm,axiom,
! [G2: pre_pr7278220950009878019t_unit,Sel2: b > real,Cf: a > real] :
( ( query_graph_a_b @ G2 @ Sel2 @ Cf )
=> ( sel_symm_a @ ( query_match_sel_a_b @ G2 @ Sel2 ) ) ) ).
% query_graph.match_sel_symm
thf(fact_234_query__graph_Opos__cards,axiom,
! [G2: pre_pr7278220950009878019t_unit,Sel2: b > real,Cf: a > real,X: a] :
( ( query_graph_a_b @ G2 @ Sel2 @ Cf )
=> ( ( member_a @ X @ ( pre_ve642382030648772252t_unit @ G2 ) )
=> ( ord_less_real @ zero_zero_real @ ( Cf @ X ) ) ) ) ).
% query_graph.pos_cards
thf(fact_235_query__graph_Osel__less__arc,axiom,
! [G2: pre_pr7278220950009878019t_unit,Sel2: b > real,Cf: a > real,X: b] :
( ( query_graph_a_b @ G2 @ Sel2 @ Cf )
=> ( ( ord_less_real @ ( Sel2 @ X ) @ one_one_real )
=> ( member_b @ X @ ( pre_ar1395965042833527383t_unit @ G2 ) ) ) ) ).
% query_graph.sel_less_arc
thf(fact_236_not__psubset__empty,axiom,
! [A2: set_b] :
~ ( ord_less_set_b @ A2 @ bot_bot_set_b ) ).
% not_psubset_empty
thf(fact_237_not__psubset__empty,axiom,
! [A2: set_pr5411798346947241657t_unit] :
~ ( ord_le2693654750756130573t_unit @ A2 @ bot_bo1839476491465656141t_unit ) ).
% not_psubset_empty
thf(fact_238_not__psubset__empty,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_239_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_240_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_241_emptyE,axiom,
! [A: b] :
~ ( member_b @ A @ bot_bot_set_b ) ).
% emptyE
thf(fact_242_emptyE,axiom,
! [A: pre_pr7278220950009878019t_unit] :
~ ( member6939884229742472986t_unit @ A @ bot_bo1839476491465656141t_unit ) ).
% emptyE
thf(fact_243_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_244_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_245_equals0D,axiom,
! [A2: set_b,A: b] :
( ( A2 = bot_bot_set_b )
=> ~ ( member_b @ A @ A2 ) ) ).
% equals0D
thf(fact_246_equals0D,axiom,
! [A2: set_pr5411798346947241657t_unit,A: pre_pr7278220950009878019t_unit] :
( ( A2 = bot_bo1839476491465656141t_unit )
=> ~ ( member6939884229742472986t_unit @ A @ A2 ) ) ).
% equals0D
thf(fact_247_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y3: set_a] :
~ ( member_set_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_248_equals0I,axiom,
! [A2: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_249_equals0I,axiom,
! [A2: set_b] :
( ! [Y3: b] :
~ ( member_b @ Y3 @ A2 )
=> ( A2 = bot_bot_set_b ) ) ).
% equals0I
thf(fact_250_equals0I,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ! [Y3: pre_pr7278220950009878019t_unit] :
~ ( member6939884229742472986t_unit @ Y3 @ A2 )
=> ( A2 = bot_bo1839476491465656141t_unit ) ) ).
% equals0I
thf(fact_251_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X2: set_a] : ( member_set_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_252_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_253_ex__in__conv,axiom,
! [A2: set_b] :
( ( ? [X2: b] : ( member_b @ X2 @ A2 ) )
= ( A2 != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_254_ex__in__conv,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ( ? [X2: pre_pr7278220950009878019t_unit] : ( member6939884229742472986t_unit @ X2 @ A2 ) )
= ( A2 != bot_bo1839476491465656141t_unit ) ) ).
% ex_in_conv
thf(fact_255_query__graph_Oremove__sel__pos,axiom,
! [G2: pre_pr7278220950009878019t_unit,Sel2: b > real,Cf: a > real,X: a,E: b] :
( ( query_graph_a_b @ G2 @ Sel2 @ Cf )
=> ( ord_less_real @ zero_zero_real @ ( query_remove_sel_a_b @ G2 @ Sel2 @ X @ E ) ) ) ).
% query_graph.remove_sel_pos
thf(fact_256_finite__psubset__induct,axiom,
! [A2: set_b,P: set_b > $o] :
( ( finite_finite_b @ A2 )
=> ( ! [A5: set_b] :
( ( finite_finite_b @ A5 )
=> ( ! [B3: set_b] :
( ( ord_less_set_b @ B3 @ A5 )
=> ( P @ B3 ) )
=> ( P @ A5 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_257_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 )
=> ( ! [B3: set_set_a] :
( ( ord_less_set_set_a @ B3 @ A5 )
=> ( P @ B3 ) )
=> ( P @ A5 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_258_finite__psubset__induct,axiom,
! [A2: set_Extended_ereal,P: set_Extended_ereal > $o] :
( ( finite7198162374296863863_ereal @ A2 )
=> ( ! [A5: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ A5 )
=> ( ! [B3: set_Extended_ereal] :
( ( ord_le5321083090456148570_ereal @ B3 @ A5 )
=> ( P @ B3 ) )
=> ( P @ A5 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_259_finite__psubset__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ! [B3: set_a] :
( ( ord_less_set_a @ B3 @ A5 )
=> ( P @ B3 ) )
=> ( P @ A5 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_260_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_set_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_261_subset__iff__psubset__eq,axiom,
( ord_le8200006823705900825t_unit
= ( ^ [A4: set_pr5411798346947241657t_unit,B4: set_pr5411798346947241657t_unit] :
( ( ord_le2693654750756130573t_unit @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_262_subset__psubset__trans,axiom,
! [A2: set_a,B2: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C3 )
=> ( ord_less_set_a @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_263_subset__psubset__trans,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit,C3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ B2 )
=> ( ( ord_le2693654750756130573t_unit @ B2 @ C3 )
=> ( ord_le2693654750756130573t_unit @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_264_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ~ ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_265_subset__not__subset__eq,axiom,
( ord_le2693654750756130573t_unit
= ( ^ [A4: set_pr5411798346947241657t_unit,B4: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A4 @ B4 )
& ~ ( ord_le8200006823705900825t_unit @ B4 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_266_psubset__subset__trans,axiom,
! [A2: set_a,B2: set_a,C3: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ord_less_set_a @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_267_psubset__subset__trans,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit,C3: set_pr5411798346947241657t_unit] :
( ( ord_le2693654750756130573t_unit @ A2 @ B2 )
=> ( ( ord_le8200006823705900825t_unit @ B2 @ C3 )
=> ( ord_le2693654750756130573t_unit @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_268_psubset__imp__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_269_psubset__imp__subset,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( ord_le2693654750756130573t_unit @ A2 @ B2 )
=> ( ord_le8200006823705900825t_unit @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_270_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_271_psubset__eq,axiom,
( ord_le2693654750756130573t_unit
= ( ^ [A4: set_pr5411798346947241657t_unit,B4: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_272_psubsetE,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_273_psubsetE,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( ord_le2693654750756130573t_unit @ A2 @ B2 )
=> ~ ( ( ord_le8200006823705900825t_unit @ A2 @ B2 )
=> ( ord_le8200006823705900825t_unit @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_274_in__mono,axiom,
! [A2: set_b,B2: set_b,X: b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( member_b @ X @ A2 )
=> ( member_b @ X @ B2 ) ) ) ).
% in_mono
thf(fact_275_in__mono,axiom,
! [A2: set_set_a,B2: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ X @ A2 )
=> ( member_set_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_276_in__mono,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_277_in__mono,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit,X: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ B2 )
=> ( ( member6939884229742472986t_unit @ X @ A2 )
=> ( member6939884229742472986t_unit @ X @ B2 ) ) ) ).
% in_mono
thf(fact_278_subsetD,axiom,
! [A2: set_b,B2: set_b,C: b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B2 ) ) ) ).
% subsetD
thf(fact_279_subsetD,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_280_subsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_281_subsetD,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit,C: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ B2 )
=> ( ( member6939884229742472986t_unit @ C @ A2 )
=> ( member6939884229742472986t_unit @ C @ B2 ) ) ) ).
% subsetD
thf(fact_282_equalityE,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_283_equalityE,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( A2 = B2 )
=> ~ ( ( ord_le8200006823705900825t_unit @ A2 @ B2 )
=> ~ ( ord_le8200006823705900825t_unit @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_284_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B4: set_b] :
! [X2: b] :
( ( member_b @ X2 @ A4 )
=> ( member_b @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_285_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
! [X2: set_a] :
( ( member_set_a @ X2 @ A4 )
=> ( member_set_a @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_286_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A4 )
=> ( member_a @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_287_subset__eq,axiom,
( ord_le8200006823705900825t_unit
= ( ^ [A4: set_pr5411798346947241657t_unit,B4: set_pr5411798346947241657t_unit] :
! [X2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X2 @ A4 )
=> ( member6939884229742472986t_unit @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_288_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_289_equalityD1,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( A2 = B2 )
=> ( ord_le8200006823705900825t_unit @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_290_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_291_equalityD2,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( A2 = B2 )
=> ( ord_le8200006823705900825t_unit @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_292_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B4: set_b] :
! [T: b] :
( ( member_b @ T @ A4 )
=> ( member_b @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_293_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A4 )
=> ( member_set_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_294_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [T: a] :
( ( member_a @ T @ A4 )
=> ( member_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_295_subset__iff,axiom,
( ord_le8200006823705900825t_unit
= ( ^ [A4: set_pr5411798346947241657t_unit,B4: set_pr5411798346947241657t_unit] :
! [T: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ T @ A4 )
=> ( member6939884229742472986t_unit @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_296_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_297_subset__refl,axiom,
! [A2: set_pr5411798346947241657t_unit] : ( ord_le8200006823705900825t_unit @ A2 @ A2 ) ).
% subset_refl
thf(fact_298_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_299_Collect__mono,axiom,
! [P: pre_pr7278220950009878019t_unit > $o,Q: pre_pr7278220950009878019t_unit > $o] :
( ! [X3: pre_pr7278220950009878019t_unit] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le8200006823705900825t_unit @ ( collec8000012497822511960t_unit @ P ) @ ( collec8000012497822511960t_unit @ Q ) ) ) ).
% Collect_mono
thf(fact_300_subset__trans,axiom,
! [A2: set_a,B2: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ord_less_eq_set_a @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_301_subset__trans,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit,C3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ B2 )
=> ( ( ord_le8200006823705900825t_unit @ B2 @ C3 )
=> ( ord_le8200006823705900825t_unit @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_302_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_303_set__eq__subset,axiom,
( ( ^ [Y4: set_pr5411798346947241657t_unit,Z: set_pr5411798346947241657t_unit] : ( Y4 = Z ) )
= ( ^ [A4: set_pr5411798346947241657t_unit,B4: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A4 @ B4 )
& ( ord_le8200006823705900825t_unit @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_304_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_305_Collect__mono__iff,axiom,
! [P: pre_pr7278220950009878019t_unit > $o,Q: pre_pr7278220950009878019t_unit > $o] :
( ( ord_le8200006823705900825t_unit @ ( collec8000012497822511960t_unit @ P ) @ ( collec8000012497822511960t_unit @ Q ) )
= ( ! [X2: pre_pr7278220950009878019t_unit] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_306_insertE,axiom,
! [A: b,B: b,A2: set_b] :
( ( member_b @ A @ ( insert_b @ B @ A2 ) )
=> ( ( A != B )
=> ( member_b @ A @ A2 ) ) ) ).
% insertE
thf(fact_307_insertE,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_308_insertE,axiom,
! [A: pre_pr7278220950009878019t_unit,B: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ A @ ( insert6864688055023459379t_unit @ B @ A2 ) )
=> ( ( A != B )
=> ( member6939884229742472986t_unit @ A @ A2 ) ) ) ).
% insertE
thf(fact_309_insertE,axiom,
! [A: set_a,B: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_310_insertI1,axiom,
! [A: b,B2: set_b] : ( member_b @ A @ ( insert_b @ A @ B2 ) ) ).
% insertI1
thf(fact_311_insertI1,axiom,
! [A: a,B2: set_a] : ( member_a @ A @ ( insert_a @ A @ B2 ) ) ).
% insertI1
thf(fact_312_insertI1,axiom,
! [A: pre_pr7278220950009878019t_unit,B2: set_pr5411798346947241657t_unit] : ( member6939884229742472986t_unit @ A @ ( insert6864688055023459379t_unit @ A @ B2 ) ) ).
% insertI1
thf(fact_313_insertI1,axiom,
! [A: set_a,B2: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B2 ) ) ).
% insertI1
thf(fact_314_insertI2,axiom,
! [A: b,B2: set_b,B: b] :
( ( member_b @ A @ B2 )
=> ( member_b @ A @ ( insert_b @ B @ B2 ) ) ) ).
% insertI2
thf(fact_315_insertI2,axiom,
! [A: a,B2: set_a,B: a] :
( ( member_a @ A @ B2 )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_316_insertI2,axiom,
! [A: pre_pr7278220950009878019t_unit,B2: set_pr5411798346947241657t_unit,B: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ A @ B2 )
=> ( member6939884229742472986t_unit @ A @ ( insert6864688055023459379t_unit @ B @ B2 ) ) ) ).
% insertI2
thf(fact_317_insertI2,axiom,
! [A: set_a,B2: set_set_a,B: set_a] :
( ( member_set_a @ A @ B2 )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_318_Set_Oset__insert,axiom,
! [X: b,A2: set_b] :
( ( member_b @ X @ A2 )
=> ~ ! [B5: set_b] :
( ( A2
= ( insert_b @ X @ B5 ) )
=> ( member_b @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_319_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B5: set_a] :
( ( A2
= ( insert_a @ X @ B5 ) )
=> ( member_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_320_Set_Oset__insert,axiom,
! [X: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ X @ A2 )
=> ~ ! [B5: set_pr5411798346947241657t_unit] :
( ( A2
= ( insert6864688055023459379t_unit @ X @ B5 ) )
=> ( member6939884229742472986t_unit @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_321_Set_Oset__insert,axiom,
! [X: set_a,A2: set_set_a] :
( ( member_set_a @ X @ A2 )
=> ~ ! [B5: set_set_a] :
( ( A2
= ( insert_set_a @ X @ B5 ) )
=> ( member_set_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_322_insert__ident,axiom,
! [X: b,A2: set_b,B2: set_b] :
( ~ ( member_b @ X @ A2 )
=> ( ~ ( member_b @ X @ B2 )
=> ( ( ( insert_b @ X @ A2 )
= ( insert_b @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_323_insert__ident,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B2 )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_324_insert__ident,axiom,
! [X: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ X @ A2 )
=> ( ~ ( member6939884229742472986t_unit @ X @ B2 )
=> ( ( ( insert6864688055023459379t_unit @ X @ A2 )
= ( insert6864688055023459379t_unit @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_325_insert__ident,axiom,
! [X: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ~ ( member_set_a @ X @ B2 )
=> ( ( ( insert_set_a @ X @ A2 )
= ( insert_set_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_326_insert__absorb,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ( ( insert_b @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_327_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_328_insert__absorb,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ A @ A2 )
=> ( ( insert6864688055023459379t_unit @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_329_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_330_insert__eq__iff,axiom,
! [A: b,A2: set_b,B: b,B2: set_b] :
( ~ ( member_b @ A @ A2 )
=> ( ~ ( member_b @ B @ B2 )
=> ( ( ( insert_b @ A @ A2 )
= ( insert_b @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C4: set_b] :
( ( A2
= ( insert_b @ B @ C4 ) )
& ~ ( member_b @ B @ C4 )
& ( B2
= ( insert_b @ A @ C4 ) )
& ~ ( member_b @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_331_insert__eq__iff,axiom,
! [A: a,A2: set_a,B: a,B2: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B @ B2 )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C4: set_a] :
( ( A2
= ( insert_a @ B @ C4 ) )
& ~ ( member_a @ B @ C4 )
& ( B2
= ( insert_a @ A @ C4 ) )
& ~ ( member_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_332_insert__eq__iff,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B: pre_pr7278220950009878019t_unit,B2: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ A @ A2 )
=> ( ~ ( member6939884229742472986t_unit @ B @ B2 )
=> ( ( ( insert6864688055023459379t_unit @ A @ A2 )
= ( insert6864688055023459379t_unit @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C4: set_pr5411798346947241657t_unit] :
( ( A2
= ( insert6864688055023459379t_unit @ B @ C4 ) )
& ~ ( member6939884229742472986t_unit @ B @ C4 )
& ( B2
= ( insert6864688055023459379t_unit @ A @ C4 ) )
& ~ ( member6939884229742472986t_unit @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_333_insert__eq__iff,axiom,
! [A: set_a,A2: set_set_a,B: set_a,B2: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ~ ( member_set_a @ B @ B2 )
=> ( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C4: set_set_a] :
( ( A2
= ( insert_set_a @ B @ C4 ) )
& ~ ( member_set_a @ B @ C4 )
& ( B2
= ( insert_set_a @ A @ C4 ) )
& ~ ( member_set_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_334_insert__commute,axiom,
! [X: a,Y: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A2 ) )
= ( insert_a @ Y @ ( insert_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_335_insert__commute,axiom,
! [X: b,Y: b,A2: set_b] :
( ( insert_b @ X @ ( insert_b @ Y @ A2 ) )
= ( insert_b @ Y @ ( insert_b @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_336_insert__commute,axiom,
! [X: pre_pr7278220950009878019t_unit,Y: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( insert6864688055023459379t_unit @ X @ ( insert6864688055023459379t_unit @ Y @ A2 ) )
= ( insert6864688055023459379t_unit @ Y @ ( insert6864688055023459379t_unit @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_337_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_338_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_339_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_340_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_341_query__graph_Oremove__sel__1,axiom,
! [G2: pre_pr7278220950009878019t_unit,Sel2: b > real,Cf: a > real,E: b,X: a] :
( ( query_graph_a_b @ G2 @ Sel2 @ Cf )
=> ( ~ ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ G2 ) )
=> ( ( query_remove_sel_a_b @ G2 @ Sel2 @ X @ E )
= one_one_real ) ) ) ).
% query_graph.remove_sel_1
thf(fact_342_query__graph_Odel__vert__remove__sel__1,axiom,
! [G2: pre_pr7278220950009878019t_unit,Sel2: b > real,Cf: a > real,E: b,X: a] :
( ( query_graph_a_b @ G2 @ Sel2 @ Cf )
=> ( ~ ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ ( pre_del_vert_a_b @ G2 @ X ) ) )
=> ( ( query_remove_sel_a_b @ G2 @ Sel2 @ X @ E )
= one_one_real ) ) ) ).
% query_graph.del_vert_remove_sel_1
thf(fact_343_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_344_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_345_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_346_finite__has__minimal2,axiom,
! [A2: set_se2139339572462915695t_unit,A: set_pr5411798346947241657t_unit] :
( ( finite519047672745605328t_unit @ A2 )
=> ( ( member5449360183034373072t_unit @ A @ A2 )
=> ? [X3: set_pr5411798346947241657t_unit] :
( ( member5449360183034373072t_unit @ X3 @ A2 )
& ( ord_le8200006823705900825t_unit @ X3 @ A )
& ! [Xa: set_pr5411798346947241657t_unit] :
( ( member5449360183034373072t_unit @ Xa @ A2 )
=> ( ( ord_le8200006823705900825t_unit @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_347_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_348_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_349_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_350_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_351_finite__has__maximal2,axiom,
! [A2: set_se2139339572462915695t_unit,A: set_pr5411798346947241657t_unit] :
( ( finite519047672745605328t_unit @ A2 )
=> ( ( member5449360183034373072t_unit @ A @ A2 )
=> ? [X3: set_pr5411798346947241657t_unit] :
( ( member5449360183034373072t_unit @ X3 @ A2 )
& ( ord_le8200006823705900825t_unit @ A @ X3 )
& ! [Xa: set_pr5411798346947241657t_unit] :
( ( member5449360183034373072t_unit @ Xa @ A2 )
=> ( ( ord_le8200006823705900825t_unit @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_352_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_353_finite_OemptyI,axiom,
finite_finite_set_a @ bot_bot_set_set_a ).
% finite.emptyI
thf(fact_354_finite_OemptyI,axiom,
finite7198162374296863863_ereal @ bot_bo8367695208629047834_ereal ).
% finite.emptyI
thf(fact_355_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_356_finite_OemptyI,axiom,
finite_finite_b @ bot_bot_set_b ).
% finite.emptyI
thf(fact_357_finite_OemptyI,axiom,
finite8852549406693098522t_unit @ bot_bo1839476491465656141t_unit ).
% finite.emptyI
thf(fact_358_infinite__imp__nonempty,axiom,
! [S: set_set_a] :
( ~ ( finite_finite_set_a @ S )
=> ( S != bot_bot_set_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_359_infinite__imp__nonempty,axiom,
! [S: set_Extended_ereal] :
( ~ ( finite7198162374296863863_ereal @ S )
=> ( S != bot_bo8367695208629047834_ereal ) ) ).
% infinite_imp_nonempty
thf(fact_360_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_361_infinite__imp__nonempty,axiom,
! [S: set_b] :
( ~ ( finite_finite_b @ S )
=> ( S != bot_bot_set_b ) ) ).
% infinite_imp_nonempty
thf(fact_362_infinite__imp__nonempty,axiom,
! [S: set_pr5411798346947241657t_unit] :
( ~ ( finite8852549406693098522t_unit @ S )
=> ( S != bot_bo1839476491465656141t_unit ) ) ).
% infinite_imp_nonempty
thf(fact_363_finite__subset,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( finite_finite_b @ B2 )
=> ( finite_finite_b @ A2 ) ) ) ).
% finite_subset
thf(fact_364_finite__subset,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( finite_finite_set_a @ B2 )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% finite_subset
thf(fact_365_finite__subset,axiom,
! [A2: set_Extended_ereal,B2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ A2 @ B2 )
=> ( ( finite7198162374296863863_ereal @ B2 )
=> ( finite7198162374296863863_ereal @ A2 ) ) ) ).
% finite_subset
thf(fact_366_finite__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_367_finite__subset,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ B2 )
=> ( ( finite8852549406693098522t_unit @ B2 )
=> ( finite8852549406693098522t_unit @ A2 ) ) ) ).
% finite_subset
thf(fact_368_infinite__super,axiom,
! [S: set_b,T2: set_b] :
( ( ord_less_eq_set_b @ S @ T2 )
=> ( ~ ( finite_finite_b @ S )
=> ~ ( finite_finite_b @ T2 ) ) ) ).
% infinite_super
thf(fact_369_infinite__super,axiom,
! [S: set_set_a,T2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S @ T2 )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ T2 ) ) ) ).
% infinite_super
thf(fact_370_infinite__super,axiom,
! [S: set_Extended_ereal,T2: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ S @ T2 )
=> ( ~ ( finite7198162374296863863_ereal @ S )
=> ~ ( finite7198162374296863863_ereal @ T2 ) ) ) ).
% infinite_super
thf(fact_371_infinite__super,axiom,
! [S: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_372_infinite__super,axiom,
! [S: set_pr5411798346947241657t_unit,T2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ S @ T2 )
=> ( ~ ( finite8852549406693098522t_unit @ S )
=> ~ ( finite8852549406693098522t_unit @ T2 ) ) ) ).
% infinite_super
thf(fact_373_rev__finite__subset,axiom,
! [B2: set_b,A2: set_b] :
( ( finite_finite_b @ B2 )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( finite_finite_b @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_374_rev__finite__subset,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_375_rev__finite__subset,axiom,
! [B2: set_Extended_ereal,A2: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ B2 )
=> ( ( ord_le1644982726543182158_ereal @ A2 @ B2 )
=> ( finite7198162374296863863_ereal @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_376_rev__finite__subset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_377_rev__finite__subset,axiom,
! [B2: set_pr5411798346947241657t_unit,A2: set_pr5411798346947241657t_unit] :
( ( finite8852549406693098522t_unit @ B2 )
=> ( ( ord_le8200006823705900825t_unit @ A2 @ B2 )
=> ( finite8852549406693098522t_unit @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_378_singletonD,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_379_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_380_singletonD,axiom,
! [B: b,A: b] :
( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_381_singletonD,axiom,
! [B: pre_pr7278220950009878019t_unit,A: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ B @ ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_382_singleton__iff,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_383_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_384_singleton__iff,axiom,
! [B: b,A: b] :
( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_385_singleton__iff,axiom,
! [B: pre_pr7278220950009878019t_unit,A: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ B @ ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_386_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_387_doubleton__eq__iff,axiom,
! [A: b,B: b,C: b,D: b] :
( ( ( insert_b @ A @ ( insert_b @ B @ bot_bot_set_b ) )
= ( insert_b @ C @ ( insert_b @ D @ bot_bot_set_b ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_388_doubleton__eq__iff,axiom,
! [A: pre_pr7278220950009878019t_unit,B: pre_pr7278220950009878019t_unit,C: pre_pr7278220950009878019t_unit,D: pre_pr7278220950009878019t_unit] :
( ( ( insert6864688055023459379t_unit @ A @ ( insert6864688055023459379t_unit @ B @ bot_bo1839476491465656141t_unit ) )
= ( insert6864688055023459379t_unit @ C @ ( insert6864688055023459379t_unit @ D @ bot_bo1839476491465656141t_unit ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_389_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_390_insert__not__empty,axiom,
! [A: b,A2: set_b] :
( ( insert_b @ A @ A2 )
!= bot_bot_set_b ) ).
% insert_not_empty
thf(fact_391_insert__not__empty,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( insert6864688055023459379t_unit @ A @ A2 )
!= bot_bo1839476491465656141t_unit ) ).
% insert_not_empty
thf(fact_392_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_393_singleton__inject,axiom,
! [A: b,B: b] :
( ( ( insert_b @ A @ bot_bot_set_b )
= ( insert_b @ B @ bot_bot_set_b ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_394_singleton__inject,axiom,
! [A: pre_pr7278220950009878019t_unit,B: pre_pr7278220950009878019t_unit] :
( ( ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit )
= ( insert6864688055023459379t_unit @ B @ bot_bo1839476491465656141t_unit ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_395_finite_OinsertI,axiom,
! [A2: set_pr5411798346947241657t_unit,A: pre_pr7278220950009878019t_unit] :
( ( finite8852549406693098522t_unit @ A2 )
=> ( finite8852549406693098522t_unit @ ( insert6864688055023459379t_unit @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_396_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_397_finite_OinsertI,axiom,
! [A2: set_b,A: b] :
( ( finite_finite_b @ A2 )
=> ( finite_finite_b @ ( insert_b @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_398_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_399_finite_OinsertI,axiom,
! [A2: set_Extended_ereal,A: extended_ereal] :
( ( finite7198162374296863863_ereal @ A2 )
=> ( finite7198162374296863863_ereal @ ( insert8967887681552722334_ereal @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_400_insert__mono,axiom,
! [C3: set_b,D2: set_b,A: b] :
( ( ord_less_eq_set_b @ C3 @ D2 )
=> ( ord_less_eq_set_b @ ( insert_b @ A @ C3 ) @ ( insert_b @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_401_insert__mono,axiom,
! [C3: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C3 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C3 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_402_insert__mono,axiom,
! [C3: set_pr5411798346947241657t_unit,D2: set_pr5411798346947241657t_unit,A: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ C3 @ D2 )
=> ( ord_le8200006823705900825t_unit @ ( insert6864688055023459379t_unit @ A @ C3 ) @ ( insert6864688055023459379t_unit @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_403_subset__insert,axiom,
! [X: b,A2: set_b,B2: set_b] :
( ~ ( member_b @ X @ A2 )
=> ( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X @ B2 ) )
= ( ord_less_eq_set_b @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_404_subset__insert,axiom,
! [X: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B2 ) )
= ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_405_subset__insert,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_406_subset__insert,axiom,
! [X: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ X @ A2 )
=> ( ( ord_le8200006823705900825t_unit @ A2 @ ( insert6864688055023459379t_unit @ X @ B2 ) )
= ( ord_le8200006823705900825t_unit @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_407_subset__insertI,axiom,
! [B2: set_b,A: b] : ( ord_less_eq_set_b @ B2 @ ( insert_b @ A @ B2 ) ) ).
% subset_insertI
thf(fact_408_subset__insertI,axiom,
! [B2: set_a,A: a] : ( ord_less_eq_set_a @ B2 @ ( insert_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_409_subset__insertI,axiom,
! [B2: set_pr5411798346947241657t_unit,A: pre_pr7278220950009878019t_unit] : ( ord_le8200006823705900825t_unit @ B2 @ ( insert6864688055023459379t_unit @ A @ B2 ) ) ).
% subset_insertI
thf(fact_410_subset__insertI2,axiom,
! [A2: set_b,B2: set_b,B: b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_411_subset__insertI2,axiom,
! [A2: set_a,B2: set_a,B: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_412_subset__insertI2,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit,B: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ B2 )
=> ( ord_le8200006823705900825t_unit @ A2 @ ( insert6864688055023459379t_unit @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_413_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_414_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_415_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_416_finite__has__maximal,axiom,
! [A2: set_se2139339572462915695t_unit] :
( ( finite519047672745605328t_unit @ A2 )
=> ( ( A2 != bot_bo1540698489285124355t_unit )
=> ? [X3: set_pr5411798346947241657t_unit] :
( ( member5449360183034373072t_unit @ X3 @ A2 )
& ! [Xa: set_pr5411798346947241657t_unit] :
( ( member5449360183034373072t_unit @ Xa @ A2 )
=> ( ( ord_le8200006823705900825t_unit @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_417_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_418_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_419_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_420_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_421_finite__has__minimal,axiom,
! [A2: set_se2139339572462915695t_unit] :
( ( finite519047672745605328t_unit @ A2 )
=> ( ( A2 != bot_bo1540698489285124355t_unit )
=> ? [X3: set_pr5411798346947241657t_unit] :
( ( member5449360183034373072t_unit @ X3 @ A2 )
& ! [Xa: set_pr5411798346947241657t_unit] :
( ( member5449360183034373072t_unit @ Xa @ A2 )
=> ( ( ord_le8200006823705900825t_unit @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_422_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_423_infinite__finite__induct,axiom,
! [P: set_set_a > $o,A2: set_set_a] :
( ! [A5: set_set_a] :
( ~ ( finite_finite_set_a @ A5 )
=> ( P @ A5 ) )
=> ( ( 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 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_424_infinite__finite__induct,axiom,
! [P: set_Extended_ereal > $o,A2: set_Extended_ereal] :
( ! [A5: set_Extended_ereal] :
( ~ ( finite7198162374296863863_ereal @ A5 )
=> ( P @ A5 ) )
=> ( ( 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 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_425_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A5: set_a] :
( ~ ( finite_finite_a @ A5 )
=> ( P @ A5 ) )
=> ( ( 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 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_426_infinite__finite__induct,axiom,
! [P: set_b > $o,A2: set_b] :
( ! [A5: set_b] :
( ~ ( finite_finite_b @ A5 )
=> ( P @ A5 ) )
=> ( ( 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 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_427_infinite__finite__induct,axiom,
! [P: set_pr5411798346947241657t_unit > $o,A2: set_pr5411798346947241657t_unit] :
( ! [A5: set_pr5411798346947241657t_unit] :
( ~ ( finite8852549406693098522t_unit @ A5 )
=> ( P @ A5 ) )
=> ( ( 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 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_428_finite__ne__induct,axiom,
! [F4: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F4 )
=> ( ( F4 != bot_bot_set_set_a )
=> ( ! [X3: set_a] : ( P @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
=> ( ! [X3: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( F3 != bot_bot_set_set_a )
=> ( ~ ( member_set_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_429_finite__ne__induct,axiom,
! [F4: set_Extended_ereal,P: set_Extended_ereal > $o] :
( ( finite7198162374296863863_ereal @ F4 )
=> ( ( F4 != 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 @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_430_finite__ne__induct,axiom,
! [F4: set_a,P: set_a > $o] :
( ( finite_finite_a @ F4 )
=> ( ( F4 != bot_bot_set_a )
=> ( ! [X3: a] : ( P @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_431_finite__ne__induct,axiom,
! [F4: set_b,P: set_b > $o] :
( ( finite_finite_b @ F4 )
=> ( ( F4 != 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 @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_432_finite__ne__induct,axiom,
! [F4: set_pr5411798346947241657t_unit,P: set_pr5411798346947241657t_unit > $o] :
( ( finite8852549406693098522t_unit @ F4 )
=> ( ( F4 != bot_bo1839476491465656141t_unit )
=> ( ! [X3: pre_pr7278220950009878019t_unit] : ( P @ ( insert6864688055023459379t_unit @ X3 @ bot_bo1839476491465656141t_unit ) )
=> ( ! [X3: pre_pr7278220950009878019t_unit,F3: set_pr5411798346947241657t_unit] :
( ( finite8852549406693098522t_unit @ F3 )
=> ( ( F3 != bot_bo1839476491465656141t_unit )
=> ( ~ ( member6939884229742472986t_unit @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert6864688055023459379t_unit @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_433_finite__induct,axiom,
! [F4: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F4 )
=> ( ( 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 @ F4 ) ) ) ) ).
% finite_induct
thf(fact_434_finite__induct,axiom,
! [F4: set_Extended_ereal,P: set_Extended_ereal > $o] :
( ( finite7198162374296863863_ereal @ F4 )
=> ( ( 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 @ F4 ) ) ) ) ).
% finite_induct
thf(fact_435_finite__induct,axiom,
! [F4: set_a,P: set_a > $o] :
( ( finite_finite_a @ F4 )
=> ( ( 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 @ F4 ) ) ) ) ).
% finite_induct
thf(fact_436_finite__induct,axiom,
! [F4: set_b,P: set_b > $o] :
( ( finite_finite_b @ F4 )
=> ( ( 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 @ F4 ) ) ) ) ).
% finite_induct
thf(fact_437_finite__induct,axiom,
! [F4: set_pr5411798346947241657t_unit,P: set_pr5411798346947241657t_unit > $o] :
( ( finite8852549406693098522t_unit @ F4 )
=> ( ( 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 @ F4 ) ) ) ) ).
% finite_induct
thf(fact_438_finite_Osimps,axiom,
( finite_finite_set_a
= ( ^ [A6: set_set_a] :
( ( A6 = bot_bot_set_set_a )
| ? [A4: set_set_a,B6: set_a] :
( ( A6
= ( insert_set_a @ B6 @ A4 ) )
& ( finite_finite_set_a @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_439_finite_Osimps,axiom,
( finite7198162374296863863_ereal
= ( ^ [A6: set_Extended_ereal] :
( ( A6 = bot_bo8367695208629047834_ereal )
| ? [A4: set_Extended_ereal,B6: extended_ereal] :
( ( A6
= ( insert8967887681552722334_ereal @ B6 @ A4 ) )
& ( finite7198162374296863863_ereal @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_440_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A6: set_a] :
( ( A6 = bot_bot_set_a )
| ? [A4: set_a,B6: a] :
( ( A6
= ( insert_a @ B6 @ A4 ) )
& ( finite_finite_a @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_441_finite_Osimps,axiom,
( finite_finite_b
= ( ^ [A6: set_b] :
( ( A6 = bot_bot_set_b )
| ? [A4: set_b,B6: b] :
( ( A6
= ( insert_b @ B6 @ A4 ) )
& ( finite_finite_b @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_442_finite_Osimps,axiom,
( finite8852549406693098522t_unit
= ( ^ [A6: set_pr5411798346947241657t_unit] :
( ( A6 = bot_bo1839476491465656141t_unit )
| ? [A4: set_pr5411798346947241657t_unit,B6: pre_pr7278220950009878019t_unit] :
( ( A6
= ( insert6864688055023459379t_unit @ B6 @ A4 ) )
& ( finite8852549406693098522t_unit @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_443_finite_Ocases,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ~ ! [A5: set_set_a] :
( ? [A3: set_a] :
( A
= ( insert_set_a @ A3 @ A5 ) )
=> ~ ( finite_finite_set_a @ A5 ) ) ) ) ).
% finite.cases
thf(fact_444_finite_Ocases,axiom,
! [A: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ A )
=> ( ( A != bot_bo8367695208629047834_ereal )
=> ~ ! [A5: set_Extended_ereal] :
( ? [A3: extended_ereal] :
( A
= ( insert8967887681552722334_ereal @ A3 @ A5 ) )
=> ~ ( finite7198162374296863863_ereal @ A5 ) ) ) ) ).
% finite.cases
thf(fact_445_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A5: set_a] :
( ? [A3: a] :
( A
= ( insert_a @ A3 @ A5 ) )
=> ~ ( finite_finite_a @ A5 ) ) ) ) ).
% finite.cases
thf(fact_446_finite_Ocases,axiom,
! [A: set_b] :
( ( finite_finite_b @ A )
=> ( ( A != bot_bot_set_b )
=> ~ ! [A5: set_b] :
( ? [A3: b] :
( A
= ( insert_b @ A3 @ A5 ) )
=> ~ ( finite_finite_b @ A5 ) ) ) ) ).
% finite.cases
thf(fact_447_finite_Ocases,axiom,
! [A: set_pr5411798346947241657t_unit] :
( ( finite8852549406693098522t_unit @ A )
=> ( ( A != bot_bo1839476491465656141t_unit )
=> ~ ! [A5: set_pr5411798346947241657t_unit] :
( ? [A3: pre_pr7278220950009878019t_unit] :
( A
= ( insert6864688055023459379t_unit @ A3 @ A5 ) )
=> ~ ( finite8852549406693098522t_unit @ A5 ) ) ) ) ).
% finite.cases
thf(fact_448_subset__singletonD,axiom,
! [A2: set_b,X: b] :
( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) )
=> ( ( A2 = bot_bot_set_b )
| ( A2
= ( insert_b @ X @ bot_bot_set_b ) ) ) ) ).
% subset_singletonD
thf(fact_449_subset__singletonD,axiom,
! [A2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_450_subset__singletonD,axiom,
! [A2: set_pr5411798346947241657t_unit,X: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ ( insert6864688055023459379t_unit @ X @ bot_bo1839476491465656141t_unit ) )
=> ( ( A2 = bot_bo1839476491465656141t_unit )
| ( A2
= ( insert6864688055023459379t_unit @ X @ bot_bo1839476491465656141t_unit ) ) ) ) ).
% subset_singletonD
thf(fact_451_subset__singleton__iff,axiom,
! [X4: set_b,A: b] :
( ( ord_less_eq_set_b @ X4 @ ( insert_b @ A @ bot_bot_set_b ) )
= ( ( X4 = bot_bot_set_b )
| ( X4
= ( insert_b @ A @ bot_bot_set_b ) ) ) ) ).
% subset_singleton_iff
thf(fact_452_subset__singleton__iff,axiom,
! [X4: set_a,A: a] :
( ( ord_less_eq_set_a @ X4 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X4 = bot_bot_set_a )
| ( X4
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_453_subset__singleton__iff,axiom,
! [X4: set_pr5411798346947241657t_unit,A: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ X4 @ ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) )
= ( ( X4 = bot_bo1839476491465656141t_unit )
| ( X4
= ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) ) ) ) ).
% subset_singleton_iff
thf(fact_454_finite__subset__induct_H,axiom,
! [F4: set_set_a,A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F4 )
=> ( ( ord_le3724670747650509150_set_a @ F4 @ A2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A3: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A3 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ F3 @ A2 )
=> ( ~ ( member_set_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_455_finite__subset__induct_H,axiom,
! [F4: set_Extended_ereal,A2: set_Extended_ereal,P: set_Extended_ereal > $o] :
( ( finite7198162374296863863_ereal @ F4 )
=> ( ( ord_le1644982726543182158_ereal @ F4 @ A2 )
=> ( ( P @ bot_bo8367695208629047834_ereal )
=> ( ! [A3: extended_ereal,F3: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ F3 )
=> ( ( member2350847679896131959_ereal @ A3 @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ F3 @ A2 )
=> ( ~ ( member2350847679896131959_ereal @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert8967887681552722334_ereal @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_456_finite__subset__induct_H,axiom,
! [F4: set_b,A2: set_b,P: set_b > $o] :
( ( finite_finite_b @ F4 )
=> ( ( ord_less_eq_set_b @ F4 @ A2 )
=> ( ( P @ bot_bot_set_b )
=> ( ! [A3: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ( member_b @ A3 @ A2 )
=> ( ( ord_less_eq_set_b @ F3 @ A2 )
=> ( ~ ( member_b @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_457_finite__subset__induct_H,axiom,
! [F4: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F4 )
=> ( ( ord_less_eq_set_a @ F4 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A3 @ A2 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ~ ( member_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_458_finite__subset__induct_H,axiom,
! [F4: set_pr5411798346947241657t_unit,A2: set_pr5411798346947241657t_unit,P: set_pr5411798346947241657t_unit > $o] :
( ( finite8852549406693098522t_unit @ F4 )
=> ( ( ord_le8200006823705900825t_unit @ F4 @ A2 )
=> ( ( P @ bot_bo1839476491465656141t_unit )
=> ( ! [A3: pre_pr7278220950009878019t_unit,F3: set_pr5411798346947241657t_unit] :
( ( finite8852549406693098522t_unit @ F3 )
=> ( ( member6939884229742472986t_unit @ A3 @ A2 )
=> ( ( ord_le8200006823705900825t_unit @ F3 @ A2 )
=> ( ~ ( member6939884229742472986t_unit @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert6864688055023459379t_unit @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_459_finite__subset__induct,axiom,
! [F4: set_set_a,A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F4 )
=> ( ( ord_le3724670747650509150_set_a @ F4 @ A2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A3: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A3 @ A2 )
=> ( ~ ( member_set_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_460_finite__subset__induct,axiom,
! [F4: set_Extended_ereal,A2: set_Extended_ereal,P: set_Extended_ereal > $o] :
( ( finite7198162374296863863_ereal @ F4 )
=> ( ( ord_le1644982726543182158_ereal @ F4 @ A2 )
=> ( ( P @ bot_bo8367695208629047834_ereal )
=> ( ! [A3: extended_ereal,F3: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ F3 )
=> ( ( member2350847679896131959_ereal @ A3 @ A2 )
=> ( ~ ( member2350847679896131959_ereal @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert8967887681552722334_ereal @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_461_finite__subset__induct,axiom,
! [F4: set_b,A2: set_b,P: set_b > $o] :
( ( finite_finite_b @ F4 )
=> ( ( ord_less_eq_set_b @ F4 @ A2 )
=> ( ( P @ bot_bot_set_b )
=> ( ! [A3: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ( member_b @ A3 @ A2 )
=> ( ~ ( member_b @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_462_finite__subset__induct,axiom,
! [F4: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F4 )
=> ( ( ord_less_eq_set_a @ F4 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A3 @ A2 )
=> ( ~ ( member_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_463_finite__subset__induct,axiom,
! [F4: set_pr5411798346947241657t_unit,A2: set_pr5411798346947241657t_unit,P: set_pr5411798346947241657t_unit > $o] :
( ( finite8852549406693098522t_unit @ F4 )
=> ( ( ord_le8200006823705900825t_unit @ F4 @ A2 )
=> ( ( P @ bot_bo1839476491465656141t_unit )
=> ( ! [A3: pre_pr7278220950009878019t_unit,F3: set_pr5411798346947241657t_unit] :
( ( finite8852549406693098522t_unit @ F3 )
=> ( ( member6939884229742472986t_unit @ A3 @ A2 )
=> ( ~ ( member6939884229742472986t_unit @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert6864688055023459379t_unit @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_464_connected__arcs__empty,axiom,
( ( digrap8783888973171253482ed_a_b @ g )
=> ( ( ( pre_ar1395965042833527383t_unit @ g )
= bot_bot_set_b )
=> ( ( ( pre_ve642382030648772252t_unit @ g )
!= bot_bot_set_a )
=> ~ ! [V2: a] :
( ( pre_ve642382030648772252t_unit @ g )
!= ( insert_a @ V2 @ bot_bot_set_a ) ) ) ) ) ).
% connected_arcs_empty
thf(fact_465_in__sccsE,axiom,
! [C: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ g ) )
=> ~ ( ( digrap5251062021860773499ph_a_b @ C @ g )
=> ( ( digrap8691851296217657702ed_a_b @ C )
=> ? [D3: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ D3 @ g )
& ( digrap8691851296217657702ed_a_b @ D3 )
& ( ord_less_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ D3 ) ) ) ) ) ) ).
% in_sccsE
thf(fact_466_verts__del__vert,axiom,
! [U: a] :
( ( pre_ve642382030648772252t_unit @ ( pre_del_vert_a_b @ g @ U ) )
= ( minus_minus_set_a @ ( pre_ve642382030648772252t_unit @ g ) @ ( insert_a @ U @ bot_bot_set_a ) ) ) ).
% verts_del_vert
thf(fact_467_finite__linorder__min__induct,axiom,
! [A2: set_Extended_ereal,P: set_Extended_ereal > $o] :
( ( finite7198162374296863863_ereal @ A2 )
=> ( ( P @ bot_bo8367695208629047834_ereal )
=> ( ! [B7: extended_ereal,A5: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ A5 )
=> ( ! [X5: extended_ereal] :
( ( member2350847679896131959_ereal @ X5 @ A5 )
=> ( ord_le1188267648640031866_ereal @ B7 @ X5 ) )
=> ( ( P @ A5 )
=> ( P @ ( insert8967887681552722334_ereal @ B7 @ A5 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_468_finite__linorder__min__induct,axiom,
! [A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ A2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [B7: real,A5: set_real] :
( ( finite_finite_real @ A5 )
=> ( ! [X5: real] :
( ( member_real @ X5 @ A5 )
=> ( ord_less_real @ B7 @ X5 ) )
=> ( ( P @ A5 )
=> ( P @ ( insert_real @ B7 @ A5 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_469_finite__linorder__min__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B7: nat,A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A5 )
=> ( ord_less_nat @ B7 @ X5 ) )
=> ( ( P @ A5 )
=> ( P @ ( insert_nat @ B7 @ A5 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_470_finite__linorder__max__induct,axiom,
! [A2: set_Extended_ereal,P: set_Extended_ereal > $o] :
( ( finite7198162374296863863_ereal @ A2 )
=> ( ( P @ bot_bo8367695208629047834_ereal )
=> ( ! [B7: extended_ereal,A5: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ A5 )
=> ( ! [X5: extended_ereal] :
( ( member2350847679896131959_ereal @ X5 @ A5 )
=> ( ord_le1188267648640031866_ereal @ X5 @ B7 ) )
=> ( ( P @ A5 )
=> ( P @ ( insert8967887681552722334_ereal @ B7 @ A5 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_471_finite__linorder__max__induct,axiom,
! [A2: set_real,P: set_real > $o] :
( ( finite_finite_real @ A2 )
=> ( ( P @ bot_bot_set_real )
=> ( ! [B7: real,A5: set_real] :
( ( finite_finite_real @ A5 )
=> ( ! [X5: real] :
( ( member_real @ X5 @ A5 )
=> ( ord_less_real @ X5 @ B7 ) )
=> ( ( P @ A5 )
=> ( P @ ( insert_real @ B7 @ A5 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_472_finite__linorder__max__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B7: nat,A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A5 )
=> ( ord_less_nat @ X5 @ B7 ) )
=> ( ( P @ A5 )
=> ( P @ ( insert_nat @ B7 @ A5 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_473_finite__ranking__induct,axiom,
! [S: set_set_a,P: set_set_a > $o,F: set_a > real] :
( ( finite_finite_set_a @ S )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X3: set_a,S2: set_set_a] :
( ( finite_finite_set_a @ S2 )
=> ( ! [Y5: set_a] :
( ( member_set_a @ Y5 @ S2 )
=> ( ord_less_eq_real @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_set_a @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_474_finite__ranking__induct,axiom,
! [S: set_Extended_ereal,P: set_Extended_ereal > $o,F: extended_ereal > real] :
( ( finite7198162374296863863_ereal @ S )
=> ( ( P @ bot_bo8367695208629047834_ereal )
=> ( ! [X3: extended_ereal,S2: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ S2 )
=> ( ! [Y5: extended_ereal] :
( ( member2350847679896131959_ereal @ Y5 @ S2 )
=> ( ord_less_eq_real @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert8967887681552722334_ereal @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_475_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F: a > real] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S2 )
=> ( ord_less_eq_real @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_476_finite__ranking__induct,axiom,
! [S: set_b,P: set_b > $o,F: b > real] :
( ( finite_finite_b @ S )
=> ( ( P @ bot_bot_set_b )
=> ( ! [X3: b,S2: set_b] :
( ( finite_finite_b @ S2 )
=> ( ! [Y5: b] :
( ( member_b @ Y5 @ S2 )
=> ( ord_less_eq_real @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_b @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_477_finite__ranking__induct,axiom,
! [S: set_pr5411798346947241657t_unit,P: set_pr5411798346947241657t_unit > $o,F: pre_pr7278220950009878019t_unit > real] :
( ( finite8852549406693098522t_unit @ S )
=> ( ( P @ bot_bo1839476491465656141t_unit )
=> ( ! [X3: pre_pr7278220950009878019t_unit,S2: set_pr5411798346947241657t_unit] :
( ( finite8852549406693098522t_unit @ S2 )
=> ( ! [Y5: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ Y5 @ S2 )
=> ( ord_less_eq_real @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert6864688055023459379t_unit @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_478_finite__ranking__induct,axiom,
! [S: set_set_a,P: set_set_a > $o,F: set_a > nat] :
( ( finite_finite_set_a @ S )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X3: set_a,S2: set_set_a] :
( ( finite_finite_set_a @ S2 )
=> ( ! [Y5: set_a] :
( ( member_set_a @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_set_a @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_479_finite__ranking__induct,axiom,
! [S: set_Extended_ereal,P: set_Extended_ereal > $o,F: extended_ereal > nat] :
( ( finite7198162374296863863_ereal @ S )
=> ( ( P @ bot_bo8367695208629047834_ereal )
=> ( ! [X3: extended_ereal,S2: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ S2 )
=> ( ! [Y5: extended_ereal] :
( ( member2350847679896131959_ereal @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert8967887681552722334_ereal @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_480_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_481_finite__ranking__induct,axiom,
! [S: set_b,P: set_b > $o,F: b > nat] :
( ( finite_finite_b @ S )
=> ( ( P @ bot_bot_set_b )
=> ( ! [X3: b,S2: set_b] :
( ( finite_finite_b @ S2 )
=> ( ! [Y5: b] :
( ( member_b @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_b @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_482_finite__ranking__induct,axiom,
! [S: set_pr5411798346947241657t_unit,P: set_pr5411798346947241657t_unit > $o,F: pre_pr7278220950009878019t_unit > nat] :
( ( finite8852549406693098522t_unit @ S )
=> ( ( P @ bot_bo1839476491465656141t_unit )
=> ( ! [X3: pre_pr7278220950009878019t_unit,S2: set_pr5411798346947241657t_unit] :
( ( finite8852549406693098522t_unit @ S2 )
=> ( ! [Y5: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert6864688055023459379t_unit @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_483_scc__disj,axiom,
! [C: pre_pr7278220950009878019t_unit,D: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ g ) )
=> ( ( member6939884229742472986t_unit @ D @ ( digraph_pre_sccs_a_b @ g ) )
=> ( ( C != D )
=> ( ( inf_inf_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ D ) )
= bot_bot_set_a ) ) ) ) ).
% scc_disj
thf(fact_484_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_485_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_486_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_487_image__eqI,axiom,
! [B: b,F: b > b,X: b,A2: set_b] :
( ( B
= ( F @ X ) )
=> ( ( member_b @ X @ A2 )
=> ( member_b @ B @ ( image_b_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_488_image__eqI,axiom,
! [B: a,F: b > a,X: b,A2: set_b] :
( ( B
= ( F @ X ) )
=> ( ( member_b @ X @ A2 )
=> ( member_a @ B @ ( image_b_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_489_image__eqI,axiom,
! [B: b,F: a > b,X: a,A2: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_b @ B @ ( image_a_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_490_image__eqI,axiom,
! [B: a,F: a > a,X: a,A2: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ B @ ( image_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_491_image__eqI,axiom,
! [B: set_a,F: b > set_a,X: b,A2: set_b] :
( ( B
= ( F @ X ) )
=> ( ( member_b @ X @ A2 )
=> ( member_set_a @ B @ ( image_b_set_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_492_image__eqI,axiom,
! [B: set_a,F: a > set_a,X: a,A2: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_set_a @ B @ ( image_a_set_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_493_image__eqI,axiom,
! [B: b,F: set_a > b,X: set_a,A2: set_set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_set_a @ X @ A2 )
=> ( member_b @ B @ ( image_set_a_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_494_image__eqI,axiom,
! [B: a,F: set_a > a,X: set_a,A2: set_set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_set_a @ X @ A2 )
=> ( member_a @ B @ ( image_set_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_495_image__eqI,axiom,
! [B: set_a,F: set_a > set_a,X: set_a,A2: set_set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_set_a @ X @ A2 )
=> ( member_set_a @ B @ ( image_set_a_set_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_496_image__eqI,axiom,
! [B: pre_pr7278220950009878019t_unit,F: b > pre_pr7278220950009878019t_unit,X: b,A2: set_b] :
( ( B
= ( F @ X ) )
=> ( ( member_b @ X @ A2 )
=> ( member6939884229742472986t_unit @ B @ ( image_4434118323594779837t_unit @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_497_IntI,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ A2 )
=> ( ( member_b @ C @ B2 )
=> ( member_b @ C @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_498_IntI,axiom,
! [C: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C @ A2 )
=> ( ( member6939884229742472986t_unit @ C @ B2 )
=> ( member6939884229742472986t_unit @ C @ ( inf_in1092213268631476299t_unit @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_499_IntI,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_500_IntI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ( member_a @ C @ B2 )
=> ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_501_Int__iff,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A2 @ B2 ) )
= ( ( member_b @ C @ A2 )
& ( member_b @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_502_Int__iff,axiom,
! [C: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C @ ( inf_in1092213268631476299t_unit @ A2 @ B2 ) )
= ( ( member6939884229742472986t_unit @ C @ A2 )
& ( member6939884229742472986t_unit @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_503_Int__iff,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B2 ) )
= ( ( member_set_a @ C @ A2 )
& ( member_set_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_504_Int__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ( member_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_505_DiffI,axiom,
! [C: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C @ A2 )
=> ( ~ ( member6939884229742472986t_unit @ C @ B2 )
=> ( member6939884229742472986t_unit @ C @ ( minus_3777555517894451474t_unit @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_506_DiffI,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( ~ ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_507_DiffI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_508_DiffI,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ A2 )
=> ( ~ ( member_b @ C @ B2 )
=> ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_509_Diff__iff,axiom,
! [C: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C @ ( minus_3777555517894451474t_unit @ A2 @ B2 ) )
= ( ( member6939884229742472986t_unit @ C @ A2 )
& ~ ( member6939884229742472986t_unit @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_510_Diff__iff,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
= ( ( member_set_a @ C @ A2 )
& ~ ( member_set_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_511_Diff__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_512_Diff__iff,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
= ( ( member_b @ C @ A2 )
& ~ ( member_b @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_513_Diff__idemp,axiom,
! [A2: set_a,B2: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_514_Diff__idemp,axiom,
! [A2: set_b,B2: set_b] :
( ( minus_minus_set_b @ ( minus_minus_set_b @ A2 @ B2 ) @ B2 )
= ( minus_minus_set_b @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_515_sccs__verts__disjoint,axiom,
! [S: set_a,T2: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ g ) )
=> ( ( member_set_a @ T2 @ ( digrap2871191568752656621ts_a_b @ g ) )
=> ( ( S != T2 )
=> ( ( inf_inf_set_a @ S @ T2 )
= bot_bot_set_a ) ) ) ) ).
% sccs_verts_disjoint
thf(fact_516_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_517_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_518_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_519_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_520_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_521_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_522_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_523_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_524_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_525_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_526_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_527_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_528_image__is__empty,axiom,
! [F: pre_pr7278220950009878019t_unit > a,A2: set_pr5411798346947241657t_unit] :
( ( ( image_4969699134812999796unit_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bo1839476491465656141t_unit ) ) ).
% image_is_empty
thf(fact_529_image__is__empty,axiom,
! [F: pre_pr7278220950009878019t_unit > b,A2: set_pr5411798346947241657t_unit] :
( ( ( image_4969699134812999797unit_b @ F @ A2 )
= bot_bot_set_b )
= ( A2 = bot_bo1839476491465656141t_unit ) ) ).
% image_is_empty
thf(fact_530_image__is__empty,axiom,
! [F: a > pre_pr7278220950009878019t_unit,A2: set_a] :
( ( ( image_5713294457175270716t_unit @ F @ A2 )
= bot_bo1839476491465656141t_unit )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_531_image__is__empty,axiom,
! [F: b > pre_pr7278220950009878019t_unit,A2: set_b] :
( ( ( image_4434118323594779837t_unit @ F @ A2 )
= bot_bo1839476491465656141t_unit )
= ( A2 = bot_bot_set_b ) ) ).
% image_is_empty
thf(fact_532_image__is__empty,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit] :
( ( ( image_7466199892558553556_set_a @ F @ A2 )
= bot_bot_set_set_a )
= ( A2 = bot_bo1839476491465656141t_unit ) ) ).
% image_is_empty
thf(fact_533_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_534_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_535_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_536_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_537_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_538_empty__is__image,axiom,
! [F: pre_pr7278220950009878019t_unit > a,A2: set_pr5411798346947241657t_unit] :
( ( bot_bot_set_a
= ( image_4969699134812999796unit_a @ F @ A2 ) )
= ( A2 = bot_bo1839476491465656141t_unit ) ) ).
% empty_is_image
thf(fact_539_empty__is__image,axiom,
! [F: pre_pr7278220950009878019t_unit > b,A2: set_pr5411798346947241657t_unit] :
( ( bot_bot_set_b
= ( image_4969699134812999797unit_b @ F @ A2 ) )
= ( A2 = bot_bo1839476491465656141t_unit ) ) ).
% empty_is_image
thf(fact_540_empty__is__image,axiom,
! [F: a > pre_pr7278220950009878019t_unit,A2: set_a] :
( ( bot_bo1839476491465656141t_unit
= ( image_5713294457175270716t_unit @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_541_empty__is__image,axiom,
! [F: b > pre_pr7278220950009878019t_unit,A2: set_b] :
( ( bot_bo1839476491465656141t_unit
= ( image_4434118323594779837t_unit @ F @ A2 ) )
= ( A2 = bot_bot_set_b ) ) ).
% empty_is_image
thf(fact_542_empty__is__image,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit] :
( ( bot_bot_set_set_a
= ( image_7466199892558553556_set_a @ F @ A2 ) )
= ( A2 = bot_bo1839476491465656141t_unit ) ) ).
% empty_is_image
thf(fact_543_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_544_image__empty,axiom,
! [F: a > b] :
( ( image_a_b @ F @ bot_bot_set_a )
= bot_bot_set_b ) ).
% image_empty
thf(fact_545_image__empty,axiom,
! [F: b > a] :
( ( image_b_a @ F @ bot_bot_set_b )
= bot_bot_set_a ) ).
% image_empty
thf(fact_546_image__empty,axiom,
! [F: b > b] :
( ( image_b_b @ F @ bot_bot_set_b )
= bot_bot_set_b ) ).
% image_empty
thf(fact_547_image__empty,axiom,
! [F: a > set_a] :
( ( image_a_set_a @ F @ bot_bot_set_a )
= bot_bot_set_set_a ) ).
% image_empty
thf(fact_548_image__empty,axiom,
! [F: a > pre_pr7278220950009878019t_unit] :
( ( image_5713294457175270716t_unit @ F @ bot_bot_set_a )
= bot_bo1839476491465656141t_unit ) ).
% image_empty
thf(fact_549_image__empty,axiom,
! [F: b > pre_pr7278220950009878019t_unit] :
( ( image_4434118323594779837t_unit @ F @ bot_bot_set_b )
= bot_bo1839476491465656141t_unit ) ).
% image_empty
thf(fact_550_image__empty,axiom,
! [F: pre_pr7278220950009878019t_unit > a] :
( ( image_4969699134812999796unit_a @ F @ bot_bo1839476491465656141t_unit )
= bot_bot_set_a ) ).
% image_empty
thf(fact_551_image__empty,axiom,
! [F: pre_pr7278220950009878019t_unit > b] :
( ( image_4969699134812999797unit_b @ F @ bot_bo1839476491465656141t_unit )
= bot_bot_set_b ) ).
% image_empty
thf(fact_552_image__empty,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit] :
( ( image_6801035452528096924t_unit @ F @ bot_bot_set_set_a )
= bot_bo1839476491465656141t_unit ) ).
% image_empty
thf(fact_553_finite__imageI,axiom,
! [F4: set_a,H2: a > a] :
( ( finite_finite_a @ F4 )
=> ( finite_finite_a @ ( image_a_a @ H2 @ F4 ) ) ) ).
% finite_imageI
thf(fact_554_finite__imageI,axiom,
! [F4: set_a,H2: a > b] :
( ( finite_finite_a @ F4 )
=> ( finite_finite_b @ ( image_a_b @ H2 @ F4 ) ) ) ).
% finite_imageI
thf(fact_555_finite__imageI,axiom,
! [F4: set_a,H2: a > extended_ereal] :
( ( finite_finite_a @ F4 )
=> ( finite7198162374296863863_ereal @ ( image_8405481351990995413_ereal @ H2 @ F4 ) ) ) ).
% finite_imageI
thf(fact_556_finite__imageI,axiom,
! [F4: set_b,H2: b > a] :
( ( finite_finite_b @ F4 )
=> ( finite_finite_a @ ( image_b_a @ H2 @ F4 ) ) ) ).
% finite_imageI
thf(fact_557_finite__imageI,axiom,
! [F4: set_b,H2: b > b] :
( ( finite_finite_b @ F4 )
=> ( finite_finite_b @ ( image_b_b @ H2 @ F4 ) ) ) ).
% finite_imageI
thf(fact_558_finite__imageI,axiom,
! [F4: set_b,H2: b > extended_ereal] :
( ( finite_finite_b @ F4 )
=> ( finite7198162374296863863_ereal @ ( image_5319725110001000852_ereal @ H2 @ F4 ) ) ) ).
% finite_imageI
thf(fact_559_finite__imageI,axiom,
! [F4: set_Extended_ereal,H2: extended_ereal > a] :
( ( finite7198162374296863863_ereal @ F4 )
=> ( finite_finite_a @ ( image_3724615099042636213real_a @ H2 @ F4 ) ) ) ).
% finite_imageI
thf(fact_560_finite__imageI,axiom,
! [F4: set_Extended_ereal,H2: extended_ereal > b] :
( ( finite7198162374296863863_ereal @ F4 )
=> ( finite_finite_b @ ( image_3724615099042636214real_b @ H2 @ F4 ) ) ) ).
% finite_imageI
thf(fact_561_finite__imageI,axiom,
! [F4: set_Extended_ereal,H2: extended_ereal > extended_ereal] :
( ( finite7198162374296863863_ereal @ F4 )
=> ( finite7198162374296863863_ereal @ ( image_6042159593519690757_ereal @ H2 @ F4 ) ) ) ).
% finite_imageI
thf(fact_562_finite__imageI,axiom,
! [F4: set_a,H2: a > set_a] :
( ( finite_finite_a @ F4 )
=> ( finite_finite_set_a @ ( image_a_set_a @ H2 @ F4 ) ) ) ).
% finite_imageI
thf(fact_563_insert__image,axiom,
! [X: b,A2: set_b,F: b > a] :
( ( member_b @ X @ A2 )
=> ( ( insert_a @ ( F @ X ) @ ( image_b_a @ F @ A2 ) )
= ( image_b_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_564_insert__image,axiom,
! [X: b,A2: set_b,F: b > b] :
( ( member_b @ X @ A2 )
=> ( ( insert_b @ ( F @ X ) @ ( image_b_b @ F @ A2 ) )
= ( image_b_b @ F @ A2 ) ) ) ).
% insert_image
thf(fact_565_insert__image,axiom,
! [X: a,A2: set_a,F: a > a] :
( ( member_a @ X @ A2 )
=> ( ( insert_a @ ( F @ X ) @ ( image_a_a @ F @ A2 ) )
= ( image_a_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_566_insert__image,axiom,
! [X: a,A2: set_a,F: a > b] :
( ( member_a @ X @ A2 )
=> ( ( insert_b @ ( F @ X ) @ ( image_a_b @ F @ A2 ) )
= ( image_a_b @ F @ A2 ) ) ) ).
% insert_image
thf(fact_567_insert__image,axiom,
! [X: a,A2: set_a,F: a > set_a] :
( ( member_a @ X @ A2 )
=> ( ( insert_set_a @ ( F @ X ) @ ( image_a_set_a @ F @ A2 ) )
= ( image_a_set_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_568_insert__image,axiom,
! [X: set_a,A2: set_set_a,F: set_a > a] :
( ( member_set_a @ X @ A2 )
=> ( ( insert_a @ ( F @ X ) @ ( image_set_a_a @ F @ A2 ) )
= ( image_set_a_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_569_insert__image,axiom,
! [X: set_a,A2: set_set_a,F: set_a > b] :
( ( member_set_a @ X @ A2 )
=> ( ( insert_b @ ( F @ X ) @ ( image_set_a_b @ F @ A2 ) )
= ( image_set_a_b @ F @ A2 ) ) ) ).
% insert_image
thf(fact_570_insert__image,axiom,
! [X: b,A2: set_b,F: b > pre_pr7278220950009878019t_unit] :
( ( member_b @ X @ A2 )
=> ( ( insert6864688055023459379t_unit @ ( F @ X ) @ ( image_4434118323594779837t_unit @ F @ A2 ) )
= ( image_4434118323594779837t_unit @ F @ A2 ) ) ) ).
% insert_image
thf(fact_571_insert__image,axiom,
! [X: a,A2: set_a,F: a > pre_pr7278220950009878019t_unit] :
( ( member_a @ X @ A2 )
=> ( ( insert6864688055023459379t_unit @ ( F @ X ) @ ( image_5713294457175270716t_unit @ F @ A2 ) )
= ( image_5713294457175270716t_unit @ F @ A2 ) ) ) ).
% insert_image
thf(fact_572_insert__image,axiom,
! [X: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > a] :
( ( member6939884229742472986t_unit @ X @ A2 )
=> ( ( insert_a @ ( F @ X ) @ ( image_4969699134812999796unit_a @ F @ A2 ) )
= ( image_4969699134812999796unit_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_573_image__insert,axiom,
! [F: a > a,A: a,B2: set_a] :
( ( image_a_a @ F @ ( insert_a @ A @ B2 ) )
= ( insert_a @ ( F @ A ) @ ( image_a_a @ F @ B2 ) ) ) ).
% image_insert
thf(fact_574_image__insert,axiom,
! [F: a > b,A: a,B2: set_a] :
( ( image_a_b @ F @ ( insert_a @ A @ B2 ) )
= ( insert_b @ ( F @ A ) @ ( image_a_b @ F @ B2 ) ) ) ).
% image_insert
thf(fact_575_image__insert,axiom,
! [F: b > a,A: b,B2: set_b] :
( ( image_b_a @ F @ ( insert_b @ A @ B2 ) )
= ( insert_a @ ( F @ A ) @ ( image_b_a @ F @ B2 ) ) ) ).
% image_insert
thf(fact_576_image__insert,axiom,
! [F: b > b,A: b,B2: set_b] :
( ( image_b_b @ F @ ( insert_b @ A @ B2 ) )
= ( insert_b @ ( F @ A ) @ ( image_b_b @ F @ B2 ) ) ) ).
% image_insert
thf(fact_577_image__insert,axiom,
! [F: a > set_a,A: a,B2: set_a] :
( ( image_a_set_a @ F @ ( insert_a @ A @ B2 ) )
= ( insert_set_a @ ( F @ A ) @ ( image_a_set_a @ F @ B2 ) ) ) ).
% image_insert
thf(fact_578_image__insert,axiom,
! [F: a > pre_pr7278220950009878019t_unit,A: a,B2: set_a] :
( ( image_5713294457175270716t_unit @ F @ ( insert_a @ A @ B2 ) )
= ( insert6864688055023459379t_unit @ ( F @ A ) @ ( image_5713294457175270716t_unit @ F @ B2 ) ) ) ).
% image_insert
thf(fact_579_image__insert,axiom,
! [F: b > pre_pr7278220950009878019t_unit,A: b,B2: set_b] :
( ( image_4434118323594779837t_unit @ F @ ( insert_b @ A @ B2 ) )
= ( insert6864688055023459379t_unit @ ( F @ A ) @ ( image_4434118323594779837t_unit @ F @ B2 ) ) ) ).
% image_insert
thf(fact_580_image__insert,axiom,
! [F: pre_pr7278220950009878019t_unit > a,A: pre_pr7278220950009878019t_unit,B2: set_pr5411798346947241657t_unit] :
( ( image_4969699134812999796unit_a @ F @ ( insert6864688055023459379t_unit @ A @ B2 ) )
= ( insert_a @ ( F @ A ) @ ( image_4969699134812999796unit_a @ F @ B2 ) ) ) ).
% image_insert
thf(fact_581_image__insert,axiom,
! [F: pre_pr7278220950009878019t_unit > b,A: pre_pr7278220950009878019t_unit,B2: set_pr5411798346947241657t_unit] :
( ( image_4969699134812999797unit_b @ F @ ( insert6864688055023459379t_unit @ A @ B2 ) )
= ( insert_b @ ( F @ A ) @ ( image_4969699134812999797unit_b @ F @ B2 ) ) ) ).
% image_insert
thf(fact_582_image__insert,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A: set_a,B2: set_set_a] :
( ( image_6801035452528096924t_unit @ F @ ( insert_set_a @ A @ B2 ) )
= ( insert6864688055023459379t_unit @ ( F @ A ) @ ( image_6801035452528096924t_unit @ F @ B2 ) ) ) ).
% image_insert
thf(fact_583_finite__Int,axiom,
! [F4: set_b,G2: set_b] :
( ( ( finite_finite_b @ F4 )
| ( finite_finite_b @ G2 ) )
=> ( finite_finite_b @ ( inf_inf_set_b @ F4 @ G2 ) ) ) ).
% finite_Int
thf(fact_584_finite__Int,axiom,
! [F4: set_set_a,G2: set_set_a] :
( ( ( finite_finite_set_a @ F4 )
| ( finite_finite_set_a @ G2 ) )
=> ( finite_finite_set_a @ ( inf_inf_set_set_a @ F4 @ G2 ) ) ) ).
% finite_Int
thf(fact_585_finite__Int,axiom,
! [F4: set_Extended_ereal,G2: set_Extended_ereal] :
( ( ( finite7198162374296863863_ereal @ F4 )
| ( finite7198162374296863863_ereal @ G2 ) )
=> ( finite7198162374296863863_ereal @ ( inf_in2779415704524776092_ereal @ F4 @ G2 ) ) ) ).
% finite_Int
thf(fact_586_finite__Int,axiom,
! [F4: set_a,G2: set_a] :
( ( ( finite_finite_a @ F4 )
| ( finite_finite_a @ G2 ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F4 @ G2 ) ) ) ).
% finite_Int
thf(fact_587_Int__subset__iff,axiom,
! [C3: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( ord_less_eq_set_a @ C3 @ A2 )
& ( ord_less_eq_set_a @ C3 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_588_Int__subset__iff,axiom,
! [C3: set_pr5411798346947241657t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ C3 @ ( inf_in1092213268631476299t_unit @ A2 @ B2 ) )
= ( ( ord_le8200006823705900825t_unit @ C3 @ A2 )
& ( ord_le8200006823705900825t_unit @ C3 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_589_Diff__cancel,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ( minus_3777555517894451474t_unit @ A2 @ A2 )
= bot_bo1839476491465656141t_unit ) ).
% Diff_cancel
thf(fact_590_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_591_Diff__cancel,axiom,
! [A2: set_b] :
( ( minus_minus_set_b @ A2 @ A2 )
= bot_bot_set_b ) ).
% Diff_cancel
thf(fact_592_empty__Diff,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ( minus_3777555517894451474t_unit @ bot_bo1839476491465656141t_unit @ A2 )
= bot_bo1839476491465656141t_unit ) ).
% empty_Diff
thf(fact_593_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_594_empty__Diff,axiom,
! [A2: set_b] :
( ( minus_minus_set_b @ bot_bot_set_b @ A2 )
= bot_bot_set_b ) ).
% empty_Diff
thf(fact_595_Diff__empty,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ( minus_3777555517894451474t_unit @ A2 @ bot_bo1839476491465656141t_unit )
= A2 ) ).
% Diff_empty
thf(fact_596_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_597_Diff__empty,axiom,
! [A2: set_b] :
( ( minus_minus_set_b @ A2 @ bot_bot_set_b )
= A2 ) ).
% Diff_empty
thf(fact_598_Int__insert__right__if1,axiom,
! [A: b,A2: set_b,B2: set_b] :
( ( member_b @ A @ A2 )
=> ( ( inf_inf_set_b @ A2 @ ( insert_b @ A @ B2 ) )
= ( insert_b @ A @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_599_Int__insert__right__if1,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ A @ A2 )
=> ( ( inf_in1092213268631476299t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ B2 ) )
= ( insert6864688055023459379t_unit @ A @ ( inf_in1092213268631476299t_unit @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_600_Int__insert__right__if1,axiom,
! [A: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B2 ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_601_Int__insert__right__if1,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_602_Int__insert__right__if0,axiom,
! [A: b,A2: set_b,B2: set_b] :
( ~ ( member_b @ A @ A2 )
=> ( ( inf_inf_set_b @ A2 @ ( insert_b @ A @ B2 ) )
= ( inf_inf_set_b @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_603_Int__insert__right__if0,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ A @ A2 )
=> ( ( inf_in1092213268631476299t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ B2 ) )
= ( inf_in1092213268631476299t_unit @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_604_Int__insert__right__if0,axiom,
! [A: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B2 ) )
= ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_605_Int__insert__right__if0,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_606_insert__inter__insert,axiom,
! [A: b,A2: set_b,B2: set_b] :
( ( inf_inf_set_b @ ( insert_b @ A @ A2 ) @ ( insert_b @ A @ B2 ) )
= ( insert_b @ A @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_607_insert__inter__insert,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A @ A2 ) @ ( insert6864688055023459379t_unit @ A @ B2 ) )
= ( insert6864688055023459379t_unit @ A @ ( inf_in1092213268631476299t_unit @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_608_insert__inter__insert,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_609_Int__insert__left__if1,axiom,
! [A: b,C3: set_b,B2: set_b] :
( ( member_b @ A @ C3 )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B2 ) @ C3 )
= ( insert_b @ A @ ( inf_inf_set_b @ B2 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_610_Int__insert__left__if1,axiom,
! [A: pre_pr7278220950009878019t_unit,C3: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ A @ C3 )
=> ( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A @ B2 ) @ C3 )
= ( insert6864688055023459379t_unit @ A @ ( inf_in1092213268631476299t_unit @ B2 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_611_Int__insert__left__if1,axiom,
! [A: set_a,C3: set_set_a,B2: set_set_a] :
( ( member_set_a @ A @ C3 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B2 ) @ C3 )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_612_Int__insert__left__if1,axiom,
! [A: a,C3: set_a,B2: set_a] :
( ( member_a @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C3 )
= ( insert_a @ A @ ( inf_inf_set_a @ B2 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_613_Int__insert__left__if0,axiom,
! [A: b,C3: set_b,B2: set_b] :
( ~ ( member_b @ A @ C3 )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B2 ) @ C3 )
= ( inf_inf_set_b @ B2 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_614_Int__insert__left__if0,axiom,
! [A: pre_pr7278220950009878019t_unit,C3: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ A @ C3 )
=> ( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A @ B2 ) @ C3 )
= ( inf_in1092213268631476299t_unit @ B2 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_615_Int__insert__left__if0,axiom,
! [A: set_a,C3: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ A @ C3 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B2 ) @ C3 )
= ( inf_inf_set_set_a @ B2 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_616_Int__insert__left__if0,axiom,
! [A: a,C3: set_a,B2: set_a] :
( ~ ( member_a @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C3 )
= ( inf_inf_set_a @ B2 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_617_finite__Diff2,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
= ( finite_finite_set_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_618_finite__Diff2,axiom,
! [B2: set_Extended_ereal,A2: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ B2 )
=> ( ( finite7198162374296863863_ereal @ ( minus_1264018925008434325_ereal @ A2 @ B2 ) )
= ( finite7198162374296863863_ereal @ A2 ) ) ) ).
% finite_Diff2
thf(fact_619_finite__Diff2,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_620_finite__Diff2,axiom,
! [B2: set_b,A2: set_b] :
( ( finite_finite_b @ B2 )
=> ( ( finite_finite_b @ ( minus_minus_set_b @ A2 @ B2 ) )
= ( finite_finite_b @ A2 ) ) ) ).
% finite_Diff2
thf(fact_621_finite__Diff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_622_finite__Diff,axiom,
! [A2: set_Extended_ereal,B2: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ A2 )
=> ( finite7198162374296863863_ereal @ ( minus_1264018925008434325_ereal @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_623_finite__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_624_finite__Diff,axiom,
! [A2: set_b,B2: set_b] :
( ( finite_finite_b @ A2 )
=> ( finite_finite_b @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_625_insert__Diff1,axiom,
! [X: pre_pr7278220950009878019t_unit,B2: set_pr5411798346947241657t_unit,A2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ X @ B2 )
=> ( ( minus_3777555517894451474t_unit @ ( insert6864688055023459379t_unit @ X @ A2 ) @ B2 )
= ( minus_3777555517894451474t_unit @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_626_insert__Diff1,axiom,
! [X: set_a,B2: set_set_a,A2: set_set_a] :
( ( member_set_a @ X @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B2 )
= ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_627_insert__Diff1,axiom,
! [X: a,B2: set_a,A2: set_a] :
( ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_628_insert__Diff1,axiom,
! [X: b,B2: set_b,A2: set_b] :
( ( member_b @ X @ B2 )
=> ( ( minus_minus_set_b @ ( insert_b @ X @ A2 ) @ B2 )
= ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_629_Diff__insert0,axiom,
! [X: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ X @ A2 )
=> ( ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ X @ B2 ) )
= ( minus_3777555517894451474t_unit @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_630_Diff__insert0,axiom,
! [X: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ B2 ) )
= ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_631_Diff__insert0,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_632_Diff__insert0,axiom,
! [X: b,A2: set_b,B2: set_b] :
( ~ ( member_b @ X @ A2 )
=> ( ( minus_minus_set_b @ A2 @ ( insert_b @ X @ B2 ) )
= ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_633_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_634_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_635_insert__disjoint_I1_J,axiom,
! [A: set_a,A2: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ B2 )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B2 )
& ( ( inf_inf_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_636_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B2 )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B2 )
& ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_637_insert__disjoint_I1_J,axiom,
! [A: b,A2: set_b,B2: set_b] :
( ( ( inf_inf_set_b @ ( insert_b @ A @ A2 ) @ B2 )
= bot_bot_set_b )
= ( ~ ( member_b @ A @ B2 )
& ( ( inf_inf_set_b @ A2 @ B2 )
= bot_bot_set_b ) ) ) ).
% insert_disjoint(1)
thf(fact_638_insert__disjoint_I1_J,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A @ A2 ) @ B2 )
= bot_bo1839476491465656141t_unit )
= ( ~ ( member6939884229742472986t_unit @ A @ B2 )
& ( ( inf_in1092213268631476299t_unit @ A2 @ B2 )
= bot_bo1839476491465656141t_unit ) ) ) ).
% insert_disjoint(1)
thf(fact_639_insert__disjoint_I2_J,axiom,
! [A: set_a,A2: set_set_a,B2: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ B2 ) )
= ( ~ ( member_set_a @ A @ B2 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_640_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B2 ) )
= ( ~ ( member_a @ A @ B2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_641_insert__disjoint_I2_J,axiom,
! [A: b,A2: set_b,B2: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ ( insert_b @ A @ A2 ) @ B2 ) )
= ( ~ ( member_b @ A @ B2 )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_642_insert__disjoint_I2_J,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( bot_bo1839476491465656141t_unit
= ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A @ A2 ) @ B2 ) )
= ( ~ ( member6939884229742472986t_unit @ A @ B2 )
& ( bot_bo1839476491465656141t_unit
= ( inf_in1092213268631476299t_unit @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_643_disjoint__insert_I1_J,axiom,
! [B2: set_set_a,A: set_a,A2: set_set_a] :
( ( ( inf_inf_set_set_a @ B2 @ ( insert_set_a @ A @ A2 ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B2 )
& ( ( inf_inf_set_set_a @ B2 @ A2 )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_644_disjoint__insert_I1_J,axiom,
! [B2: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B2 @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B2 )
& ( ( inf_inf_set_a @ B2 @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_645_disjoint__insert_I1_J,axiom,
! [B2: set_b,A: b,A2: set_b] :
( ( ( inf_inf_set_b @ B2 @ ( insert_b @ A @ A2 ) )
= bot_bot_set_b )
= ( ~ ( member_b @ A @ B2 )
& ( ( inf_inf_set_b @ B2 @ A2 )
= bot_bot_set_b ) ) ) ).
% disjoint_insert(1)
thf(fact_646_disjoint__insert_I1_J,axiom,
! [B2: set_pr5411798346947241657t_unit,A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( ( inf_in1092213268631476299t_unit @ B2 @ ( insert6864688055023459379t_unit @ A @ A2 ) )
= bot_bo1839476491465656141t_unit )
= ( ~ ( member6939884229742472986t_unit @ A @ B2 )
& ( ( inf_in1092213268631476299t_unit @ B2 @ A2 )
= bot_bo1839476491465656141t_unit ) ) ) ).
% disjoint_insert(1)
thf(fact_647_disjoint__insert_I2_J,axiom,
! [A2: set_set_a,B: set_a,B2: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ B @ B2 ) ) )
= ( ~ ( member_set_a @ B @ A2 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_648_disjoint__insert_I2_J,axiom,
! [A2: set_a,B: a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B @ B2 ) ) )
= ( ~ ( member_a @ B @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_649_disjoint__insert_I2_J,axiom,
! [A2: set_b,B: b,B2: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ A2 @ ( insert_b @ B @ B2 ) ) )
= ( ~ ( member_b @ B @ A2 )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_650_disjoint__insert_I2_J,axiom,
! [A2: set_pr5411798346947241657t_unit,B: pre_pr7278220950009878019t_unit,B2: set_pr5411798346947241657t_unit] :
( ( bot_bo1839476491465656141t_unit
= ( inf_in1092213268631476299t_unit @ A2 @ ( insert6864688055023459379t_unit @ B @ B2 ) ) )
= ( ~ ( member6939884229742472986t_unit @ B @ A2 )
& ( bot_bo1839476491465656141t_unit
= ( inf_in1092213268631476299t_unit @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_651_Diff__eq__empty__iff,axiom,
! [A2: set_b,B2: set_b] :
( ( ( minus_minus_set_b @ A2 @ B2 )
= bot_bot_set_b )
= ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_652_Diff__eq__empty__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( minus_minus_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_653_Diff__eq__empty__iff,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( ( minus_3777555517894451474t_unit @ A2 @ B2 )
= bot_bo1839476491465656141t_unit )
= ( ord_le8200006823705900825t_unit @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_654_insert__Diff__single,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( insert6864688055023459379t_unit @ A @ ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) ) )
= ( insert6864688055023459379t_unit @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_655_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_656_insert__Diff__single,axiom,
! [A: b,A2: set_b] :
( ( insert_b @ A @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
= ( insert_b @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_657_finite__Diff__insert,axiom,
! [A2: set_pr5411798346947241657t_unit,A: pre_pr7278220950009878019t_unit,B2: set_pr5411798346947241657t_unit] :
( ( finite8852549406693098522t_unit @ ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ B2 ) ) )
= ( finite8852549406693098522t_unit @ ( minus_3777555517894451474t_unit @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_658_finite__Diff__insert,axiom,
! [A2: set_set_a,A: set_a,B2: set_set_a] :
( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ B2 ) ) )
= ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_659_finite__Diff__insert,axiom,
! [A2: set_Extended_ereal,A: extended_ereal,B2: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ ( minus_1264018925008434325_ereal @ A2 @ ( insert8967887681552722334_ereal @ A @ B2 ) ) )
= ( finite7198162374296863863_ereal @ ( minus_1264018925008434325_ereal @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_660_finite__Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_661_finite__Diff__insert,axiom,
! [A2: set_b,A: b,B2: set_b] :
( ( finite_finite_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ B2 ) ) )
= ( finite_finite_b @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_662_Diff__disjoint,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ A2 @ ( minus_3777555517894451474t_unit @ B2 @ A2 ) )
= bot_bo1839476491465656141t_unit ) ).
% Diff_disjoint
thf(fact_663_Diff__disjoint,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B2 @ A2 ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_664_Diff__disjoint,axiom,
! [A2: set_b,B2: set_b] :
( ( inf_inf_set_b @ A2 @ ( minus_minus_set_b @ B2 @ A2 ) )
= bot_bot_set_b ) ).
% Diff_disjoint
thf(fact_665_spanning__tree__imp__connected,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digrap5718416180170401981ee_a_b @ H @ g )
=> ( digrap8783888973171253482ed_a_b @ g ) ) ).
% spanning_tree_imp_connected
thf(fact_666_inj__on__insert,axiom,
! [F: a > b,A: a,A2: set_a] :
( ( inj_on_a_b @ F @ ( insert_a @ A @ A2 ) )
= ( ( inj_on_a_b @ F @ A2 )
& ~ ( member_b @ ( F @ A ) @ ( image_a_b @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_667_inj__on__insert,axiom,
! [F: a > a,A: a,A2: set_a] :
( ( inj_on_a_a @ F @ ( insert_a @ A @ A2 ) )
= ( ( inj_on_a_a @ F @ A2 )
& ~ ( member_a @ ( F @ A ) @ ( image_a_a @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_668_inj__on__insert,axiom,
! [F: b > b,A: b,A2: set_b] :
( ( inj_on_b_b @ F @ ( insert_b @ A @ A2 ) )
= ( ( inj_on_b_b @ F @ A2 )
& ~ ( member_b @ ( F @ A ) @ ( image_b_b @ F @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_669_inj__on__insert,axiom,
! [F: b > a,A: b,A2: set_b] :
( ( inj_on_b_a @ F @ ( insert_b @ A @ A2 ) )
= ( ( inj_on_b_a @ F @ A2 )
& ~ ( member_a @ ( F @ A ) @ ( image_b_a @ F @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_670_inj__on__insert,axiom,
! [F: a > set_a,A: a,A2: set_a] :
( ( inj_on_a_set_a @ F @ ( insert_a @ A @ A2 ) )
= ( ( inj_on_a_set_a @ F @ A2 )
& ~ ( member_set_a @ ( F @ A ) @ ( image_a_set_a @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_671_inj__on__insert,axiom,
! [F: b > set_a,A: b,A2: set_b] :
( ( inj_on_b_set_a @ F @ ( insert_b @ A @ A2 ) )
= ( ( inj_on_b_set_a @ F @ A2 )
& ~ ( member_set_a @ ( F @ A ) @ ( image_b_set_a @ F @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_672_inj__on__insert,axiom,
! [F: b > product_prod_a_a,A: b,A2: set_b] :
( ( inj_on8302219782909765977od_a_a @ F @ ( insert_b @ A @ A2 ) )
= ( ( inj_on8302219782909765977od_a_a @ F @ A2 )
& ~ ( member1426531477525435216od_a_a @ ( F @ A ) @ ( image_6761185482258179565od_a_a @ F @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_673_inj__on__insert,axiom,
! [F: pre_pr7278220950009878019t_unit > b,A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( inj_on4336934664824710153unit_b @ F @ ( insert6864688055023459379t_unit @ A @ A2 ) )
= ( ( inj_on4336934664824710153unit_b @ F @ A2 )
& ~ ( member_b @ ( F @ A ) @ ( image_4969699134812999797unit_b @ F @ ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_674_inj__on__insert,axiom,
! [F: pre_pr7278220950009878019t_unit > a,A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( inj_on4336934664824710152unit_a @ F @ ( insert6864688055023459379t_unit @ A @ A2 ) )
= ( ( inj_on4336934664824710152unit_a @ F @ A2 )
& ~ ( member_a @ ( F @ A ) @ ( image_4969699134812999796unit_a @ F @ ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_675_inj__on__insert,axiom,
! [F: a > pre_pr7278220950009878019t_unit,A: a,A2: set_a] :
( ( inj_on5080529987186981072t_unit @ F @ ( insert_a @ A @ A2 ) )
= ( ( inj_on5080529987186981072t_unit @ F @ A2 )
& ~ ( member6939884229742472986t_unit @ ( F @ A ) @ ( image_5713294457175270716t_unit @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_676_in__sccsI,axiom,
! [C: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ C @ g )
=> ( ( digrap8691851296217657702ed_a_b @ C )
=> ( ~ ? [C5: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ C5 @ g )
& ( digrap8691851296217657702ed_a_b @ C5 )
& ( ord_less_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ C5 ) ) )
=> ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ g ) ) ) ) ) ).
% in_sccsI
thf(fact_677_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_678_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_679_image__diff__subset,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ ( image_7466199892558553556_set_a @ F @ A2 ) @ ( image_7466199892558553556_set_a @ F @ B2 ) ) @ ( image_7466199892558553556_set_a @ F @ ( minus_3777555517894451474t_unit @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_680_image__diff__subset,axiom,
! [F: a > set_a,A2: set_a,B2: set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ ( image_a_set_a @ F @ A2 ) @ ( image_a_set_a @ F @ B2 ) ) @ ( image_a_set_a @ F @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_681_image__diff__subset,axiom,
! [F: a > b,A2: set_a,B2: set_a] : ( ord_less_eq_set_b @ ( minus_minus_set_b @ ( image_a_b @ F @ A2 ) @ ( image_a_b @ F @ B2 ) ) @ ( image_a_b @ F @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_682_image__diff__subset,axiom,
! [F: b > b,A2: set_b,B2: set_b] : ( ord_less_eq_set_b @ ( minus_minus_set_b @ ( image_b_b @ F @ A2 ) @ ( image_b_b @ F @ B2 ) ) @ ( image_b_b @ F @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_683_image__diff__subset,axiom,
! [F: a > a,A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B2 ) ) @ ( image_a_a @ F @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_684_image__diff__subset,axiom,
! [F: b > a,A2: set_b,B2: set_b] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_b_a @ F @ A2 ) @ ( image_b_a @ F @ B2 ) ) @ ( image_b_a @ F @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_685_image__diff__subset,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a,B2: set_set_a] : ( ord_le8200006823705900825t_unit @ ( minus_3777555517894451474t_unit @ ( image_6801035452528096924t_unit @ F @ A2 ) @ ( image_6801035452528096924t_unit @ F @ B2 ) ) @ ( image_6801035452528096924t_unit @ F @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_686_image__diff__subset,axiom,
! [F: a > pre_pr7278220950009878019t_unit,A2: set_a,B2: set_a] : ( ord_le8200006823705900825t_unit @ ( minus_3777555517894451474t_unit @ ( image_5713294457175270716t_unit @ F @ A2 ) @ ( image_5713294457175270716t_unit @ F @ B2 ) ) @ ( image_5713294457175270716t_unit @ F @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_687_image__diff__subset,axiom,
! [F: b > pre_pr7278220950009878019t_unit,A2: set_b,B2: set_b] : ( ord_le8200006823705900825t_unit @ ( minus_3777555517894451474t_unit @ ( image_4434118323594779837t_unit @ F @ A2 ) @ ( image_4434118323594779837t_unit @ F @ B2 ) ) @ ( image_4434118323594779837t_unit @ F @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_688_Int__Diff__disjoint,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ ( inf_in1092213268631476299t_unit @ A2 @ B2 ) @ ( minus_3777555517894451474t_unit @ A2 @ B2 ) )
= bot_bo1839476491465656141t_unit ) ).
% Int_Diff_disjoint
thf(fact_689_Int__Diff__disjoint,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ A2 @ B2 ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_690_Int__Diff__disjoint,axiom,
! [A2: set_b,B2: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ ( minus_minus_set_b @ A2 @ B2 ) )
= bot_bot_set_b ) ).
% Int_Diff_disjoint
thf(fact_691_image__Int__subset,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] : ( ord_le3724670747650509150_set_a @ ( image_7466199892558553556_set_a @ F @ ( inf_in1092213268631476299t_unit @ A2 @ B2 ) ) @ ( inf_inf_set_set_a @ ( image_7466199892558553556_set_a @ F @ A2 ) @ ( image_7466199892558553556_set_a @ F @ B2 ) ) ) ).
% image_Int_subset
thf(fact_692_image__Int__subset,axiom,
! [F: a > set_a,A2: set_a,B2: set_a] : ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ ( inf_inf_set_a @ A2 @ B2 ) ) @ ( inf_inf_set_set_a @ ( image_a_set_a @ F @ A2 ) @ ( image_a_set_a @ F @ B2 ) ) ) ).
% image_Int_subset
thf(fact_693_image__Int__subset,axiom,
! [F: b > a,A2: set_b,B2: set_b] : ( ord_less_eq_set_a @ ( image_b_a @ F @ ( inf_inf_set_b @ A2 @ B2 ) ) @ ( inf_inf_set_a @ ( image_b_a @ F @ A2 ) @ ( image_b_a @ F @ B2 ) ) ) ).
% image_Int_subset
thf(fact_694_image__Int__subset,axiom,
! [F: a > a,A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F @ ( inf_inf_set_a @ A2 @ B2 ) ) @ ( inf_inf_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B2 ) ) ) ).
% image_Int_subset
thf(fact_695_image__Int__subset,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a,B2: set_set_a] : ( ord_le8200006823705900825t_unit @ ( image_6801035452528096924t_unit @ F @ ( inf_inf_set_set_a @ A2 @ B2 ) ) @ ( inf_in1092213268631476299t_unit @ ( image_6801035452528096924t_unit @ F @ A2 ) @ ( image_6801035452528096924t_unit @ F @ B2 ) ) ) ).
% image_Int_subset
thf(fact_696_image__Int__subset,axiom,
! [F: a > pre_pr7278220950009878019t_unit,A2: set_a,B2: set_a] : ( ord_le8200006823705900825t_unit @ ( image_5713294457175270716t_unit @ F @ ( inf_inf_set_a @ A2 @ B2 ) ) @ ( inf_in1092213268631476299t_unit @ ( image_5713294457175270716t_unit @ F @ A2 ) @ ( image_5713294457175270716t_unit @ F @ B2 ) ) ) ).
% image_Int_subset
thf(fact_697_Diff__triv,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( ( inf_in1092213268631476299t_unit @ A2 @ B2 )
= bot_bo1839476491465656141t_unit )
=> ( ( minus_3777555517894451474t_unit @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_698_Diff__triv,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_699_Diff__triv,axiom,
! [A2: set_b,B2: set_b] :
( ( ( inf_inf_set_b @ A2 @ B2 )
= bot_bot_set_b )
=> ( ( minus_minus_set_b @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_700_psubsetD,axiom,
! [A2: set_b,B2: set_b,C: b] :
( ( ord_less_set_b @ A2 @ B2 )
=> ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_701_psubsetD,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit,C: pre_pr7278220950009878019t_unit] :
( ( ord_le2693654750756130573t_unit @ A2 @ B2 )
=> ( ( member6939884229742472986t_unit @ C @ A2 )
=> ( member6939884229742472986t_unit @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_702_psubsetD,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_703_psubsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_704_psubset__trans,axiom,
! [A2: set_a,B2: set_a,C3: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C3 )
=> ( ord_less_set_a @ A2 @ C3 ) ) ) ).
% psubset_trans
thf(fact_705_psubset__imp__ex__mem,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( ord_le2693654750756130573t_unit @ A2 @ B2 )
=> ? [B7: pre_pr7278220950009878019t_unit] : ( member6939884229742472986t_unit @ B7 @ ( minus_3777555517894451474t_unit @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_706_psubset__imp__ex__mem,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ? [B7: set_a] : ( member_set_a @ B7 @ ( minus_5736297505244876581_set_a @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_707_psubset__imp__ex__mem,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_set_b @ A2 @ B2 )
=> ? [B7: b] : ( member_b @ B7 @ ( minus_minus_set_b @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_708_psubset__imp__ex__mem,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ? [B7: a] : ( member_a @ B7 @ ( minus_minus_set_a @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_709_IntE,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A2 @ B2 ) )
=> ~ ( ( member_b @ C @ A2 )
=> ~ ( member_b @ C @ B2 ) ) ) ).
% IntE
thf(fact_710_IntE,axiom,
! [C: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C @ ( inf_in1092213268631476299t_unit @ A2 @ B2 ) )
=> ~ ( ( member6939884229742472986t_unit @ C @ A2 )
=> ~ ( member6939884229742472986t_unit @ C @ B2 ) ) ) ).
% IntE
thf(fact_711_IntE,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B2 ) )
=> ~ ( ( member_set_a @ C @ A2 )
=> ~ ( member_set_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_712_IntE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ~ ( member_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_713_DiffE,axiom,
! [C: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C @ ( minus_3777555517894451474t_unit @ A2 @ B2 ) )
=> ~ ( ( member6939884229742472986t_unit @ C @ A2 )
=> ( member6939884229742472986t_unit @ C @ B2 ) ) ) ).
% DiffE
thf(fact_714_DiffE,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ~ ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_715_DiffE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_716_DiffE,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
=> ~ ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B2 ) ) ) ).
% DiffE
thf(fact_717_IntD1,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A2 @ B2 ) )
=> ( member_b @ C @ A2 ) ) ).
% IntD1
thf(fact_718_IntD1,axiom,
! [C: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C @ ( inf_in1092213268631476299t_unit @ A2 @ B2 ) )
=> ( member6939884229742472986t_unit @ C @ A2 ) ) ).
% IntD1
thf(fact_719_IntD1,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B2 ) )
=> ( member_set_a @ C @ A2 ) ) ).
% IntD1
thf(fact_720_IntD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ A2 ) ) ).
% IntD1
thf(fact_721_IntD2,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A2 @ B2 ) )
=> ( member_b @ C @ B2 ) ) ).
% IntD2
thf(fact_722_IntD2,axiom,
! [C: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C @ ( inf_in1092213268631476299t_unit @ A2 @ B2 ) )
=> ( member6939884229742472986t_unit @ C @ B2 ) ) ).
% IntD2
thf(fact_723_IntD2,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B2 ) )
=> ( member_set_a @ C @ B2 ) ) ).
% IntD2
thf(fact_724_IntD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ B2 ) ) ).
% IntD2
thf(fact_725_DiffD1,axiom,
! [C: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C @ ( minus_3777555517894451474t_unit @ A2 @ B2 ) )
=> ( member6939884229742472986t_unit @ C @ A2 ) ) ).
% DiffD1
thf(fact_726_DiffD1,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ( member_set_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_727_DiffD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_728_DiffD1,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
=> ( member_b @ C @ A2 ) ) ).
% DiffD1
thf(fact_729_DiffD2,axiom,
! [C: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C @ ( minus_3777555517894451474t_unit @ A2 @ B2 ) )
=> ~ ( member6939884229742472986t_unit @ C @ B2 ) ) ).
% DiffD2
thf(fact_730_DiffD2,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ~ ( member_set_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_731_DiffD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( member_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_732_DiffD2,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
=> ~ ( member_b @ C @ B2 ) ) ).
% DiffD2
thf(fact_733_imageI,axiom,
! [X: b,A2: set_b,F: b > b] :
( ( member_b @ X @ A2 )
=> ( member_b @ ( F @ X ) @ ( image_b_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_734_imageI,axiom,
! [X: b,A2: set_b,F: b > a] :
( ( member_b @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( image_b_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_735_imageI,axiom,
! [X: a,A2: set_a,F: a > b] :
( ( member_a @ X @ A2 )
=> ( member_b @ ( F @ X ) @ ( image_a_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_736_imageI,axiom,
! [X: a,A2: set_a,F: a > a] :
( ( member_a @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_737_imageI,axiom,
! [X: b,A2: set_b,F: b > set_a] :
( ( member_b @ X @ A2 )
=> ( member_set_a @ ( F @ X ) @ ( image_b_set_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_738_imageI,axiom,
! [X: a,A2: set_a,F: a > set_a] :
( ( member_a @ X @ A2 )
=> ( member_set_a @ ( F @ X ) @ ( image_a_set_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_739_imageI,axiom,
! [X: set_a,A2: set_set_a,F: set_a > b] :
( ( member_set_a @ X @ A2 )
=> ( member_b @ ( F @ X ) @ ( image_set_a_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_740_imageI,axiom,
! [X: set_a,A2: set_set_a,F: set_a > a] :
( ( member_set_a @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( image_set_a_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_741_imageI,axiom,
! [X: set_a,A2: set_set_a,F: set_a > set_a] :
( ( member_set_a @ X @ A2 )
=> ( member_set_a @ ( F @ X ) @ ( image_set_a_set_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_742_imageI,axiom,
! [X: b,A2: set_b,F: b > pre_pr7278220950009878019t_unit] :
( ( member_b @ X @ A2 )
=> ( member6939884229742472986t_unit @ ( F @ X ) @ ( image_4434118323594779837t_unit @ F @ A2 ) ) ) ).
% imageI
thf(fact_743_Int__Diff,axiom,
! [A2: set_a,B2: set_a,C3: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C3 )
= ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B2 @ C3 ) ) ) ).
% Int_Diff
thf(fact_744_Int__Diff,axiom,
! [A2: set_b,B2: set_b,C3: set_b] :
( ( minus_minus_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ C3 )
= ( inf_inf_set_b @ A2 @ ( minus_minus_set_b @ B2 @ C3 ) ) ) ).
% Int_Diff
thf(fact_745_Diff__Int2,axiom,
! [A2: set_a,C3: set_a,B2: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C3 ) @ ( inf_inf_set_a @ B2 @ C3 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C3 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_746_Diff__Int2,axiom,
! [A2: set_b,C3: set_b,B2: set_b] :
( ( minus_minus_set_b @ ( inf_inf_set_b @ A2 @ C3 ) @ ( inf_inf_set_b @ B2 @ C3 ) )
= ( minus_minus_set_b @ ( inf_inf_set_b @ A2 @ C3 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_747_Int__assoc,axiom,
! [A2: set_a,B2: set_a,C3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C3 )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C3 ) ) ) ).
% Int_assoc
thf(fact_748_image__iff,axiom,
! [Z2: a,F: b > a,A2: set_b] :
( ( member_a @ Z2 @ ( image_b_a @ F @ A2 ) )
= ( ? [X2: b] :
( ( member_b @ X2 @ A2 )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_749_image__iff,axiom,
! [Z2: pre_pr7278220950009878019t_unit,F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a] :
( ( member6939884229742472986t_unit @ Z2 @ ( image_6801035452528096924t_unit @ F @ A2 ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_750_image__iff,axiom,
! [Z2: set_a,F: a > set_a,A2: set_a] :
( ( member_set_a @ Z2 @ ( image_a_set_a @ F @ A2 ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_751_image__iff,axiom,
! [Z2: set_a,F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit] :
( ( member_set_a @ Z2 @ ( image_7466199892558553556_set_a @ F @ A2 ) )
= ( ? [X2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X2 @ A2 )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_752_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_753_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_754_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_755_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_756_bex__imageD,axiom,
! [F: b > a,A2: set_b,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( image_b_a @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X3: b] :
( ( member_b @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_757_image__cong,axiom,
! [M2: set_b,N2: set_b,F: b > a,G: b > a] :
( ( M2 = N2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_b_a @ F @ M2 )
= ( image_b_a @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_758_image__cong,axiom,
! [M2: set_a,N2: set_a,F: a > set_a,G: a > set_a] :
( ( M2 = N2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_a_set_a @ F @ M2 )
= ( image_a_set_a @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_759_image__cong,axiom,
! [M2: set_pr5411798346947241657t_unit,N2: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > set_a,G: pre_pr7278220950009878019t_unit > set_a] :
( ( M2 = N2 )
=> ( ! [X3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_7466199892558553556_set_a @ F @ M2 )
= ( image_7466199892558553556_set_a @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_760_image__cong,axiom,
! [M2: set_set_a,N2: set_set_a,F: set_a > pre_pr7278220950009878019t_unit,G: set_a > pre_pr7278220950009878019t_unit] :
( ( M2 = N2 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_6801035452528096924t_unit @ F @ M2 )
= ( image_6801035452528096924t_unit @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_761_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_762_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_763_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_764_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_765_ball__imageD,axiom,
! [F: b > a,A2: set_b,P: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ ( image_b_a @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X5: b] :
( ( member_b @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_766_Diff__Diff__Int,axiom,
! [A2: set_a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_767_Diff__Diff__Int,axiom,
! [A2: set_b,B2: set_b] :
( ( minus_minus_set_b @ A2 @ ( minus_minus_set_b @ A2 @ B2 ) )
= ( inf_inf_set_b @ A2 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_768_rev__image__eqI,axiom,
! [X: b,A2: set_b,B: b,F: b > b] :
( ( member_b @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_b @ B @ ( image_b_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_769_rev__image__eqI,axiom,
! [X: b,A2: set_b,B: a,F: b > a] :
( ( member_b @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_b_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_770_rev__image__eqI,axiom,
! [X: a,A2: set_a,B: b,F: a > b] :
( ( member_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_b @ B @ ( image_a_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_771_rev__image__eqI,axiom,
! [X: a,A2: set_a,B: a,F: a > a] :
( ( member_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_772_rev__image__eqI,axiom,
! [X: b,A2: set_b,B: set_a,F: b > set_a] :
( ( member_b @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_a @ B @ ( image_b_set_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_773_rev__image__eqI,axiom,
! [X: a,A2: set_a,B: set_a,F: a > set_a] :
( ( member_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_a @ B @ ( image_a_set_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_774_rev__image__eqI,axiom,
! [X: set_a,A2: set_set_a,B: b,F: set_a > b] :
( ( member_set_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_b @ B @ ( image_set_a_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_775_rev__image__eqI,axiom,
! [X: set_a,A2: set_set_a,B: a,F: set_a > a] :
( ( member_set_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_set_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_776_rev__image__eqI,axiom,
! [X: set_a,A2: set_set_a,B: set_a,F: set_a > set_a] :
( ( member_set_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_set_a @ B @ ( image_set_a_set_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_777_rev__image__eqI,axiom,
! [X: b,A2: set_b,B: pre_pr7278220950009878019t_unit,F: b > pre_pr7278220950009878019t_unit] :
( ( member_b @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member6939884229742472986t_unit @ B @ ( image_4434118323594779837t_unit @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_778_Int__left__absorb,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% Int_left_absorb
thf(fact_779_Diff__Int__distrib,axiom,
! [C3: set_a,A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ C3 @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C3 @ A2 ) @ ( inf_inf_set_a @ C3 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_780_Diff__Int__distrib,axiom,
! [C3: set_b,A2: set_b,B2: set_b] :
( ( inf_inf_set_b @ C3 @ ( minus_minus_set_b @ A2 @ B2 ) )
= ( minus_minus_set_b @ ( inf_inf_set_b @ C3 @ A2 ) @ ( inf_inf_set_b @ C3 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_781_Int__left__commute,axiom,
! [A2: set_a,B2: set_a,C3: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C3 ) )
= ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A2 @ C3 ) ) ) ).
% Int_left_commute
thf(fact_782_Diff__Int__distrib2,axiom,
! [A2: set_a,B2: set_a,C3: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ C3 )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C3 ) @ ( inf_inf_set_a @ B2 @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_783_Diff__Int__distrib2,axiom,
! [A2: set_b,B2: set_b,C3: set_b] :
( ( inf_inf_set_b @ ( minus_minus_set_b @ A2 @ B2 ) @ C3 )
= ( minus_minus_set_b @ ( inf_inf_set_b @ A2 @ C3 ) @ ( inf_inf_set_b @ B2 @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_784_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_785_inj__on__image__Int,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,C3: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( inj_on6007161714731792944t_unit @ F @ C3 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ C3 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C3 )
=> ( ( image_6801035452528096924t_unit @ F @ ( inf_inf_set_set_a @ A2 @ B2 ) )
= ( inf_in1092213268631476299t_unit @ ( image_6801035452528096924t_unit @ F @ A2 ) @ ( image_6801035452528096924t_unit @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_786_inj__on__image__Int,axiom,
! [F: b > product_prod_a_a,C3: set_b,A2: set_b,B2: set_b] :
( ( inj_on8302219782909765977od_a_a @ F @ C3 )
=> ( ( ord_less_eq_set_b @ A2 @ C3 )
=> ( ( ord_less_eq_set_b @ B2 @ C3 )
=> ( ( image_6761185482258179565od_a_a @ F @ ( inf_inf_set_b @ A2 @ B2 ) )
= ( inf_in8905007599844390133od_a_a @ ( image_6761185482258179565od_a_a @ F @ A2 ) @ ( image_6761185482258179565od_a_a @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_787_inj__on__image__Int,axiom,
! [F: b > a,C3: set_b,A2: set_b,B2: set_b] :
( ( inj_on_b_a @ F @ C3 )
=> ( ( ord_less_eq_set_b @ A2 @ C3 )
=> ( ( ord_less_eq_set_b @ B2 @ C3 )
=> ( ( image_b_a @ F @ ( inf_inf_set_b @ A2 @ B2 ) )
= ( inf_inf_set_a @ ( image_b_a @ F @ A2 ) @ ( image_b_a @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_788_inj__on__image__Int,axiom,
! [F: a > set_a,C3: set_a,A2: set_a,B2: set_a] :
( ( inj_on_a_set_a @ F @ C3 )
=> ( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ( image_a_set_a @ F @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( inf_inf_set_set_a @ ( image_a_set_a @ F @ A2 ) @ ( image_a_set_a @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_789_inj__on__image__Int,axiom,
! [F: a > a,C3: set_a,A2: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ C3 )
=> ( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ( image_a_a @ F @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_790_inj__on__image__Int,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,C3: set_pr5411798346947241657t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( inj_on6672326154762249576_set_a @ F @ C3 )
=> ( ( ord_le8200006823705900825t_unit @ A2 @ C3 )
=> ( ( ord_le8200006823705900825t_unit @ B2 @ C3 )
=> ( ( image_7466199892558553556_set_a @ F @ ( inf_in1092213268631476299t_unit @ A2 @ B2 ) )
= ( inf_inf_set_set_a @ ( image_7466199892558553556_set_a @ F @ A2 ) @ ( image_7466199892558553556_set_a @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_791_inj__on__image__Int,axiom,
! [F: pre_pr7278220950009878019t_unit > a,C3: set_pr5411798346947241657t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( inj_on4336934664824710152unit_a @ F @ C3 )
=> ( ( ord_le8200006823705900825t_unit @ A2 @ C3 )
=> ( ( ord_le8200006823705900825t_unit @ B2 @ C3 )
=> ( ( image_4969699134812999796unit_a @ F @ ( inf_in1092213268631476299t_unit @ A2 @ B2 ) )
= ( inf_inf_set_a @ ( image_4969699134812999796unit_a @ F @ A2 ) @ ( image_4969699134812999796unit_a @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_792_inj__on__image__set__diff,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,C3: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( inj_on6007161714731792944t_unit @ F @ C3 )
=> ( ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) @ C3 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C3 )
=> ( ( image_6801035452528096924t_unit @ F @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
= ( minus_3777555517894451474t_unit @ ( image_6801035452528096924t_unit @ F @ A2 ) @ ( image_6801035452528096924t_unit @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_793_inj__on__image__set__diff,axiom,
! [F: b > product_prod_a_a,C3: set_b,A2: set_b,B2: set_b] :
( ( inj_on8302219782909765977od_a_a @ F @ C3 )
=> ( ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ B2 ) @ C3 )
=> ( ( ord_less_eq_set_b @ B2 @ C3 )
=> ( ( image_6761185482258179565od_a_a @ F @ ( minus_minus_set_b @ A2 @ B2 ) )
= ( minus_6817036919807184750od_a_a @ ( image_6761185482258179565od_a_a @ F @ A2 ) @ ( image_6761185482258179565od_a_a @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_794_inj__on__image__set__diff,axiom,
! [F: b > a,C3: set_b,A2: set_b,B2: set_b] :
( ( inj_on_b_a @ F @ C3 )
=> ( ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ B2 ) @ C3 )
=> ( ( ord_less_eq_set_b @ B2 @ C3 )
=> ( ( image_b_a @ F @ ( minus_minus_set_b @ A2 @ B2 ) )
= ( minus_minus_set_a @ ( image_b_a @ F @ A2 ) @ ( image_b_a @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_795_inj__on__image__set__diff,axiom,
! [F: b > b,C3: set_b,A2: set_b,B2: set_b] :
( ( inj_on_b_b @ F @ C3 )
=> ( ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ B2 ) @ C3 )
=> ( ( ord_less_eq_set_b @ B2 @ C3 )
=> ( ( image_b_b @ F @ ( minus_minus_set_b @ A2 @ B2 ) )
= ( minus_minus_set_b @ ( image_b_b @ F @ A2 ) @ ( image_b_b @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_796_inj__on__image__set__diff,axiom,
! [F: a > set_a,C3: set_a,A2: set_a,B2: set_a] :
( ( inj_on_a_set_a @ F @ C3 )
=> ( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ( image_a_set_a @ F @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_5736297505244876581_set_a @ ( image_a_set_a @ F @ A2 ) @ ( image_a_set_a @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_797_inj__on__image__set__diff,axiom,
! [F: a > a,C3: set_a,A2: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ C3 )
=> ( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ( image_a_a @ F @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_minus_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_798_inj__on__image__set__diff,axiom,
! [F: a > b,C3: set_a,A2: set_a,B2: set_a] :
( ( inj_on_a_b @ F @ C3 )
=> ( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ( image_a_b @ F @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_minus_set_b @ ( image_a_b @ F @ A2 ) @ ( image_a_b @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_799_inj__on__image__set__diff,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,C3: set_pr5411798346947241657t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( inj_on6672326154762249576_set_a @ F @ C3 )
=> ( ( ord_le8200006823705900825t_unit @ ( minus_3777555517894451474t_unit @ A2 @ B2 ) @ C3 )
=> ( ( ord_le8200006823705900825t_unit @ B2 @ C3 )
=> ( ( image_7466199892558553556_set_a @ F @ ( minus_3777555517894451474t_unit @ A2 @ B2 ) )
= ( minus_5736297505244876581_set_a @ ( image_7466199892558553556_set_a @ F @ A2 ) @ ( image_7466199892558553556_set_a @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_800_inj__on__image__set__diff,axiom,
! [F: pre_pr7278220950009878019t_unit > a,C3: set_pr5411798346947241657t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( inj_on4336934664824710152unit_a @ F @ C3 )
=> ( ( ord_le8200006823705900825t_unit @ ( minus_3777555517894451474t_unit @ A2 @ B2 ) @ C3 )
=> ( ( ord_le8200006823705900825t_unit @ B2 @ C3 )
=> ( ( image_4969699134812999796unit_a @ F @ ( minus_3777555517894451474t_unit @ A2 @ B2 ) )
= ( minus_minus_set_a @ ( image_4969699134812999796unit_a @ F @ A2 ) @ ( image_4969699134812999796unit_a @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_801_inj__on__image__set__diff,axiom,
! [F: pre_pr7278220950009878019t_unit > b,C3: set_pr5411798346947241657t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( inj_on4336934664824710153unit_b @ F @ C3 )
=> ( ( ord_le8200006823705900825t_unit @ ( minus_3777555517894451474t_unit @ A2 @ B2 ) @ C3 )
=> ( ( ord_le8200006823705900825t_unit @ B2 @ C3 )
=> ( ( image_4969699134812999797unit_b @ F @ ( minus_3777555517894451474t_unit @ A2 @ B2 ) )
= ( minus_minus_set_b @ ( image_4969699134812999797unit_b @ F @ A2 ) @ ( image_4969699134812999797unit_b @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_802_all__subset__image,axiom,
! [F: a > set_a,A2: set_a,P: set_set_a > $o] :
( ( ! [B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ ( image_a_set_a @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A2 )
=> ( P @ ( image_a_set_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_803_all__subset__image,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,P: set_set_a > $o] :
( ( ! [B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ ( image_7466199892558553556_set_a @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ B4 @ A2 )
=> ( P @ ( image_7466199892558553556_set_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_804_all__subset__image,axiom,
! [F: b > a,A2: set_b,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_b_a @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_b] :
( ( ord_less_eq_set_b @ B4 @ A2 )
=> ( P @ ( image_b_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_805_all__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A2 )
=> ( P @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_806_all__subset__image,axiom,
! [F: pre_pr7278220950009878019t_unit > a,A2: set_pr5411798346947241657t_unit,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_4969699134812999796unit_a @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ B4 @ A2 )
=> ( P @ ( image_4969699134812999796unit_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_807_all__subset__image,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a,P: set_pr5411798346947241657t_unit > $o] :
( ( ! [B4: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ B4 @ ( image_6801035452528096924t_unit @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A2 )
=> ( P @ ( image_6801035452528096924t_unit @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_808_all__subset__image,axiom,
! [F: a > pre_pr7278220950009878019t_unit,A2: set_a,P: set_pr5411798346947241657t_unit > $o] :
( ( ! [B4: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ B4 @ ( image_5713294457175270716t_unit @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A2 )
=> ( P @ ( image_5713294457175270716t_unit @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_809_all__subset__image,axiom,
! [F: pre_pr7278220950009878019t_unit > pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,P: set_pr5411798346947241657t_unit > $o] :
( ( ! [B4: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ B4 @ ( image_7933780498232994317t_unit @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ B4 @ A2 )
=> ( P @ ( image_7933780498232994317t_unit @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_810_subset__image__iff,axiom,
! [B2: set_set_a,F: a > set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ ( image_a_set_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B2
= ( image_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_811_subset__image__iff,axiom,
! [B2: set_set_a,F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit] :
( ( ord_le3724670747650509150_set_a @ B2 @ ( image_7466199892558553556_set_a @ F @ A2 ) )
= ( ? [AA: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ AA @ A2 )
& ( B2
= ( image_7466199892558553556_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_812_subset__image__iff,axiom,
! [B2: set_a,F: b > a,A2: set_b] :
( ( ord_less_eq_set_a @ B2 @ ( image_b_a @ F @ A2 ) )
= ( ? [AA: set_b] :
( ( ord_less_eq_set_b @ AA @ A2 )
& ( B2
= ( image_b_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_813_subset__image__iff,axiom,
! [B2: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B2
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_814_subset__image__iff,axiom,
! [B2: set_a,F: pre_pr7278220950009878019t_unit > a,A2: set_pr5411798346947241657t_unit] :
( ( ord_less_eq_set_a @ B2 @ ( image_4969699134812999796unit_a @ F @ A2 ) )
= ( ? [AA: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ AA @ A2 )
& ( B2
= ( image_4969699134812999796unit_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_815_subset__image__iff,axiom,
! [B2: set_pr5411798346947241657t_unit,F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a] :
( ( ord_le8200006823705900825t_unit @ B2 @ ( image_6801035452528096924t_unit @ F @ A2 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A2 )
& ( B2
= ( image_6801035452528096924t_unit @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_816_subset__image__iff,axiom,
! [B2: set_pr5411798346947241657t_unit,F: a > pre_pr7278220950009878019t_unit,A2: set_a] :
( ( ord_le8200006823705900825t_unit @ B2 @ ( image_5713294457175270716t_unit @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B2
= ( image_5713294457175270716t_unit @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_817_subset__image__iff,axiom,
! [B2: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ B2 @ ( image_7933780498232994317t_unit @ F @ A2 ) )
= ( ? [AA: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ AA @ A2 )
& ( B2
= ( image_7933780498232994317t_unit @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_818_image__subset__iff,axiom,
! [F: a > set_a,A2: set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ B2 )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_set_a @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_819_image__subset__iff,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_7466199892558553556_set_a @ F @ A2 ) @ B2 )
= ( ! [X2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X2 @ A2 )
=> ( member_set_a @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_820_image__subset__iff,axiom,
! [F: b > a,A2: set_b,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_b_a @ F @ A2 ) @ B2 )
= ( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_821_image__subset__iff,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a,B2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ ( image_6801035452528096924t_unit @ F @ A2 ) @ B2 )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( member6939884229742472986t_unit @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_822_subset__imageE,axiom,
! [B2: set_set_a,F: a > set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ ( image_a_set_a @ F @ A2 ) )
=> ~ ! [C6: set_a] :
( ( ord_less_eq_set_a @ C6 @ A2 )
=> ( B2
!= ( image_a_set_a @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_823_subset__imageE,axiom,
! [B2: set_set_a,F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit] :
( ( ord_le3724670747650509150_set_a @ B2 @ ( image_7466199892558553556_set_a @ F @ A2 ) )
=> ~ ! [C6: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ C6 @ A2 )
=> ( B2
!= ( image_7466199892558553556_set_a @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_824_subset__imageE,axiom,
! [B2: set_a,F: b > a,A2: set_b] :
( ( ord_less_eq_set_a @ B2 @ ( image_b_a @ F @ A2 ) )
=> ~ ! [C6: set_b] :
( ( ord_less_eq_set_b @ C6 @ A2 )
=> ( B2
!= ( image_b_a @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_825_subset__imageE,axiom,
! [B2: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
=> ~ ! [C6: set_a] :
( ( ord_less_eq_set_a @ C6 @ A2 )
=> ( B2
!= ( image_a_a @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_826_subset__imageE,axiom,
! [B2: set_a,F: pre_pr7278220950009878019t_unit > a,A2: set_pr5411798346947241657t_unit] :
( ( ord_less_eq_set_a @ B2 @ ( image_4969699134812999796unit_a @ F @ A2 ) )
=> ~ ! [C6: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ C6 @ A2 )
=> ( B2
!= ( image_4969699134812999796unit_a @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_827_subset__imageE,axiom,
! [B2: set_pr5411798346947241657t_unit,F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a] :
( ( ord_le8200006823705900825t_unit @ B2 @ ( image_6801035452528096924t_unit @ F @ A2 ) )
=> ~ ! [C6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C6 @ A2 )
=> ( B2
!= ( image_6801035452528096924t_unit @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_828_subset__imageE,axiom,
! [B2: set_pr5411798346947241657t_unit,F: a > pre_pr7278220950009878019t_unit,A2: set_a] :
( ( ord_le8200006823705900825t_unit @ B2 @ ( image_5713294457175270716t_unit @ F @ A2 ) )
=> ~ ! [C6: set_a] :
( ( ord_less_eq_set_a @ C6 @ A2 )
=> ( B2
!= ( image_5713294457175270716t_unit @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_829_subset__imageE,axiom,
! [B2: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ B2 @ ( image_7933780498232994317t_unit @ F @ A2 ) )
=> ~ ! [C6: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ C6 @ A2 )
=> ( B2
!= ( image_7933780498232994317t_unit @ F @ C6 ) ) ) ) ).
% subset_imageE
thf(fact_830_image__subsetI,axiom,
! [A2: set_b,F: b > b,B2: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( member_b @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_b @ ( image_b_b @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_831_image__subsetI,axiom,
! [A2: set_a,F: a > b,B2: set_b] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_b @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_b @ ( image_a_b @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_832_image__subsetI,axiom,
! [A2: set_b,F: b > a,B2: set_a] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( member_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_b_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_833_image__subsetI,axiom,
! [A2: set_a,F: a > a,B2: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_834_image__subsetI,axiom,
! [A2: set_b,F: b > set_a,B2: set_set_a] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( member_set_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_b_set_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_835_image__subsetI,axiom,
! [A2: set_a,F: a > set_a,B2: set_set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_set_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_836_image__subsetI,axiom,
! [A2: set_set_a,F: set_a > b,B2: set_b] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( member_b @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_b @ ( image_set_a_b @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_837_image__subsetI,axiom,
! [A2: set_set_a,F: set_a > a,B2: set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( member_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_838_image__subsetI,axiom,
! [A2: set_set_a,F: set_a > set_a,B2: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( member_set_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_839_image__subsetI,axiom,
! [A2: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > b,B2: set_b] :
( ! [X3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ A2 )
=> ( member_b @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq_set_b @ ( image_4969699134812999797unit_b @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_840_image__mono,axiom,
! [A2: set_b,B2: set_b,F: b > a] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( image_b_a @ F @ A2 ) @ ( image_b_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_841_image__mono,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_a > pre_pr7278220950009878019t_unit] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_le8200006823705900825t_unit @ ( image_6801035452528096924t_unit @ F @ A2 ) @ ( image_6801035452528096924t_unit @ F @ B2 ) ) ) ).
% image_mono
thf(fact_842_image__mono,axiom,
! [A2: set_a,B2: set_a,F: a > set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ ( image_a_set_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_843_image__mono,axiom,
! [A2: set_a,B2: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_844_image__mono,axiom,
! [A2: set_a,B2: set_a,F: a > pre_pr7278220950009878019t_unit] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_le8200006823705900825t_unit @ ( image_5713294457175270716t_unit @ F @ A2 ) @ ( image_5713294457175270716t_unit @ F @ B2 ) ) ) ).
% image_mono
thf(fact_845_image__mono,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > set_a] :
( ( ord_le8200006823705900825t_unit @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( image_7466199892558553556_set_a @ F @ A2 ) @ ( image_7466199892558553556_set_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_846_image__mono,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > a] :
( ( ord_le8200006823705900825t_unit @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( image_4969699134812999796unit_a @ F @ A2 ) @ ( image_4969699134812999796unit_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_847_image__mono,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ B2 )
=> ( ord_le8200006823705900825t_unit @ ( image_7933780498232994317t_unit @ F @ A2 ) @ ( image_7933780498232994317t_unit @ F @ B2 ) ) ) ).
% image_mono
thf(fact_848_diff__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_849_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_850_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_851_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_852_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [A6: real,B6: real] :
( ( minus_minus_real @ A6 @ B6 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_853_diff__strict__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_854_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_855_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_856_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_857_image__strict__mono,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,B2: set_set_a,A2: set_set_a] :
( ( inj_on6007161714731792944t_unit @ F @ B2 )
=> ( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ord_le2693654750756130573t_unit @ ( image_6801035452528096924t_unit @ F @ A2 ) @ ( image_6801035452528096924t_unit @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_858_image__strict__mono,axiom,
! [F: b > product_prod_a_a,B2: set_b,A2: set_b] :
( ( inj_on8302219782909765977od_a_a @ F @ B2 )
=> ( ( ord_less_set_b @ A2 @ B2 )
=> ( ord_le6819997720685908915od_a_a @ ( image_6761185482258179565od_a_a @ F @ A2 ) @ ( image_6761185482258179565od_a_a @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_859_image__strict__mono,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,B2: set_pr5411798346947241657t_unit,A2: set_pr5411798346947241657t_unit] :
( ( inj_on6672326154762249576_set_a @ F @ B2 )
=> ( ( ord_le2693654750756130573t_unit @ A2 @ B2 )
=> ( ord_less_set_set_a @ ( image_7466199892558553556_set_a @ F @ A2 ) @ ( image_7466199892558553556_set_a @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_860_image__strict__mono,axiom,
! [F: b > a,B2: set_b,A2: set_b] :
( ( inj_on_b_a @ F @ B2 )
=> ( ( ord_less_set_b @ A2 @ B2 )
=> ( ord_less_set_a @ ( image_b_a @ F @ A2 ) @ ( image_b_a @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_861_image__strict__mono,axiom,
! [F: a > set_a,B2: set_a,A2: set_a] :
( ( inj_on_a_set_a @ F @ B2 )
=> ( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_set_set_a @ ( image_a_set_a @ F @ A2 ) @ ( image_a_set_a @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_862_image__strict__mono,axiom,
! [F: a > a,B2: set_a,A2: set_a] :
( ( inj_on_a_a @ F @ B2 )
=> ( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_863_inj__on__image__iff,axiom,
! [A2: set_b,G: b > product_prod_a_a,F: b > b] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ! [Xa2: b] :
( ( member_b @ Xa2 @ A2 )
=> ( ( ( G @ ( F @ X3 ) )
= ( G @ ( F @ Xa2 ) ) )
= ( ( G @ X3 )
= ( G @ Xa2 ) ) ) ) )
=> ( ( inj_on_b_b @ F @ A2 )
=> ( ( inj_on8302219782909765977od_a_a @ G @ ( image_b_b @ F @ A2 ) )
= ( inj_on8302219782909765977od_a_a @ G @ A2 ) ) ) ) ).
% inj_on_image_iff
thf(fact_864_inj__on__image__iff,axiom,
! [A2: set_pr5411798346947241657t_unit,G: pre_pr7278220950009878019t_unit > set_a,F: pre_pr7278220950009878019t_unit > pre_pr7278220950009878019t_unit] :
( ! [X3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ A2 )
=> ! [Xa2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ Xa2 @ A2 )
=> ( ( ( G @ ( F @ X3 ) )
= ( G @ ( F @ Xa2 ) ) )
= ( ( G @ X3 )
= ( G @ Xa2 ) ) ) ) )
=> ( ( inj_on6227784922168685433t_unit @ F @ A2 )
=> ( ( inj_on6672326154762249576_set_a @ G @ ( image_7933780498232994317t_unit @ F @ A2 ) )
= ( inj_on6672326154762249576_set_a @ G @ A2 ) ) ) ) ).
% inj_on_image_iff
thf(fact_865_disjoint__iff__not__equal,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ B2 )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_866_disjoint__iff__not__equal,axiom,
! [A2: set_b,B2: set_b] :
( ( ( inf_inf_set_b @ A2 @ B2 )
= bot_bot_set_b )
= ( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ! [Y2: b] :
( ( member_b @ Y2 @ B2 )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_867_disjoint__iff__not__equal,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( ( inf_in1092213268631476299t_unit @ A2 @ B2 )
= bot_bo1839476491465656141t_unit )
= ( ! [X2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X2 @ A2 )
=> ! [Y2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ Y2 @ B2 )
=> ( X2 != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_868_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_869_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_870_Int__empty__right,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ A2 @ bot_bo1839476491465656141t_unit )
= bot_bo1839476491465656141t_unit ) ).
% Int_empty_right
thf(fact_871_Int__empty__left,axiom,
! [B2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B2 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_872_Int__empty__left,axiom,
! [B2: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ B2 )
= bot_bot_set_b ) ).
% Int_empty_left
thf(fact_873_Int__empty__left,axiom,
! [B2: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ bot_bo1839476491465656141t_unit @ B2 )
= bot_bo1839476491465656141t_unit ) ).
% Int_empty_left
thf(fact_874_disjoint__iff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ~ ( member_set_a @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_875_disjoint__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ~ ( member_a @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_876_disjoint__iff,axiom,
! [A2: set_b,B2: set_b] :
( ( ( inf_inf_set_b @ A2 @ B2 )
= bot_bot_set_b )
= ( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ~ ( member_b @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_877_disjoint__iff,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( ( inf_in1092213268631476299t_unit @ A2 @ B2 )
= bot_bo1839476491465656141t_unit )
= ( ! [X2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X2 @ A2 )
=> ~ ( member6939884229742472986t_unit @ X2 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_878_Int__emptyI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ~ ( member_set_a @ X3 @ B2 ) )
=> ( ( inf_inf_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_879_Int__emptyI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B2 ) )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_880_Int__emptyI,axiom,
! [A2: set_b,B2: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ~ ( member_b @ X3 @ B2 ) )
=> ( ( inf_inf_set_b @ A2 @ B2 )
= bot_bot_set_b ) ) ).
% Int_emptyI
thf(fact_881_Int__emptyI,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ! [X3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ A2 )
=> ~ ( member6939884229742472986t_unit @ X3 @ B2 ) )
=> ( ( inf_in1092213268631476299t_unit @ A2 @ B2 )
= bot_bo1839476491465656141t_unit ) ) ).
% Int_emptyI
thf(fact_882_Int__Collect__mono,axiom,
! [A2: set_b,B2: set_b,P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ! [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 @ B2 @ ( collect_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_883_Int__Collect__mono,axiom,
! [A2: set_set_a,B2: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ! [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 @ B2 @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_884_Int__Collect__mono,axiom,
! [A2: set_a,B2: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [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 @ B2 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_885_Int__Collect__mono,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit,P: pre_pr7278220950009878019t_unit > $o,Q: pre_pr7278220950009878019t_unit > $o] :
( ( ord_le8200006823705900825t_unit @ A2 @ B2 )
=> ( ! [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 @ B2 @ ( collec8000012497822511960t_unit @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_886_Int__greatest,axiom,
! [C3: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( ( ord_less_eq_set_a @ C3 @ B2 )
=> ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_887_Int__greatest,axiom,
! [C3: set_pr5411798346947241657t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ C3 @ A2 )
=> ( ( ord_le8200006823705900825t_unit @ C3 @ B2 )
=> ( ord_le8200006823705900825t_unit @ C3 @ ( inf_in1092213268631476299t_unit @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_888_Int__absorb2,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_889_Int__absorb2,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ B2 )
=> ( ( inf_in1092213268631476299t_unit @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_890_Int__absorb1,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_891_Int__absorb1,axiom,
! [B2: set_pr5411798346947241657t_unit,A2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ B2 @ A2 )
=> ( ( inf_in1092213268631476299t_unit @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_892_Int__lower2,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_893_Int__lower2,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] : ( ord_le8200006823705900825t_unit @ ( inf_in1092213268631476299t_unit @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_894_Int__lower1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_895_Int__lower1,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] : ( ord_le8200006823705900825t_unit @ ( inf_in1092213268631476299t_unit @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_896_Int__mono,axiom,
! [A2: set_a,C3: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_897_Int__mono,axiom,
! [A2: set_pr5411798346947241657t_unit,C3: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit,D2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ C3 )
=> ( ( ord_le8200006823705900825t_unit @ B2 @ D2 )
=> ( ord_le8200006823705900825t_unit @ ( inf_in1092213268631476299t_unit @ A2 @ B2 ) @ ( inf_in1092213268631476299t_unit @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_898_Int__insert__right,axiom,
! [A: b,A2: set_b,B2: set_b] :
( ( ( member_b @ A @ A2 )
=> ( ( inf_inf_set_b @ A2 @ ( insert_b @ A @ B2 ) )
= ( insert_b @ A @ ( inf_inf_set_b @ A2 @ B2 ) ) ) )
& ( ~ ( member_b @ A @ A2 )
=> ( ( inf_inf_set_b @ A2 @ ( insert_b @ A @ B2 ) )
= ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_899_Int__insert__right,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( ( member6939884229742472986t_unit @ A @ A2 )
=> ( ( inf_in1092213268631476299t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ B2 ) )
= ( insert6864688055023459379t_unit @ A @ ( inf_in1092213268631476299t_unit @ A2 @ B2 ) ) ) )
& ( ~ ( member6939884229742472986t_unit @ A @ A2 )
=> ( ( inf_in1092213268631476299t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ B2 ) )
= ( inf_in1092213268631476299t_unit @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_900_Int__insert__right,axiom,
! [A: set_a,A2: set_set_a,B2: set_set_a] :
( ( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B2 ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B2 ) ) ) )
& ( ~ ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B2 ) )
= ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_901_Int__insert__right,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B2 ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_902_Int__insert__left,axiom,
! [A: b,C3: set_b,B2: set_b] :
( ( ( member_b @ A @ C3 )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B2 ) @ C3 )
= ( insert_b @ A @ ( inf_inf_set_b @ B2 @ C3 ) ) ) )
& ( ~ ( member_b @ A @ C3 )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B2 ) @ C3 )
= ( inf_inf_set_b @ B2 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_903_Int__insert__left,axiom,
! [A: pre_pr7278220950009878019t_unit,C3: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( ( member6939884229742472986t_unit @ A @ C3 )
=> ( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A @ B2 ) @ C3 )
= ( insert6864688055023459379t_unit @ A @ ( inf_in1092213268631476299t_unit @ B2 @ C3 ) ) ) )
& ( ~ ( member6939884229742472986t_unit @ A @ C3 )
=> ( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A @ B2 ) @ C3 )
= ( inf_in1092213268631476299t_unit @ B2 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_904_Int__insert__left,axiom,
! [A: set_a,C3: set_set_a,B2: set_set_a] :
( ( ( member_set_a @ A @ C3 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B2 ) @ C3 )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B2 @ C3 ) ) ) )
& ( ~ ( member_set_a @ A @ C3 )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B2 ) @ C3 )
= ( inf_inf_set_set_a @ B2 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_905_Int__insert__left,axiom,
! [A: a,C3: set_a,B2: set_a] :
( ( ( member_a @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C3 )
= ( insert_a @ A @ ( inf_inf_set_a @ B2 @ C3 ) ) ) )
& ( ~ ( member_a @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C3 )
= ( inf_inf_set_a @ B2 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_906_Diff__infinite__finite,axiom,
! [T2: set_set_a,S: set_set_a] :
( ( finite_finite_set_a @ T2 )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_907_Diff__infinite__finite,axiom,
! [T2: set_Extended_ereal,S: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ T2 )
=> ( ~ ( finite7198162374296863863_ereal @ S )
=> ~ ( finite7198162374296863863_ereal @ ( minus_1264018925008434325_ereal @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_908_Diff__infinite__finite,axiom,
! [T2: set_a,S: set_a] :
( ( finite_finite_a @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_909_Diff__infinite__finite,axiom,
! [T2: set_b,S: set_b] :
( ( finite_finite_b @ T2 )
=> ( ~ ( finite_finite_b @ S )
=> ~ ( finite_finite_b @ ( minus_minus_set_b @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_910_double__diff,axiom,
! [A2: set_b,B2: set_b,C3: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( ord_less_eq_set_b @ B2 @ C3 )
=> ( ( minus_minus_set_b @ B2 @ ( minus_minus_set_b @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_911_double__diff,axiom,
! [A2: set_a,B2: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ( minus_minus_set_a @ B2 @ ( minus_minus_set_a @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_912_double__diff,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit,C3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ B2 )
=> ( ( ord_le8200006823705900825t_unit @ B2 @ C3 )
=> ( ( minus_3777555517894451474t_unit @ B2 @ ( minus_3777555517894451474t_unit @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_913_Diff__subset,axiom,
! [A2: set_b,B2: set_b] : ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_914_Diff__subset,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_915_Diff__subset,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] : ( ord_le8200006823705900825t_unit @ ( minus_3777555517894451474t_unit @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_916_Diff__mono,axiom,
! [A2: set_b,C3: set_b,D2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ C3 )
=> ( ( ord_less_eq_set_b @ D2 @ B2 )
=> ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ B2 ) @ ( minus_minus_set_b @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_917_Diff__mono,axiom,
! [A2: set_a,C3: set_a,D2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ D2 @ B2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_918_Diff__mono,axiom,
! [A2: set_pr5411798346947241657t_unit,C3: set_pr5411798346947241657t_unit,D2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ C3 )
=> ( ( ord_le8200006823705900825t_unit @ D2 @ B2 )
=> ( ord_le8200006823705900825t_unit @ ( minus_3777555517894451474t_unit @ A2 @ B2 ) @ ( minus_3777555517894451474t_unit @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_919_insert__Diff__if,axiom,
! [X: pre_pr7278220950009878019t_unit,B2: set_pr5411798346947241657t_unit,A2: set_pr5411798346947241657t_unit] :
( ( ( member6939884229742472986t_unit @ X @ B2 )
=> ( ( minus_3777555517894451474t_unit @ ( insert6864688055023459379t_unit @ X @ A2 ) @ B2 )
= ( minus_3777555517894451474t_unit @ A2 @ B2 ) ) )
& ( ~ ( member6939884229742472986t_unit @ X @ B2 )
=> ( ( minus_3777555517894451474t_unit @ ( insert6864688055023459379t_unit @ X @ A2 ) @ B2 )
= ( insert6864688055023459379t_unit @ X @ ( minus_3777555517894451474t_unit @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_920_insert__Diff__if,axiom,
! [X: set_a,B2: set_set_a,A2: set_set_a] :
( ( ( member_set_a @ X @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B2 )
= ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) )
& ( ~ ( member_set_a @ X @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B2 )
= ( insert_set_a @ X @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_921_insert__Diff__if,axiom,
! [X: a,B2: set_a,A2: set_a] :
( ( ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) )
& ( ~ ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_922_insert__Diff__if,axiom,
! [X: b,B2: set_b,A2: set_b] :
( ( ( member_b @ X @ B2 )
=> ( ( minus_minus_set_b @ ( insert_b @ X @ A2 ) @ B2 )
= ( minus_minus_set_b @ A2 @ B2 ) ) )
& ( ~ ( member_b @ X @ B2 )
=> ( ( minus_minus_set_b @ ( insert_b @ X @ A2 ) @ B2 )
= ( insert_b @ X @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_923_inj__on__Int,axiom,
! [F: b > product_prod_a_a,A2: set_b,B2: set_b] :
( ( ( inj_on8302219782909765977od_a_a @ F @ A2 )
| ( inj_on8302219782909765977od_a_a @ F @ B2 ) )
=> ( inj_on8302219782909765977od_a_a @ F @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ).
% inj_on_Int
thf(fact_924_inj__on__Int,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( ( inj_on6672326154762249576_set_a @ F @ A2 )
| ( inj_on6672326154762249576_set_a @ F @ B2 ) )
=> ( inj_on6672326154762249576_set_a @ F @ ( inf_in1092213268631476299t_unit @ A2 @ B2 ) ) ) ).
% inj_on_Int
thf(fact_925_inj__on__diff,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( inj_on6672326154762249576_set_a @ F @ A2 )
=> ( inj_on6672326154762249576_set_a @ F @ ( minus_3777555517894451474t_unit @ A2 @ B2 ) ) ) ).
% inj_on_diff
thf(fact_926_inj__on__diff,axiom,
! [F: b > product_prod_a_a,A2: set_b,B2: set_b] :
( ( inj_on8302219782909765977od_a_a @ F @ A2 )
=> ( inj_on8302219782909765977od_a_a @ F @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% inj_on_diff
thf(fact_927_finite__surj,axiom,
! [A2: set_a,B2: set_b,F: a > b] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_b @ B2 @ ( image_a_b @ F @ A2 ) )
=> ( finite_finite_b @ B2 ) ) ) ).
% finite_surj
thf(fact_928_finite__surj,axiom,
! [A2: set_a,B2: set_Extended_ereal,F: a > extended_ereal] :
( ( finite_finite_a @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ B2 @ ( image_8405481351990995413_ereal @ F @ A2 ) )
=> ( finite7198162374296863863_ereal @ B2 ) ) ) ).
% finite_surj
thf(fact_929_finite__surj,axiom,
! [A2: set_b,B2: set_b,F: b > b] :
( ( finite_finite_b @ A2 )
=> ( ( ord_less_eq_set_b @ B2 @ ( image_b_b @ F @ A2 ) )
=> ( finite_finite_b @ B2 ) ) ) ).
% finite_surj
thf(fact_930_finite__surj,axiom,
! [A2: set_b,B2: set_Extended_ereal,F: b > extended_ereal] :
( ( finite_finite_b @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ B2 @ ( image_5319725110001000852_ereal @ F @ A2 ) )
=> ( finite7198162374296863863_ereal @ B2 ) ) ) ).
% finite_surj
thf(fact_931_finite__surj,axiom,
! [A2: set_Extended_ereal,B2: set_b,F: extended_ereal > b] :
( ( finite7198162374296863863_ereal @ A2 )
=> ( ( ord_less_eq_set_b @ B2 @ ( image_3724615099042636214real_b @ F @ A2 ) )
=> ( finite_finite_b @ B2 ) ) ) ).
% finite_surj
thf(fact_932_finite__surj,axiom,
! [A2: set_Extended_ereal,B2: set_Extended_ereal,F: extended_ereal > extended_ereal] :
( ( finite7198162374296863863_ereal @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ B2 @ ( image_6042159593519690757_ereal @ F @ A2 ) )
=> ( finite7198162374296863863_ereal @ B2 ) ) ) ).
% finite_surj
thf(fact_933_finite__surj,axiom,
! [A2: set_a,B2: set_a,F: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
=> ( finite_finite_a @ B2 ) ) ) ).
% finite_surj
thf(fact_934_finite__surj,axiom,
! [A2: set_b,B2: set_a,F: b > a] :
( ( finite_finite_b @ A2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_b_a @ F @ A2 ) )
=> ( finite_finite_a @ B2 ) ) ) ).
% finite_surj
thf(fact_935_finite__surj,axiom,
! [A2: set_Extended_ereal,B2: set_a,F: extended_ereal > a] :
( ( finite7198162374296863863_ereal @ A2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_3724615099042636213real_a @ F @ A2 ) )
=> ( finite_finite_a @ B2 ) ) ) ).
% finite_surj
thf(fact_936_finite__surj,axiom,
! [A2: set_a,B2: set_set_a,F: a > set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ ( image_a_set_a @ F @ A2 ) )
=> ( finite_finite_set_a @ B2 ) ) ) ).
% finite_surj
thf(fact_937_finite__subset__image,axiom,
! [B2: set_b,F: b > b,A2: set_b] :
( ( finite_finite_b @ B2 )
=> ( ( ord_less_eq_set_b @ B2 @ ( image_b_b @ F @ A2 ) )
=> ? [C6: set_b] :
( ( ord_less_eq_set_b @ C6 @ A2 )
& ( finite_finite_b @ C6 )
& ( B2
= ( image_b_b @ F @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_938_finite__subset__image,axiom,
! [B2: set_b,F: extended_ereal > b,A2: set_Extended_ereal] :
( ( finite_finite_b @ B2 )
=> ( ( ord_less_eq_set_b @ B2 @ ( image_3724615099042636214real_b @ F @ A2 ) )
=> ? [C6: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ C6 @ A2 )
& ( finite7198162374296863863_ereal @ C6 )
& ( B2
= ( image_3724615099042636214real_b @ F @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_939_finite__subset__image,axiom,
! [B2: set_Extended_ereal,F: b > extended_ereal,A2: set_b] :
( ( finite7198162374296863863_ereal @ B2 )
=> ( ( ord_le1644982726543182158_ereal @ B2 @ ( image_5319725110001000852_ereal @ F @ A2 ) )
=> ? [C6: set_b] :
( ( ord_less_eq_set_b @ C6 @ A2 )
& ( finite_finite_b @ C6 )
& ( B2
= ( image_5319725110001000852_ereal @ F @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_940_finite__subset__image,axiom,
! [B2: set_Extended_ereal,F: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ B2 )
=> ( ( ord_le1644982726543182158_ereal @ B2 @ ( image_6042159593519690757_ereal @ F @ A2 ) )
=> ? [C6: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ C6 @ A2 )
& ( finite7198162374296863863_ereal @ C6 )
& ( B2
= ( image_6042159593519690757_ereal @ F @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_941_finite__subset__image,axiom,
! [B2: set_b,F: a > b,A2: set_a] :
( ( finite_finite_b @ B2 )
=> ( ( ord_less_eq_set_b @ B2 @ ( image_a_b @ F @ A2 ) )
=> ? [C6: set_a] :
( ( ord_less_eq_set_a @ C6 @ A2 )
& ( finite_finite_a @ C6 )
& ( B2
= ( image_a_b @ F @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_942_finite__subset__image,axiom,
! [B2: set_Extended_ereal,F: a > extended_ereal,A2: set_a] :
( ( finite7198162374296863863_ereal @ B2 )
=> ( ( ord_le1644982726543182158_ereal @ B2 @ ( image_8405481351990995413_ereal @ F @ A2 ) )
=> ? [C6: set_a] :
( ( ord_less_eq_set_a @ C6 @ A2 )
& ( finite_finite_a @ C6 )
& ( B2
= ( image_8405481351990995413_ereal @ F @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_943_finite__subset__image,axiom,
! [B2: set_a,F: b > a,A2: set_b] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_b_a @ F @ A2 ) )
=> ? [C6: set_b] :
( ( ord_less_eq_set_b @ C6 @ A2 )
& ( finite_finite_b @ C6 )
& ( B2
= ( image_b_a @ F @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_944_finite__subset__image,axiom,
! [B2: set_a,F: extended_ereal > a,A2: set_Extended_ereal] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_3724615099042636213real_a @ F @ A2 ) )
=> ? [C6: set_Extended_ereal] :
( ( ord_le1644982726543182158_ereal @ C6 @ A2 )
& ( finite7198162374296863863_ereal @ C6 )
& ( B2
= ( image_3724615099042636213real_a @ F @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_945_finite__subset__image,axiom,
! [B2: set_a,F: a > a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
=> ? [C6: set_a] :
( ( ord_less_eq_set_a @ C6 @ A2 )
& ( finite_finite_a @ C6 )
& ( B2
= ( image_a_a @ F @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_946_finite__subset__image,axiom,
! [B2: set_b,F: set_a > b,A2: set_set_a] :
( ( finite_finite_b @ B2 )
=> ( ( ord_less_eq_set_b @ B2 @ ( image_set_a_b @ F @ A2 ) )
=> ? [C6: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C6 @ A2 )
& ( finite_finite_set_a @ C6 )
& ( B2
= ( image_set_a_b @ F @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_947_ex__finite__subset__image,axiom,
! [F: b > b,A2: set_b,P: set_b > $o] :
( ( ? [B4: set_b] :
( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ ( image_b_b @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_b] :
( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ A2 )
& ( P @ ( image_b_b @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_948_ex__finite__subset__image,axiom,
! [F: extended_ereal > b,A2: set_Extended_ereal,P: set_b > $o] :
( ( ? [B4: set_b] :
( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ ( image_3724615099042636214real_b @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ B4 )
& ( ord_le1644982726543182158_ereal @ B4 @ A2 )
& ( P @ ( image_3724615099042636214real_b @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_949_ex__finite__subset__image,axiom,
! [F: b > extended_ereal,A2: set_b,P: set_Extended_ereal > $o] :
( ( ? [B4: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ B4 )
& ( ord_le1644982726543182158_ereal @ B4 @ ( image_5319725110001000852_ereal @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_b] :
( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ A2 )
& ( P @ ( image_5319725110001000852_ereal @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_950_ex__finite__subset__image,axiom,
! [F: extended_ereal > extended_ereal,A2: set_Extended_ereal,P: set_Extended_ereal > $o] :
( ( ? [B4: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ B4 )
& ( ord_le1644982726543182158_ereal @ B4 @ ( image_6042159593519690757_ereal @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ B4 )
& ( ord_le1644982726543182158_ereal @ B4 @ A2 )
& ( P @ ( image_6042159593519690757_ereal @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_951_ex__finite__subset__image,axiom,
! [F: a > b,A2: set_a,P: set_b > $o] :
( ( ? [B4: set_b] :
( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ ( image_a_b @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 )
& ( P @ ( image_a_b @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_952_ex__finite__subset__image,axiom,
! [F: a > extended_ereal,A2: set_a,P: set_Extended_ereal > $o] :
( ( ? [B4: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ B4 )
& ( ord_le1644982726543182158_ereal @ B4 @ ( image_8405481351990995413_ereal @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 )
& ( P @ ( image_8405481351990995413_ereal @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_953_ex__finite__subset__image,axiom,
! [F: b > a,A2: set_b,P: set_a > $o] :
( ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_b_a @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_b] :
( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ A2 )
& ( P @ ( image_b_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_954_ex__finite__subset__image,axiom,
! [F: extended_ereal > a,A2: set_Extended_ereal,P: set_a > $o] :
( ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_3724615099042636213real_a @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ B4 )
& ( ord_le1644982726543182158_ereal @ B4 @ A2 )
& ( P @ ( image_3724615099042636213real_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_955_ex__finite__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 )
& ( P @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_956_ex__finite__subset__image,axiom,
! [F: set_a > b,A2: set_set_a,P: set_b > $o] :
( ( ? [B4: set_b] :
( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ ( image_set_a_b @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_set_a] :
( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A2 )
& ( P @ ( image_set_a_b @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_957_all__finite__subset__image,axiom,
! [F: b > b,A2: set_b,P: set_b > $o] :
( ( ! [B4: set_b] :
( ( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ ( image_b_b @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_b] :
( ( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ A2 ) )
=> ( P @ ( image_b_b @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_958_all__finite__subset__image,axiom,
! [F: extended_ereal > b,A2: set_Extended_ereal,P: set_b > $o] :
( ( ! [B4: set_b] :
( ( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ ( image_3724615099042636214real_b @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_Extended_ereal] :
( ( ( finite7198162374296863863_ereal @ B4 )
& ( ord_le1644982726543182158_ereal @ B4 @ A2 ) )
=> ( P @ ( image_3724615099042636214real_b @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_959_all__finite__subset__image,axiom,
! [F: b > extended_ereal,A2: set_b,P: set_Extended_ereal > $o] :
( ( ! [B4: set_Extended_ereal] :
( ( ( finite7198162374296863863_ereal @ B4 )
& ( ord_le1644982726543182158_ereal @ B4 @ ( image_5319725110001000852_ereal @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_b] :
( ( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ A2 ) )
=> ( P @ ( image_5319725110001000852_ereal @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_960_all__finite__subset__image,axiom,
! [F: extended_ereal > extended_ereal,A2: set_Extended_ereal,P: set_Extended_ereal > $o] :
( ( ! [B4: set_Extended_ereal] :
( ( ( finite7198162374296863863_ereal @ B4 )
& ( ord_le1644982726543182158_ereal @ B4 @ ( image_6042159593519690757_ereal @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_Extended_ereal] :
( ( ( finite7198162374296863863_ereal @ B4 )
& ( ord_le1644982726543182158_ereal @ B4 @ A2 ) )
=> ( P @ ( image_6042159593519690757_ereal @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_961_all__finite__subset__image,axiom,
! [F: a > b,A2: set_a,P: set_b > $o] :
( ( ! [B4: set_b] :
( ( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ ( image_a_b @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 ) )
=> ( P @ ( image_a_b @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_962_all__finite__subset__image,axiom,
! [F: a > extended_ereal,A2: set_a,P: set_Extended_ereal > $o] :
( ( ! [B4: set_Extended_ereal] :
( ( ( finite7198162374296863863_ereal @ B4 )
& ( ord_le1644982726543182158_ereal @ B4 @ ( image_8405481351990995413_ereal @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 ) )
=> ( P @ ( image_8405481351990995413_ereal @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_963_all__finite__subset__image,axiom,
! [F: b > a,A2: set_b,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_b_a @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_b] :
( ( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ A2 ) )
=> ( P @ ( image_b_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_964_all__finite__subset__image,axiom,
! [F: extended_ereal > a,A2: set_Extended_ereal,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_3724615099042636213real_a @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_Extended_ereal] :
( ( ( finite7198162374296863863_ereal @ B4 )
& ( ord_le1644982726543182158_ereal @ B4 @ A2 ) )
=> ( P @ ( image_3724615099042636213real_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_965_all__finite__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 ) )
=> ( P @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_966_all__finite__subset__image,axiom,
! [F: set_a > b,A2: set_set_a,P: set_b > $o] :
( ( ! [B4: set_b] :
( ( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ ( image_set_a_b @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_a] :
( ( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A2 ) )
=> ( P @ ( image_set_a_b @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_967_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A6: real,B6: real] : ( ord_less_eq_real @ ( minus_minus_real @ A6 @ B6 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_968_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A6: real,B6: real] : ( ord_less_real @ ( minus_minus_real @ A6 @ B6 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_969_finite__image__iff,axiom,
! [F: a > a,A2: set_a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ( finite_finite_a @ ( image_a_a @ F @ A2 ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_image_iff
thf(fact_970_finite__image__iff,axiom,
! [F: b > a,A2: set_b] :
( ( inj_on_b_a @ F @ A2 )
=> ( ( finite_finite_a @ ( image_b_a @ F @ A2 ) )
= ( finite_finite_b @ A2 ) ) ) ).
% finite_image_iff
thf(fact_971_finite__image__iff,axiom,
! [F: extended_ereal > a,A2: set_Extended_ereal] :
( ( inj_on8242634198667403041real_a @ F @ A2 )
=> ( ( finite_finite_a @ ( image_3724615099042636213real_a @ F @ A2 ) )
= ( finite7198162374296863863_ereal @ A2 ) ) ) ).
% finite_image_iff
thf(fact_972_finite__image__iff,axiom,
! [F: a > b,A2: set_a] :
( ( inj_on_a_b @ F @ A2 )
=> ( ( finite_finite_b @ ( image_a_b @ F @ A2 ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_image_iff
thf(fact_973_finite__image__iff,axiom,
! [F: b > b,A2: set_b] :
( ( inj_on_b_b @ F @ A2 )
=> ( ( finite_finite_b @ ( image_b_b @ F @ A2 ) )
= ( finite_finite_b @ A2 ) ) ) ).
% finite_image_iff
thf(fact_974_finite__image__iff,axiom,
! [F: extended_ereal > b,A2: set_Extended_ereal] :
( ( inj_on8242634198667403042real_b @ F @ A2 )
=> ( ( finite_finite_b @ ( image_3724615099042636214real_b @ F @ A2 ) )
= ( finite7198162374296863863_ereal @ A2 ) ) ) ).
% finite_image_iff
thf(fact_975_finite__image__iff,axiom,
! [F: a > extended_ereal,A2: set_a] :
( ( inj_on3700128414760986433_ereal @ F @ A2 )
=> ( ( finite7198162374296863863_ereal @ ( image_8405481351990995413_ereal @ F @ A2 ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_image_iff
thf(fact_976_finite__image__iff,axiom,
! [F: b > extended_ereal,A2: set_b] :
( ( inj_on614372172770991872_ereal @ F @ A2 )
=> ( ( finite7198162374296863863_ereal @ ( image_5319725110001000852_ereal @ F @ A2 ) )
= ( finite_finite_b @ A2 ) ) ) ).
% finite_image_iff
thf(fact_977_finite__image__iff,axiom,
! [F: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F @ A2 )
=> ( ( finite7198162374296863863_ereal @ ( image_6042159593519690757_ereal @ F @ A2 ) )
= ( finite7198162374296863863_ereal @ A2 ) ) ) ).
% finite_image_iff
thf(fact_978_finite__image__iff,axiom,
! [F: set_a > a,A2: set_set_a] :
( ( inj_on_set_a_a @ F @ A2 )
=> ( ( finite_finite_a @ ( image_set_a_a @ F @ A2 ) )
= ( finite_finite_set_a @ A2 ) ) ) ).
% finite_image_iff
thf(fact_979_finite__imageD,axiom,
! [F: a > a,A2: set_a] :
( ( finite_finite_a @ ( image_a_a @ F @ A2 ) )
=> ( ( inj_on_a_a @ F @ A2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_imageD
thf(fact_980_finite__imageD,axiom,
! [F: b > a,A2: set_b] :
( ( finite_finite_a @ ( image_b_a @ F @ A2 ) )
=> ( ( inj_on_b_a @ F @ A2 )
=> ( finite_finite_b @ A2 ) ) ) ).
% finite_imageD
thf(fact_981_finite__imageD,axiom,
! [F: extended_ereal > a,A2: set_Extended_ereal] :
( ( finite_finite_a @ ( image_3724615099042636213real_a @ F @ A2 ) )
=> ( ( inj_on8242634198667403041real_a @ F @ A2 )
=> ( finite7198162374296863863_ereal @ A2 ) ) ) ).
% finite_imageD
thf(fact_982_finite__imageD,axiom,
! [F: a > b,A2: set_a] :
( ( finite_finite_b @ ( image_a_b @ F @ A2 ) )
=> ( ( inj_on_a_b @ F @ A2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_imageD
thf(fact_983_finite__imageD,axiom,
! [F: b > b,A2: set_b] :
( ( finite_finite_b @ ( image_b_b @ F @ A2 ) )
=> ( ( inj_on_b_b @ F @ A2 )
=> ( finite_finite_b @ A2 ) ) ) ).
% finite_imageD
thf(fact_984_finite__imageD,axiom,
! [F: extended_ereal > b,A2: set_Extended_ereal] :
( ( finite_finite_b @ ( image_3724615099042636214real_b @ F @ A2 ) )
=> ( ( inj_on8242634198667403042real_b @ F @ A2 )
=> ( finite7198162374296863863_ereal @ A2 ) ) ) ).
% finite_imageD
thf(fact_985_finite__imageD,axiom,
! [F: a > extended_ereal,A2: set_a] :
( ( finite7198162374296863863_ereal @ ( image_8405481351990995413_ereal @ F @ A2 ) )
=> ( ( inj_on3700128414760986433_ereal @ F @ A2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_imageD
thf(fact_986_finite__imageD,axiom,
! [F: b > extended_ereal,A2: set_b] :
( ( finite7198162374296863863_ereal @ ( image_5319725110001000852_ereal @ F @ A2 ) )
=> ( ( inj_on614372172770991872_ereal @ F @ A2 )
=> ( finite_finite_b @ A2 ) ) ) ).
% finite_imageD
thf(fact_987_finite__imageD,axiom,
! [F: extended_ereal > extended_ereal,A2: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ ( image_6042159593519690757_ereal @ F @ A2 ) )
=> ( ( inj_on7162434037990268785_ereal @ F @ A2 )
=> ( finite7198162374296863863_ereal @ A2 ) ) ) ).
% finite_imageD
thf(fact_988_finite__imageD,axiom,
! [F: set_a > a,A2: set_set_a] :
( ( finite_finite_a @ ( image_set_a_a @ F @ A2 ) )
=> ( ( inj_on_set_a_a @ F @ A2 )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% finite_imageD
thf(fact_989_inj__on__image__eq__iff,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,C3: set_set_a,A2: set_set_a,B2: set_set_a] :
( ( inj_on6007161714731792944t_unit @ F @ C3 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ C3 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C3 )
=> ( ( ( image_6801035452528096924t_unit @ F @ A2 )
= ( image_6801035452528096924t_unit @ F @ B2 ) )
= ( A2 = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_990_inj__on__image__eq__iff,axiom,
! [F: b > a,C3: set_b,A2: set_b,B2: set_b] :
( ( inj_on_b_a @ F @ C3 )
=> ( ( ord_less_eq_set_b @ A2 @ C3 )
=> ( ( ord_less_eq_set_b @ B2 @ C3 )
=> ( ( ( image_b_a @ F @ A2 )
= ( image_b_a @ F @ B2 ) )
= ( A2 = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_991_inj__on__image__eq__iff,axiom,
! [F: b > product_prod_a_a,C3: set_b,A2: set_b,B2: set_b] :
( ( inj_on8302219782909765977od_a_a @ F @ C3 )
=> ( ( ord_less_eq_set_b @ A2 @ C3 )
=> ( ( ord_less_eq_set_b @ B2 @ C3 )
=> ( ( ( image_6761185482258179565od_a_a @ F @ A2 )
= ( image_6761185482258179565od_a_a @ F @ B2 ) )
= ( A2 = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_992_inj__on__image__eq__iff,axiom,
! [F: a > set_a,C3: set_a,A2: set_a,B2: set_a] :
( ( inj_on_a_set_a @ F @ C3 )
=> ( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ( ( image_a_set_a @ F @ A2 )
= ( image_a_set_a @ F @ B2 ) )
= ( A2 = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_993_inj__on__image__eq__iff,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,C3: set_pr5411798346947241657t_unit,A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit] :
( ( inj_on6672326154762249576_set_a @ F @ C3 )
=> ( ( ord_le8200006823705900825t_unit @ A2 @ C3 )
=> ( ( ord_le8200006823705900825t_unit @ B2 @ C3 )
=> ( ( ( image_7466199892558553556_set_a @ F @ A2 )
= ( image_7466199892558553556_set_a @ F @ B2 ) )
= ( A2 = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_994_inj__on__image__mem__iff,axiom,
! [F: b > b,B2: set_b,A: b,A2: set_b] :
( ( inj_on_b_b @ F @ B2 )
=> ( ( member_b @ A @ B2 )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( member_b @ ( F @ A ) @ ( image_b_b @ F @ A2 ) )
= ( member_b @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_995_inj__on__image__mem__iff,axiom,
! [F: b > a,B2: set_b,A: b,A2: set_b] :
( ( inj_on_b_a @ F @ B2 )
=> ( ( member_b @ A @ B2 )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( member_a @ ( F @ A ) @ ( image_b_a @ F @ A2 ) )
= ( member_b @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_996_inj__on__image__mem__iff,axiom,
! [F: a > b,B2: set_a,A: a,A2: set_a] :
( ( inj_on_a_b @ F @ B2 )
=> ( ( member_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_b @ ( F @ A ) @ ( image_a_b @ F @ A2 ) )
= ( member_a @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_997_inj__on__image__mem__iff,axiom,
! [F: a > a,B2: set_a,A: a,A2: set_a] :
( ( inj_on_a_a @ F @ B2 )
=> ( ( member_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ ( F @ A ) @ ( image_a_a @ F @ A2 ) )
= ( member_a @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_998_inj__on__image__mem__iff,axiom,
! [F: b > set_a,B2: set_b,A: b,A2: set_b] :
( ( inj_on_b_set_a @ F @ B2 )
=> ( ( member_b @ A @ B2 )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( member_set_a @ ( F @ A ) @ ( image_b_set_a @ F @ A2 ) )
= ( member_b @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_999_inj__on__image__mem__iff,axiom,
! [F: set_a > b,B2: set_set_a,A: set_a,A2: set_set_a] :
( ( inj_on_set_a_b @ F @ B2 )
=> ( ( member_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_b @ ( F @ A ) @ ( image_set_a_b @ F @ A2 ) )
= ( member_set_a @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1000_inj__on__image__mem__iff,axiom,
! [F: set_a > a,B2: set_set_a,A: set_a,A2: set_set_a] :
( ( inj_on_set_a_a @ F @ B2 )
=> ( ( member_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_a @ ( F @ A ) @ ( image_set_a_a @ F @ A2 ) )
= ( member_set_a @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1001_inj__on__image__mem__iff,axiom,
! [F: a > set_a,B2: set_a,A: a,A2: set_a] :
( ( inj_on_a_set_a @ F @ B2 )
=> ( ( member_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_set_a @ ( F @ A ) @ ( image_a_set_a @ F @ A2 ) )
= ( member_a @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1002_inj__on__image__mem__iff,axiom,
! [F: b > product_prod_a_a,B2: set_b,A: b,A2: set_b] :
( ( inj_on8302219782909765977od_a_a @ F @ B2 )
=> ( ( member_b @ A @ B2 )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( member1426531477525435216od_a_a @ ( F @ A ) @ ( image_6761185482258179565od_a_a @ F @ A2 ) )
= ( member_b @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1003_inj__on__image__mem__iff,axiom,
! [F: set_a > set_a,B2: set_set_a,A: set_a,A2: set_set_a] :
( ( inj_on_set_a_set_a @ F @ B2 )
=> ( ( member_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ ( F @ A ) @ ( image_set_a_set_a @ F @ A2 ) )
= ( member_set_a @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1004_inj__img__insertE,axiom,
! [F: b > b,A2: set_b,X: b,B2: set_b] :
( ( inj_on_b_b @ F @ A2 )
=> ( ~ ( member_b @ X @ B2 )
=> ( ( ( insert_b @ X @ B2 )
= ( image_b_b @ F @ A2 ) )
=> ~ ! [X6: b,A7: set_b] :
( ~ ( member_b @ X6 @ A7 )
=> ( ( A2
= ( insert_b @ X6 @ A7 ) )
=> ( ( X
= ( F @ X6 ) )
=> ( B2
!= ( image_b_b @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1005_inj__img__insertE,axiom,
! [F: a > b,A2: set_a,X: b,B2: set_b] :
( ( inj_on_a_b @ F @ A2 )
=> ( ~ ( member_b @ X @ B2 )
=> ( ( ( insert_b @ X @ B2 )
= ( image_a_b @ F @ A2 ) )
=> ~ ! [X6: a,A7: set_a] :
( ~ ( member_a @ X6 @ A7 )
=> ( ( A2
= ( insert_a @ X6 @ A7 ) )
=> ( ( X
= ( F @ X6 ) )
=> ( B2
!= ( image_a_b @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1006_inj__img__insertE,axiom,
! [F: b > a,A2: set_b,X: a,B2: set_a] :
( ( inj_on_b_a @ F @ A2 )
=> ( ~ ( member_a @ X @ B2 )
=> ( ( ( insert_a @ X @ B2 )
= ( image_b_a @ F @ A2 ) )
=> ~ ! [X6: b,A7: set_b] :
( ~ ( member_b @ X6 @ A7 )
=> ( ( A2
= ( insert_b @ X6 @ A7 ) )
=> ( ( X
= ( F @ X6 ) )
=> ( B2
!= ( image_b_a @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1007_inj__img__insertE,axiom,
! [F: a > a,A2: set_a,X: a,B2: set_a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ~ ( member_a @ X @ B2 )
=> ( ( ( insert_a @ X @ B2 )
= ( image_a_a @ F @ A2 ) )
=> ~ ! [X6: a,A7: set_a] :
( ~ ( member_a @ X6 @ A7 )
=> ( ( A2
= ( insert_a @ X6 @ A7 ) )
=> ( ( X
= ( F @ X6 ) )
=> ( B2
!= ( image_a_a @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1008_inj__img__insertE,axiom,
! [F: set_a > b,A2: set_set_a,X: b,B2: set_b] :
( ( inj_on_set_a_b @ F @ A2 )
=> ( ~ ( member_b @ X @ B2 )
=> ( ( ( insert_b @ X @ B2 )
= ( image_set_a_b @ F @ A2 ) )
=> ~ ! [X6: set_a,A7: set_set_a] :
( ~ ( member_set_a @ X6 @ A7 )
=> ( ( A2
= ( insert_set_a @ X6 @ A7 ) )
=> ( ( X
= ( F @ X6 ) )
=> ( B2
!= ( image_set_a_b @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1009_inj__img__insertE,axiom,
! [F: set_a > a,A2: set_set_a,X: a,B2: set_a] :
( ( inj_on_set_a_a @ F @ A2 )
=> ( ~ ( member_a @ X @ B2 )
=> ( ( ( insert_a @ X @ B2 )
= ( image_set_a_a @ F @ A2 ) )
=> ~ ! [X6: set_a,A7: set_set_a] :
( ~ ( member_set_a @ X6 @ A7 )
=> ( ( A2
= ( insert_set_a @ X6 @ A7 ) )
=> ( ( X
= ( F @ X6 ) )
=> ( B2
!= ( image_set_a_a @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1010_inj__img__insertE,axiom,
! [F: b > set_a,A2: set_b,X: set_a,B2: set_set_a] :
( ( inj_on_b_set_a @ F @ A2 )
=> ( ~ ( member_set_a @ X @ B2 )
=> ( ( ( insert_set_a @ X @ B2 )
= ( image_b_set_a @ F @ A2 ) )
=> ~ ! [X6: b,A7: set_b] :
( ~ ( member_b @ X6 @ A7 )
=> ( ( A2
= ( insert_b @ X6 @ A7 ) )
=> ( ( X
= ( F @ X6 ) )
=> ( B2
!= ( image_b_set_a @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1011_inj__img__insertE,axiom,
! [F: a > set_a,A2: set_a,X: set_a,B2: set_set_a] :
( ( inj_on_a_set_a @ F @ A2 )
=> ( ~ ( member_set_a @ X @ B2 )
=> ( ( ( insert_set_a @ X @ B2 )
= ( image_a_set_a @ F @ A2 ) )
=> ~ ! [X6: a,A7: set_a] :
( ~ ( member_a @ X6 @ A7 )
=> ( ( A2
= ( insert_a @ X6 @ A7 ) )
=> ( ( X
= ( F @ X6 ) )
=> ( B2
!= ( image_a_set_a @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1012_inj__img__insertE,axiom,
! [F: b > product_prod_a_a,A2: set_b,X: product_prod_a_a,B2: set_Product_prod_a_a] :
( ( inj_on8302219782909765977od_a_a @ F @ A2 )
=> ( ~ ( member1426531477525435216od_a_a @ X @ B2 )
=> ( ( ( insert4534936382041156343od_a_a @ X @ B2 )
= ( image_6761185482258179565od_a_a @ F @ A2 ) )
=> ~ ! [X6: b,A7: set_b] :
( ~ ( member_b @ X6 @ A7 )
=> ( ( A2
= ( insert_b @ X6 @ A7 ) )
=> ( ( X
= ( F @ X6 ) )
=> ( B2
!= ( image_6761185482258179565od_a_a @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1013_inj__img__insertE,axiom,
! [F: set_a > set_a,A2: set_set_a,X: set_a,B2: set_set_a] :
( ( inj_on_set_a_set_a @ F @ A2 )
=> ( ~ ( member_set_a @ X @ B2 )
=> ( ( ( insert_set_a @ X @ B2 )
= ( image_set_a_set_a @ F @ A2 ) )
=> ~ ! [X6: set_a,A7: set_set_a] :
( ~ ( member_set_a @ X6 @ A7 )
=> ( ( A2
= ( insert_set_a @ X6 @ A7 ) )
=> ( ( X
= ( F @ X6 ) )
=> ( B2
!= ( image_set_a_set_a @ F @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1014_Diff__insert__absorb,axiom,
! [X: set_a,A2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_1015_Diff__insert__absorb,axiom,
! [X: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ X @ A2 )
=> ( ( minus_3777555517894451474t_unit @ ( insert6864688055023459379t_unit @ X @ A2 ) @ ( insert6864688055023459379t_unit @ X @ bot_bo1839476491465656141t_unit ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_1016_Diff__insert__absorb,axiom,
! [X: a,A2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_1017_Diff__insert__absorb,axiom,
! [X: b,A2: set_b] :
( ~ ( member_b @ X @ A2 )
=> ( ( minus_minus_set_b @ ( insert_b @ X @ A2 ) @ ( insert_b @ X @ bot_bot_set_b ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_1018_Diff__insert2,axiom,
! [A2: set_pr5411798346947241657t_unit,A: pre_pr7278220950009878019t_unit,B2: set_pr5411798346947241657t_unit] :
( ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ B2 ) )
= ( minus_3777555517894451474t_unit @ ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_1019_Diff__insert2,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_1020_Diff__insert2,axiom,
! [A2: set_b,A: b,B2: set_b] :
( ( minus_minus_set_b @ A2 @ ( insert_b @ A @ B2 ) )
= ( minus_minus_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_1021_insert__Diff,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( insert_set_a @ A @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_1022_insert__Diff,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ A @ A2 )
=> ( ( insert6864688055023459379t_unit @ A @ ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_1023_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_1024_insert__Diff,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ( ( insert_b @ A @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_1025_Diff__insert,axiom,
! [A2: set_pr5411798346947241657t_unit,A: pre_pr7278220950009878019t_unit,B2: set_pr5411798346947241657t_unit] :
( ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ B2 ) )
= ( minus_3777555517894451474t_unit @ ( minus_3777555517894451474t_unit @ A2 @ B2 ) @ ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) ) ) ).
% Diff_insert
thf(fact_1026_Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_1027_Diff__insert,axiom,
! [A2: set_b,A: b,B2: set_b] :
( ( minus_minus_set_b @ A2 @ ( insert_b @ A @ B2 ) )
= ( minus_minus_set_b @ ( minus_minus_set_b @ A2 @ B2 ) @ ( insert_b @ A @ bot_bot_set_b ) ) ) ).
% Diff_insert
thf(fact_1028_subset__Diff__insert,axiom,
! [A2: set_set_a,B2: set_set_a,X: set_a,C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B2 @ ( insert_set_a @ X @ C3 ) ) )
= ( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B2 @ C3 ) )
& ~ ( member_set_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1029_subset__Diff__insert,axiom,
! [A2: set_b,B2: set_b,X: b,C3: set_b] :
( ( ord_less_eq_set_b @ A2 @ ( minus_minus_set_b @ B2 @ ( insert_b @ X @ C3 ) ) )
= ( ( ord_less_eq_set_b @ A2 @ ( minus_minus_set_b @ B2 @ C3 ) )
& ~ ( member_b @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1030_subset__Diff__insert,axiom,
! [A2: set_a,B2: set_a,X: a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ ( insert_a @ X @ C3 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ C3 ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1031_subset__Diff__insert,axiom,
! [A2: set_pr5411798346947241657t_unit,B2: set_pr5411798346947241657t_unit,X: pre_pr7278220950009878019t_unit,C3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ ( minus_3777555517894451474t_unit @ B2 @ ( insert6864688055023459379t_unit @ X @ C3 ) ) )
= ( ( ord_le8200006823705900825t_unit @ A2 @ ( minus_3777555517894451474t_unit @ B2 @ C3 ) )
& ~ ( member6939884229742472986t_unit @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1032_finite__surj__inj,axiom,
! [A2: set_b,F: b > b] :
( ( finite_finite_b @ A2 )
=> ( ( ord_less_eq_set_b @ A2 @ ( image_b_b @ F @ A2 ) )
=> ( inj_on_b_b @ F @ A2 ) ) ) ).
% finite_surj_inj
thf(fact_1033_finite__surj__inj,axiom,
! [A2: set_set_a,F: set_a > set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ ( image_set_a_set_a @ F @ A2 ) )
=> ( inj_on_set_a_set_a @ F @ A2 ) ) ) ).
% finite_surj_inj
thf(fact_1034_finite__surj__inj,axiom,
! [A2: set_Extended_ereal,F: extended_ereal > extended_ereal] :
( ( finite7198162374296863863_ereal @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ A2 @ ( image_6042159593519690757_ereal @ F @ A2 ) )
=> ( inj_on7162434037990268785_ereal @ F @ A2 ) ) ) ).
% finite_surj_inj
thf(fact_1035_finite__surj__inj,axiom,
! [A2: set_a,F: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( image_a_a @ F @ A2 ) )
=> ( inj_on_a_a @ F @ A2 ) ) ) ).
% finite_surj_inj
thf(fact_1036_finite__surj__inj,axiom,
! [A2: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > pre_pr7278220950009878019t_unit] :
( ( finite8852549406693098522t_unit @ A2 )
=> ( ( ord_le8200006823705900825t_unit @ A2 @ ( image_7933780498232994317t_unit @ F @ A2 ) )
=> ( inj_on6227784922168685433t_unit @ F @ A2 ) ) ) ).
% finite_surj_inj
thf(fact_1037_inj__on__finite,axiom,
! [F: a > b,A2: set_a,B2: set_b] :
( ( inj_on_a_b @ F @ A2 )
=> ( ( ord_less_eq_set_b @ ( image_a_b @ F @ A2 ) @ B2 )
=> ( ( finite_finite_b @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_1038_inj__on__finite,axiom,
! [F: b > b,A2: set_b,B2: set_b] :
( ( inj_on_b_b @ F @ A2 )
=> ( ( ord_less_eq_set_b @ ( image_b_b @ F @ A2 ) @ B2 )
=> ( ( finite_finite_b @ B2 )
=> ( finite_finite_b @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_1039_inj__on__finite,axiom,
! [F: extended_ereal > b,A2: set_Extended_ereal,B2: set_b] :
( ( inj_on8242634198667403042real_b @ F @ A2 )
=> ( ( ord_less_eq_set_b @ ( image_3724615099042636214real_b @ F @ A2 ) @ B2 )
=> ( ( finite_finite_b @ B2 )
=> ( finite7198162374296863863_ereal @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_1040_inj__on__finite,axiom,
! [F: a > extended_ereal,A2: set_a,B2: set_Extended_ereal] :
( ( inj_on3700128414760986433_ereal @ F @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ ( image_8405481351990995413_ereal @ F @ A2 ) @ B2 )
=> ( ( finite7198162374296863863_ereal @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_1041_inj__on__finite,axiom,
! [F: b > extended_ereal,A2: set_b,B2: set_Extended_ereal] :
( ( inj_on614372172770991872_ereal @ F @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ ( image_5319725110001000852_ereal @ F @ A2 ) @ B2 )
=> ( ( finite7198162374296863863_ereal @ B2 )
=> ( finite_finite_b @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_1042_inj__on__finite,axiom,
! [F: extended_ereal > extended_ereal,A2: set_Extended_ereal,B2: set_Extended_ereal] :
( ( inj_on7162434037990268785_ereal @ F @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F @ A2 ) @ B2 )
=> ( ( finite7198162374296863863_ereal @ B2 )
=> ( finite7198162374296863863_ereal @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_1043_inj__on__finite,axiom,
! [F: a > a,A2: set_a,B2: set_a] :
( ( inj_on_a_a @ F @ A2 )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_1044_inj__on__finite,axiom,
! [F: b > a,A2: set_b,B2: set_a] :
( ( inj_on_b_a @ F @ A2 )
=> ( ( ord_less_eq_set_a @ ( image_b_a @ F @ A2 ) @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_b @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_1045_inj__on__finite,axiom,
! [F: extended_ereal > a,A2: set_Extended_ereal,B2: set_a] :
( ( inj_on8242634198667403041real_a @ F @ A2 )
=> ( ( ord_less_eq_set_a @ ( image_3724615099042636213real_a @ F @ A2 ) @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite7198162374296863863_ereal @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_1046_inj__on__finite,axiom,
! [F: set_a > b,A2: set_set_a,B2: set_b] :
( ( inj_on_set_a_b @ F @ A2 )
=> ( ( ord_less_eq_set_b @ ( image_set_a_b @ F @ A2 ) @ B2 )
=> ( ( finite_finite_b @ B2 )
=> ( finite_finite_set_a @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_1047_endo__inj__surj,axiom,
! [A2: set_b,F: b > b] :
( ( finite_finite_b @ A2 )
=> ( ( ord_less_eq_set_b @ ( image_b_b @ F @ A2 ) @ A2 )
=> ( ( inj_on_b_b @ F @ A2 )
=> ( ( image_b_b @ F @ A2 )
= A2 ) ) ) ) ).
% endo_inj_surj
thf(fact_1048_endo__inj__surj,axiom,
! [A2: set_set_a,F: set_a > set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A2 ) @ A2 )
=> ( ( inj_on_set_a_set_a @ F @ A2 )
=> ( ( image_set_a_set_a @ F @ A2 )
= A2 ) ) ) ) ).
% endo_inj_surj
thf(fact_1049_endo__inj__surj,axiom,
! [A2: set_Extended_ereal,F: extended_ereal > extended_ereal] :
( ( finite7198162374296863863_ereal @ A2 )
=> ( ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F @ A2 ) @ A2 )
=> ( ( inj_on7162434037990268785_ereal @ F @ A2 )
=> ( ( image_6042159593519690757_ereal @ F @ A2 )
= A2 ) ) ) ) ).
% endo_inj_surj
thf(fact_1050_endo__inj__surj,axiom,
! [A2: set_a,F: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ A2 )
=> ( ( inj_on_a_a @ F @ A2 )
=> ( ( image_a_a @ F @ A2 )
= A2 ) ) ) ) ).
% endo_inj_surj
thf(fact_1051_endo__inj__surj,axiom,
! [A2: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > pre_pr7278220950009878019t_unit] :
( ( finite8852549406693098522t_unit @ A2 )
=> ( ( ord_le8200006823705900825t_unit @ ( image_7933780498232994317t_unit @ F @ A2 ) @ A2 )
=> ( ( inj_on6227784922168685433t_unit @ F @ A2 )
=> ( ( image_7933780498232994317t_unit @ F @ A2 )
= A2 ) ) ) ) ).
% endo_inj_surj
thf(fact_1052_finite__induct__select,axiom,
! [S: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ S )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [T3: set_set_a] :
( ( ord_less_set_set_a @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ ( minus_5736297505244876581_set_a @ S @ T3 ) )
& ( P @ ( insert_set_a @ X5 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1053_finite__induct__select,axiom,
! [S: set_Extended_ereal,P: set_Extended_ereal > $o] :
( ( finite7198162374296863863_ereal @ S )
=> ( ( P @ bot_bo8367695208629047834_ereal )
=> ( ! [T3: set_Extended_ereal] :
( ( ord_le5321083090456148570_ereal @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X5: extended_ereal] :
( ( member2350847679896131959_ereal @ X5 @ ( minus_1264018925008434325_ereal @ S @ T3 ) )
& ( P @ ( insert8967887681552722334_ereal @ X5 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1054_finite__induct__select,axiom,
! [S: set_pr5411798346947241657t_unit,P: set_pr5411798346947241657t_unit > $o] :
( ( finite8852549406693098522t_unit @ S )
=> ( ( P @ bot_bo1839476491465656141t_unit )
=> ( ! [T3: set_pr5411798346947241657t_unit] :
( ( ord_le2693654750756130573t_unit @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X5: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X5 @ ( minus_3777555517894451474t_unit @ S @ T3 ) )
& ( P @ ( insert6864688055023459379t_unit @ X5 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1055_finite__induct__select,axiom,
! [S: set_b,P: set_b > $o] :
( ( finite_finite_b @ S )
=> ( ( P @ bot_bot_set_b )
=> ( ! [T3: set_b] :
( ( ord_less_set_b @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X5: b] :
( ( member_b @ X5 @ ( minus_minus_set_b @ S @ T3 ) )
& ( P @ ( insert_b @ X5 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1056_finite__induct__select,axiom,
! [S: set_a,P: set_a > $o] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [T3: set_a] :
( ( ord_less_set_a @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X5: a] :
( ( member_a @ X5 @ ( minus_minus_set_a @ S @ T3 ) )
& ( P @ ( insert_a @ X5 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1057_infinite__remove,axiom,
! [S: set_set_a,A: set_a] :
( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ S @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ).
% infinite_remove
thf(fact_1058_infinite__remove,axiom,
! [S: set_Extended_ereal,A: extended_ereal] :
( ~ ( finite7198162374296863863_ereal @ S )
=> ~ ( finite7198162374296863863_ereal @ ( minus_1264018925008434325_ereal @ S @ ( insert8967887681552722334_ereal @ A @ bot_bo8367695208629047834_ereal ) ) ) ) ).
% infinite_remove
thf(fact_1059_infinite__remove,axiom,
! [S: set_pr5411798346947241657t_unit,A: pre_pr7278220950009878019t_unit] :
( ~ ( finite8852549406693098522t_unit @ S )
=> ~ ( finite8852549406693098522t_unit @ ( minus_3777555517894451474t_unit @ S @ ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) ) ) ) ).
% infinite_remove
thf(fact_1060_infinite__remove,axiom,
! [S: set_a,A: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_1061_infinite__remove,axiom,
! [S: set_b,A: b] :
( ~ ( finite_finite_b @ S )
=> ~ ( finite_finite_b @ ( minus_minus_set_b @ S @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ).
% infinite_remove
thf(fact_1062_infinite__coinduct,axiom,
! [X4: set_set_a > $o,A2: set_set_a] :
( ( X4 @ A2 )
=> ( ! [A5: set_set_a] :
( ( X4 @ A5 )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ A5 )
& ( ( X4 @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) )
| ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) ) ) ) )
=> ~ ( finite_finite_set_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_1063_infinite__coinduct,axiom,
! [X4: set_Extended_ereal > $o,A2: set_Extended_ereal] :
( ( X4 @ A2 )
=> ( ! [A5: set_Extended_ereal] :
( ( X4 @ A5 )
=> ? [X5: extended_ereal] :
( ( member2350847679896131959_ereal @ X5 @ A5 )
& ( ( X4 @ ( minus_1264018925008434325_ereal @ A5 @ ( insert8967887681552722334_ereal @ X5 @ bot_bo8367695208629047834_ereal ) ) )
| ~ ( finite7198162374296863863_ereal @ ( minus_1264018925008434325_ereal @ A5 @ ( insert8967887681552722334_ereal @ X5 @ bot_bo8367695208629047834_ereal ) ) ) ) ) )
=> ~ ( finite7198162374296863863_ereal @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_1064_infinite__coinduct,axiom,
! [X4: set_pr5411798346947241657t_unit > $o,A2: set_pr5411798346947241657t_unit] :
( ( X4 @ A2 )
=> ( ! [A5: set_pr5411798346947241657t_unit] :
( ( X4 @ A5 )
=> ? [X5: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X5 @ A5 )
& ( ( X4 @ ( minus_3777555517894451474t_unit @ A5 @ ( insert6864688055023459379t_unit @ X5 @ bot_bo1839476491465656141t_unit ) ) )
| ~ ( finite8852549406693098522t_unit @ ( minus_3777555517894451474t_unit @ A5 @ ( insert6864688055023459379t_unit @ X5 @ bot_bo1839476491465656141t_unit ) ) ) ) ) )
=> ~ ( finite8852549406693098522t_unit @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_1065_infinite__coinduct,axiom,
! [X4: set_a > $o,A2: set_a] :
( ( X4 @ A2 )
=> ( ! [A5: set_a] :
( ( X4 @ A5 )
=> ? [X5: a] :
( ( member_a @ X5 @ A5 )
& ( ( X4 @ ( minus_minus_set_a @ A5 @ ( insert_a @ X5 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A5 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_1066_infinite__coinduct,axiom,
! [X4: set_b > $o,A2: set_b] :
( ( X4 @ A2 )
=> ( ! [A5: set_b] :
( ( X4 @ A5 )
=> ? [X5: b] :
( ( member_b @ X5 @ A5 )
& ( ( X4 @ ( minus_minus_set_b @ A5 @ ( insert_b @ X5 @ bot_bot_set_b ) ) )
| ~ ( finite_finite_b @ ( minus_minus_set_b @ A5 @ ( insert_b @ X5 @ bot_bot_set_b ) ) ) ) ) )
=> ~ ( finite_finite_b @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_1067_finite__empty__induct,axiom,
! [A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: set_a,A5: set_set_a] :
( ( finite_finite_set_a @ A5 )
=> ( ( member_set_a @ A3 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ A3 @ bot_bot_set_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1068_finite__empty__induct,axiom,
! [A2: set_Extended_ereal,P: set_Extended_ereal > $o] :
( ( finite7198162374296863863_ereal @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: extended_ereal,A5: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ A5 )
=> ( ( member2350847679896131959_ereal @ A3 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_1264018925008434325_ereal @ A5 @ ( insert8967887681552722334_ereal @ A3 @ bot_bo8367695208629047834_ereal ) ) ) ) ) )
=> ( P @ bot_bo8367695208629047834_ereal ) ) ) ) ).
% finite_empty_induct
thf(fact_1069_finite__empty__induct,axiom,
! [A2: set_pr5411798346947241657t_unit,P: set_pr5411798346947241657t_unit > $o] :
( ( finite8852549406693098522t_unit @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: pre_pr7278220950009878019t_unit,A5: set_pr5411798346947241657t_unit] :
( ( finite8852549406693098522t_unit @ A5 )
=> ( ( member6939884229742472986t_unit @ A3 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_3777555517894451474t_unit @ A5 @ ( insert6864688055023459379t_unit @ A3 @ bot_bo1839476491465656141t_unit ) ) ) ) ) )
=> ( P @ bot_bo1839476491465656141t_unit ) ) ) ) ).
% finite_empty_induct
thf(fact_1070_finite__empty__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: a,A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( member_a @ A3 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_minus_set_a @ A5 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1071_finite__empty__induct,axiom,
! [A2: set_b,P: set_b > $o] :
( ( finite_finite_b @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: b,A5: set_b] :
( ( finite_finite_b @ A5 )
=> ( ( member_b @ A3 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_minus_set_b @ A5 @ ( insert_b @ A3 @ bot_bot_set_b ) ) ) ) ) )
=> ( P @ bot_bot_set_b ) ) ) ) ).
% finite_empty_induct
thf(fact_1072_Diff__single__insert,axiom,
! [A2: set_b,X: b,B2: set_b] :
( ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) ) @ B2 )
=> ( ord_less_eq_set_b @ A2 @ ( insert_b @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_1073_Diff__single__insert,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_1074_Diff__single__insert,axiom,
! [A2: set_pr5411798346947241657t_unit,X: pre_pr7278220950009878019t_unit,B2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ X @ bot_bo1839476491465656141t_unit ) ) @ B2 )
=> ( ord_le8200006823705900825t_unit @ A2 @ ( insert6864688055023459379t_unit @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_1075_subset__insert__iff,axiom,
! [A2: set_set_a,X: set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B2 ) )
= ( ( ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B2 ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1076_subset__insert__iff,axiom,
! [A2: set_b,X: b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X @ B2 ) )
= ( ( ( member_b @ X @ A2 )
=> ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) ) @ B2 ) )
& ( ~ ( member_b @ X @ A2 )
=> ( ord_less_eq_set_b @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1077_subset__insert__iff,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1078_subset__insert__iff,axiom,
! [A2: set_pr5411798346947241657t_unit,X: pre_pr7278220950009878019t_unit,B2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ ( insert6864688055023459379t_unit @ X @ B2 ) )
= ( ( ( member6939884229742472986t_unit @ X @ A2 )
=> ( ord_le8200006823705900825t_unit @ ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ X @ bot_bo1839476491465656141t_unit ) ) @ B2 ) )
& ( ~ ( member6939884229742472986t_unit @ X @ A2 )
=> ( ord_le8200006823705900825t_unit @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1079_psubset__insert__iff,axiom,
! [A2: set_set_a,X: set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ ( insert_set_a @ X @ B2 ) )
= ( ( ( member_set_a @ X @ B2 )
=> ( ord_less_set_set_a @ A2 @ B2 ) )
& ( ~ ( member_set_a @ X @ B2 )
=> ( ( ( member_set_a @ X @ A2 )
=> ( ord_less_set_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B2 ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1080_psubset__insert__iff,axiom,
! [A2: set_b,X: b,B2: set_b] :
( ( ord_less_set_b @ A2 @ ( insert_b @ X @ B2 ) )
= ( ( ( member_b @ X @ B2 )
=> ( ord_less_set_b @ A2 @ B2 ) )
& ( ~ ( member_b @ X @ B2 )
=> ( ( ( member_b @ X @ A2 )
=> ( ord_less_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) ) @ B2 ) )
& ( ~ ( member_b @ X @ A2 )
=> ( ord_less_eq_set_b @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1081_psubset__insert__iff,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ( ( member_a @ X @ B2 )
=> ( ord_less_set_a @ A2 @ B2 ) )
& ( ~ ( member_a @ X @ B2 )
=> ( ( ( member_a @ X @ A2 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1082_psubset__insert__iff,axiom,
! [A2: set_pr5411798346947241657t_unit,X: pre_pr7278220950009878019t_unit,B2: set_pr5411798346947241657t_unit] :
( ( ord_le2693654750756130573t_unit @ A2 @ ( insert6864688055023459379t_unit @ X @ B2 ) )
= ( ( ( member6939884229742472986t_unit @ X @ B2 )
=> ( ord_le2693654750756130573t_unit @ A2 @ B2 ) )
& ( ~ ( member6939884229742472986t_unit @ X @ B2 )
=> ( ( ( member6939884229742472986t_unit @ X @ A2 )
=> ( ord_le2693654750756130573t_unit @ ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ X @ bot_bo1839476491465656141t_unit ) ) @ B2 ) )
& ( ~ ( member6939884229742472986t_unit @ X @ A2 )
=> ( ord_le8200006823705900825t_unit @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1083_remove__induct,axiom,
! [P: set_set_a > $o,B2: set_set_a] :
( ( P @ bot_bot_set_set_a )
=> ( ( ~ ( finite_finite_set_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A5: set_set_a] :
( ( finite_finite_set_a @ A5 )
=> ( ( A5 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A5 @ B2 )
=> ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A5 )
=> ( P @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1084_remove__induct,axiom,
! [P: set_Extended_ereal > $o,B2: set_Extended_ereal] :
( ( P @ bot_bo8367695208629047834_ereal )
=> ( ( ~ ( finite7198162374296863863_ereal @ B2 )
=> ( P @ B2 ) )
=> ( ! [A5: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ A5 )
=> ( ( A5 != bot_bo8367695208629047834_ereal )
=> ( ( ord_le1644982726543182158_ereal @ A5 @ B2 )
=> ( ! [X5: extended_ereal] :
( ( member2350847679896131959_ereal @ X5 @ A5 )
=> ( P @ ( minus_1264018925008434325_ereal @ A5 @ ( insert8967887681552722334_ereal @ X5 @ bot_bo8367695208629047834_ereal ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1085_remove__induct,axiom,
! [P: set_b > $o,B2: set_b] :
( ( P @ bot_bot_set_b )
=> ( ( ~ ( finite_finite_b @ B2 )
=> ( P @ B2 ) )
=> ( ! [A5: set_b] :
( ( finite_finite_b @ A5 )
=> ( ( A5 != bot_bot_set_b )
=> ( ( ord_less_eq_set_b @ A5 @ B2 )
=> ( ! [X5: b] :
( ( member_b @ X5 @ A5 )
=> ( P @ ( minus_minus_set_b @ A5 @ ( insert_b @ X5 @ bot_bot_set_b ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1086_remove__induct,axiom,
! [P: set_a > $o,B2: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( A5 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A5 @ B2 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A5 )
=> ( P @ ( minus_minus_set_a @ A5 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1087_remove__induct,axiom,
! [P: set_pr5411798346947241657t_unit > $o,B2: set_pr5411798346947241657t_unit] :
( ( P @ bot_bo1839476491465656141t_unit )
=> ( ( ~ ( finite8852549406693098522t_unit @ B2 )
=> ( P @ B2 ) )
=> ( ! [A5: set_pr5411798346947241657t_unit] :
( ( finite8852549406693098522t_unit @ A5 )
=> ( ( A5 != bot_bo1839476491465656141t_unit )
=> ( ( ord_le8200006823705900825t_unit @ A5 @ B2 )
=> ( ! [X5: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X5 @ A5 )
=> ( P @ ( minus_3777555517894451474t_unit @ A5 @ ( insert6864688055023459379t_unit @ X5 @ bot_bo1839476491465656141t_unit ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1088_finite__remove__induct,axiom,
! [B2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ B2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A5: set_set_a] :
( ( finite_finite_set_a @ A5 )
=> ( ( A5 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A5 @ B2 )
=> ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A5 )
=> ( P @ ( minus_5736297505244876581_set_a @ A5 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1089_finite__remove__induct,axiom,
! [B2: set_Extended_ereal,P: set_Extended_ereal > $o] :
( ( finite7198162374296863863_ereal @ B2 )
=> ( ( P @ bot_bo8367695208629047834_ereal )
=> ( ! [A5: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ A5 )
=> ( ( A5 != bot_bo8367695208629047834_ereal )
=> ( ( ord_le1644982726543182158_ereal @ A5 @ B2 )
=> ( ! [X5: extended_ereal] :
( ( member2350847679896131959_ereal @ X5 @ A5 )
=> ( P @ ( minus_1264018925008434325_ereal @ A5 @ ( insert8967887681552722334_ereal @ X5 @ bot_bo8367695208629047834_ereal ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1090_finite__remove__induct,axiom,
! [B2: set_b,P: set_b > $o] :
( ( finite_finite_b @ B2 )
=> ( ( P @ bot_bot_set_b )
=> ( ! [A5: set_b] :
( ( finite_finite_b @ A5 )
=> ( ( A5 != bot_bot_set_b )
=> ( ( ord_less_eq_set_b @ A5 @ B2 )
=> ( ! [X5: b] :
( ( member_b @ X5 @ A5 )
=> ( P @ ( minus_minus_set_b @ A5 @ ( insert_b @ X5 @ bot_bot_set_b ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1091_finite__remove__induct,axiom,
! [B2: set_a,P: set_a > $o] :
( ( finite_finite_a @ B2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( A5 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A5 @ B2 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A5 )
=> ( P @ ( minus_minus_set_a @ A5 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1092_finite__remove__induct,axiom,
! [B2: set_pr5411798346947241657t_unit,P: set_pr5411798346947241657t_unit > $o] :
( ( finite8852549406693098522t_unit @ B2 )
=> ( ( P @ bot_bo1839476491465656141t_unit )
=> ( ! [A5: set_pr5411798346947241657t_unit] :
( ( finite8852549406693098522t_unit @ A5 )
=> ( ( A5 != bot_bo1839476491465656141t_unit )
=> ( ( ord_le8200006823705900825t_unit @ A5 @ B2 )
=> ( ! [X5: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X5 @ A5 )
=> ( P @ ( minus_3777555517894451474t_unit @ A5 @ ( insert6864688055023459379t_unit @ X5 @ bot_bo1839476491465656141t_unit ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1093_pre__digraph_Oarcs__del__arc,axiom,
! [G2: pre_pr7278220950009878019t_unit,A: b] :
( ( pre_ar1395965042833527383t_unit @ ( pre_del_arc_a_b @ G2 @ A ) )
= ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ G2 ) @ ( insert_b @ A @ bot_bot_set_b ) ) ) ).
% pre_digraph.arcs_del_arc
thf(fact_1094_pre__digraph_Oin__sccsE,axiom,
! [C: pre_pr7278220950009878019t_unit,G2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ G2 ) )
=> ~ ( ( digrap5251062021860773499ph_a_b @ C @ G2 )
=> ( ( digrap8691851296217657702ed_a_b @ C )
=> ? [D3: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ D3 @ G2 )
& ( digrap8691851296217657702ed_a_b @ D3 )
& ( ord_less_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ D3 ) ) ) ) ) ) ).
% pre_digraph.in_sccsE
thf(fact_1095_pre__digraph_Oin__sccsI,axiom,
! [C: pre_pr7278220950009878019t_unit,G2: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ C @ G2 )
=> ( ( digrap8691851296217657702ed_a_b @ C )
=> ( ~ ? [C5: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ C5 @ G2 )
& ( digrap8691851296217657702ed_a_b @ C5 )
& ( ord_less_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ C5 ) ) )
=> ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ G2 ) ) ) ) ) ).
% pre_digraph.in_sccsI
thf(fact_1096_pre__digraph_Overts__del__vert,axiom,
! [G2: pre_pr7278220950009878019t_unit,U: a] :
( ( pre_ve642382030648772252t_unit @ ( pre_del_vert_a_b @ G2 @ U ) )
= ( minus_minus_set_a @ ( pre_ve642382030648772252t_unit @ G2 ) @ ( insert_a @ U @ bot_bot_set_a ) ) ) ).
% pre_digraph.verts_del_vert
thf(fact_1097_infinite__growing,axiom,
! [X4: set_Extended_ereal] :
( ( X4 != bot_bo8367695208629047834_ereal )
=> ( ! [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ X4 )
=> ? [Xa: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa @ X4 )
& ( ord_le1188267648640031866_ereal @ X3 @ Xa ) ) )
=> ~ ( finite7198162374296863863_ereal @ X4 ) ) ) ).
% infinite_growing
thf(fact_1098_infinite__growing,axiom,
! [X4: set_real] :
( ( X4 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X4 )
=> ? [Xa: real] :
( ( member_real @ Xa @ X4 )
& ( ord_less_real @ X3 @ Xa ) ) )
=> ~ ( finite_finite_real @ X4 ) ) ) ).
% infinite_growing
thf(fact_1099_infinite__growing,axiom,
! [X4: set_nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X4 )
& ( ord_less_nat @ X3 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X4 ) ) ) ).
% infinite_growing
thf(fact_1100_ex__min__if__finite,axiom,
! [S: set_Extended_ereal] :
( ( finite7198162374296863863_ereal @ S )
=> ( ( S != bot_bo8367695208629047834_ereal )
=> ? [X3: extended_ereal] :
( ( member2350847679896131959_ereal @ X3 @ S )
& ~ ? [Xa: extended_ereal] :
( ( member2350847679896131959_ereal @ Xa @ S )
& ( ord_le1188267648640031866_ereal @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1101_ex__min__if__finite,axiom,
! [S: set_real] :
( ( finite_finite_real @ S )
=> ( ( S != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ S )
& ~ ? [Xa: real] :
( ( member_real @ Xa @ S )
& ( ord_less_real @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1102_ex__min__if__finite,axiom,
! [S: set_set_a] :
( ( finite_finite_set_a @ S )
=> ( ( S != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ S )
& ~ ? [Xa: set_a] :
( ( member_set_a @ Xa @ S )
& ( ord_less_set_a @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1103_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S )
& ( ord_less_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1104_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_1105_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_1106_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_1107_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_b] :
( ( inf_inf_set_b @ X @ bot_bot_set_b )
= bot_bot_set_b ) ).
% boolean_algebra.conj_zero_right
thf(fact_1108_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ X @ bot_bo1839476491465656141t_unit )
= bot_bo1839476491465656141t_unit ) ).
% boolean_algebra.conj_zero_right
thf(fact_1109_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_1110_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ X )
= bot_bot_set_b ) ).
% boolean_algebra.conj_zero_left
thf(fact_1111_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ bot_bo1839476491465656141t_unit @ X )
= bot_bo1839476491465656141t_unit ) ).
% boolean_algebra.conj_zero_left
thf(fact_1112_inf__bot__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_1113_inf__bot__right,axiom,
! [X: set_b] :
( ( inf_inf_set_b @ X @ bot_bot_set_b )
= bot_bot_set_b ) ).
% inf_bot_right
thf(fact_1114_inf__bot__right,axiom,
! [X: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ X @ bot_bo1839476491465656141t_unit )
= bot_bo1839476491465656141t_unit ) ).
% inf_bot_right
thf(fact_1115_inf__bot__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_1116_inf__bot__left,axiom,
! [X: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ X )
= bot_bot_set_b ) ).
% inf_bot_left
thf(fact_1117_inf__bot__left,axiom,
! [X: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ bot_bo1839476491465656141t_unit @ X )
= bot_bo1839476491465656141t_unit ) ).
% inf_bot_left
thf(fact_1118_inf_Obounded__iff,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( inf_inf_real @ B @ C ) )
= ( ( ord_less_eq_real @ A @ B )
& ( ord_less_eq_real @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1119_inf_Obounded__iff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( ( ord_less_eq_set_a @ A @ B )
& ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1120_inf_Obounded__iff,axiom,
! [A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit,C: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A @ ( inf_in1092213268631476299t_unit @ B @ C ) )
= ( ( ord_le8200006823705900825t_unit @ A @ B )
& ( ord_le8200006823705900825t_unit @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1121_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_1122_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_1123_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_1124_le__inf__iff,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z2 ) )
= ( ( ord_less_eq_real @ X @ Y )
& ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_1125_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z2 ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_1126_le__inf__iff,axiom,
! [X: set_pr5411798346947241657t_unit,Y: set_pr5411798346947241657t_unit,Z2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ X @ ( inf_in1092213268631476299t_unit @ Y @ Z2 ) )
= ( ( ord_le8200006823705900825t_unit @ X @ Y )
& ( ord_le8200006823705900825t_unit @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_1127_le__inf__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_1128_sym__arcs,axiom,
symmetric_a_b @ g ).
% sym_arcs
thf(fact_1129_sym__digraphI,axiom,
( ( symmetric_a_b @ g )
=> ( sym_digraph_a_b @ g ) ) ).
% sym_digraphI
thf(fact_1130_graphI,axiom,
( ( symmetric_a_b @ g )
=> ( graph_a_b @ g ) ) ).
% graphI
thf(fact_1131_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_1132_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_1133_bot__set__def,axiom,
( bot_bot_set_b
= ( collect_b @ bot_bot_b_o ) ) ).
% bot_set_def
thf(fact_1134_bot__set__def,axiom,
( bot_bo1839476491465656141t_unit
= ( collec8000012497822511960t_unit @ bot_bo8537066411596906360unit_o ) ) ).
% bot_set_def
thf(fact_1135_induced__graph__imp__symmetric,axiom,
! [G2: pre_pr7278220950009878019t_unit,H: pre_pr7278220950009878019t_unit] :
( ( symmetric_a_b @ G2 )
=> ( ( digrap5251062021860773499ph_a_b @ H @ G2 )
=> ( symmetric_a_b @ H ) ) ) ).
% induced_graph_imp_symmetric
thf(fact_1136_sym__digraph_Osym__arcs,axiom,
! [G2: pre_pr7278220950009878019t_unit] :
( ( sym_digraph_a_b @ G2 )
=> ( symmetric_a_b @ G2 ) ) ).
% sym_digraph.sym_arcs
thf(fact_1137_in__image__insert__iff,axiom,
! [B2: set_se2139339572462915695t_unit,X: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ! [C6: set_pr5411798346947241657t_unit] :
( ( member5449360183034373072t_unit @ C6 @ B2 )
=> ~ ( member6939884229742472986t_unit @ X @ C6 ) )
=> ( ( member5449360183034373072t_unit @ A2 @ ( image_4554393186639800441t_unit @ ( insert6864688055023459379t_unit @ X ) @ B2 ) )
= ( ( member6939884229742472986t_unit @ X @ A2 )
& ( member5449360183034373072t_unit @ ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ X @ bot_bo1839476491465656141t_unit ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1138_in__image__insert__iff,axiom,
! [B2: set_set_a,X: a,A2: set_a] :
( ! [C6: set_a] :
( ( member_set_a @ C6 @ B2 )
=> ~ ( member_a @ X @ C6 ) )
=> ( ( member_set_a @ A2 @ ( image_set_a_set_a @ ( insert_a @ X ) @ B2 ) )
= ( ( member_a @ X @ A2 )
& ( member_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1139_in__image__insert__iff,axiom,
! [B2: set_set_b,X: b,A2: set_b] :
( ! [C6: set_b] :
( ( member_set_b @ C6 @ B2 )
=> ~ ( member_b @ X @ C6 ) )
=> ( ( member_set_b @ A2 @ ( image_set_b_set_b @ ( insert_b @ X ) @ B2 ) )
= ( ( member_b @ X @ A2 )
& ( member_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_1140_euler__imp__connected,axiom,
! [U: a,P2: list_b,V: a] :
( ( pre_euler_trail_a_b @ g @ U @ P2 @ V )
=> ( digrap8783888973171253482ed_a_b @ g ) ) ).
% euler_imp_connected
thf(fact_1141_closed__euler2_I1_J,axiom,
! [U: a,P2: list_b] :
( ( pre_euler_trail_a_b @ g @ U @ P2 @ U )
=> ( digrap8783888973171253482ed_a_b @ g ) ) ).
% closed_euler2(1)
thf(fact_1142_unvis__insert,axiom,
! [U: a,X: a,U2: set_a] :
( ( graph_2016941059203891550ts_a_b @ g @ U @ ( insert_a @ X @ U2 ) )
= ( minus_minus_set_a @ ( graph_2016941059203891550ts_a_b @ g @ U @ U2 ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% unvis_insert
thf(fact_1143_not__elem__no__in__arcs,axiom,
! [V: a] :
( ~ ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( in_arcs_a_b @ g @ V )
= bot_bot_set_b ) ) ).
% not_elem_no_in_arcs
thf(fact_1144_unvis__finite,axiom,
! [U: a,U2: set_a] : ( finite_finite_a @ ( graph_2016941059203891550ts_a_b @ g @ U @ U2 ) ) ).
% unvis_finite
thf(fact_1145_disj__unvis__vis,axiom,
! [U: a,U2: set_a] :
( ( inf_inf_set_a @ ( graph_2016941059203891550ts_a_b @ g @ U @ U2 ) @ U2 )
= bot_bot_set_a ) ).
% disj_unvis_vis
thf(fact_1146_finite__in__arcs,axiom,
! [V: a] : ( finite_finite_b @ ( in_arcs_a_b @ g @ V ) ) ).
% finite_in_arcs
thf(fact_1147_arcs__del__vert2,axiom,
! [V: a] :
( ( pre_ar1395965042833527383t_unit @ ( pre_del_vert_a_b @ g @ V ) )
= ( minus_minus_set_b @ ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ g ) @ ( in_arcs_a_b @ g @ V ) ) @ ( out_arcs_a_b @ g @ V ) ) ) ).
% arcs_del_vert2
thf(fact_1148_some__unvis__vert_I1_J,axiom,
! [U: a,U2: set_a,X: a,W: b > real] :
( ( ( graph_2016941059203891550ts_a_b @ g @ U @ U2 )
!= bot_bot_set_a )
=> ( ( X
= ( graph_3614428260325061028rt_a_b @ g @ W @ U @ U2 ) )
=> ( member_a @ X @ ( graph_2016941059203891550ts_a_b @ g @ U @ U2 ) ) ) ) ).
% some_unvis_vert(1)
thf(fact_1149_nearest__vert__not__mem,axiom,
! [U: a,U2: set_a,W: b > real] :
( ( ( graph_2016941059203891550ts_a_b @ g @ U @ U2 )
!= bot_bot_set_a )
=> ~ ( member_a @ ( graph_3614428260325061028rt_a_b @ g @ W @ U @ U2 ) @ U2 ) ) ).
% nearest_vert_not_mem
thf(fact_1150_nearest__vert__unvis,axiom,
! [U: a,U2: set_a,W: b > real] :
( ( ( graph_2016941059203891550ts_a_b @ g @ U @ U2 )
!= bot_bot_set_a )
=> ( member_a @ ( graph_3614428260325061028rt_a_b @ g @ W @ U @ U2 ) @ ( graph_2016941059203891550ts_a_b @ g @ U @ U2 ) ) ) ).
% nearest_vert_unvis
thf(fact_1151_not__elem__no__out__arcs,axiom,
! [V: a] :
( ~ ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( out_arcs_a_b @ g @ V )
= bot_bot_set_b ) ) ).
% not_elem_no_out_arcs
thf(fact_1152_finite__out__arcs,axiom,
! [V: a] : ( finite_finite_b @ ( out_arcs_a_b @ g @ V ) ) ).
% finite_out_arcs
thf(fact_1153_leaf__def,axiom,
! [V: a] :
( ( shorte1213025427933718126af_a_b @ g @ V )
= ( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) )
& ( ( out_arcs_a_b @ g @ V )
= bot_bot_set_b ) ) ) ).
% leaf_def
thf(fact_1154_in__arcs__del__arc__iff,axiom,
! [A: b,U: a] :
( ( ( ( pre_he5236287464308401016t_unit @ g @ A )
= U )
=> ( ( in_arcs_a_b @ ( pre_del_arc_a_b @ g @ A ) @ U )
= ( minus_minus_set_b @ ( in_arcs_a_b @ g @ U ) @ ( insert_b @ A @ bot_bot_set_b ) ) ) )
& ( ( ( pre_he5236287464308401016t_unit @ g @ A )
!= U )
=> ( ( in_arcs_a_b @ ( pre_del_arc_a_b @ g @ A ) @ U )
= ( in_arcs_a_b @ g @ U ) ) ) ) ).
% in_arcs_del_arc_iff
thf(fact_1155_out__arcs__del__arc__iff,axiom,
! [A: b,U: a] :
( ( ( ( pre_ta4931606617599662728t_unit @ g @ A )
= U )
=> ( ( out_arcs_a_b @ ( pre_del_arc_a_b @ g @ A ) @ U )
= ( minus_minus_set_b @ ( out_arcs_a_b @ g @ U ) @ ( insert_b @ A @ bot_bot_set_b ) ) ) )
& ( ( ( pre_ta4931606617599662728t_unit @ g @ A )
!= U )
=> ( ( out_arcs_a_b @ ( pre_del_arc_a_b @ g @ A ) @ U )
= ( out_arcs_a_b @ g @ U ) ) ) ) ).
% out_arcs_del_arc_iff
thf(fact_1156_tail__del__vert,axiom,
! [U: a] :
( ( pre_ta4931606617599662728t_unit @ ( pre_del_vert_a_b @ g @ U ) )
= ( pre_ta4931606617599662728t_unit @ g ) ) ).
% tail_del_vert
thf(fact_1157_head__del__vert,axiom,
! [U: a] :
( ( pre_he5236287464308401016t_unit @ ( pre_del_vert_a_b @ g @ U ) )
= ( pre_he5236287464308401016t_unit @ g ) ) ).
% head_del_vert
thf(fact_1158_tail__add__vert,axiom,
! [U: a] :
( ( pre_ta4931606617599662728t_unit @ ( pre_add_vert_a_b @ g @ U ) )
= ( pre_ta4931606617599662728t_unit @ g ) ) ).
% tail_add_vert
thf(fact_1159_head__add__vert,axiom,
! [U: a] :
( ( pre_he5236287464308401016t_unit @ ( pre_add_vert_a_b @ g @ U ) )
= ( pre_he5236287464308401016t_unit @ g ) ) ).
% head_add_vert
thf(fact_1160_tail__in__verts,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( member_a @ ( pre_ta4931606617599662728t_unit @ g @ E ) @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% tail_in_verts
thf(fact_1161_head__in__verts,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( member_a @ ( pre_he5236287464308401016t_unit @ g @ E ) @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% head_in_verts
thf(fact_1162_arc__ends__eq__impl__arc__eq,axiom,
! [E1: b,E2: b] :
( ( member_b @ E1 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( ( pre_he5236287464308401016t_unit @ g @ E1 )
= ( pre_he5236287464308401016t_unit @ g @ E2 ) )
=> ( ( ( pre_ta4931606617599662728t_unit @ g @ E1 )
= ( pre_ta4931606617599662728t_unit @ g @ E2 ) )
=> ( E1 = E2 ) ) ) ) ) ).
% arc_ends_eq_impl_arc_eq
thf(fact_1163_no__loops,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( pre_ta4931606617599662728t_unit @ g @ E )
!= ( pre_he5236287464308401016t_unit @ g @ E ) ) ) ).
% no_loops
thf(fact_1164_no__multi__alt,axiom,
! [E1: b,E2: b] :
( ( member_b @ E1 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( E1 != E2 )
=> ( ( ( pre_he5236287464308401016t_unit @ g @ E1 )
!= ( pre_he5236287464308401016t_unit @ g @ E2 ) )
| ( ( pre_ta4931606617599662728t_unit @ g @ E1 )
!= ( pre_ta4931606617599662728t_unit @ g @ E2 ) ) ) ) ) ) ).
% no_multi_alt
thf(fact_1165_symmetric__arcs,axiom,
! [X: b] :
( ( member_b @ X @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ? [Y3: b] :
( ( ( pre_he5236287464308401016t_unit @ g @ X )
= ( pre_ta4931606617599662728t_unit @ g @ Y3 ) )
& ( ( pre_ta4931606617599662728t_unit @ g @ X )
= ( pre_he5236287464308401016t_unit @ g @ Y3 ) ) ) ) ).
% symmetric_arcs
thf(fact_1166_arc__ends__eq__impl__sel__eq,axiom,
! [X: b,Y: b] :
( ( ( pre_he5236287464308401016t_unit @ g @ X )
= ( pre_he5236287464308401016t_unit @ g @ Y ) )
=> ( ( ( pre_ta4931606617599662728t_unit @ g @ X )
= ( pre_ta4931606617599662728t_unit @ g @ Y ) )
=> ( ( sel @ X )
= ( sel @ Y ) ) ) ) ).
% arc_ends_eq_impl_sel_eq
thf(fact_1167_sel__sym,axiom,
! [E_1: b,E_2: b] :
( ( ( pre_ta4931606617599662728t_unit @ g @ E_1 )
= ( pre_he5236287464308401016t_unit @ g @ E_2 ) )
=> ( ( ( pre_he5236287464308401016t_unit @ g @ E_1 )
= ( pre_ta4931606617599662728t_unit @ g @ E_2 ) )
=> ( ( sel @ E_1 )
= ( sel @ E_2 ) ) ) ) ).
% sel_sym
thf(fact_1168_out__arcs__add__arc__iff,axiom,
! [A: b,U: a] :
( ( ( ( pre_ta4931606617599662728t_unit @ g @ A )
= U )
=> ( ( out_arcs_a_b @ ( pre_add_arc_a_b @ g @ A ) @ U )
= ( insert_b @ A @ ( out_arcs_a_b @ g @ U ) ) ) )
& ( ( ( pre_ta4931606617599662728t_unit @ g @ A )
!= U )
=> ( ( out_arcs_a_b @ ( pre_add_arc_a_b @ g @ A ) @ U )
= ( out_arcs_a_b @ g @ U ) ) ) ) ).
% out_arcs_add_arc_iff
thf(fact_1169_in__arcs__add__arc__iff,axiom,
! [A: b,U: a] :
( ( ( ( pre_he5236287464308401016t_unit @ g @ A )
= U )
=> ( ( in_arcs_a_b @ ( pre_add_arc_a_b @ g @ A ) @ U )
= ( insert_b @ A @ ( in_arcs_a_b @ g @ U ) ) ) )
& ( ( ( pre_he5236287464308401016t_unit @ g @ A )
!= U )
=> ( ( in_arcs_a_b @ ( pre_add_arc_a_b @ g @ A ) @ U )
= ( in_arcs_a_b @ g @ U ) ) ) ) ).
% in_arcs_add_arc_iff
thf(fact_1170_remove__sel__sym,axiom,
! [E_1: b,E_2: b,X: a] :
( ( ( pre_ta4931606617599662728t_unit @ g @ E_1 )
= ( pre_he5236287464308401016t_unit @ g @ E_2 ) )
=> ( ( ( pre_he5236287464308401016t_unit @ g @ E_1 )
= ( pre_ta4931606617599662728t_unit @ g @ E_2 ) )
=> ( ( query_remove_sel_a_b @ g @ sel @ X @ E_1 )
= ( query_remove_sel_a_b @ g @ sel @ X @ E_2 ) ) ) ) ).
% remove_sel_sym
thf(fact_1171_matching__sel__def,axiom,
! [F: a > a > real] :
( ( query_5351540782772103094el_a_b @ g @ sel @ F )
= ( ! [X2: a,Y2: a] :
( ? [E3: b] :
( ( ( pre_ta4931606617599662728t_unit @ g @ E3 )
= X2 )
& ( ( pre_he5236287464308401016t_unit @ g @ E3 )
= Y2 )
& ( ( F @ X2 @ Y2 )
= ( sel @ E3 ) ) )
| ( ~ ? [E3: b] :
( ( ( pre_ta4931606617599662728t_unit @ g @ E3 )
= X2 )
& ( ( pre_he5236287464308401016t_unit @ g @ E3 )
= Y2 ) )
& ( ( F @ X2 @ Y2 )
= one_one_real ) ) ) ) ) ).
% matching_sel_def
thf(fact_1172_matching__sel__simp__if__arc,axiom,
! [Sf: a > a > real,E: b] :
( ( query_5351540782772103094el_a_b @ g @ sel @ Sf )
=> ( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( Sf @ ( pre_ta4931606617599662728t_unit @ g @ E ) @ ( pre_he5236287464308401016t_unit @ g @ E ) )
= ( sel @ E ) ) ) ) ).
% matching_sel_simp_if_arc
thf(fact_1173_matching__sel__alt,axiom,
! [F: a > a > real] :
( ( query_5351540782772103094el_a_b @ g @ sel @ F )
= ( ! [X2: a,Y2: a] :
( ? [Z3: b] :
( ( member_b @ Z3 @ ( pre_ar1395965042833527383t_unit @ g ) )
& ( ( pre_ta4931606617599662728t_unit @ g @ Z3 )
= X2 )
& ( ( pre_he5236287464308401016t_unit @ g @ Z3 )
= Y2 )
& ( ( F @ X2 @ Y2 )
= ( sel @ Z3 ) ) )
| ( ~ ? [E3: b] :
( ( member_b @ E3 @ ( pre_ar1395965042833527383t_unit @ g ) )
& ( ( pre_ta4931606617599662728t_unit @ g @ E3 )
= X2 )
& ( ( pre_he5236287464308401016t_unit @ g @ E3 )
= Y2 ) )
& ( ( F @ X2 @ Y2 )
= one_one_real ) ) ) ) ) ).
% matching_sel_alt
thf(fact_1174_matching__sel__alt__aux1,axiom,
! [F: a > a > real] :
( ( query_5351540782772103094el_a_b @ g @ sel @ F )
=> ! [X5: a,Y5: a] :
( ? [Xa2: b] :
( ( member_b @ Xa2 @ ( pre_ar1395965042833527383t_unit @ g ) )
& ( ( pre_ta4931606617599662728t_unit @ g @ Xa2 )
= X5 )
& ( ( pre_he5236287464308401016t_unit @ g @ Xa2 )
= Y5 )
& ( ( F @ X5 @ Y5 )
= ( sel @ Xa2 ) ) )
| ( ~ ? [E4: b] :
( ( member_b @ E4 @ ( pre_ar1395965042833527383t_unit @ g ) )
& ( ( pre_ta4931606617599662728t_unit @ g @ E4 )
= X5 )
& ( ( pre_he5236287464308401016t_unit @ g @ E4 )
= Y5 ) )
& ( ( F @ X5 @ Y5 )
= one_one_real ) ) ) ) ).
% matching_sel_alt_aux1
thf(fact_1175_matching__sel__alt__aux2,axiom,
! [F: a > a > real] :
( ! [X3: a,Y3: a] :
( ? [Xa: b] :
( ( member_b @ Xa @ ( pre_ar1395965042833527383t_unit @ g ) )
& ( ( pre_ta4931606617599662728t_unit @ g @ Xa )
= X3 )
& ( ( pre_he5236287464308401016t_unit @ g @ Xa )
= Y3 )
& ( ( F @ X3 @ Y3 )
= ( sel @ Xa ) ) )
| ( ~ ? [E5: b] :
( ( member_b @ E5 @ ( pre_ar1395965042833527383t_unit @ g ) )
& ( ( pre_ta4931606617599662728t_unit @ g @ E5 )
= X3 )
& ( ( pre_he5236287464308401016t_unit @ g @ E5 )
= Y3 ) )
& ( ( F @ X3 @ Y3 )
= one_one_real ) ) )
=> ( query_5351540782772103094el_a_b @ g @ sel @ F ) ) ).
% matching_sel_alt_aux2
thf(fact_1176_matching__sel__simp__if__not1,axiom,
! [Sf: a > a > real,X: a,Y: a] :
( ( query_5351540782772103094el_a_b @ g @ sel @ Sf )
=> ( ( ( Sf @ X @ Y )
!= one_one_real )
=> ? [X3: b] :
( ( member_b @ X3 @ ( pre_ar1395965042833527383t_unit @ g ) )
& ( ( pre_ta4931606617599662728t_unit @ g @ X3 )
= X )
& ( ( pre_he5236287464308401016t_unit @ g @ X3 )
= Y )
& ( ( Sf @ X @ Y )
= ( sel @ X3 ) ) ) ) ) ).
% matching_sel_simp_if_not1
thf(fact_1177_tail__del__arc,axiom,
! [A: b] :
( ( pre_ta4931606617599662728t_unit @ ( pre_del_arc_a_b @ g @ A ) )
= ( pre_ta4931606617599662728t_unit @ g ) ) ).
% tail_del_arc
thf(fact_1178_head__del__arc,axiom,
! [A: b] :
( ( pre_he5236287464308401016t_unit @ ( pre_del_arc_a_b @ g @ A ) )
= ( pre_he5236287464308401016t_unit @ g ) ) ).
% head_del_arc
thf(fact_1179_tail__add__arc,axiom,
! [A: b] :
( ( pre_ta4931606617599662728t_unit @ ( pre_add_arc_a_b @ g @ A ) )
= ( pre_ta4931606617599662728t_unit @ g ) ) ).
% tail_add_arc
thf(fact_1180_head__add__arc,axiom,
! [A: b] :
( ( pre_he5236287464308401016t_unit @ ( pre_add_arc_a_b @ g @ A ) )
= ( pre_he5236287464308401016t_unit @ g ) ) ).
% head_add_arc
thf(fact_1181_verts__add__arc,axiom,
! [A: b] :
( ( member_a @ ( pre_ta4931606617599662728t_unit @ g @ A ) @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( member_a @ ( pre_he5236287464308401016t_unit @ g @ A ) @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( pre_ve642382030648772252t_unit @ ( pre_add_arc_a_b @ g @ A ) )
= ( pre_ve642382030648772252t_unit @ g ) ) ) ) ).
% verts_add_arc
thf(fact_1182_del__add__arc__collapse,axiom,
! [A: b] :
( ( member_a @ ( pre_ta4931606617599662728t_unit @ g @ A ) @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( member_a @ ( pre_he5236287464308401016t_unit @ g @ A ) @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( pre_del_arc_a_b @ ( pre_add_arc_a_b @ g @ A ) @ A )
= ( pre_del_arc_a_b @ g @ A ) ) ) ) ).
% del_add_arc_collapse
thf(fact_1183_connected__verts,axiom,
( ( digrap8783888973171253482ed_a_b @ g )
=> ( ( ( pre_ar1395965042833527383t_unit @ g )
!= bot_bot_set_b )
=> ( ( pre_ve642382030648772252t_unit @ g )
= ( sup_sup_set_a @ ( image_b_a @ ( pre_ta4931606617599662728t_unit @ g ) @ ( pre_ar1395965042833527383t_unit @ g ) ) @ ( image_b_a @ ( pre_he5236287464308401016t_unit @ g ) @ ( pre_ar1395965042833527383t_unit @ g ) ) ) ) ) ) ).
% connected_verts
thf(fact_1184_verts__add__arc__conv,axiom,
! [A: b] :
( ( pre_ve642382030648772252t_unit @ ( pre_add_arc_a_b @ g @ A ) )
= ( sup_sup_set_a @ ( pre_ve642382030648772252t_unit @ g ) @ ( insert_a @ ( pre_ta4931606617599662728t_unit @ g @ A ) @ ( insert_a @ ( pre_he5236287464308401016t_unit @ g @ A ) @ bot_bot_set_a ) ) ) ) ).
% verts_add_arc_conv
thf(fact_1185_bidirected__digraphI,axiom,
! [Arev: b > b] :
( ! [A3: b] :
( ~ ( member_b @ A3 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( Arev @ A3 )
= A3 ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( Arev @ A3 )
!= A3 ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( Arev @ ( Arev @ A3 ) )
= A3 ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( pre_ta4931606617599662728t_unit @ g @ ( Arev @ A3 ) )
= ( pre_he5236287464308401016t_unit @ g @ A3 ) ) )
=> ( bidire6463457107099887885ph_a_b @ g @ Arev ) ) ) ) ) ).
% bidirected_digraphI
thf(fact_1186_in__degree__del__arc__iff,axiom,
! [A: b,U: a] :
( ( ( ( ( pre_he5236287464308401016t_unit @ g @ A )
= U )
& ( member_b @ A @ ( pre_ar1395965042833527383t_unit @ g ) ) )
=> ( ( in_degree_a_b @ ( pre_del_arc_a_b @ g @ A ) @ U )
= ( minus_minus_nat @ ( in_degree_a_b @ g @ U ) @ one_one_nat ) ) )
& ( ~ ( ( ( pre_he5236287464308401016t_unit @ g @ A )
= U )
& ( member_b @ A @ ( pre_ar1395965042833527383t_unit @ g ) ) )
=> ( ( in_degree_a_b @ ( pre_del_arc_a_b @ g @ A ) @ U )
= ( in_degree_a_b @ g @ U ) ) ) ) ).
% in_degree_del_arc_iff
thf(fact_1187_not__elem__in__0,axiom,
! [V: a] :
( ~ ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( in_degree_a_b @ g @ V )
= zero_zero_nat ) ) ).
% not_elem_in_0
thf(fact_1188_in__degree__add__arc__iff,axiom,
! [A: b,U: a] :
( ( ( ( ( pre_he5236287464308401016t_unit @ g @ A )
= U )
& ~ ( member_b @ A @ ( pre_ar1395965042833527383t_unit @ g ) ) )
=> ( ( in_degree_a_b @ ( pre_add_arc_a_b @ g @ A ) @ U )
= ( plus_plus_nat @ ( in_degree_a_b @ g @ U ) @ one_one_nat ) ) )
& ( ~ ( ( ( pre_he5236287464308401016t_unit @ g @ A )
= U )
& ~ ( member_b @ A @ ( pre_ar1395965042833527383t_unit @ g ) ) )
=> ( ( in_degree_a_b @ ( pre_add_arc_a_b @ g @ A ) @ U )
= ( in_degree_a_b @ g @ U ) ) ) ) ).
% in_degree_add_arc_iff
thf(fact_1189_out__degree__del__arc__iff,axiom,
! [A: b,U: a] :
( ( ( ( ( pre_ta4931606617599662728t_unit @ g @ A )
= U )
& ( member_b @ A @ ( pre_ar1395965042833527383t_unit @ g ) ) )
=> ( ( out_degree_a_b @ ( pre_del_arc_a_b @ g @ A ) @ U )
= ( minus_minus_nat @ ( out_degree_a_b @ g @ U ) @ one_one_nat ) ) )
& ( ~ ( ( ( pre_ta4931606617599662728t_unit @ g @ A )
= U )
& ( member_b @ A @ ( pre_ar1395965042833527383t_unit @ g ) ) )
=> ( ( out_degree_a_b @ ( pre_del_arc_a_b @ g @ A ) @ U )
= ( out_degree_a_b @ g @ U ) ) ) ) ).
% out_degree_del_arc_iff
thf(fact_1190_not__elem__out__0,axiom,
! [V: a] :
( ~ ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( out_degree_a_b @ g @ V )
= zero_zero_nat ) ) ).
% not_elem_out_0
thf(fact_1191_closed__euler__imp__eq__degree,axiom,
! [U: a,P2: list_b,V: a] :
( ( pre_euler_trail_a_b @ g @ U @ P2 @ U )
=> ( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( in_degree_a_b @ g @ V )
= ( out_degree_a_b @ g @ V ) ) ) ) ).
% closed_euler_imp_eq_degree
thf(fact_1192_closed__euler2_I2_J,axiom,
! [Ua: a,P2: list_b,U: a] :
( ( pre_euler_trail_a_b @ g @ Ua @ P2 @ Ua )
=> ( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( in_degree_a_b @ g @ U )
= ( out_degree_a_b @ g @ U ) ) ) ) ).
% closed_euler2(2)
thf(fact_1193_closed__euler,axiom,
( ( ? [U3: a,P3: list_b] : ( pre_euler_trail_a_b @ g @ U3 @ P3 @ U3 ) )
= ( ( digrap8783888973171253482ed_a_b @ g )
& ! [X2: a] :
( ( member_a @ X2 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( in_degree_a_b @ g @ X2 )
= ( out_degree_a_b @ g @ X2 ) ) ) ) ) ).
% closed_euler
thf(fact_1194_closed__euler1,axiom,
( ( digrap8783888973171253482ed_a_b @ g )
=> ( ! [U4: a] :
( ( member_a @ U4 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( in_degree_a_b @ g @ U4 )
= ( out_degree_a_b @ g @ U4 ) ) )
=> ? [U4: a,P4: list_b] : ( pre_euler_trail_a_b @ g @ U4 @ P4 @ U4 ) ) ) ).
% closed_euler1
thf(fact_1195_out__degree__add__arc__iff,axiom,
! [A: b,U: a] :
( ( ( ( ( pre_ta4931606617599662728t_unit @ g @ A )
= U )
& ~ ( member_b @ A @ ( pre_ar1395965042833527383t_unit @ g ) ) )
=> ( ( out_degree_a_b @ ( pre_add_arc_a_b @ g @ A ) @ U )
= ( plus_plus_nat @ ( out_degree_a_b @ g @ U ) @ one_one_nat ) ) )
& ( ~ ( ( ( pre_ta4931606617599662728t_unit @ g @ A )
= U )
& ~ ( member_b @ A @ ( pre_ar1395965042833527383t_unit @ g ) ) )
=> ( ( out_degree_a_b @ ( pre_add_arc_a_b @ g @ A ) @ U )
= ( out_degree_a_b @ g @ U ) ) ) ) ).
% out_degree_add_arc_iff
thf(fact_1196_open__euler2,axiom,
! [U: a,P2: list_b,V: a] :
( ( pre_euler_trail_a_b @ g @ U @ P2 @ V )
=> ( ( U != V )
=> ( ( digrap8783888973171253482ed_a_b @ g )
& ! [X5: a] :
( ( member_a @ X5 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( U != X5 )
=> ( ( V != X5 )
=> ( ( in_degree_a_b @ g @ X5 )
= ( out_degree_a_b @ g @ X5 ) ) ) ) )
& ( ( plus_plus_nat @ ( in_degree_a_b @ g @ U ) @ one_one_nat )
= ( out_degree_a_b @ g @ U ) )
& ( ( plus_plus_nat @ ( out_degree_a_b @ g @ V ) @ one_one_nat )
= ( in_degree_a_b @ g @ V ) ) ) ) ) ).
% open_euler2
thf(fact_1197_open__euler1,axiom,
! [U: a,V: a] :
( ( digrap8783888973171253482ed_a_b @ g )
=> ( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ! [W2: a] :
( ( member_a @ W2 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( U != W2 )
=> ( ( V != W2 )
=> ( ( in_degree_a_b @ g @ W2 )
= ( out_degree_a_b @ g @ W2 ) ) ) ) )
=> ( ( ( plus_plus_nat @ ( in_degree_a_b @ g @ U ) @ one_one_nat )
= ( out_degree_a_b @ g @ U ) )
=> ( ( ( plus_plus_nat @ ( out_degree_a_b @ g @ V ) @ one_one_nat )
= ( in_degree_a_b @ g @ V ) )
=> ? [P4: list_b] : ( pre_euler_trail_a_b @ g @ U @ P4 @ V ) ) ) ) ) ) ) ).
% open_euler1
thf(fact_1198_open__euler,axiom,
( ( ? [U3: a,P3: list_b,V3: a] :
( ( pre_euler_trail_a_b @ g @ U3 @ P3 @ V3 )
& ( U3 != V3 ) ) )
= ( ( digrap8783888973171253482ed_a_b @ g )
& ? [U3: a] :
( ( member_a @ U3 @ ( pre_ve642382030648772252t_unit @ g ) )
& ? [V3: a] :
( ( member_a @ V3 @ ( pre_ve642382030648772252t_unit @ g ) )
& ! [X2: a] :
( ( member_a @ X2 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( U3 != X2 )
=> ( ( V3 != X2 )
=> ( ( in_degree_a_b @ g @ X2 )
= ( out_degree_a_b @ g @ X2 ) ) ) ) )
& ( ( plus_plus_nat @ ( in_degree_a_b @ g @ U3 ) @ one_one_nat )
= ( out_degree_a_b @ g @ U3 ) )
& ( ( plus_plus_nat @ ( out_degree_a_b @ g @ V3 ) @ one_one_nat )
= ( in_degree_a_b @ g @ V3 ) ) ) ) ) ) ).
% open_euler
thf(fact_1199_arc__set__balanced__all,axiom,
! [U: a,V: a] :
( ( pre_ar5931435604406180204ed_a_b @ g @ U @ ( pre_ar1395965042833527383t_unit @ g ) @ V )
= ( ( ( U = V )
=> ! [X2: a] :
( ( member_a @ X2 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( in_degree_a_b @ g @ X2 )
= ( out_degree_a_b @ g @ X2 ) ) ) )
& ( ( U != V )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( ( X2 != U )
& ( X2 != V ) )
=> ( ( in_degree_a_b @ g @ X2 )
= ( out_degree_a_b @ g @ X2 ) ) ) )
& ( ( plus_plus_nat @ ( in_degree_a_b @ g @ U ) @ one_one_nat )
= ( out_degree_a_b @ g @ U ) )
& ( ( plus_plus_nat @ ( out_degree_a_b @ g @ V ) @ one_one_nat )
= ( in_degree_a_b @ g @ V ) ) ) ) ) ) ).
% arc_set_balanced_all
thf(fact_1200_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1201_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1202_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1203_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1204_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1205_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1206_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1207_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1208_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1209_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1210_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1211_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1212_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1213_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1214_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_1215_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1216_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1217_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1218_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1219_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1220_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_1221_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1222_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1223_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1224_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1225_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1226_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1227_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1228_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1229_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1230_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1231_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_1232_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1233_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1234_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1235_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1236_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1237_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1238_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1239_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1240_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1241_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1242_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1243_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1244_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1245_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1246_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1247_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1248_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1249_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1250_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1251_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1252_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1253_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1254_out__degree__0__only__self,axiom,
! [V: a,X: a] :
( ( finite_finite_b @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ( ( out_degree_a_b @ g @ V )
= zero_zero_nat )
=> ( ( reachable_a_b @ g @ V @ X )
=> ( X = V ) ) ) ) ).
% out_degree_0_only_self
thf(fact_1255_symmetric__reachable_H,axiom,
! [V: a,W: a] :
( ( reachable_a_b @ g @ V @ W )
=> ( reachable_a_b @ g @ W @ V ) ) ).
% symmetric_reachable'
thf(fact_1256_reachable__trans,axiom,
! [U: a,V: a,W: a] :
( ( reachable_a_b @ g @ U @ V )
=> ( ( reachable_a_b @ g @ V @ W )
=> ( reachable_a_b @ g @ U @ W ) ) ) ).
% reachable_trans
thf(fact_1257_reachable__in__verts_I1_J,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ g @ U @ V )
=> ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% reachable_in_verts(1)
thf(fact_1258_reachable__in__verts_I2_J,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ g @ U @ V )
=> ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% reachable_in_verts(2)
thf(fact_1259_scc__ofI__reachable_H,axiom,
! [V: a,U: a] :
( ( reachable_a_b @ g @ V @ U )
=> ( member_a @ U @ ( digrap2937667069914300949of_a_b @ g @ V ) ) ) ).
% scc_ofI_reachable'
thf(fact_1260_scc__ofI__reachable,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ g @ U @ V )
=> ( member_a @ U @ ( digrap2937667069914300949of_a_b @ g @ V ) ) ) ).
% scc_ofI_reachable
thf(fact_1261_k__nh__reachable,axiom,
! [U: a,W: b > real,V: a,K: real] :
( ( member_a @ U @ ( graph_3921080825633621230od_a_b @ g @ W @ V @ K ) )
=> ( reachable_a_b @ g @ V @ U ) ) ).
% k_nh_reachable
thf(fact_1262_in__sccs__verts__conv__reachable,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ g ) )
= ( ( S != bot_bot_set_a )
& ! [X2: a] :
( ( member_a @ X2 @ S )
=> ! [Y2: a] :
( ( member_a @ Y2 @ S )
=> ( reachable_a_b @ g @ X2 @ Y2 ) ) )
& ! [X2: a] :
( ( member_a @ X2 @ S )
=> ! [V3: a] :
( ~ ( member_a @ V3 @ S )
=> ( ~ ( reachable_a_b @ g @ X2 @ V3 )
| ~ ( reachable_a_b @ g @ V3 @ X2 ) ) ) ) ) ) ).
% in_sccs_verts_conv_reachable
thf(fact_1263_reachable__induce__ss,axiom,
! [S: set_a,U: a,V: a,T2: set_a] :
( ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ g @ S ) @ U @ V )
=> ( ( ord_less_eq_set_a @ S @ T2 )
=> ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ g @ T2 ) @ U @ V ) ) ) ).
% reachable_induce_ss
thf(fact_1264_last__branch__alt,axiom,
! [X: a] :
( ( member_a @ X @ ( graph_1747835947655717337ts_a_b @ g ) )
=> ! [Z4: a] :
( ( ( reachable_a_b @ g @ X @ Z4 )
& ( Z4 != X ) )
=> ~ ( member_a @ Z4 @ ( graph_4596510882073158607ts_a_b @ g ) ) ) ) ).
% last_branch_alt
thf(fact_1265_last__merge__alt,axiom,
! [X: a] :
( ( member_a @ X @ ( graph_2659413520663303054ts_a_b @ g ) )
=> ! [Z4: a] :
( ( ( reachable_a_b @ g @ X @ Z4 )
& ( Z4 != X ) )
=> ~ ( member_a @ Z4 @ ( graph_2957805489637798020ts_a_b @ g ) ) ) ) ).
% last_merge_alt
thf(fact_1266_connected__iff__reachable,axiom,
( ( digrap8783888973171253482ed_a_b @ g )
= ( ! [X2: a] :
( ( member_a @ X2 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ! [Y2: a] :
( ( member_a @ Y2 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( reachable_a_b @ g @ X2 @ Y2 ) ) )
& ( ( pre_ve642382030648772252t_unit @ g )
!= bot_bot_set_a ) ) ) ).
% connected_iff_reachable
thf(fact_1267_verts__reachable__connected,axiom,
( ( ( pre_ve642382030648772252t_unit @ g )
!= bot_bot_set_a )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ! [Xa2: a] :
( ( member_a @ Xa2 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( reachable_a_b @ g @ X3 @ Xa2 ) ) )
=> ( digrap8783888973171253482ed_a_b @ g ) ) ) ).
% verts_reachable_connected
thf(fact_1268_reachable__induce__subgraphD,axiom,
! [S: set_a,U: a,V: a] :
( ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ g @ S ) @ U @ V )
=> ( ( ord_less_eq_set_a @ S @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( reachable_a_b @ g @ U @ V ) ) ) ).
% reachable_induce_subgraphD
thf(fact_1269_nearest__vert__reachable,axiom,
! [U: a,U2: set_a,W: b > real] :
( ( ( graph_2016941059203891550ts_a_b @ g @ U @ U2 )
!= bot_bot_set_a )
=> ( reachable_a_b @ g @ U @ ( graph_3614428260325061028rt_a_b @ g @ W @ U @ U2 ) ) ) ).
% nearest_vert_reachable
thf(fact_1270_reachable__refl,axiom,
! [V: a] :
( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( reachable_a_b @ g @ V @ V ) ) ).
% reachable_refl
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_real @ ( query_remove_sel_a_b @ g @ sel @ x @ e ) @ one_one_real ).
%------------------------------------------------------------------------------