TPTP Problem File: SLH0038^1.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/0008_Directed_Tree_Additions/prob_00806_035514__15038130_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1461 ( 607 unt; 186 typ; 0 def)
% Number of atoms : 3304 (1275 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 10300 ( 419 ~; 40 |; 259 &;8323 @)
% ( 0 <=>;1259 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 25 ( 24 usr)
% Number of type conns : 615 ( 615 >; 0 *; 0 +; 0 <<)
% Number of symbols : 165 ( 162 usr; 15 con; 0-5 aty)
% Number of variables : 3153 ( 112 ^;2931 !; 110 ?;3153 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:05:59.014
%------------------------------------------------------------------------------
% Could-be-implicit typings (24)
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% Explicit typings (162)
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Real__Oreal,type,
inf_inf_real: real > real > real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J_J,type,
inf_in1092213268631476299t_unit: set_pr5411798346947241657t_unit > set_pr5411798346947241657t_unit > set_pr5411798346947241657t_unit ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
inf_inf_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__b_J,type,
inf_inf_set_b: set_b > set_b > set_b ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
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__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__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__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
ord_less_eq_set_b: set_b > set_b > $o ).
thf(sy_c_Set_OCollect_001t__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J,type,
collec8000012497822511960t_unit: ( pre_pr7278220950009878019t_unit > $o ) > set_pr5411798346947241657t_unit ).
thf(sy_c_Set_OCollect_001t__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__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_It__Set__Oset_Itf__a_J_J_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
image_1042221919965026181_set_a: ( set_set_a > set_set_a ) > set_set_set_a > set_set_set_a ).
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__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__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__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Oinsert_001tf__b,type,
insert_b: b > set_b > set_b ).
thf(sy_c_Shortest__Path_Owf__digraph_O_092_060mu_062_001tf__a_001tf__b,type,
shortest_wf_mu_a_b: pre_pr7278220950009878019t_unit > ( b > real ) > a > a > extended_ereal ).
thf(sy_c_Shortest__Path__Tree_Opre__digraph_Oleaf_001t__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J_001tf__a,type,
shorte2370036064065676172unit_a: pre_pr4248724542568774193t_unit > pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Shortest__Path__Tree_Opre__digraph_Oleaf_001t__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J_001tf__b,type,
shorte2370036064065676173unit_b: pre_pr8199616178187362672t_unit > pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Shortest__Path__Tree_Opre__digraph_Oleaf_001t__Set__Oset_Itf__a_J_001tf__a,type,
shorte1274571178419068173et_a_a: pre_pr3647964229410195492t_unit > set_a > $o ).
thf(sy_c_Shortest__Path__Tree_Opre__digraph_Oleaf_001t__Set__Oset_Itf__a_J_001tf__b,type,
shorte1274571178419068174et_a_b: pre_pr7598855865028783971t_unit > set_a > $o ).
thf(sy_c_Shortest__Path__Tree_Opre__digraph_Oleaf_001tf__a_001t__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J,type,
shorte3113631386427947092t_unit: pre_pr4380274127475126449t_unit > a > $o ).
thf(sy_c_Shortest__Path__Tree_Opre__digraph_Oleaf_001tf__a_001tf__a,type,
shorte1213025427933718125af_a_a: pre_pr3327329314391289540t_unit > a > $o ).
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_Opre__digraph_Oleaf_001tf__b_001t__Digraph__Opre____digraph__Opre____digraph____ext_Itf__a_Mtf__b_Mt__Product____Type__Ounit_J,type,
shorte1834455252847456213t_unit: pre_pr4845345393412534256t_unit > b > $o ).
thf(sy_c_Shortest__Path__Tree_Opre__digraph_Oleaf_001tf__b_001tf__a,type,
shorte7648941882815817900af_b_a: pre_pr3994228789931197893t_unit > b > $o ).
thf(sy_c_Shortest__Path__Tree_Opre__digraph_Oleaf_001tf__b_001tf__b,type,
shorte7648941882815817901af_b_b: pre_pr7945120425549786372t_unit > b > $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__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_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $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 ).
% Relevant facts (1271)
thf(fact_0_card__gt0,axiom,
ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( pre_ve642382030648772252t_unit @ g ) ) ).
% card_gt0
thf(fact_1_pseudo__graph__axioms,axiom,
pseudo_graph_a_b @ g ).
% pseudo_graph_axioms
thf(fact_2_loopfree__digraph__axioms,axiom,
loopfree_digraph_a_b @ g ).
% loopfree_digraph_axioms
thf(fact_3_nomulti__digraph__axioms,axiom,
nomulti_digraph_a_b @ g ).
% nomulti_digraph_axioms
thf(fact_4_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_5_digraph__axioms,axiom,
digraph_a_b @ g ).
% digraph_axioms
thf(fact_6_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_7_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_8_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_9_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_10_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_11_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_12_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_13_sym__digraph__axioms,axiom,
sym_digraph_a_b @ g ).
% sym_digraph_axioms
thf(fact_14_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_15_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_16_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_17_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_18_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_19_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_20_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_21_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_22_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_23_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_24_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_25_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_26_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_27_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_28_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_29_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_30_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_31_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_32_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_33_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_34_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_35_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_36_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_37_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_38_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_39_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_40_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_41_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_42_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_43_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_44_mem__Collect__eq,axiom,
! [A: b,P: b > $o] :
( ( member_b @ A @ ( collect_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ( collec8000012497822511960t_unit
@ ^ [X3: pre_pr7278220950009878019t_unit] : ( member6939884229742472986t_unit @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A2: set_b] :
( ( collect_b
@ ^ [X3: b] : ( member_b @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_49_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_50_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_51_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_52_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_53_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_54_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_55_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_56_lift__Suc__mono__less__iff,axiom,
! [F: nat > set_a,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_set_a @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_set_a @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_57_lift__Suc__mono__less__iff,axiom,
! [F: nat > extended_ereal,N: nat,M: nat] :
( ! [N2: nat] : ( ord_le1188267648640031866_ereal @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_le1188267648640031866_ereal @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_58_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_59_lift__Suc__mono__less,axiom,
! [F: nat > set_a,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_set_a @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_set_a @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_60_lift__Suc__mono__less,axiom,
! [F: nat > extended_ereal,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_le1188267648640031866_ereal @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_le1188267648640031866_ereal @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_61_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J: nat] :
( ( M
= ( suc @ J ) )
& ( ord_less_nat @ J @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_62_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_63_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_64_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_65_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_66_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_67_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_68_strict__inc__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ! [I3: nat] :
( ( J2
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_69_less__Suc__induct,axiom,
! [I2: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ K2 )
=> ( ( P @ I3 @ J3 )
=> ( ( P @ J3 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I2 @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_70_less__trans__Suc,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_71_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_72_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_73_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_74_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ I ) ) ) ) ).
% All_less_Suc
thf(fact_75_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_76_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_77_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ N )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_78_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_79_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_80_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_81_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_82_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_83_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_84_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_85_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_86_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_87_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_88_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_89_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X4: nat,Y3: nat] :
( ( P @ X4 @ Y3 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_90_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_91_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_92_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_93_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_94_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_95_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_96_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_97_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_98_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_99_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_100_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_101_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_102_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_103_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_104_diff__commute,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_105_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_106_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_107_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_108_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_109_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_110_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I2: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_111_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_112_merge__in__supergraph,axiom,
! [C: pre_pr7278220950009878019t_unit,X: a] :
( ( shorte3657265928840388360ph_a_b @ C @ g )
=> ( ( member_a @ X @ ( graph_2957805489637798020ts_a_b @ C ) )
=> ( member_a @ X @ ( graph_2957805489637798020ts_a_b @ g ) ) ) ) ).
% merge_in_supergraph
thf(fact_113_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_114_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_115_branch__in__supergraph,axiom,
! [C: pre_pr7278220950009878019t_unit,X: a] :
( ( shorte3657265928840388360ph_a_b @ C @ g )
=> ( ( member_a @ X @ ( graph_4596510882073158607ts_a_b @ C ) )
=> ( member_a @ X @ ( graph_4596510882073158607ts_a_b @ g ) ) ) ) ).
% branch_in_supergraph
thf(fact_116_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_117_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_118_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_119_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_120_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_121_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_122_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_123_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_124_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_125_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_126_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_127_zero__reorient,axiom,
! [X: extended_ereal] :
( ( zero_z2744965634713055877_ereal = X )
= ( X = zero_z2744965634713055877_ereal ) ) ).
% zero_reorient
thf(fact_128_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_129_one__reorient,axiom,
! [X: extended_ereal] :
( ( one_on4623092294121504201_ereal = X )
= ( X = one_on4623092294121504201_ereal ) ) ).
% one_reorient
thf(fact_130_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_131_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_132_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_133_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_134_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_135_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_136_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_137_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [A3: real,B2: real] :
( ( minus_minus_real @ A3 @ B2 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_138_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_139_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_140_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_141_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_142_digraphI__induced,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ H @ g )
=> ( digraph_a_b @ H ) ) ).
% digraphI_induced
thf(fact_143_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_144_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_145_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_146_sccs__verts__subsets,axiom,
! [S2: set_a] :
( ( member_set_a @ S2 @ ( digrap2871191568752656621ts_a_b @ g ) )
=> ( ord_less_eq_set_a @ S2 @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% sccs_verts_subsets
thf(fact_147_last__merge__alt,axiom,
! [X: a] :
( ( member_a @ X @ ( graph_2659413520663303054ts_a_b @ g ) )
=> ! [Z2: a] :
( ( ( reachable_a_b @ g @ X @ Z2 )
& ( Z2 != X ) )
=> ~ ( member_a @ Z2 @ ( graph_2957805489637798020ts_a_b @ g ) ) ) ) ).
% last_merge_alt
thf(fact_148_last__branch__alt,axiom,
! [X: a] :
( ( member_a @ X @ ( graph_1747835947655717337ts_a_b @ g ) )
=> ! [Z2: a] :
( ( ( reachable_a_b @ g @ X @ Z2 )
& ( Z2 != X ) )
=> ~ ( member_a @ Z2 @ ( graph_4596510882073158607ts_a_b @ g ) ) ) ) ).
% last_branch_alt
thf(fact_149_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_150_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_151_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_152_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_153_symmetric__reachable_H,axiom,
! [V: a,W: a] :
( ( reachable_a_b @ g @ V @ W )
=> ( reachable_a_b @ g @ W @ V ) ) ).
% symmetric_reachable'
thf(fact_154_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_155_induced__subgraph__refl,axiom,
digrap5251062021860773499ph_a_b @ g @ g ).
% induced_subgraph_refl
thf(fact_156_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_157_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_158_verts__nempty,axiom,
( ( pre_ve642382030648772252t_unit @ g )
!= bot_bot_set_a ) ).
% verts_nempty
thf(fact_159_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_160_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_161_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_162_in__sccs__verts__conv__reachable,axiom,
! [S2: set_a] :
( ( member_set_a @ S2 @ ( digrap2871191568752656621ts_a_b @ g ) )
= ( ( S2 != bot_bot_set_a )
& ! [X3: a] :
( ( member_a @ X3 @ S2 )
=> ! [Y5: a] :
( ( member_a @ Y5 @ S2 )
=> ( reachable_a_b @ g @ X3 @ Y5 ) ) )
& ! [X3: a] :
( ( member_a @ X3 @ S2 )
=> ! [V2: a] :
( ~ ( member_a @ V2 @ S2 )
=> ( ~ ( reachable_a_b @ g @ X3 @ V2 )
| ~ ( reachable_a_b @ g @ V2 @ X3 ) ) ) ) ) ) ).
% in_sccs_verts_conv_reachable
thf(fact_163_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_164_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_165_reachable__refl,axiom,
! [V: a] :
( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( reachable_a_b @ g @ V @ V ) ) ).
% reachable_refl
thf(fact_166_is__chain__def,axiom,
( ( graph_3890552050688490787in_a_b @ g )
= ( ( graph_4596510882073158607ts_a_b @ g )
= bot_bot_set_a ) ) ).
% is_chain_def
thf(fact_167_is__chain_H__def,axiom,
( ( graph_8150681439568091980in_a_b @ g )
= ( ( graph_2957805489637798020ts_a_b @ g )
= bot_bot_set_a ) ) ).
% is_chain'_def
thf(fact_168_lift__Suc__antimono__le,axiom,
! [F: nat > real,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_169_lift__Suc__antimono__le,axiom,
! [F: nat > set_pr5411798346947241657t_unit,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_le8200006823705900825t_unit @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le8200006823705900825t_unit @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_170_lift__Suc__antimono__le,axiom,
! [F: nat > set_b,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_b @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_b @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_171_lift__Suc__antimono__le,axiom,
! [F: nat > set_a,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_a @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_172_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_173_lift__Suc__mono__le,axiom,
! [F: nat > real,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_174_lift__Suc__mono__le,axiom,
! [F: nat > set_pr5411798346947241657t_unit,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_le8200006823705900825t_unit @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le8200006823705900825t_unit @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_175_lift__Suc__mono__le,axiom,
! [F: nat > set_b,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_b @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_b @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_176_lift__Suc__mono__le,axiom,
! [F: nat > set_a,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_a @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_177_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_178_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_179_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_180_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_181_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_182_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_183_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C2: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C2 @ D ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C2 @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_184_diff__right__mono,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ C2 ) ) ) ).
% diff_right_mono
thf(fact_185_diff__left__mono,axiom,
! [B: real,A: real,C2: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C2 @ A ) @ ( minus_minus_real @ C2 @ B ) ) ) ).
% diff_left_mono
thf(fact_186_diff__mono,axiom,
! [A: real,B: real,D: real,C2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D @ C2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_187_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_188_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_189_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_190_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_191_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_192_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_193_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_194_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_195_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_196_zero__neq__one,axiom,
zero_z2744965634713055877_ereal != one_on4623092294121504201_ereal ).
% zero_neq_one
thf(fact_197_card_Oempty,axiom,
( ( finite_card_set_a @ bot_bot_set_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_198_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_199_card_Oempty,axiom,
( ( finite_card_b @ bot_bot_set_b )
= zero_zero_nat ) ).
% card.empty
thf(fact_200_card_Oempty,axiom,
( ( finite2416455745057997787t_unit @ bot_bo1839476491465656141t_unit )
= zero_zero_nat ) ).
% card.empty
thf(fact_201_Diff__eq__empty__iff,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ( minus_3777555517894451474t_unit @ A2 @ B3 )
= bot_bo1839476491465656141t_unit )
= ( ord_le8200006823705900825t_unit @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_202_Diff__eq__empty__iff,axiom,
! [A2: set_b,B3: set_b] :
( ( ( minus_minus_set_b @ A2 @ B3 )
= bot_bot_set_b )
= ( ord_less_eq_set_b @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_203_Diff__eq__empty__iff,axiom,
! [A2: set_a,B3: set_a] :
( ( ( minus_minus_set_a @ A2 @ B3 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_204_subset__empty,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ bot_bo1839476491465656141t_unit )
= ( A2 = bot_bo1839476491465656141t_unit ) ) ).
% subset_empty
thf(fact_205_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_206_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_207_empty__subsetI,axiom,
! [A2: set_pr5411798346947241657t_unit] : ( ord_le8200006823705900825t_unit @ bot_bo1839476491465656141t_unit @ A2 ) ).
% empty_subsetI
thf(fact_208_empty__subsetI,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A2 ) ).
% empty_subsetI
thf(fact_209_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_210_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_211_image__empty,axiom,
! [F: a > b] :
( ( image_a_b @ F @ bot_bot_set_a )
= bot_bot_set_b ) ).
% image_empty
thf(fact_212_image__empty,axiom,
! [F: b > a] :
( ( image_b_a @ F @ bot_bot_set_b )
= bot_bot_set_a ) ).
% image_empty
thf(fact_213_image__empty,axiom,
! [F: b > b] :
( ( image_b_b @ F @ bot_bot_set_b )
= bot_bot_set_b ) ).
% image_empty
thf(fact_214_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_215_image__empty,axiom,
! [F: a > pre_pr7278220950009878019t_unit] :
( ( image_5713294457175270716t_unit @ F @ bot_bot_set_a )
= bot_bo1839476491465656141t_unit ) ).
% image_empty
thf(fact_216_image__empty,axiom,
! [F: b > pre_pr7278220950009878019t_unit] :
( ( image_4434118323594779837t_unit @ F @ bot_bot_set_b )
= bot_bo1839476491465656141t_unit ) ).
% image_empty
thf(fact_217_image__empty,axiom,
! [F: pre_pr7278220950009878019t_unit > a] :
( ( image_4969699134812999796unit_a @ F @ bot_bo1839476491465656141t_unit )
= bot_bot_set_a ) ).
% image_empty
thf(fact_218_image__empty,axiom,
! [F: pre_pr7278220950009878019t_unit > b] :
( ( image_4969699134812999797unit_b @ F @ bot_bo1839476491465656141t_unit )
= bot_bot_set_b ) ).
% image_empty
thf(fact_219_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_220_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_221_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_222_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_223_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_224_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_225_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_226_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_227_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_228_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_229_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_230_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_231_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_232_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_233_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_234_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_235_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_236_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_237_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_238_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_239_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_240_reachable__induce__subgraphD,axiom,
! [S2: set_a,U: a,V: a] :
( ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ g @ S2 ) @ U @ V )
=> ( ( ord_less_eq_set_a @ S2 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( reachable_a_b @ g @ U @ V ) ) ) ).
% reachable_induce_subgraphD
thf(fact_241_k__nh__subs__nnvs,axiom,
! [W: b > real,U: a,N: nat,U2: set_a,K: real] :
( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ U2 )
=> ( ( ord_less_nat @ ( finite_card_a @ ( graph_3921080825633621230od_a_b @ g @ W @ U @ K ) ) @ ( finite_card_a @ U2 ) )
=> ( ord_less_eq_set_a @ ( graph_3921080825633621230od_a_b @ g @ W @ U @ K ) @ U2 ) ) ) ).
% k_nh_subs_nnvs
thf(fact_242_nnvs__mem,axiom,
! [W: b > real,U: a,N: nat,U2: set_a] :
( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ U2 )
=> ( member_a @ U @ U2 ) ) ).
% nnvs_mem
thf(fact_243_source__mem__nnvs,axiom,
! [W: b > real,U: a,N: nat,U2: set_a] :
( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ U2 )
=> ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% source_mem_nnvs
thf(fact_244_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_245_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_246_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_247_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_248_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_249_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_250_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_251_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_252_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_253_image__eqI,axiom,
! [B: pre_pr7278220950009878019t_unit,F: a > pre_pr7278220950009878019t_unit,X: a,A2: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member6939884229742472986t_unit @ B @ ( image_5713294457175270716t_unit @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_254_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_255_empty__Collect__eq,axiom,
! [P: b > $o] :
( ( bot_bot_set_b
= ( collect_b @ P ) )
= ( ! [X3: b] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_256_empty__Collect__eq,axiom,
! [P: pre_pr7278220950009878019t_unit > $o] :
( ( bot_bo1839476491465656141t_unit
= ( collec8000012497822511960t_unit @ P ) )
= ( ! [X3: pre_pr7278220950009878019t_unit] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_257_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_258_Collect__empty__eq,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( ! [X3: b] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_259_Collect__empty__eq,axiom,
! [P: pre_pr7278220950009878019t_unit > $o] :
( ( ( collec8000012497822511960t_unit @ P )
= bot_bo1839476491465656141t_unit )
= ( ! [X3: pre_pr7278220950009878019t_unit] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_260_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_261_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_262_all__not__in__conv,axiom,
! [A2: set_b] :
( ( ! [X3: b] :
~ ( member_b @ X3 @ A2 ) )
= ( A2 = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_263_all__not__in__conv,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ( ! [X3: pre_pr7278220950009878019t_unit] :
~ ( member6939884229742472986t_unit @ X3 @ A2 ) )
= ( A2 = bot_bo1839476491465656141t_unit ) ) ).
% all_not_in_conv
thf(fact_264_empty__iff,axiom,
! [C2: set_a] :
~ ( member_set_a @ C2 @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_265_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_266_empty__iff,axiom,
! [C2: b] :
~ ( member_b @ C2 @ bot_bot_set_b ) ).
% empty_iff
thf(fact_267_empty__iff,axiom,
! [C2: pre_pr7278220950009878019t_unit] :
~ ( member6939884229742472986t_unit @ C2 @ bot_bo1839476491465656141t_unit ) ).
% empty_iff
thf(fact_268_subset__antisym,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ B3 )
=> ( ( ord_le8200006823705900825t_unit @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_269_subset__antisym,axiom,
! [A2: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ( ord_less_eq_set_b @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_270_subset__antisym,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_271_subsetI,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ( member_set_a @ X4 @ B3 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_272_subsetI,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ! [X4: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X4 @ A2 )
=> ( member6939884229742472986t_unit @ X4 @ B3 ) )
=> ( ord_le8200006823705900825t_unit @ A2 @ B3 ) ) ).
% subsetI
thf(fact_273_subsetI,axiom,
! [A2: set_b,B3: set_b] :
( ! [X4: b] :
( ( member_b @ X4 @ A2 )
=> ( member_b @ X4 @ B3 ) )
=> ( ord_less_eq_set_b @ A2 @ B3 ) ) ).
% subsetI
thf(fact_274_subsetI,axiom,
! [A2: set_a,B3: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( member_a @ X4 @ B3 ) )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_275_psubsetI,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_le2693654750756130573t_unit @ A2 @ B3 ) ) ) ).
% psubsetI
thf(fact_276_psubsetI,axiom,
! [A2: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_set_b @ A2 @ B3 ) ) ) ).
% psubsetI
thf(fact_277_psubsetI,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_set_a @ A2 @ B3 ) ) ) ).
% psubsetI
thf(fact_278_DiffI,axiom,
! [C2: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C2 @ A2 )
=> ( ~ ( member_set_a @ C2 @ B3 )
=> ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_279_DiffI,axiom,
! [C2: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C2 @ A2 )
=> ( ~ ( member6939884229742472986t_unit @ C2 @ B3 )
=> ( member6939884229742472986t_unit @ C2 @ ( minus_3777555517894451474t_unit @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_280_DiffI,axiom,
! [C2: a,A2: set_a,B3: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ~ ( member_a @ C2 @ B3 )
=> ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_281_DiffI,axiom,
! [C2: b,A2: set_b,B3: set_b] :
( ( member_b @ C2 @ A2 )
=> ( ~ ( member_b @ C2 @ B3 )
=> ( member_b @ C2 @ ( minus_minus_set_b @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_282_Diff__iff,axiom,
! [C2: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) )
= ( ( member_set_a @ C2 @ A2 )
& ~ ( member_set_a @ C2 @ B3 ) ) ) ).
% Diff_iff
thf(fact_283_Diff__iff,axiom,
! [C2: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( minus_3777555517894451474t_unit @ A2 @ B3 ) )
= ( ( member6939884229742472986t_unit @ C2 @ A2 )
& ~ ( member6939884229742472986t_unit @ C2 @ B3 ) ) ) ).
% Diff_iff
thf(fact_284_Diff__iff,axiom,
! [C2: a,A2: set_a,B3: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B3 ) )
= ( ( member_a @ C2 @ A2 )
& ~ ( member_a @ C2 @ B3 ) ) ) ).
% Diff_iff
thf(fact_285_Diff__iff,axiom,
! [C2: b,A2: set_b,B3: set_b] :
( ( member_b @ C2 @ ( minus_minus_set_b @ A2 @ B3 ) )
= ( ( member_b @ C2 @ A2 )
& ~ ( member_b @ C2 @ B3 ) ) ) ).
% Diff_iff
thf(fact_286_Diff__idemp,axiom,
! [A2: set_a,B3: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ B3 )
= ( minus_minus_set_a @ A2 @ B3 ) ) ).
% Diff_idemp
thf(fact_287_Diff__idemp,axiom,
! [A2: set_b,B3: set_b] :
( ( minus_minus_set_b @ ( minus_minus_set_b @ A2 @ B3 ) @ B3 )
= ( minus_minus_set_b @ A2 @ B3 ) ) ).
% Diff_idemp
thf(fact_288_nnvs__subs__verts,axiom,
! [W: b > real,U: a,N: nat,U2: set_a] :
( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ U2 )
=> ( ord_less_eq_set_a @ U2 @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
% nnvs_subs_verts
thf(fact_289_reachable__induce__ss,axiom,
! [S2: set_a,U: a,V: a,T2: set_a] :
( ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ g @ S2 ) @ U @ V )
=> ( ( ord_less_eq_set_a @ S2 @ T2 )
=> ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ g @ T2 ) @ U @ V ) ) ) ).
% reachable_induce_ss
thf(fact_290_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_291_nnvs__card__le__n,axiom,
! [W: b > real,U: a,N: nat,U2: set_a] :
( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ U2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ U2 ) @ ( suc @ N ) ) ) ).
% nnvs_card_le_n
thf(fact_292_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_293_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_294_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_295_Diff__empty,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ( minus_3777555517894451474t_unit @ A2 @ bot_bo1839476491465656141t_unit )
= A2 ) ).
% Diff_empty
thf(fact_296_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_297_Diff__empty,axiom,
! [A2: set_b] :
( ( minus_minus_set_b @ A2 @ bot_bot_set_b )
= A2 ) ).
% Diff_empty
thf(fact_298_empty__Diff,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ( minus_3777555517894451474t_unit @ bot_bo1839476491465656141t_unit @ A2 )
= bot_bo1839476491465656141t_unit ) ).
% empty_Diff
thf(fact_299_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_300_empty__Diff,axiom,
! [A2: set_b] :
( ( minus_minus_set_b @ bot_bot_set_b @ A2 )
= bot_bot_set_b ) ).
% empty_Diff
thf(fact_301_Diff__cancel,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ( minus_3777555517894451474t_unit @ A2 @ A2 )
= bot_bo1839476491465656141t_unit ) ).
% Diff_cancel
thf(fact_302_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_303_Diff__cancel,axiom,
! [A2: set_b] :
( ( minus_minus_set_b @ A2 @ A2 )
= bot_bot_set_b ) ).
% Diff_cancel
thf(fact_304_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_305_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_306_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_307_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_308_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 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_309_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_310_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_311_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_312_le__trans,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_313_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_314_DiffE,axiom,
! [C2: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) )
=> ~ ( ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ B3 ) ) ) ).
% DiffE
thf(fact_315_DiffE,axiom,
! [C2: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( minus_3777555517894451474t_unit @ A2 @ B3 ) )
=> ~ ( ( member6939884229742472986t_unit @ C2 @ A2 )
=> ( member6939884229742472986t_unit @ C2 @ B3 ) ) ) ).
% DiffE
thf(fact_316_DiffE,axiom,
! [C2: a,A2: set_a,B3: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B3 ) )
=> ~ ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B3 ) ) ) ).
% DiffE
thf(fact_317_DiffE,axiom,
! [C2: b,A2: set_b,B3: set_b] :
( ( member_b @ C2 @ ( minus_minus_set_b @ A2 @ B3 ) )
=> ~ ( ( member_b @ C2 @ A2 )
=> ( member_b @ C2 @ B3 ) ) ) ).
% DiffE
thf(fact_318_DiffD1,axiom,
! [C2: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) )
=> ( member_set_a @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_319_DiffD1,axiom,
! [C2: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( minus_3777555517894451474t_unit @ A2 @ B3 ) )
=> ( member6939884229742472986t_unit @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_320_DiffD1,axiom,
! [C2: a,A2: set_a,B3: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B3 ) )
=> ( member_a @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_321_DiffD1,axiom,
! [C2: b,A2: set_b,B3: set_b] :
( ( member_b @ C2 @ ( minus_minus_set_b @ A2 @ B3 ) )
=> ( member_b @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_322_DiffD2,axiom,
! [C2: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C2 @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) )
=> ~ ( member_set_a @ C2 @ B3 ) ) ).
% DiffD2
thf(fact_323_DiffD2,axiom,
! [C2: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( minus_3777555517894451474t_unit @ A2 @ B3 ) )
=> ~ ( member6939884229742472986t_unit @ C2 @ B3 ) ) ).
% DiffD2
thf(fact_324_DiffD2,axiom,
! [C2: a,A2: set_a,B3: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B3 ) )
=> ~ ( member_a @ C2 @ B3 ) ) ).
% DiffD2
thf(fact_325_DiffD2,axiom,
! [C2: b,A2: set_b,B3: set_b] :
( ( member_b @ C2 @ ( minus_minus_set_b @ A2 @ B3 ) )
=> ~ ( member_b @ C2 @ B3 ) ) ).
% DiffD2
thf(fact_326_psubset__imp__ex__mem,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B3 )
=> ? [B4: set_a] : ( member_set_a @ B4 @ ( minus_5736297505244876581_set_a @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_327_psubset__imp__ex__mem,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ord_le2693654750756130573t_unit @ A2 @ B3 )
=> ? [B4: pre_pr7278220950009878019t_unit] : ( member6939884229742472986t_unit @ B4 @ ( minus_3777555517894451474t_unit @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_328_psubset__imp__ex__mem,axiom,
! [A2: set_b,B3: set_b] :
( ( ord_less_set_b @ A2 @ B3 )
=> ? [B4: b] : ( member_b @ B4 @ ( minus_minus_set_b @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_329_psubset__imp__ex__mem,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ? [B4: a] : ( member_a @ B4 @ ( minus_minus_set_a @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_330_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_331_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_332_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_333_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_334_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_335_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_336_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_337_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M3: nat] :
( M6
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_338_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_339_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_340_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_341_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_342_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_343_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y3: nat,Z3: nat] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_344_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J2: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_345_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_346_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_347_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
| ( M4 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_348_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_349_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
& ( M4 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_350_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_351_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_352_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_353_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_354_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_355_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_356_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_357_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_358_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_359_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_360_dec__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( P @ I2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_361_inc__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( P @ J2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_362_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_363_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_364_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_365_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_366_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_367_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_368_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_369_diff__less__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_370_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_371_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_372_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_373_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_374_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_375_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_376_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_377_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_378_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_379_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_380_rev__image__eqI,axiom,
! [X: a,A2: set_a,B: pre_pr7278220950009878019t_unit,F: a > pre_pr7278220950009878019t_unit] :
( ( member_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member6939884229742472986t_unit @ B @ ( image_5713294457175270716t_unit @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_381_ball__imageD,axiom,
! [F: a > set_a,A2: set_a,P: set_a > $o] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ ( image_a_set_a @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_382_ball__imageD,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a,P: pre_pr7278220950009878019t_unit > $o] :
( ! [X4: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X4 @ ( image_6801035452528096924t_unit @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_383_ball__imageD,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,P: set_a > $o] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ ( image_7466199892558553556_set_a @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X5: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_384_image__cong,axiom,
! [M7: set_a,N5: set_a,F: a > set_a,G: a > set_a] :
( ( M7 = N5 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ N5 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_a_set_a @ F @ M7 )
= ( image_a_set_a @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_385_image__cong,axiom,
! [M7: set_set_a,N5: set_set_a,F: set_a > pre_pr7278220950009878019t_unit,G: set_a > pre_pr7278220950009878019t_unit] :
( ( M7 = N5 )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ N5 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_6801035452528096924t_unit @ F @ M7 )
= ( image_6801035452528096924t_unit @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_386_image__cong,axiom,
! [M7: set_pr5411798346947241657t_unit,N5: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > set_a,G: pre_pr7278220950009878019t_unit > set_a] :
( ( M7 = N5 )
=> ( ! [X4: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X4 @ N5 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_7466199892558553556_set_a @ F @ M7 )
= ( image_7466199892558553556_set_a @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_387_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 ) )
=> ? [X4: a] :
( ( member_a @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_388_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 ) )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_389_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 ) )
=> ? [X4: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_390_image__iff,axiom,
! [Z4: set_a,F: a > set_a,A2: set_a] :
( ( member_set_a @ Z4 @ ( image_a_set_a @ F @ A2 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( Z4
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_391_image__iff,axiom,
! [Z4: set_a,F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit] :
( ( member_set_a @ Z4 @ ( image_7466199892558553556_set_a @ F @ A2 ) )
= ( ? [X3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ A2 )
& ( Z4
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_392_image__iff,axiom,
! [Z4: pre_pr7278220950009878019t_unit,F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a] :
( ( member6939884229742472986t_unit @ Z4 @ ( image_6801035452528096924t_unit @ F @ A2 ) )
= ( ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( Z4
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_393_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_394_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_395_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_396_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_397_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_398_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_399_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_400_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_401_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_402_imageI,axiom,
! [X: a,A2: set_a,F: a > pre_pr7278220950009878019t_unit] :
( ( member_a @ X @ A2 )
=> ( member6939884229742472986t_unit @ ( F @ X ) @ ( image_5713294457175270716t_unit @ F @ A2 ) ) ) ).
% imageI
thf(fact_403_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_404_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_405_ex__in__conv,axiom,
! [A2: set_b] :
( ( ? [X3: b] : ( member_b @ X3 @ A2 ) )
= ( A2 != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_406_ex__in__conv,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ( ? [X3: pre_pr7278220950009878019t_unit] : ( member6939884229742472986t_unit @ X3 @ A2 ) )
= ( A2 != bot_bo1839476491465656141t_unit ) ) ).
% ex_in_conv
thf(fact_407_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y3: set_a] :
~ ( member_set_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_408_equals0I,axiom,
! [A2: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_409_equals0I,axiom,
! [A2: set_b] :
( ! [Y3: b] :
~ ( member_b @ Y3 @ A2 )
=> ( A2 = bot_bot_set_b ) ) ).
% equals0I
thf(fact_410_equals0I,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ! [Y3: pre_pr7278220950009878019t_unit] :
~ ( member6939884229742472986t_unit @ Y3 @ A2 )
=> ( A2 = bot_bo1839476491465656141t_unit ) ) ).
% equals0I
thf(fact_411_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_412_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_413_equals0D,axiom,
! [A2: set_b,A: b] :
( ( A2 = bot_bot_set_b )
=> ~ ( member_b @ A @ A2 ) ) ).
% equals0D
thf(fact_414_equals0D,axiom,
! [A2: set_pr5411798346947241657t_unit,A: pre_pr7278220950009878019t_unit] :
( ( A2 = bot_bo1839476491465656141t_unit )
=> ~ ( member6939884229742472986t_unit @ A @ A2 ) ) ).
% equals0D
thf(fact_415_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_416_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_417_emptyE,axiom,
! [A: b] :
~ ( member_b @ A @ bot_bot_set_b ) ).
% emptyE
thf(fact_418_emptyE,axiom,
! [A: pre_pr7278220950009878019t_unit] :
~ ( member6939884229742472986t_unit @ A @ bot_bo1839476491465656141t_unit ) ).
% emptyE
thf(fact_419_not__psubset__empty,axiom,
! [A2: set_b] :
~ ( ord_less_set_b @ A2 @ bot_bot_set_b ) ).
% not_psubset_empty
thf(fact_420_not__psubset__empty,axiom,
! [A2: set_pr5411798346947241657t_unit] :
~ ( ord_le2693654750756130573t_unit @ A2 @ bot_bo1839476491465656141t_unit ) ).
% not_psubset_empty
thf(fact_421_not__psubset__empty,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_422_Collect__mono__iff,axiom,
! [P: pre_pr7278220950009878019t_unit > $o,Q: pre_pr7278220950009878019t_unit > $o] :
( ( ord_le8200006823705900825t_unit @ ( collec8000012497822511960t_unit @ P ) @ ( collec8000012497822511960t_unit @ Q ) )
= ( ! [X3: pre_pr7278220950009878019t_unit] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_423_Collect__mono__iff,axiom,
! [P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) )
= ( ! [X3: b] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_424_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_425_set__eq__subset,axiom,
( ( ^ [Y4: set_pr5411798346947241657t_unit,Z: set_pr5411798346947241657t_unit] : ( Y4 = Z ) )
= ( ^ [A4: set_pr5411798346947241657t_unit,B5: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A4 @ B5 )
& ( ord_le8200006823705900825t_unit @ B5 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_426_set__eq__subset,axiom,
( ( ^ [Y4: set_b,Z: set_b] : ( Y4 = Z ) )
= ( ^ [A4: set_b,B5: set_b] :
( ( ord_less_eq_set_b @ A4 @ B5 )
& ( ord_less_eq_set_b @ B5 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_427_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [A4: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_428_subset__trans,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit,C: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ B3 )
=> ( ( ord_le8200006823705900825t_unit @ B3 @ C )
=> ( ord_le8200006823705900825t_unit @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_429_subset__trans,axiom,
! [A2: set_b,B3: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ( ord_less_eq_set_b @ B3 @ C )
=> ( ord_less_eq_set_b @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_430_subset__trans,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_431_Collect__mono,axiom,
! [P: pre_pr7278220950009878019t_unit > $o,Q: pre_pr7278220950009878019t_unit > $o] :
( ! [X4: pre_pr7278220950009878019t_unit] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le8200006823705900825t_unit @ ( collec8000012497822511960t_unit @ P ) @ ( collec8000012497822511960t_unit @ Q ) ) ) ).
% Collect_mono
thf(fact_432_Collect__mono,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X4: b] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% Collect_mono
thf(fact_433_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_434_subset__refl,axiom,
! [A2: set_pr5411798346947241657t_unit] : ( ord_le8200006823705900825t_unit @ A2 @ A2 ) ).
% subset_refl
thf(fact_435_subset__refl,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ A2 @ A2 ) ).
% subset_refl
thf(fact_436_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_437_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] :
! [T3: set_a] :
( ( member_set_a @ T3 @ A4 )
=> ( member_set_a @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_438_subset__iff,axiom,
( ord_le8200006823705900825t_unit
= ( ^ [A4: set_pr5411798346947241657t_unit,B5: set_pr5411798346947241657t_unit] :
! [T3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ T3 @ A4 )
=> ( member6939884229742472986t_unit @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_439_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B5: set_b] :
! [T3: b] :
( ( member_b @ T3 @ A4 )
=> ( member_b @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_440_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B5: set_a] :
! [T3: a] :
( ( member_a @ T3 @ A4 )
=> ( member_a @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_441_equalityD2,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( A2 = B3 )
=> ( ord_le8200006823705900825t_unit @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_442_equalityD2,axiom,
! [A2: set_b,B3: set_b] :
( ( A2 = B3 )
=> ( ord_less_eq_set_b @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_443_equalityD2,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_444_equalityD1,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( A2 = B3 )
=> ( ord_le8200006823705900825t_unit @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_445_equalityD1,axiom,
! [A2: set_b,B3: set_b] :
( ( A2 = B3 )
=> ( ord_less_eq_set_b @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_446_equalityD1,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_447_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B5: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A4 )
=> ( member_set_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_448_subset__eq,axiom,
( ord_le8200006823705900825t_unit
= ( ^ [A4: set_pr5411798346947241657t_unit,B5: set_pr5411798346947241657t_unit] :
! [X3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ A4 )
=> ( member6939884229742472986t_unit @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_449_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B5: set_b] :
! [X3: b] :
( ( member_b @ X3 @ A4 )
=> ( member_b @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_450_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B5: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( member_a @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_451_equalityE,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( A2 = B3 )
=> ~ ( ( ord_le8200006823705900825t_unit @ A2 @ B3 )
=> ~ ( ord_le8200006823705900825t_unit @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_452_equalityE,axiom,
! [A2: set_b,B3: set_b] :
( ( A2 = B3 )
=> ~ ( ( ord_less_eq_set_b @ A2 @ B3 )
=> ~ ( ord_less_eq_set_b @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_453_equalityE,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ~ ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_454_subsetD,axiom,
! [A2: set_set_a,B3: set_set_a,C2: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_455_subsetD,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit,C2: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ B3 )
=> ( ( member6939884229742472986t_unit @ C2 @ A2 )
=> ( member6939884229742472986t_unit @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_456_subsetD,axiom,
! [A2: set_b,B3: set_b,C2: b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ( member_b @ C2 @ A2 )
=> ( member_b @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_457_subsetD,axiom,
! [A2: set_a,B3: set_a,C2: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_458_in__mono,axiom,
! [A2: set_set_a,B3: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( member_set_a @ X @ A2 )
=> ( member_set_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_459_in__mono,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit,X: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ B3 )
=> ( ( member6939884229742472986t_unit @ X @ A2 )
=> ( member6939884229742472986t_unit @ X @ B3 ) ) ) ).
% in_mono
thf(fact_460_in__mono,axiom,
! [A2: set_b,B3: set_b,X: b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ( member_b @ X @ A2 )
=> ( member_b @ X @ B3 ) ) ) ).
% in_mono
thf(fact_461_in__mono,axiom,
! [A2: set_a,B3: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_462_Diff__mono,axiom,
! [A2: set_pr5411798346947241657t_unit,C: set_pr5411798346947241657t_unit,D2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ C )
=> ( ( ord_le8200006823705900825t_unit @ D2 @ B3 )
=> ( ord_le8200006823705900825t_unit @ ( minus_3777555517894451474t_unit @ A2 @ B3 ) @ ( minus_3777555517894451474t_unit @ C @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_463_Diff__mono,axiom,
! [A2: set_b,C: set_b,D2: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A2 @ C )
=> ( ( ord_less_eq_set_b @ D2 @ B3 )
=> ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ B3 ) @ ( minus_minus_set_b @ C @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_464_Diff__mono,axiom,
! [A2: set_a,C: set_a,D2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ D2 @ B3 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ ( minus_minus_set_a @ C @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_465_Diff__subset,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] : ( ord_le8200006823705900825t_unit @ ( minus_3777555517894451474t_unit @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_466_Diff__subset,axiom,
! [A2: set_b,B3: set_b] : ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_467_Diff__subset,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_468_double__diff,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit,C: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ B3 )
=> ( ( ord_le8200006823705900825t_unit @ B3 @ C )
=> ( ( minus_3777555517894451474t_unit @ B3 @ ( minus_3777555517894451474t_unit @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_469_double__diff,axiom,
! [A2: set_b,B3: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ( ord_less_eq_set_b @ B3 @ C )
=> ( ( minus_minus_set_b @ B3 @ ( minus_minus_set_b @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_470_double__diff,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ( minus_minus_set_a @ B3 @ ( minus_minus_set_a @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_471_subset__iff__psubset__eq,axiom,
( ord_le8200006823705900825t_unit
= ( ^ [A4: set_pr5411798346947241657t_unit,B5: set_pr5411798346947241657t_unit] :
( ( ord_le2693654750756130573t_unit @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_472_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B5: set_b] :
( ( ord_less_set_b @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_473_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B5: set_a] :
( ( ord_less_set_a @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_474_subset__psubset__trans,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit,C: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ B3 )
=> ( ( ord_le2693654750756130573t_unit @ B3 @ C )
=> ( ord_le2693654750756130573t_unit @ A2 @ C ) ) ) ).
% subset_psubset_trans
thf(fact_475_subset__psubset__trans,axiom,
! [A2: set_b,B3: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ( ord_less_set_b @ B3 @ C )
=> ( ord_less_set_b @ A2 @ C ) ) ) ).
% subset_psubset_trans
thf(fact_476_subset__psubset__trans,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_set_a @ B3 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% subset_psubset_trans
thf(fact_477_subset__not__subset__eq,axiom,
( ord_le2693654750756130573t_unit
= ( ^ [A4: set_pr5411798346947241657t_unit,B5: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A4 @ B5 )
& ~ ( ord_le8200006823705900825t_unit @ B5 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_478_subset__not__subset__eq,axiom,
( ord_less_set_b
= ( ^ [A4: set_b,B5: set_b] :
( ( ord_less_eq_set_b @ A4 @ B5 )
& ~ ( ord_less_eq_set_b @ B5 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_479_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
& ~ ( ord_less_eq_set_a @ B5 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_480_psubset__subset__trans,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit,C: set_pr5411798346947241657t_unit] :
( ( ord_le2693654750756130573t_unit @ A2 @ B3 )
=> ( ( ord_le8200006823705900825t_unit @ B3 @ C )
=> ( ord_le2693654750756130573t_unit @ A2 @ C ) ) ) ).
% psubset_subset_trans
thf(fact_481_psubset__subset__trans,axiom,
! [A2: set_b,B3: set_b,C: set_b] :
( ( ord_less_set_b @ A2 @ B3 )
=> ( ( ord_less_eq_set_b @ B3 @ C )
=> ( ord_less_set_b @ A2 @ C ) ) ) ).
% psubset_subset_trans
thf(fact_482_psubset__subset__trans,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% psubset_subset_trans
thf(fact_483_psubset__imp__subset,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ord_le2693654750756130573t_unit @ A2 @ B3 )
=> ( ord_le8200006823705900825t_unit @ A2 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_484_psubset__imp__subset,axiom,
! [A2: set_b,B3: set_b] :
( ( ord_less_set_b @ A2 @ B3 )
=> ( ord_less_eq_set_b @ A2 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_485_psubset__imp__subset,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_486_psubset__eq,axiom,
( ord_le2693654750756130573t_unit
= ( ^ [A4: set_pr5411798346947241657t_unit,B5: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_487_psubset__eq,axiom,
( ord_less_set_b
= ( ^ [A4: set_b,B5: set_b] :
( ( ord_less_eq_set_b @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_488_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_489_psubsetE,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ord_le2693654750756130573t_unit @ A2 @ B3 )
=> ~ ( ( ord_le8200006823705900825t_unit @ A2 @ B3 )
=> ( ord_le8200006823705900825t_unit @ B3 @ A2 ) ) ) ).
% psubsetE
thf(fact_490_psubsetE,axiom,
! [A2: set_b,B3: set_b] :
( ( ord_less_set_b @ A2 @ B3 )
=> ~ ( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ord_less_eq_set_b @ B3 @ A2 ) ) ) ).
% psubsetE
thf(fact_491_psubsetE,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ).
% psubsetE
thf(fact_492_all__subset__image,axiom,
! [F: b > b,A2: set_b,P: set_b > $o] :
( ( ! [B5: set_b] :
( ( ord_less_eq_set_b @ B5 @ ( image_b_b @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_b] :
( ( ord_less_eq_set_b @ B5 @ A2 )
=> ( P @ ( image_b_b @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_493_all__subset__image,axiom,
! [F: a > b,A2: set_a,P: set_b > $o] :
( ( ! [B5: set_b] :
( ( ord_less_eq_set_b @ B5 @ ( image_a_b @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A2 )
=> ( P @ ( image_a_b @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_494_all__subset__image,axiom,
! [F: b > a,A2: set_b,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ ( image_b_a @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_b] :
( ( ord_less_eq_set_b @ B5 @ A2 )
=> ( P @ ( image_b_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_495_all__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A2 )
=> ( P @ ( image_a_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_496_all__subset__image,axiom,
! [F: a > set_a,A2: set_a,P: set_set_a > $o] :
( ( ! [B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ ( image_a_set_a @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A2 )
=> ( P @ ( image_a_set_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_497_all__subset__image,axiom,
! [F: b > pre_pr7278220950009878019t_unit,A2: set_b,P: set_pr5411798346947241657t_unit > $o] :
( ( ! [B5: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ B5 @ ( image_4434118323594779837t_unit @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_b] :
( ( ord_less_eq_set_b @ B5 @ A2 )
=> ( P @ ( image_4434118323594779837t_unit @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_498_all__subset__image,axiom,
! [F: a > pre_pr7278220950009878019t_unit,A2: set_a,P: set_pr5411798346947241657t_unit > $o] :
( ( ! [B5: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ B5 @ ( image_5713294457175270716t_unit @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A2 )
=> ( P @ ( image_5713294457175270716t_unit @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_499_all__subset__image,axiom,
! [F: pre_pr7278220950009878019t_unit > b,A2: set_pr5411798346947241657t_unit,P: set_b > $o] :
( ( ! [B5: set_b] :
( ( ord_less_eq_set_b @ B5 @ ( image_4969699134812999797unit_b @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ B5 @ A2 )
=> ( P @ ( image_4969699134812999797unit_b @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_500_all__subset__image,axiom,
! [F: pre_pr7278220950009878019t_unit > a,A2: set_pr5411798346947241657t_unit,P: set_a > $o] :
( ( ! [B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ ( image_4969699134812999796unit_a @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ B5 @ A2 )
=> ( P @ ( image_4969699134812999796unit_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_501_all__subset__image,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,P: set_set_a > $o] :
( ( ! [B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ ( image_7466199892558553556_set_a @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ B5 @ A2 )
=> ( P @ ( image_7466199892558553556_set_a @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_502_subset__image__iff,axiom,
! [B3: set_b,F: b > b,A2: set_b] :
( ( ord_less_eq_set_b @ B3 @ ( image_b_b @ F @ A2 ) )
= ( ? [AA: set_b] :
( ( ord_less_eq_set_b @ AA @ A2 )
& ( B3
= ( image_b_b @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_503_subset__image__iff,axiom,
! [B3: set_b,F: a > b,A2: set_a] :
( ( ord_less_eq_set_b @ B3 @ ( image_a_b @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B3
= ( image_a_b @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_504_subset__image__iff,axiom,
! [B3: set_a,F: b > a,A2: set_b] :
( ( ord_less_eq_set_a @ B3 @ ( image_b_a @ F @ A2 ) )
= ( ? [AA: set_b] :
( ( ord_less_eq_set_b @ AA @ A2 )
& ( B3
= ( image_b_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_505_subset__image__iff,axiom,
! [B3: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B3
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_506_subset__image__iff,axiom,
! [B3: set_set_a,F: a > set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_a_set_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B3
= ( image_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_507_subset__image__iff,axiom,
! [B3: set_pr5411798346947241657t_unit,F: b > pre_pr7278220950009878019t_unit,A2: set_b] :
( ( ord_le8200006823705900825t_unit @ B3 @ ( image_4434118323594779837t_unit @ F @ A2 ) )
= ( ? [AA: set_b] :
( ( ord_less_eq_set_b @ AA @ A2 )
& ( B3
= ( image_4434118323594779837t_unit @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_508_subset__image__iff,axiom,
! [B3: set_pr5411798346947241657t_unit,F: a > pre_pr7278220950009878019t_unit,A2: set_a] :
( ( ord_le8200006823705900825t_unit @ B3 @ ( image_5713294457175270716t_unit @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B3
= ( image_5713294457175270716t_unit @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_509_subset__image__iff,axiom,
! [B3: set_b,F: pre_pr7278220950009878019t_unit > b,A2: set_pr5411798346947241657t_unit] :
( ( ord_less_eq_set_b @ B3 @ ( image_4969699134812999797unit_b @ F @ A2 ) )
= ( ? [AA: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ AA @ A2 )
& ( B3
= ( image_4969699134812999797unit_b @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_510_subset__image__iff,axiom,
! [B3: set_a,F: pre_pr7278220950009878019t_unit > a,A2: set_pr5411798346947241657t_unit] :
( ( ord_less_eq_set_a @ B3 @ ( image_4969699134812999796unit_a @ F @ A2 ) )
= ( ? [AA: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ AA @ A2 )
& ( B3
= ( image_4969699134812999796unit_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_511_subset__image__iff,axiom,
! [B3: set_set_a,F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_7466199892558553556_set_a @ F @ A2 ) )
= ( ? [AA: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ AA @ A2 )
& ( B3
= ( image_7466199892558553556_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_512_image__subset__iff,axiom,
! [F: a > set_a,A2: set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ B3 )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_set_a @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_513_image__subset__iff,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_7466199892558553556_set_a @ F @ A2 ) @ B3 )
= ( ! [X3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ A2 )
=> ( member_set_a @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_514_image__subset__iff,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a,B3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ ( image_6801035452528096924t_unit @ F @ A2 ) @ B3 )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( member6939884229742472986t_unit @ ( F @ X3 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_515_subset__imageE,axiom,
! [B3: set_b,F: b > b,A2: set_b] :
( ( ord_less_eq_set_b @ B3 @ ( image_b_b @ F @ A2 ) )
=> ~ ! [C3: set_b] :
( ( ord_less_eq_set_b @ C3 @ A2 )
=> ( B3
!= ( image_b_b @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_516_subset__imageE,axiom,
! [B3: set_b,F: a > b,A2: set_a] :
( ( ord_less_eq_set_b @ B3 @ ( image_a_b @ F @ A2 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( B3
!= ( image_a_b @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_517_subset__imageE,axiom,
! [B3: set_a,F: b > a,A2: set_b] :
( ( ord_less_eq_set_a @ B3 @ ( image_b_a @ F @ A2 ) )
=> ~ ! [C3: set_b] :
( ( ord_less_eq_set_b @ C3 @ A2 )
=> ( B3
!= ( image_b_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_518_subset__imageE,axiom,
! [B3: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A2 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( B3
!= ( image_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_519_subset__imageE,axiom,
! [B3: set_set_a,F: a > set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_a_set_a @ F @ A2 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( B3
!= ( image_a_set_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_520_subset__imageE,axiom,
! [B3: set_pr5411798346947241657t_unit,F: b > pre_pr7278220950009878019t_unit,A2: set_b] :
( ( ord_le8200006823705900825t_unit @ B3 @ ( image_4434118323594779837t_unit @ F @ A2 ) )
=> ~ ! [C3: set_b] :
( ( ord_less_eq_set_b @ C3 @ A2 )
=> ( B3
!= ( image_4434118323594779837t_unit @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_521_subset__imageE,axiom,
! [B3: set_pr5411798346947241657t_unit,F: a > pre_pr7278220950009878019t_unit,A2: set_a] :
( ( ord_le8200006823705900825t_unit @ B3 @ ( image_5713294457175270716t_unit @ F @ A2 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( B3
!= ( image_5713294457175270716t_unit @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_522_subset__imageE,axiom,
! [B3: set_b,F: pre_pr7278220950009878019t_unit > b,A2: set_pr5411798346947241657t_unit] :
( ( ord_less_eq_set_b @ B3 @ ( image_4969699134812999797unit_b @ F @ A2 ) )
=> ~ ! [C3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ C3 @ A2 )
=> ( B3
!= ( image_4969699134812999797unit_b @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_523_subset__imageE,axiom,
! [B3: set_a,F: pre_pr7278220950009878019t_unit > a,A2: set_pr5411798346947241657t_unit] :
( ( ord_less_eq_set_a @ B3 @ ( image_4969699134812999796unit_a @ F @ A2 ) )
=> ~ ! [C3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ C3 @ A2 )
=> ( B3
!= ( image_4969699134812999796unit_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_524_subset__imageE,axiom,
! [B3: set_set_a,F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_7466199892558553556_set_a @ F @ A2 ) )
=> ~ ! [C3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ C3 @ A2 )
=> ( B3
!= ( image_7466199892558553556_set_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_525_image__subsetI,axiom,
! [A2: set_a,F: a > b,B3: set_b] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( member_b @ ( F @ X4 ) @ B3 ) )
=> ( ord_less_eq_set_b @ ( image_a_b @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_526_image__subsetI,axiom,
! [A2: set_b,F: b > b,B3: set_b] :
( ! [X4: b] :
( ( member_b @ X4 @ A2 )
=> ( member_b @ ( F @ X4 ) @ B3 ) )
=> ( ord_less_eq_set_b @ ( image_b_b @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_527_image__subsetI,axiom,
! [A2: set_a,F: a > a,B3: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( member_a @ ( F @ X4 ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_528_image__subsetI,axiom,
! [A2: set_b,F: b > a,B3: set_a] :
( ! [X4: b] :
( ( member_b @ X4 @ A2 )
=> ( member_a @ ( F @ X4 ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_b_a @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_529_image__subsetI,axiom,
! [A2: set_a,F: a > set_a,B3: set_set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( member_set_a @ ( F @ X4 ) @ B3 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_530_image__subsetI,axiom,
! [A2: set_b,F: b > set_a,B3: set_set_a] :
( ! [X4: b] :
( ( member_b @ X4 @ A2 )
=> ( member_set_a @ ( F @ X4 ) @ B3 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_b_set_a @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_531_image__subsetI,axiom,
! [A2: set_set_a,F: set_a > b,B3: set_b] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ( member_b @ ( F @ X4 ) @ B3 ) )
=> ( ord_less_eq_set_b @ ( image_set_a_b @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_532_image__subsetI,axiom,
! [A2: set_set_a,F: set_a > a,B3: set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ( member_a @ ( F @ X4 ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_533_image__subsetI,axiom,
! [A2: set_set_a,F: set_a > set_a,B3: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ( member_set_a @ ( F @ X4 ) @ B3 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_534_image__subsetI,axiom,
! [A2: set_a,F: a > pre_pr7278220950009878019t_unit,B3: set_pr5411798346947241657t_unit] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( member6939884229742472986t_unit @ ( F @ X4 ) @ B3 ) )
=> ( ord_le8200006823705900825t_unit @ ( image_5713294457175270716t_unit @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_535_image__mono,axiom,
! [A2: set_b,B3: set_b,F: b > b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ord_less_eq_set_b @ ( image_b_b @ F @ A2 ) @ ( image_b_b @ F @ B3 ) ) ) ).
% image_mono
thf(fact_536_image__mono,axiom,
! [A2: set_b,B3: set_b,F: b > a] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ord_less_eq_set_a @ ( image_b_a @ F @ A2 ) @ ( image_b_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_537_image__mono,axiom,
! [A2: set_a,B3: set_a,F: a > b] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_b @ ( image_a_b @ F @ A2 ) @ ( image_a_b @ F @ B3 ) ) ) ).
% image_mono
thf(fact_538_image__mono,axiom,
! [A2: set_a,B3: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_539_image__mono,axiom,
! [A2: set_a,B3: set_a,F: a > set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ ( image_a_set_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_540_image__mono,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > b] :
( ( ord_le8200006823705900825t_unit @ A2 @ B3 )
=> ( ord_less_eq_set_b @ ( image_4969699134812999797unit_b @ F @ A2 ) @ ( image_4969699134812999797unit_b @ F @ B3 ) ) ) ).
% image_mono
thf(fact_541_image__mono,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > a] :
( ( ord_le8200006823705900825t_unit @ A2 @ B3 )
=> ( ord_less_eq_set_a @ ( image_4969699134812999796unit_a @ F @ A2 ) @ ( image_4969699134812999796unit_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_542_image__mono,axiom,
! [A2: set_b,B3: set_b,F: b > pre_pr7278220950009878019t_unit] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ord_le8200006823705900825t_unit @ ( image_4434118323594779837t_unit @ F @ A2 ) @ ( image_4434118323594779837t_unit @ F @ B3 ) ) ) ).
% image_mono
thf(fact_543_image__mono,axiom,
! [A2: set_a,B3: set_a,F: a > pre_pr7278220950009878019t_unit] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ord_le8200006823705900825t_unit @ ( image_5713294457175270716t_unit @ F @ A2 ) @ ( image_5713294457175270716t_unit @ F @ B3 ) ) ) ).
% image_mono
thf(fact_544_image__mono,axiom,
! [A2: set_set_a,B3: set_set_a,F: set_a > pre_pr7278220950009878019t_unit] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ord_le8200006823705900825t_unit @ ( image_6801035452528096924t_unit @ F @ A2 ) @ ( image_6801035452528096924t_unit @ F @ B3 ) ) ) ).
% image_mono
thf(fact_545_image__diff__subset,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ ( image_7466199892558553556_set_a @ F @ A2 ) @ ( image_7466199892558553556_set_a @ F @ B3 ) ) @ ( image_7466199892558553556_set_a @ F @ ( minus_3777555517894451474t_unit @ A2 @ B3 ) ) ) ).
% image_diff_subset
thf(fact_546_image__diff__subset,axiom,
! [F: a > set_a,A2: set_a,B3: set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ ( image_a_set_a @ F @ A2 ) @ ( image_a_set_a @ F @ B3 ) ) @ ( image_a_set_a @ F @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ).
% image_diff_subset
thf(fact_547_image__diff__subset,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a,B3: set_set_a] : ( ord_le8200006823705900825t_unit @ ( minus_3777555517894451474t_unit @ ( image_6801035452528096924t_unit @ F @ A2 ) @ ( image_6801035452528096924t_unit @ F @ B3 ) ) @ ( image_6801035452528096924t_unit @ F @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) ) ) ).
% image_diff_subset
thf(fact_548_image__diff__subset,axiom,
! [F: a > pre_pr7278220950009878019t_unit,A2: set_a,B3: set_a] : ( ord_le8200006823705900825t_unit @ ( minus_3777555517894451474t_unit @ ( image_5713294457175270716t_unit @ F @ A2 ) @ ( image_5713294457175270716t_unit @ F @ B3 ) ) @ ( image_5713294457175270716t_unit @ F @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ).
% image_diff_subset
thf(fact_549_image__diff__subset,axiom,
! [F: b > pre_pr7278220950009878019t_unit,A2: set_b,B3: set_b] : ( ord_le8200006823705900825t_unit @ ( minus_3777555517894451474t_unit @ ( image_4434118323594779837t_unit @ F @ A2 ) @ ( image_4434118323594779837t_unit @ F @ B3 ) ) @ ( image_4434118323594779837t_unit @ F @ ( minus_minus_set_b @ A2 @ B3 ) ) ) ).
% image_diff_subset
thf(fact_550_image__diff__subset,axiom,
! [F: a > b,A2: set_a,B3: set_a] : ( ord_less_eq_set_b @ ( minus_minus_set_b @ ( image_a_b @ F @ A2 ) @ ( image_a_b @ F @ B3 ) ) @ ( image_a_b @ F @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ).
% image_diff_subset
thf(fact_551_image__diff__subset,axiom,
! [F: b > b,A2: set_b,B3: set_b] : ( ord_less_eq_set_b @ ( minus_minus_set_b @ ( image_b_b @ F @ A2 ) @ ( image_b_b @ F @ B3 ) ) @ ( image_b_b @ F @ ( minus_minus_set_b @ A2 @ B3 ) ) ) ).
% image_diff_subset
thf(fact_552_image__diff__subset,axiom,
! [F: a > a,A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B3 ) ) @ ( image_a_a @ F @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ).
% image_diff_subset
thf(fact_553_image__diff__subset,axiom,
! [F: b > a,A2: set_b,B3: set_b] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_b_a @ F @ A2 ) @ ( image_b_a @ F @ B3 ) ) @ ( image_b_a @ F @ ( minus_minus_set_b @ A2 @ B3 ) ) ) ).
% image_diff_subset
thf(fact_554_nnvs__subs__k__nh,axiom,
! [W: b > real,U: a,N: nat,U2: set_a,K: real] :
( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ U2 )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ ( graph_3921080825633621230od_a_b @ g @ W @ U @ K ) ) )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ U2 @ ( insert_a @ U @ bot_bot_set_a ) ) @ ( graph_3921080825633621230od_a_b @ g @ W @ U @ K ) ) ) ) ).
% nnvs_subs_k_nh
thf(fact_555_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_556_nnvs__card__ge__n,axiom,
! [W: b > real,U: a,N: nat,U2: set_a] :
( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ U2 )
=> ( ( ( graph_2016941059203891550ts_a_b @ g @ U @ U2 )
!= bot_bot_set_a )
=> ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_a @ U2 ) ) ) ) ).
% nnvs_card_ge_n
thf(fact_557_nnvs__card__eq__n,axiom,
! [W: b > real,U: a,N: nat,U2: set_a] :
( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ U2 )
=> ( ( ( graph_2016941059203891550ts_a_b @ g @ U @ U2 )
!= bot_bot_set_a )
=> ( ( finite_card_a @ U2 )
= ( suc @ N ) ) ) ) ).
% nnvs_card_eq_n
thf(fact_558_zero__nnvs,axiom,
! [U: a,Uu: b > real] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( graph_3148032005746981223ts_a_b @ g @ Uu @ U @ zero_zero_nat @ ( insert_a @ U @ bot_bot_set_a ) ) ) ).
% zero_nnvs
thf(fact_559_connected__iff__reachable,axiom,
( ( digrap8783888973171253482ed_a_b @ g )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( reachable_a_b @ g @ X3 @ Y5 ) ) )
& ( ( pre_ve642382030648772252t_unit @ g )
!= bot_bot_set_a ) ) ) ).
% connected_iff_reachable
thf(fact_560_verts__reachable__connected,axiom,
( ( ( pre_ve642382030648772252t_unit @ g )
!= bot_bot_set_a )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ! [Xa: a] :
( ( member_a @ Xa @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( reachable_a_b @ g @ X4 @ Xa ) ) )
=> ( digrap8783888973171253482ed_a_b @ g ) ) ) ).
% verts_reachable_connected
thf(fact_561_n__nnvs__vis,axiom,
! [W: b > real,U: a,N: nat,U2: set_a] :
( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ U2 )
=> ( ( ( graph_2016941059203891550ts_a_b @ g @ U @ U2 )
= bot_bot_set_a )
=> ( graph_3148032005746981223ts_a_b @ g @ W @ U @ ( suc @ N ) @ U2 ) ) ) ).
% n_nnvs_vis
thf(fact_562_connected,axiom,
digrap8783888973171253482ed_a_b @ g ).
% connected
thf(fact_563_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_564_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_565_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_566_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_567_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_568_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_569_insertCI,axiom,
! [A: a,B3: set_a,B: a] :
( ( ~ ( member_a @ A @ B3 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_570_insertCI,axiom,
! [A: set_a,B3: set_set_a,B: set_a] :
( ( ~ ( member_set_a @ A @ B3 )
=> ( A = B ) )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_571_insertCI,axiom,
! [A: pre_pr7278220950009878019t_unit,B3: set_pr5411798346947241657t_unit,B: pre_pr7278220950009878019t_unit] :
( ( ~ ( member6939884229742472986t_unit @ A @ B3 )
=> ( A = B ) )
=> ( member6939884229742472986t_unit @ A @ ( insert6864688055023459379t_unit @ B @ B3 ) ) ) ).
% insertCI
thf(fact_572_insertCI,axiom,
! [A: b,B3: set_b,B: b] :
( ( ~ ( member_b @ A @ B3 )
=> ( A = B ) )
=> ( member_b @ A @ ( insert_b @ B @ B3 ) ) ) ).
% insertCI
thf(fact_573_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_574_image__insert,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A: set_a,B3: set_set_a] :
( ( image_6801035452528096924t_unit @ F @ ( insert_set_a @ A @ B3 ) )
= ( insert6864688055023459379t_unit @ ( F @ A ) @ ( image_6801035452528096924t_unit @ F @ B3 ) ) ) ).
% image_insert
thf(fact_575_image__insert,axiom,
! [F: a > set_a,A: a,B3: set_a] :
( ( image_a_set_a @ F @ ( insert_a @ A @ B3 ) )
= ( insert_set_a @ ( F @ A ) @ ( image_a_set_a @ F @ B3 ) ) ) ).
% image_insert
thf(fact_576_image__insert,axiom,
! [F: a > a,A: a,B3: set_a] :
( ( image_a_a @ F @ ( insert_a @ A @ B3 ) )
= ( insert_a @ ( F @ A ) @ ( image_a_a @ F @ B3 ) ) ) ).
% image_insert
thf(fact_577_image__insert,axiom,
! [F: a > pre_pr7278220950009878019t_unit,A: a,B3: set_a] :
( ( image_5713294457175270716t_unit @ F @ ( insert_a @ A @ B3 ) )
= ( insert6864688055023459379t_unit @ ( F @ A ) @ ( image_5713294457175270716t_unit @ F @ B3 ) ) ) ).
% image_insert
thf(fact_578_image__insert,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A: pre_pr7278220950009878019t_unit,B3: set_pr5411798346947241657t_unit] :
( ( image_7466199892558553556_set_a @ F @ ( insert6864688055023459379t_unit @ A @ B3 ) )
= ( insert_set_a @ ( F @ A ) @ ( image_7466199892558553556_set_a @ F @ B3 ) ) ) ).
% image_insert
thf(fact_579_image__insert,axiom,
! [F: pre_pr7278220950009878019t_unit > a,A: pre_pr7278220950009878019t_unit,B3: set_pr5411798346947241657t_unit] :
( ( image_4969699134812999796unit_a @ F @ ( insert6864688055023459379t_unit @ A @ B3 ) )
= ( insert_a @ ( F @ A ) @ ( image_4969699134812999796unit_a @ F @ B3 ) ) ) ).
% image_insert
thf(fact_580_image__insert,axiom,
! [F: pre_pr7278220950009878019t_unit > pre_pr7278220950009878019t_unit,A: pre_pr7278220950009878019t_unit,B3: set_pr5411798346947241657t_unit] :
( ( image_7933780498232994317t_unit @ F @ ( insert6864688055023459379t_unit @ A @ B3 ) )
= ( insert6864688055023459379t_unit @ ( F @ A ) @ ( image_7933780498232994317t_unit @ F @ B3 ) ) ) ).
% image_insert
thf(fact_581_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_582_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_583_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_584_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_585_insert__image,axiom,
! [X: set_a,A2: set_set_a,F: set_a > pre_pr7278220950009878019t_unit] :
( ( member_set_a @ X @ A2 )
=> ( ( insert6864688055023459379t_unit @ ( F @ X ) @ ( image_6801035452528096924t_unit @ F @ A2 ) )
= ( image_6801035452528096924t_unit @ F @ A2 ) ) ) ).
% insert_image
thf(fact_586_insert__image,axiom,
! [X: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > set_a] :
( ( member6939884229742472986t_unit @ X @ A2 )
=> ( ( insert_set_a @ ( F @ X ) @ ( image_7466199892558553556_set_a @ F @ A2 ) )
= ( image_7466199892558553556_set_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_587_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_588_insert__image,axiom,
! [X: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X @ A2 )
=> ( ( insert6864688055023459379t_unit @ ( F @ X ) @ ( image_7933780498232994317t_unit @ F @ A2 ) )
= ( image_7933780498232994317t_unit @ F @ A2 ) ) ) ).
% insert_image
thf(fact_589_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_590_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_591_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_592_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_593_singletonI,axiom,
! [A: b] : ( member_b @ A @ ( insert_b @ A @ bot_bot_set_b ) ) ).
% singletonI
thf(fact_594_singletonI,axiom,
! [A: pre_pr7278220950009878019t_unit] : ( member6939884229742472986t_unit @ A @ ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) ) ).
% singletonI
thf(fact_595_insert__subset,axiom,
! [X: set_a,A2: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A2 ) @ B3 )
= ( ( member_set_a @ X @ B3 )
& ( ord_le3724670747650509150_set_a @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_596_insert__subset,axiom,
! [X: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ ( insert6864688055023459379t_unit @ X @ A2 ) @ B3 )
= ( ( member6939884229742472986t_unit @ X @ B3 )
& ( ord_le8200006823705900825t_unit @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_597_insert__subset,axiom,
! [X: b,A2: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ ( insert_b @ X @ A2 ) @ B3 )
= ( ( member_b @ X @ B3 )
& ( ord_less_eq_set_b @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_598_insert__subset,axiom,
! [X: a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B3 )
= ( ( member_a @ X @ B3 )
& ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_599_insert__Diff1,axiom,
! [X: set_a,B3: set_set_a,A2: set_set_a] :
( ( member_set_a @ X @ B3 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B3 )
= ( minus_5736297505244876581_set_a @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_600_insert__Diff1,axiom,
! [X: pre_pr7278220950009878019t_unit,B3: set_pr5411798346947241657t_unit,A2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ X @ B3 )
=> ( ( minus_3777555517894451474t_unit @ ( insert6864688055023459379t_unit @ X @ A2 ) @ B3 )
= ( minus_3777555517894451474t_unit @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_601_insert__Diff1,axiom,
! [X: a,B3: set_a,A2: set_a] :
( ( member_a @ X @ B3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B3 )
= ( minus_minus_set_a @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_602_insert__Diff1,axiom,
! [X: b,B3: set_b,A2: set_b] :
( ( member_b @ X @ B3 )
=> ( ( minus_minus_set_b @ ( insert_b @ X @ A2 ) @ B3 )
= ( minus_minus_set_b @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_603_Diff__insert0,axiom,
! [X: set_a,A2: set_set_a,B3: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ B3 ) )
= ( minus_5736297505244876581_set_a @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_604_Diff__insert0,axiom,
! [X: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ X @ A2 )
=> ( ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ X @ B3 ) )
= ( minus_3777555517894451474t_unit @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_605_Diff__insert0,axiom,
! [X: a,A2: set_a,B3: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B3 ) )
= ( minus_minus_set_a @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_606_Diff__insert0,axiom,
! [X: b,A2: set_b,B3: set_b] :
( ~ ( member_b @ X @ A2 )
=> ( ( minus_minus_set_b @ A2 @ ( insert_b @ X @ B3 ) )
= ( minus_minus_set_b @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_607_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_608_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_609_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_610_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_611_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_612_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_613_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_614_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_615_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_616_spanning__tree__imp__connected,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digrap5718416180170401981ee_a_b @ H @ g )
=> ( digrap8783888973171253482ed_a_b @ g ) ) ).
% spanning_tree_imp_connected
thf(fact_617_card__Diff__insert,axiom,
! [A: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ~ ( member_set_a @ A @ B3 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ B3 ) ) )
= ( minus_minus_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_618_card__Diff__insert,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ A @ A2 )
=> ( ~ ( member6939884229742472986t_unit @ A @ B3 )
=> ( ( finite2416455745057997787t_unit @ ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ B3 ) ) )
= ( minus_minus_nat @ ( finite2416455745057997787t_unit @ ( minus_3777555517894451474t_unit @ A2 @ B3 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_619_card__Diff__insert,axiom,
! [A: a,A2: set_a,B3: set_a] :
( ( member_a @ A @ A2 )
=> ( ~ ( member_a @ A @ B3 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B3 ) ) )
= ( minus_minus_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B3 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_620_card__Diff__insert,axiom,
! [A: b,A2: set_b,B3: set_b] :
( ( member_b @ A @ A2 )
=> ( ~ ( member_b @ A @ B3 )
=> ( ( finite_card_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ B3 ) ) )
= ( minus_minus_nat @ ( finite_card_b @ ( minus_minus_set_b @ A2 @ B3 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_621_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_622_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B6: set_a] :
( ( A2
= ( insert_a @ A @ B6 ) )
& ~ ( member_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_623_mk__disjoint__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ? [B6: set_set_a] :
( ( A2
= ( insert_set_a @ A @ B6 ) )
& ~ ( member_set_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_624_mk__disjoint__insert,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ A @ A2 )
=> ? [B6: set_pr5411798346947241657t_unit] :
( ( A2
= ( insert6864688055023459379t_unit @ A @ B6 ) )
& ~ ( member6939884229742472986t_unit @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_625_mk__disjoint__insert,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ? [B6: set_b] :
( ( A2
= ( insert_b @ A @ B6 ) )
& ~ ( member_b @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_626_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_627_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_628_insert__eq__iff,axiom,
! [A: a,A2: set_a,B: a,B3: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B @ B3 )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C4: set_a] :
( ( A2
= ( insert_a @ B @ C4 ) )
& ~ ( member_a @ B @ C4 )
& ( B3
= ( insert_a @ A @ C4 ) )
& ~ ( member_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_629_insert__eq__iff,axiom,
! [A: set_a,A2: set_set_a,B: set_a,B3: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ~ ( member_set_a @ B @ B3 )
=> ( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C4: set_set_a] :
( ( A2
= ( insert_set_a @ B @ C4 ) )
& ~ ( member_set_a @ B @ C4 )
& ( B3
= ( insert_set_a @ A @ C4 ) )
& ~ ( member_set_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_630_insert__eq__iff,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B: pre_pr7278220950009878019t_unit,B3: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ A @ A2 )
=> ( ~ ( member6939884229742472986t_unit @ B @ B3 )
=> ( ( ( insert6864688055023459379t_unit @ A @ A2 )
= ( insert6864688055023459379t_unit @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C4: set_pr5411798346947241657t_unit] :
( ( A2
= ( insert6864688055023459379t_unit @ B @ C4 ) )
& ~ ( member6939884229742472986t_unit @ B @ C4 )
& ( B3
= ( insert6864688055023459379t_unit @ A @ C4 ) )
& ~ ( member6939884229742472986t_unit @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_631_insert__eq__iff,axiom,
! [A: b,A2: set_b,B: b,B3: set_b] :
( ~ ( member_b @ A @ A2 )
=> ( ~ ( member_b @ B @ B3 )
=> ( ( ( insert_b @ A @ A2 )
= ( insert_b @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C4: set_b] :
( ( A2
= ( insert_b @ B @ C4 ) )
& ~ ( member_b @ B @ C4 )
& ( B3
= ( insert_b @ A @ C4 ) )
& ~ ( member_b @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_632_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_633_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_634_insert__absorb,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ A @ A2 )
=> ( ( insert6864688055023459379t_unit @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_635_insert__absorb,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ( ( insert_b @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_636_insert__ident,axiom,
! [X: a,A2: set_a,B3: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B3 )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_637_insert__ident,axiom,
! [X: set_a,A2: set_set_a,B3: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ~ ( member_set_a @ X @ B3 )
=> ( ( ( insert_set_a @ X @ A2 )
= ( insert_set_a @ X @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_638_insert__ident,axiom,
! [X: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ X @ A2 )
=> ( ~ ( member6939884229742472986t_unit @ X @ B3 )
=> ( ( ( insert6864688055023459379t_unit @ X @ A2 )
= ( insert6864688055023459379t_unit @ X @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_639_insert__ident,axiom,
! [X: b,A2: set_b,B3: set_b] :
( ~ ( member_b @ X @ A2 )
=> ( ~ ( member_b @ X @ B3 )
=> ( ( ( insert_b @ X @ A2 )
= ( insert_b @ X @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_640_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B6: set_a] :
( ( A2
= ( insert_a @ X @ B6 ) )
=> ( member_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_641_Set_Oset__insert,axiom,
! [X: set_a,A2: set_set_a] :
( ( member_set_a @ X @ A2 )
=> ~ ! [B6: set_set_a] :
( ( A2
= ( insert_set_a @ X @ B6 ) )
=> ( member_set_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_642_Set_Oset__insert,axiom,
! [X: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ X @ A2 )
=> ~ ! [B6: set_pr5411798346947241657t_unit] :
( ( A2
= ( insert6864688055023459379t_unit @ X @ B6 ) )
=> ( member6939884229742472986t_unit @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_643_Set_Oset__insert,axiom,
! [X: b,A2: set_b] :
( ( member_b @ X @ A2 )
=> ~ ! [B6: set_b] :
( ( A2
= ( insert_b @ X @ B6 ) )
=> ( member_b @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_644_insertI2,axiom,
! [A: a,B3: set_a,B: a] :
( ( member_a @ A @ B3 )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_645_insertI2,axiom,
! [A: set_a,B3: set_set_a,B: set_a] :
( ( member_set_a @ A @ B3 )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_646_insertI2,axiom,
! [A: pre_pr7278220950009878019t_unit,B3: set_pr5411798346947241657t_unit,B: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ A @ B3 )
=> ( member6939884229742472986t_unit @ A @ ( insert6864688055023459379t_unit @ B @ B3 ) ) ) ).
% insertI2
thf(fact_647_insertI2,axiom,
! [A: b,B3: set_b,B: b] :
( ( member_b @ A @ B3 )
=> ( member_b @ A @ ( insert_b @ B @ B3 ) ) ) ).
% insertI2
thf(fact_648_insertI1,axiom,
! [A: a,B3: set_a] : ( member_a @ A @ ( insert_a @ A @ B3 ) ) ).
% insertI1
thf(fact_649_insertI1,axiom,
! [A: set_a,B3: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B3 ) ) ).
% insertI1
thf(fact_650_insertI1,axiom,
! [A: pre_pr7278220950009878019t_unit,B3: set_pr5411798346947241657t_unit] : ( member6939884229742472986t_unit @ A @ ( insert6864688055023459379t_unit @ A @ B3 ) ) ).
% insertI1
thf(fact_651_insertI1,axiom,
! [A: b,B3: set_b] : ( member_b @ A @ ( insert_b @ A @ B3 ) ) ).
% insertI1
thf(fact_652_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_653_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_654_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_655_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_656_psubset__trans,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ( ( ord_less_set_a @ B3 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% psubset_trans
thf(fact_657_psubsetD,axiom,
! [A2: set_set_a,B3: set_set_a,C2: set_a] :
( ( ord_less_set_set_a @ A2 @ B3 )
=> ( ( member_set_a @ C2 @ A2 )
=> ( member_set_a @ C2 @ B3 ) ) ) ).
% psubsetD
thf(fact_658_psubsetD,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit,C2: pre_pr7278220950009878019t_unit] :
( ( ord_le2693654750756130573t_unit @ A2 @ B3 )
=> ( ( member6939884229742472986t_unit @ C2 @ A2 )
=> ( member6939884229742472986t_unit @ C2 @ B3 ) ) ) ).
% psubsetD
thf(fact_659_psubsetD,axiom,
! [A2: set_b,B3: set_b,C2: b] :
( ( ord_less_set_b @ A2 @ B3 )
=> ( ( member_b @ C2 @ A2 )
=> ( member_b @ C2 @ B3 ) ) ) ).
% psubsetD
thf(fact_660_psubsetD,axiom,
! [A2: set_a,B3: set_a,C2: a] :
( ( ord_less_set_a @ A2 @ B3 )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B3 ) ) ) ).
% psubsetD
thf(fact_661_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_662_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_663_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_664_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_665_insert__not__empty,axiom,
! [A: b,A2: set_b] :
( ( insert_b @ A @ A2 )
!= bot_bot_set_b ) ).
% insert_not_empty
thf(fact_666_insert__not__empty,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( insert6864688055023459379t_unit @ A @ A2 )
!= bot_bo1839476491465656141t_unit ) ).
% insert_not_empty
thf(fact_667_doubleton__eq__iff,axiom,
! [A: a,B: a,C2: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C2 @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C2 )
& ( B = D ) )
| ( ( A = D )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_668_doubleton__eq__iff,axiom,
! [A: b,B: b,C2: b,D: b] :
( ( ( insert_b @ A @ ( insert_b @ B @ bot_bot_set_b ) )
= ( insert_b @ C2 @ ( insert_b @ D @ bot_bot_set_b ) ) )
= ( ( ( A = C2 )
& ( B = D ) )
| ( ( A = D )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_669_doubleton__eq__iff,axiom,
! [A: pre_pr7278220950009878019t_unit,B: pre_pr7278220950009878019t_unit,C2: pre_pr7278220950009878019t_unit,D: pre_pr7278220950009878019t_unit] :
( ( ( insert6864688055023459379t_unit @ A @ ( insert6864688055023459379t_unit @ B @ bot_bo1839476491465656141t_unit ) )
= ( insert6864688055023459379t_unit @ C2 @ ( insert6864688055023459379t_unit @ D @ bot_bo1839476491465656141t_unit ) ) )
= ( ( ( A = C2 )
& ( B = D ) )
| ( ( A = D )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_670_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_671_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_672_singleton__iff,axiom,
! [B: b,A: b] :
( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_673_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_674_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_675_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_676_singletonD,axiom,
! [B: b,A: b] :
( ( member_b @ B @ ( insert_b @ A @ bot_bot_set_b ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_677_singletonD,axiom,
! [B: pre_pr7278220950009878019t_unit,A: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ B @ ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_678_subset__insertI2,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit,B: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ B3 )
=> ( ord_le8200006823705900825t_unit @ A2 @ ( insert6864688055023459379t_unit @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_679_subset__insertI2,axiom,
! [A2: set_b,B3: set_b,B: b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ord_less_eq_set_b @ A2 @ ( insert_b @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_680_subset__insertI2,axiom,
! [A2: set_a,B3: set_a,B: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_681_subset__insertI,axiom,
! [B3: set_pr5411798346947241657t_unit,A: pre_pr7278220950009878019t_unit] : ( ord_le8200006823705900825t_unit @ B3 @ ( insert6864688055023459379t_unit @ A @ B3 ) ) ).
% subset_insertI
thf(fact_682_subset__insertI,axiom,
! [B3: set_b,A: b] : ( ord_less_eq_set_b @ B3 @ ( insert_b @ A @ B3 ) ) ).
% subset_insertI
thf(fact_683_subset__insertI,axiom,
! [B3: set_a,A: a] : ( ord_less_eq_set_a @ B3 @ ( insert_a @ A @ B3 ) ) ).
% subset_insertI
thf(fact_684_subset__insert,axiom,
! [X: set_a,A2: set_set_a,B3: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B3 ) )
= ( ord_le3724670747650509150_set_a @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_685_subset__insert,axiom,
! [X: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ X @ A2 )
=> ( ( ord_le8200006823705900825t_unit @ A2 @ ( insert6864688055023459379t_unit @ X @ B3 ) )
= ( ord_le8200006823705900825t_unit @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_686_subset__insert,axiom,
! [X: b,A2: set_b,B3: set_b] :
( ~ ( member_b @ X @ A2 )
=> ( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X @ B3 ) )
= ( ord_less_eq_set_b @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_687_subset__insert,axiom,
! [X: a,A2: set_a,B3: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B3 ) )
= ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_688_insert__mono,axiom,
! [C: set_pr5411798346947241657t_unit,D2: set_pr5411798346947241657t_unit,A: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ C @ D2 )
=> ( ord_le8200006823705900825t_unit @ ( insert6864688055023459379t_unit @ A @ C ) @ ( insert6864688055023459379t_unit @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_689_insert__mono,axiom,
! [C: set_b,D2: set_b,A: b] :
( ( ord_less_eq_set_b @ C @ D2 )
=> ( ord_less_eq_set_b @ ( insert_b @ A @ C ) @ ( insert_b @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_690_insert__mono,axiom,
! [C: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_691_insert__Diff__if,axiom,
! [X: set_a,B3: set_set_a,A2: set_set_a] :
( ( ( member_set_a @ X @ B3 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B3 )
= ( minus_5736297505244876581_set_a @ A2 @ B3 ) ) )
& ( ~ ( member_set_a @ X @ B3 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B3 )
= ( insert_set_a @ X @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_692_insert__Diff__if,axiom,
! [X: pre_pr7278220950009878019t_unit,B3: set_pr5411798346947241657t_unit,A2: set_pr5411798346947241657t_unit] :
( ( ( member6939884229742472986t_unit @ X @ B3 )
=> ( ( minus_3777555517894451474t_unit @ ( insert6864688055023459379t_unit @ X @ A2 ) @ B3 )
= ( minus_3777555517894451474t_unit @ A2 @ B3 ) ) )
& ( ~ ( member6939884229742472986t_unit @ X @ B3 )
=> ( ( minus_3777555517894451474t_unit @ ( insert6864688055023459379t_unit @ X @ A2 ) @ B3 )
= ( insert6864688055023459379t_unit @ X @ ( minus_3777555517894451474t_unit @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_693_insert__Diff__if,axiom,
! [X: a,B3: set_a,A2: set_a] :
( ( ( member_a @ X @ B3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B3 )
= ( minus_minus_set_a @ A2 @ B3 ) ) )
& ( ~ ( member_a @ X @ B3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B3 )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_694_insert__Diff__if,axiom,
! [X: b,B3: set_b,A2: set_b] :
( ( ( member_b @ X @ B3 )
=> ( ( minus_minus_set_b @ ( insert_b @ X @ A2 ) @ B3 )
= ( minus_minus_set_b @ A2 @ B3 ) ) )
& ( ~ ( member_b @ X @ B3 )
=> ( ( minus_minus_set_b @ ( insert_b @ X @ A2 ) @ B3 )
= ( insert_b @ X @ ( minus_minus_set_b @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_695_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_696_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_697_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_698_subset__singleton__iff,axiom,
! [X6: set_pr5411798346947241657t_unit,A: pre_pr7278220950009878019t_unit] :
( ( ord_le8200006823705900825t_unit @ X6 @ ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) )
= ( ( X6 = bot_bo1839476491465656141t_unit )
| ( X6
= ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) ) ) ) ).
% subset_singleton_iff
thf(fact_699_subset__singleton__iff,axiom,
! [X6: set_b,A: b] :
( ( ord_less_eq_set_b @ X6 @ ( insert_b @ A @ bot_bot_set_b ) )
= ( ( X6 = bot_bot_set_b )
| ( X6
= ( insert_b @ A @ bot_bot_set_b ) ) ) ) ).
% subset_singleton_iff
thf(fact_700_subset__singleton__iff,axiom,
! [X6: set_a,A: a] :
( ( ord_less_eq_set_a @ X6 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X6 = bot_bot_set_a )
| ( X6
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_701_Diff__insert,axiom,
! [A2: set_pr5411798346947241657t_unit,A: pre_pr7278220950009878019t_unit,B3: set_pr5411798346947241657t_unit] :
( ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ B3 ) )
= ( minus_3777555517894451474t_unit @ ( minus_3777555517894451474t_unit @ A2 @ B3 ) @ ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) ) ) ).
% Diff_insert
thf(fact_702_Diff__insert,axiom,
! [A2: set_a,A: a,B3: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_703_Diff__insert,axiom,
! [A2: set_b,A: b,B3: set_b] :
( ( minus_minus_set_b @ A2 @ ( insert_b @ A @ B3 ) )
= ( minus_minus_set_b @ ( minus_minus_set_b @ A2 @ B3 ) @ ( insert_b @ A @ bot_bot_set_b ) ) ) ).
% Diff_insert
thf(fact_704_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_705_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_706_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_707_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_708_Diff__insert2,axiom,
! [A2: set_pr5411798346947241657t_unit,A: pre_pr7278220950009878019t_unit,B3: set_pr5411798346947241657t_unit] :
( ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ B3 ) )
= ( minus_3777555517894451474t_unit @ ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ bot_bo1839476491465656141t_unit ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_709_Diff__insert2,axiom,
! [A2: set_a,A: a,B3: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_710_Diff__insert2,axiom,
! [A2: set_b,A: b,B3: set_b] :
( ( minus_minus_set_b @ A2 @ ( insert_b @ A @ B3 ) )
= ( minus_minus_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_711_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_712_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_713_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_714_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_715_in__image__insert__iff,axiom,
! [B3: set_set_set_a,X: set_a,A2: set_set_a] :
( ! [C3: set_set_a] :
( ( member_set_set_a @ C3 @ B3 )
=> ~ ( member_set_a @ X @ C3 ) )
=> ( ( member_set_set_a @ A2 @ ( image_1042221919965026181_set_a @ ( insert_set_a @ X ) @ B3 ) )
= ( ( member_set_a @ X @ A2 )
& ( member_set_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B3 ) ) ) ) ).
% in_image_insert_iff
thf(fact_716_in__image__insert__iff,axiom,
! [B3: set_se2139339572462915695t_unit,X: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ! [C3: set_pr5411798346947241657t_unit] :
( ( member5449360183034373072t_unit @ C3 @ B3 )
=> ~ ( member6939884229742472986t_unit @ X @ C3 ) )
=> ( ( member5449360183034373072t_unit @ A2 @ ( image_4554393186639800441t_unit @ ( insert6864688055023459379t_unit @ X ) @ B3 ) )
= ( ( member6939884229742472986t_unit @ X @ A2 )
& ( member5449360183034373072t_unit @ ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ X @ bot_bo1839476491465656141t_unit ) ) @ B3 ) ) ) ) ).
% in_image_insert_iff
thf(fact_717_in__image__insert__iff,axiom,
! [B3: set_set_a,X: a,A2: set_a] :
( ! [C3: set_a] :
( ( member_set_a @ C3 @ B3 )
=> ~ ( member_a @ X @ C3 ) )
=> ( ( member_set_a @ A2 @ ( image_set_a_set_a @ ( insert_a @ X ) @ B3 ) )
= ( ( member_a @ X @ A2 )
& ( member_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B3 ) ) ) ) ).
% in_image_insert_iff
thf(fact_718_in__image__insert__iff,axiom,
! [B3: set_set_b,X: b,A2: set_b] :
( ! [C3: set_b] :
( ( member_set_b @ C3 @ B3 )
=> ~ ( member_b @ X @ C3 ) )
=> ( ( member_set_b @ A2 @ ( image_set_b_set_b @ ( insert_b @ X ) @ B3 ) )
= ( ( member_b @ X @ A2 )
& ( member_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) ) @ B3 ) ) ) ) ).
% in_image_insert_iff
thf(fact_719_subset__Diff__insert,axiom,
! [A2: set_set_a,B3: set_set_a,X: set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B3 @ ( insert_set_a @ X @ C ) ) )
= ( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B3 @ C ) )
& ~ ( member_set_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_720_subset__Diff__insert,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit,X: pre_pr7278220950009878019t_unit,C: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ ( minus_3777555517894451474t_unit @ B3 @ ( insert6864688055023459379t_unit @ X @ C ) ) )
= ( ( ord_le8200006823705900825t_unit @ A2 @ ( minus_3777555517894451474t_unit @ B3 @ C ) )
& ~ ( member6939884229742472986t_unit @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_721_subset__Diff__insert,axiom,
! [A2: set_b,B3: set_b,X: b,C: set_b] :
( ( ord_less_eq_set_b @ A2 @ ( minus_minus_set_b @ B3 @ ( insert_b @ X @ C ) ) )
= ( ( ord_less_eq_set_b @ A2 @ ( minus_minus_set_b @ B3 @ C ) )
& ~ ( member_b @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_722_subset__Diff__insert,axiom,
! [A2: set_a,B3: set_a,X: a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B3 @ ( insert_a @ X @ C ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B3 @ C ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_723_card__insert__le,axiom,
! [A2: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ ( insert_a @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_724_card__insert__le,axiom,
! [A2: set_set_a,X: set_a] : ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ ( insert_set_a @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_725_card__insert__le,axiom,
! [A2: set_pr5411798346947241657t_unit,X: pre_pr7278220950009878019t_unit] : ( ord_less_eq_nat @ ( finite2416455745057997787t_unit @ A2 ) @ ( finite2416455745057997787t_unit @ ( insert6864688055023459379t_unit @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_726_psubset__insert__iff,axiom,
! [A2: set_set_a,X: set_a,B3: set_set_a] :
( ( ord_less_set_set_a @ A2 @ ( insert_set_a @ X @ B3 ) )
= ( ( ( member_set_a @ X @ B3 )
=> ( ord_less_set_set_a @ A2 @ B3 ) )
& ( ~ ( member_set_a @ X @ B3 )
=> ( ( ( member_set_a @ X @ A2 )
=> ( ord_less_set_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B3 ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_727_psubset__insert__iff,axiom,
! [A2: set_pr5411798346947241657t_unit,X: pre_pr7278220950009878019t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ord_le2693654750756130573t_unit @ A2 @ ( insert6864688055023459379t_unit @ X @ B3 ) )
= ( ( ( member6939884229742472986t_unit @ X @ B3 )
=> ( ord_le2693654750756130573t_unit @ A2 @ B3 ) )
& ( ~ ( member6939884229742472986t_unit @ X @ B3 )
=> ( ( ( member6939884229742472986t_unit @ X @ A2 )
=> ( ord_le2693654750756130573t_unit @ ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ X @ bot_bo1839476491465656141t_unit ) ) @ B3 ) )
& ( ~ ( member6939884229742472986t_unit @ X @ A2 )
=> ( ord_le8200006823705900825t_unit @ A2 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_728_psubset__insert__iff,axiom,
! [A2: set_b,X: b,B3: set_b] :
( ( ord_less_set_b @ A2 @ ( insert_b @ X @ B3 ) )
= ( ( ( member_b @ X @ B3 )
=> ( ord_less_set_b @ A2 @ B3 ) )
& ( ~ ( member_b @ X @ B3 )
=> ( ( ( member_b @ X @ A2 )
=> ( ord_less_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) ) @ B3 ) )
& ( ~ ( member_b @ X @ A2 )
=> ( ord_less_eq_set_b @ A2 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_729_psubset__insert__iff,axiom,
! [A2: set_a,X: a,B3: set_a] :
( ( ord_less_set_a @ A2 @ ( insert_a @ X @ B3 ) )
= ( ( ( member_a @ X @ B3 )
=> ( ord_less_set_a @ A2 @ B3 ) )
& ( ~ ( member_a @ X @ B3 )
=> ( ( ( member_a @ X @ A2 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B3 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_730_Diff__single__insert,axiom,
! [A2: set_pr5411798346947241657t_unit,X: pre_pr7278220950009878019t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ X @ bot_bo1839476491465656141t_unit ) ) @ B3 )
=> ( ord_le8200006823705900825t_unit @ A2 @ ( insert6864688055023459379t_unit @ X @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_731_Diff__single__insert,axiom,
! [A2: set_b,X: b,B3: set_b] :
( ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) ) @ B3 )
=> ( ord_less_eq_set_b @ A2 @ ( insert_b @ X @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_732_Diff__single__insert,axiom,
! [A2: set_a,X: a,B3: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B3 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_733_subset__insert__iff,axiom,
! [A2: set_set_a,X: set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B3 ) )
= ( ( ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B3 ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_734_subset__insert__iff,axiom,
! [A2: set_pr5411798346947241657t_unit,X: pre_pr7278220950009878019t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ ( insert6864688055023459379t_unit @ X @ B3 ) )
= ( ( ( member6939884229742472986t_unit @ X @ A2 )
=> ( ord_le8200006823705900825t_unit @ ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ X @ bot_bo1839476491465656141t_unit ) ) @ B3 ) )
& ( ~ ( member6939884229742472986t_unit @ X @ A2 )
=> ( ord_le8200006823705900825t_unit @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_735_subset__insert__iff,axiom,
! [A2: set_b,X: b,B3: set_b] :
( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X @ B3 ) )
= ( ( ( member_b @ X @ A2 )
=> ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) ) @ B3 ) )
& ( ~ ( member_b @ X @ A2 )
=> ( ord_less_eq_set_b @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_736_subset__insert__iff,axiom,
! [A2: set_a,X: a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B3 ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B3 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_737_card__1__singletonE,axiom,
! [A2: set_set_a] :
( ( ( finite_card_set_a @ A2 )
= one_one_nat )
=> ~ ! [X4: set_a] :
( A2
!= ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ) ).
% card_1_singletonE
thf(fact_738_card__1__singletonE,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= one_one_nat )
=> ~ ! [X4: a] :
( A2
!= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ).
% card_1_singletonE
thf(fact_739_card__1__singletonE,axiom,
! [A2: set_b] :
( ( ( finite_card_b @ A2 )
= one_one_nat )
=> ~ ! [X4: b] :
( A2
!= ( insert_b @ X4 @ bot_bot_set_b ) ) ) ).
% card_1_singletonE
thf(fact_740_card__1__singletonE,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ( ( finite2416455745057997787t_unit @ A2 )
= one_one_nat )
=> ~ ! [X4: pre_pr7278220950009878019t_unit] :
( A2
!= ( insert6864688055023459379t_unit @ X4 @ bot_bo1839476491465656141t_unit ) ) ) ).
% card_1_singletonE
thf(fact_741_card__1__singleton__iff,axiom,
! [A2: set_set_a] :
( ( ( finite_card_set_a @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: set_a] :
( A2
= ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) ) ) ).
% card_1_singleton_iff
thf(fact_742_card__1__singleton__iff,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: a] :
( A2
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% card_1_singleton_iff
thf(fact_743_card__1__singleton__iff,axiom,
! [A2: set_b] :
( ( ( finite_card_b @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: b] :
( A2
= ( insert_b @ X3 @ bot_bot_set_b ) ) ) ) ).
% card_1_singleton_iff
thf(fact_744_card__1__singleton__iff,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ( ( finite2416455745057997787t_unit @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: pre_pr7278220950009878019t_unit] :
( A2
= ( insert6864688055023459379t_unit @ X3 @ bot_bo1839476491465656141t_unit ) ) ) ) ).
% card_1_singleton_iff
thf(fact_745_card__eq__SucD,axiom,
! [A2: set_set_a,K: nat] :
( ( ( finite_card_set_a @ A2 )
= ( suc @ K ) )
=> ? [B4: set_a,B6: set_set_a] :
( ( A2
= ( insert_set_a @ B4 @ B6 ) )
& ~ ( member_set_a @ B4 @ B6 )
& ( ( finite_card_set_a @ B6 )
= K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bot_set_set_a ) ) ) ) ).
% card_eq_SucD
thf(fact_746_card__eq__SucD,axiom,
! [A2: set_a,K: nat] :
( ( ( finite_card_a @ A2 )
= ( suc @ K ) )
=> ? [B4: a,B6: set_a] :
( ( A2
= ( insert_a @ B4 @ B6 ) )
& ~ ( member_a @ B4 @ B6 )
& ( ( finite_card_a @ B6 )
= K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bot_set_a ) ) ) ) ).
% card_eq_SucD
thf(fact_747_card__eq__SucD,axiom,
! [A2: set_b,K: nat] :
( ( ( finite_card_b @ A2 )
= ( suc @ K ) )
=> ? [B4: b,B6: set_b] :
( ( A2
= ( insert_b @ B4 @ B6 ) )
& ~ ( member_b @ B4 @ B6 )
& ( ( finite_card_b @ B6 )
= K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bot_set_b ) ) ) ) ).
% card_eq_SucD
thf(fact_748_card__eq__SucD,axiom,
! [A2: set_pr5411798346947241657t_unit,K: nat] :
( ( ( finite2416455745057997787t_unit @ A2 )
= ( suc @ K ) )
=> ? [B4: pre_pr7278220950009878019t_unit,B6: set_pr5411798346947241657t_unit] :
( ( A2
= ( insert6864688055023459379t_unit @ B4 @ B6 ) )
& ~ ( member6939884229742472986t_unit @ B4 @ B6 )
& ( ( finite2416455745057997787t_unit @ B6 )
= K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bo1839476491465656141t_unit ) ) ) ) ).
% card_eq_SucD
thf(fact_749_card__Suc__eq,axiom,
! [A2: set_set_a,K: nat] :
( ( ( finite_card_set_a @ A2 )
= ( suc @ K ) )
= ( ? [B2: set_a,B5: set_set_a] :
( ( A2
= ( insert_set_a @ B2 @ B5 ) )
& ~ ( member_set_a @ B2 @ B5 )
& ( ( finite_card_set_a @ B5 )
= K )
& ( ( K = zero_zero_nat )
=> ( B5 = bot_bot_set_set_a ) ) ) ) ) ).
% card_Suc_eq
thf(fact_750_card__Suc__eq,axiom,
! [A2: set_a,K: nat] :
( ( ( finite_card_a @ A2 )
= ( suc @ K ) )
= ( ? [B2: a,B5: set_a] :
( ( A2
= ( insert_a @ B2 @ B5 ) )
& ~ ( member_a @ B2 @ B5 )
& ( ( finite_card_a @ B5 )
= K )
& ( ( K = zero_zero_nat )
=> ( B5 = bot_bot_set_a ) ) ) ) ) ).
% card_Suc_eq
thf(fact_751_card__Suc__eq,axiom,
! [A2: set_b,K: nat] :
( ( ( finite_card_b @ A2 )
= ( suc @ K ) )
= ( ? [B2: b,B5: set_b] :
( ( A2
= ( insert_b @ B2 @ B5 ) )
& ~ ( member_b @ B2 @ B5 )
& ( ( finite_card_b @ B5 )
= K )
& ( ( K = zero_zero_nat )
=> ( B5 = bot_bot_set_b ) ) ) ) ) ).
% card_Suc_eq
thf(fact_752_card__Suc__eq,axiom,
! [A2: set_pr5411798346947241657t_unit,K: nat] :
( ( ( finite2416455745057997787t_unit @ A2 )
= ( suc @ K ) )
= ( ? [B2: pre_pr7278220950009878019t_unit,B5: set_pr5411798346947241657t_unit] :
( ( A2
= ( insert6864688055023459379t_unit @ B2 @ B5 ) )
& ~ ( member6939884229742472986t_unit @ B2 @ B5 )
& ( ( finite2416455745057997787t_unit @ B5 )
= K )
& ( ( K = zero_zero_nat )
=> ( B5 = bot_bo1839476491465656141t_unit ) ) ) ) ) ).
% card_Suc_eq
thf(fact_753_pre__digraph_Osccs__verts_Ocong,axiom,
digrap2871191568752656621ts_a_b = digrap2871191568752656621ts_a_b ).
% pre_digraph.sccs_verts.cong
thf(fact_754_card__Diff1__le,axiom,
! [A2: set_set_a,X: set_a] : ( ord_less_eq_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) @ ( finite_card_set_a @ A2 ) ) ).
% card_Diff1_le
thf(fact_755_card__Diff1__le,axiom,
! [A2: set_pr5411798346947241657t_unit,X: pre_pr7278220950009878019t_unit] : ( ord_less_eq_nat @ ( finite2416455745057997787t_unit @ ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ X @ bot_bo1839476491465656141t_unit ) ) ) @ ( finite2416455745057997787t_unit @ A2 ) ) ).
% card_Diff1_le
thf(fact_756_card__Diff1__le,axiom,
! [A2: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ).
% card_Diff1_le
thf(fact_757_card__Diff1__le,axiom,
! [A2: set_b,X: b] : ( ord_less_eq_nat @ ( finite_card_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) ) ) @ ( finite_card_b @ A2 ) ) ).
% card_Diff1_le
thf(fact_758_pre__digraph_Oscc__of_Ocong,axiom,
digrap2937667069914300949of_a_b = digrap2937667069914300949of_a_b ).
% pre_digraph.scc_of.cong
thf(fact_759_card__Diff__singleton__if,axiom,
! [X: set_a,A2: set_set_a] :
( ( ( member_set_a @ X @ A2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) )
= ( finite_card_set_a @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_760_card__Diff__singleton__if,axiom,
! [X: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( ( member6939884229742472986t_unit @ X @ A2 )
=> ( ( finite2416455745057997787t_unit @ ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ X @ bot_bo1839476491465656141t_unit ) ) )
= ( minus_minus_nat @ ( finite2416455745057997787t_unit @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member6939884229742472986t_unit @ X @ A2 )
=> ( ( finite2416455745057997787t_unit @ ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ X @ bot_bo1839476491465656141t_unit ) ) )
= ( finite2416455745057997787t_unit @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_761_card__Diff__singleton__if,axiom,
! [X: a,A2: set_a] :
( ( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( finite_card_a @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_762_card__Diff__singleton__if,axiom,
! [X: b,A2: set_b] :
( ( ( member_b @ X @ A2 )
=> ( ( finite_card_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) ) )
= ( minus_minus_nat @ ( finite_card_b @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_b @ X @ A2 )
=> ( ( finite_card_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) ) )
= ( finite_card_b @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_763_card__Diff__singleton,axiom,
! [X: set_a,A2: set_set_a] :
( ( member_set_a @ X @ A2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_764_card__Diff__singleton,axiom,
! [X: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ X @ A2 )
=> ( ( finite2416455745057997787t_unit @ ( minus_3777555517894451474t_unit @ A2 @ ( insert6864688055023459379t_unit @ X @ bot_bo1839476491465656141t_unit ) ) )
= ( minus_minus_nat @ ( finite2416455745057997787t_unit @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_765_card__Diff__singleton,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_766_card__Diff__singleton,axiom,
! [X: b,A2: set_b] :
( ( member_b @ X @ A2 )
=> ( ( finite_card_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ X @ bot_bot_set_b ) ) )
= ( minus_minus_nat @ ( finite_card_b @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_767_card__insert__le__m1,axiom,
! [N: nat,Y: set_a,X: a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( insert_a @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_768_card__insert__le__m1,axiom,
! [N: nat,Y: set_set_a,X: set_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ ( insert_set_a @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_769_card__insert__le__m1,axiom,
! [N: nat,Y: set_pr5411798346947241657t_unit,X: pre_pr7278220950009878019t_unit] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite2416455745057997787t_unit @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite2416455745057997787t_unit @ ( insert6864688055023459379t_unit @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_770_induced__eq__verts__imp__eq,axiom,
! [G2: pre_pr7278220950009878019t_unit,H: pre_pr7278220950009878019t_unit,G3: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ G2 @ H )
=> ( ( digrap5251062021860773499ph_a_b @ G3 @ H )
=> ( ( ( pre_ve642382030648772252t_unit @ G2 )
= ( pre_ve642382030648772252t_unit @ G3 ) )
=> ( G2 = G3 ) ) ) ) ).
% induced_eq_verts_imp_eq
thf(fact_771_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_772_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_773_sym__digraph_Oscc__ofI__reachable,axiom,
! [G2: pre_pr7278220950009878019t_unit,U: a,V: a] :
( ( sym_digraph_a_b @ G2 )
=> ( ( reachable_a_b @ G2 @ U @ V )
=> ( member_a @ U @ ( digrap2937667069914300949of_a_b @ G2 @ V ) ) ) ) ).
% sym_digraph.scc_ofI_reachable
thf(fact_774_sym__digraph_Oscc__ofI__reachable_H,axiom,
! [G2: pre_pr7278220950009878019t_unit,V: a,U: a] :
( ( sym_digraph_a_b @ G2 )
=> ( ( reachable_a_b @ G2 @ V @ U )
=> ( member_a @ U @ ( digrap2937667069914300949of_a_b @ G2 @ V ) ) ) ) ).
% sym_digraph.scc_ofI_reachable'
thf(fact_775_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_776_nnvs__ind__cases,axiom,
! [W: b > real,U: a,N: nat,U2: set_a] :
( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ U2 )
=> ( ( ( N = zero_zero_nat )
=> ( ( U2
= ( insert_a @ U @ bot_bot_set_a ) )
=> ~ ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) ) ) )
=> ( ! [N2: nat] :
( ( N
= ( suc @ N2 ) )
=> ! [U3: set_a] :
( ( U2
= ( insert_a @ ( graph_3614428260325061028rt_a_b @ g @ W @ U @ U3 ) @ U3 ) )
=> ( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N2 @ U3 )
=> ( ( graph_2016941059203891550ts_a_b @ g @ U @ U3 )
= bot_bot_set_a ) ) ) )
=> ~ ! [N2: nat] :
( ( N
= ( suc @ N2 ) )
=> ( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N2 @ U2 )
=> ( ( graph_2016941059203891550ts_a_b @ g @ U @ U2 )
!= bot_bot_set_a ) ) ) ) ) ) ).
% nnvs_ind_cases
thf(fact_777_n__nearest__verts_Osimps,axiom,
! [A1: b > real,A22: a,A32: nat,A42: set_a] :
( ( graph_3148032005746981223ts_a_b @ g @ A1 @ A22 @ A32 @ A42 )
= ( ? [U4: a,Uu2: b > real] :
( ( A1 = Uu2 )
& ( A22 = U4 )
& ( A32 = zero_zero_nat )
& ( A42
= ( insert_a @ U4 @ bot_bot_set_a ) )
& ( member_a @ U4 @ ( pre_ve642382030648772252t_unit @ g ) ) )
| ? [W2: b > real,U4: a,N4: nat,U5: set_a] :
( ( A1 = W2 )
& ( A22 = U4 )
& ( A32
= ( suc @ N4 ) )
& ( A42
= ( insert_a @ ( graph_3614428260325061028rt_a_b @ g @ W2 @ U4 @ U5 ) @ U5 ) )
& ( graph_3148032005746981223ts_a_b @ g @ W2 @ U4 @ N4 @ U5 )
& ( ( graph_2016941059203891550ts_a_b @ g @ U4 @ U5 )
!= bot_bot_set_a ) )
| ? [W2: b > real,U4: a,N4: nat,U5: set_a] :
( ( A1 = W2 )
& ( A22 = U4 )
& ( A32
= ( suc @ N4 ) )
& ( A42 = U5 )
& ( graph_3148032005746981223ts_a_b @ g @ W2 @ U4 @ N4 @ U5 )
& ( ( graph_2016941059203891550ts_a_b @ g @ U4 @ U5 )
= bot_bot_set_a ) ) ) ) ).
% n_nearest_verts.simps
thf(fact_778_n__nearest__verts_Ocases,axiom,
! [A1: b > real,A22: a,A32: nat,A42: set_a] :
( ( graph_3148032005746981223ts_a_b @ g @ A1 @ A22 @ A32 @ A42 )
=> ( ( ( A32 = zero_zero_nat )
=> ( ( A42
= ( insert_a @ A22 @ bot_bot_set_a ) )
=> ~ ( member_a @ A22 @ ( pre_ve642382030648772252t_unit @ g ) ) ) )
=> ( ! [N2: nat] :
( ( A32
= ( suc @ N2 ) )
=> ! [U3: set_a] :
( ( A42
= ( insert_a @ ( graph_3614428260325061028rt_a_b @ g @ A1 @ A22 @ U3 ) @ U3 ) )
=> ( ( graph_3148032005746981223ts_a_b @ g @ A1 @ A22 @ N2 @ U3 )
=> ( ( graph_2016941059203891550ts_a_b @ g @ A22 @ U3 )
= bot_bot_set_a ) ) ) )
=> ~ ! [N2: nat] :
( ( A32
= ( suc @ N2 ) )
=> ( ( graph_3148032005746981223ts_a_b @ g @ A1 @ A22 @ N2 @ A42 )
=> ( ( graph_2016941059203891550ts_a_b @ g @ A22 @ A42 )
!= bot_bot_set_a ) ) ) ) ) ) ).
% n_nearest_verts.cases
thf(fact_779_n__nnvs__unvis,axiom,
! [W: b > real,U: a,N: nat,U2: set_a] :
( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ U2 )
=> ( ( ( graph_2016941059203891550ts_a_b @ g @ U @ U2 )
!= bot_bot_set_a )
=> ( graph_3148032005746981223ts_a_b @ g @ W @ U @ ( suc @ N ) @ ( insert_a @ ( graph_3614428260325061028rt_a_b @ g @ W @ U @ U2 ) @ U2 ) ) ) ) ).
% n_nnvs_unvis
thf(fact_780_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_781_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_782_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_783_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_784_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_785_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_786_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_787_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_788_connected__arcs__empty,axiom,
( ( digrap8783888973171253482ed_a_b @ g )
=> ( ( ( pre_ar1395965042833527383t_unit @ g )
= bot_bot_set_b )
=> ( ( ( pre_ve642382030648772252t_unit @ g )
!= bot_bot_set_a )
=> ~ ! [V3: a] :
( ( pre_ve642382030648772252t_unit @ g )
!= ( insert_a @ V3 @ bot_bot_set_a ) ) ) ) ) ).
% connected_arcs_empty
thf(fact_789_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_790_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_791_mem__card1__singleton,axiom,
! [U: set_a,U2: set_set_a] :
( ( member_set_a @ U @ U2 )
=> ( ( ( finite_card_set_a @ U2 )
= one_one_nat )
=> ( U2
= ( insert_set_a @ U @ bot_bot_set_set_a ) ) ) ) ).
% mem_card1_singleton
thf(fact_792_mem__card1__singleton,axiom,
! [U: a,U2: set_a] :
( ( member_a @ U @ U2 )
=> ( ( ( finite_card_a @ U2 )
= one_one_nat )
=> ( U2
= ( insert_a @ U @ bot_bot_set_a ) ) ) ) ).
% mem_card1_singleton
thf(fact_793_mem__card1__singleton,axiom,
! [U: b,U2: set_b] :
( ( member_b @ U @ U2 )
=> ( ( ( finite_card_b @ U2 )
= one_one_nat )
=> ( U2
= ( insert_b @ U @ bot_bot_set_b ) ) ) ) ).
% mem_card1_singleton
thf(fact_794_mem__card1__singleton,axiom,
! [U: pre_pr7278220950009878019t_unit,U2: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ U @ U2 )
=> ( ( ( finite2416455745057997787t_unit @ U2 )
= one_one_nat )
=> ( U2
= ( insert6864688055023459379t_unit @ U @ bot_bo1839476491465656141t_unit ) ) ) ) ).
% mem_card1_singleton
thf(fact_795_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_796_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_797_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_798_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_799_strongly__connectedI,axiom,
! [G2: pre_pr7278220950009878019t_unit] :
( ( ( pre_ve642382030648772252t_unit @ G2 )
!= bot_bot_set_a )
=> ( ! [U6: a,V3: a] :
( ( member_a @ U6 @ ( pre_ve642382030648772252t_unit @ G2 ) )
=> ( ( member_a @ V3 @ ( pre_ve642382030648772252t_unit @ G2 ) )
=> ( reachable_a_b @ G2 @ U6 @ V3 ) ) )
=> ( digrap8691851296217657702ed_a_b @ G2 ) ) ) ).
% strongly_connectedI
thf(fact_800_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_801_strongly__connectedE,axiom,
! [G2: pre_pr7278220950009878019t_unit] :
( ( digrap8691851296217657702ed_a_b @ G2 )
=> ! [U7: a,V4: a] :
( ( ( member_a @ U7 @ ( pre_ve642382030648772252t_unit @ G2 ) )
& ( member_a @ V4 @ ( pre_ve642382030648772252t_unit @ G2 ) ) )
=> ( reachable_a_b @ G2 @ U7 @ V4 ) ) ) ).
% strongly_connectedE
thf(fact_802_strongly__connected__def,axiom,
( digrap8691851296217657702ed_a_b
= ( ^ [G4: pre_pr7278220950009878019t_unit] :
( ( ( pre_ve642382030648772252t_unit @ G4 )
!= bot_bot_set_a )
& ! [X3: a] :
( ( member_a @ X3 @ ( pre_ve642382030648772252t_unit @ G4 ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( pre_ve642382030648772252t_unit @ G4 ) )
=> ( reachable_a_b @ G4 @ X3 @ Y5 ) ) ) ) ) ) ).
% strongly_connected_def
thf(fact_803_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_804_reachable__in__vertsE,axiom,
! [G2: pre_pr7278220950009878019t_unit,U: a,V: a] :
( ( reachable_a_b @ G2 @ U @ V )
=> ~ ( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ G2 ) )
=> ~ ( member_a @ V @ ( pre_ve642382030648772252t_unit @ G2 ) ) ) ) ).
% reachable_in_vertsE
thf(fact_805_sym__digraph_Osymmetric__reachable_H,axiom,
! [G2: pre_pr7278220950009878019t_unit,V: a,W: a] :
( ( sym_digraph_a_b @ G2 )
=> ( ( reachable_a_b @ G2 @ V @ W )
=> ( reachable_a_b @ G2 @ W @ V ) ) ) ).
% sym_digraph.symmetric_reachable'
thf(fact_806_digraph_Oaxioms_I2_J,axiom,
! [G2: pre_pr7278220950009878019t_unit] :
( ( digraph_a_b @ G2 )
=> ( loopfree_digraph_a_b @ G2 ) ) ).
% digraph.axioms(2)
thf(fact_807_digraph_Oaxioms_I3_J,axiom,
! [G2: pre_pr7278220950009878019t_unit] :
( ( digraph_a_b @ G2 )
=> ( nomulti_digraph_a_b @ G2 ) ) ).
% digraph.axioms(3)
thf(fact_808_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_809_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_810_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_811_pre__digraph_Oarcs__del__vert2,axiom,
! [G2: pre_pr7278220950009878019t_unit,V: a] :
( ( pre_ar1395965042833527383t_unit @ ( pre_del_vert_a_b @ G2 @ V ) )
= ( minus_minus_set_b @ ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ G2 ) @ ( in_arcs_a_b @ G2 @ V ) ) @ ( out_arcs_a_b @ G2 @ V ) ) ) ).
% pre_digraph.arcs_del_vert2
thf(fact_812_in__sccsE,axiom,
! [C2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( digraph_pre_sccs_a_b @ g ) )
=> ~ ( ( digrap5251062021860773499ph_a_b @ C2 @ g )
=> ( ( digrap8691851296217657702ed_a_b @ C2 )
=> ? [D3: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ D3 @ g )
& ( digrap8691851296217657702ed_a_b @ D3 )
& ( ord_less_set_a @ ( pre_ve642382030648772252t_unit @ C2 ) @ ( pre_ve642382030648772252t_unit @ D3 ) ) ) ) ) ) ).
% in_sccsE
thf(fact_813_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_814_in__sccs__verts__conv,axiom,
! [S2: set_a] :
( ( member_set_a @ S2 @ ( digrap2871191568752656621ts_a_b @ g ) )
= ( member6939884229742472986t_unit @ ( digrap7873285959652527175ph_a_b @ g @ S2 ) @ ( digraph_pre_sccs_a_b @ g ) ) ) ).
% in_sccs_verts_conv
thf(fact_815_scc__for__vert__ex,axiom,
! [U: a] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
=> ? [C5: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C5 @ ( digraph_pre_sccs_a_b @ g ) )
& ( member_a @ U @ ( pre_ve642382030648772252t_unit @ C5 ) ) ) ) ).
% scc_for_vert_ex
thf(fact_816_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_817_in__sccs__imp__induced,axiom,
! [C2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( digraph_pre_sccs_a_b @ g ) )
=> ( digrap5251062021860773499ph_a_b @ C2 @ g ) ) ).
% in_sccs_imp_induced
thf(fact_818_card__sccs__verts,axiom,
( ( finite_card_set_a @ ( digrap2871191568752656621ts_a_b @ g ) )
= ( finite2416455745057997787t_unit @ ( digraph_pre_sccs_a_b @ g ) ) ) ).
% card_sccs_verts
thf(fact_819_exists__scc,axiom,
( ( ( pre_ve642382030648772252t_unit @ g )
!= bot_bot_set_a )
=> ? [C5: pre_pr7278220950009878019t_unit] : ( member6939884229742472986t_unit @ C5 @ ( digraph_pre_sccs_a_b @ g ) ) ) ).
% exists_scc
thf(fact_820_in__sccs__subset__imp__eq,axiom,
! [C2: pre_pr7278220950009878019t_unit,D: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( digraph_pre_sccs_a_b @ g ) )
=> ( ( member6939884229742472986t_unit @ D @ ( digraph_pre_sccs_a_b @ g ) )
=> ( ( ord_less_eq_set_a @ ( pre_ve642382030648772252t_unit @ C2 ) @ ( pre_ve642382030648772252t_unit @ D ) )
=> ( C2 = D ) ) ) ) ).
% in_sccs_subset_imp_eq
thf(fact_821_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_822_in__sccs__vertsI__sccs,axiom,
! [S2: set_a] :
( ( member_set_a @ S2 @ ( image_7466199892558553556_set_a @ pre_ve642382030648772252t_unit @ ( digraph_pre_sccs_a_b @ g ) ) )
=> ( member_set_a @ S2 @ ( digrap2871191568752656621ts_a_b @ g ) ) ) ).
% in_sccs_vertsI_sccs
thf(fact_823_in__verts__sccsD__sccs,axiom,
! [S2: set_a] :
( ( member_set_a @ S2 @ ( digrap2871191568752656621ts_a_b @ g ) )
=> ( member6939884229742472986t_unit @ ( digrap7873285959652527175ph_a_b @ g @ S2 ) @ ( digraph_pre_sccs_a_b @ g ) ) ) ).
% in_verts_sccsD_sccs
thf(fact_824_sym__arcs,axiom,
symmetric_a_b @ g ).
% sym_arcs
thf(fact_825_sym__digraphI,axiom,
( ( symmetric_a_b @ g )
=> ( sym_digraph_a_b @ g ) ) ).
% sym_digraphI
thf(fact_826_graph__is__union__sccs,axiom,
( ( digrap6752193522309670266on_a_b @ g @ ( digraph_pre_sccs_a_b @ g ) )
= g ) ).
% graph_is_union_sccs
thf(fact_827_scc__decomp__unique,axiom,
! [S2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ S2 @ ( digraph_pre_sccs_a_b @ g ) )
=> ( ( ( pre_ve642382030648772252t_unit @ ( digrap6752193522309670266on_a_b @ g @ S2 ) )
= ( pre_ve642382030648772252t_unit @ g ) )
=> ( S2
= ( digraph_pre_sccs_a_b @ g ) ) ) ) ).
% scc_decomp_unique
thf(fact_828_in__sccsI,axiom,
! [C2: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ C2 @ g )
=> ( ( digrap8691851296217657702ed_a_b @ C2 )
=> ( ~ ? [C6: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ C6 @ g )
& ( digrap8691851296217657702ed_a_b @ C6 )
& ( ord_less_set_a @ ( pre_ve642382030648772252t_unit @ C2 ) @ ( pre_ve642382030648772252t_unit @ C6 ) ) )
=> ( member6939884229742472986t_unit @ C2 @ ( digraph_pre_sccs_a_b @ g ) ) ) ) ) ).
% in_sccsI
thf(fact_829_pre__digraph_Osccs_Ocong,axiom,
digraph_pre_sccs_a_b = digraph_pre_sccs_a_b ).
% pre_digraph.sccs.cong
thf(fact_830_pre__digraph_Oin__sccs__imp__induced,axiom,
! [C2: pre_pr7278220950009878019t_unit,G2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( digraph_pre_sccs_a_b @ G2 ) )
=> ( digrap5251062021860773499ph_a_b @ C2 @ G2 ) ) ).
% pre_digraph.in_sccs_imp_induced
thf(fact_831_symmetric__reachable,axiom,
! [G2: pre_pr7278220950009878019t_unit,V: a,W: a] :
( ( symmetric_a_b @ G2 )
=> ( ( reachable_a_b @ G2 @ V @ W )
=> ( reachable_a_b @ G2 @ W @ V ) ) ) ).
% symmetric_reachable
thf(fact_832_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_833_sym__digraph_Osym__arcs,axiom,
! [G2: pre_pr7278220950009878019t_unit] :
( ( sym_digraph_a_b @ G2 )
=> ( symmetric_a_b @ G2 ) ) ).
% sym_digraph.sym_arcs
thf(fact_834_pre__digraph_Oin__sccs__subset__imp__eq,axiom,
! [C2: pre_pr7278220950009878019t_unit,G2: pre_pr7278220950009878019t_unit,D: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( digraph_pre_sccs_a_b @ G2 ) )
=> ( ( member6939884229742472986t_unit @ D @ ( digraph_pre_sccs_a_b @ G2 ) )
=> ( ( ord_less_eq_set_a @ ( pre_ve642382030648772252t_unit @ C2 ) @ ( pre_ve642382030648772252t_unit @ D ) )
=> ( C2 = D ) ) ) ) ).
% pre_digraph.in_sccs_subset_imp_eq
thf(fact_835_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 ) )
=> ? [C5: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C5 @ ( digraph_pre_sccs_a_b @ G2 ) )
& ( member_a @ U @ ( pre_ve642382030648772252t_unit @ C5 ) ) ) ) ) ).
% sym_digraph.scc_for_vert_ex
thf(fact_836_sym__digraph_Oexists__scc,axiom,
! [G2: pre_pr7278220950009878019t_unit] :
( ( sym_digraph_a_b @ G2 )
=> ( ( ( pre_ve642382030648772252t_unit @ G2 )
!= bot_bot_set_a )
=> ? [C5: pre_pr7278220950009878019t_unit] : ( member6939884229742472986t_unit @ C5 @ ( digraph_pre_sccs_a_b @ G2 ) ) ) ) ).
% sym_digraph.exists_scc
thf(fact_837_pre__digraph_Oin__sccsE,axiom,
! [C2: pre_pr7278220950009878019t_unit,G2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( digraph_pre_sccs_a_b @ G2 ) )
=> ~ ( ( digrap5251062021860773499ph_a_b @ C2 @ G2 )
=> ( ( digrap8691851296217657702ed_a_b @ C2 )
=> ? [D3: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ D3 @ G2 )
& ( digrap8691851296217657702ed_a_b @ D3 )
& ( ord_less_set_a @ ( pre_ve642382030648772252t_unit @ C2 ) @ ( pre_ve642382030648772252t_unit @ D3 ) ) ) ) ) ) ).
% pre_digraph.in_sccsE
thf(fact_838_pre__digraph_Oin__sccsI,axiom,
! [C2: pre_pr7278220950009878019t_unit,G2: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ C2 @ G2 )
=> ( ( digrap8691851296217657702ed_a_b @ C2 )
=> ( ~ ? [C6: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ C6 @ G2 )
& ( digrap8691851296217657702ed_a_b @ C6 )
& ( ord_less_set_a @ ( pre_ve642382030648772252t_unit @ C2 ) @ ( pre_ve642382030648772252t_unit @ C6 ) ) )
=> ( member6939884229742472986t_unit @ C2 @ ( digraph_pre_sccs_a_b @ G2 ) ) ) ) ) ).
% pre_digraph.in_sccsI
thf(fact_839_pre__digraph_Oleaf__def,axiom,
( shorte1213025427933718125af_a_a
= ( ^ [G4: pre_pr3327329314391289540t_unit,V2: a] :
( ( member_a @ V2 @ ( pre_ve5914862431884959581t_unit @ G4 ) )
& ( ( out_arcs_a_a @ G4 @ V2 )
= bot_bot_set_a ) ) ) ) ).
% pre_digraph.leaf_def
thf(fact_840_pre__digraph_Oleaf__def,axiom,
( shorte7648941882815817900af_b_a
= ( ^ [G4: pre_pr3994228789931197893t_unit,V2: b] :
( ( member_b @ V2 @ ( pre_ve2160112157898065374t_unit @ G4 ) )
& ( ( out_arcs_b_a @ G4 @ V2 )
= bot_bot_set_a ) ) ) ) ).
% pre_digraph.leaf_def
thf(fact_841_pre__digraph_Oleaf__def,axiom,
( shorte7648941882815817901af_b_b
= ( ^ [G4: pre_pr7945120425549786372t_unit,V2: b] :
( ( member_b @ V2 @ ( pre_ve6111003793516653853t_unit @ G4 ) )
& ( ( out_arcs_b_b @ G4 @ V2 )
= bot_bot_set_b ) ) ) ) ).
% pre_digraph.leaf_def
thf(fact_842_pre__digraph_Oleaf__def,axiom,
( shorte1213025427933718126af_a_b
= ( ^ [G4: pre_pr7278220950009878019t_unit,V2: a] :
( ( member_a @ V2 @ ( pre_ve642382030648772252t_unit @ G4 ) )
& ( ( out_arcs_a_b @ G4 @ V2 )
= bot_bot_set_b ) ) ) ) ).
% pre_digraph.leaf_def
thf(fact_843_pre__digraph_Oleaf__def,axiom,
( shorte1274571178419068173et_a_a
= ( ^ [G4: pre_pr3647964229410195492t_unit,V2: set_a] :
( ( member_set_a @ V2 @ ( pre_ve2608818176351713469t_unit @ G4 ) )
& ( ( out_arcs_set_a_a @ G4 @ V2 )
= bot_bot_set_a ) ) ) ) ).
% pre_digraph.leaf_def
thf(fact_844_pre__digraph_Oleaf__def,axiom,
( shorte1274571178419068174et_a_b
= ( ^ [G4: pre_pr7598855865028783971t_unit,V2: set_a] :
( ( member_set_a @ V2 @ ( pre_ve6559709811970301948t_unit @ G4 ) )
& ( ( out_arcs_set_a_b @ G4 @ V2 )
= bot_bot_set_b ) ) ) ) ).
% pre_digraph.leaf_def
thf(fact_845_pre__digraph_Oleaf__def,axiom,
( shorte2370036064065676172unit_a
= ( ^ [G4: pre_pr4248724542568774193t_unit,V2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ V2 @ ( pre_ve6338409166847992002t_unit @ G4 ) )
& ( ( out_ar3636660665668796167unit_a @ G4 @ V2 )
= bot_bot_set_a ) ) ) ) ).
% pre_digraph.leaf_def
thf(fact_846_pre__digraph_Oleaf__def,axiom,
( shorte2370036064065676173unit_b
= ( ^ [G4: pre_pr8199616178187362672t_unit,V2: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ V2 @ ( pre_ve1065928765611804673t_unit @ G4 ) )
& ( ( out_ar3636660665668796168unit_b @ G4 @ V2 )
= bot_bot_set_b ) ) ) ) ).
% pre_digraph.leaf_def
thf(fact_847_pre__digraph_Oleaf__def,axiom,
( shorte3113631386427947092t_unit
= ( ^ [G4: pre_pr4380274127475126449t_unit,V2: a] :
( ( member_a @ V2 @ ( pre_ve984463806674367994t_unit @ G4 ) )
& ( ( out_ar4380255988031067087t_unit @ G4 @ V2 )
= bot_bo1839476491465656141t_unit ) ) ) ) ).
% pre_digraph.leaf_def
thf(fact_848_pre__digraph_Oleaf__def,axiom,
( shorte1834455252847456213t_unit
= ( ^ [G4: pre_pr4845345393412534256t_unit,V2: b] :
( ( member_b @ V2 @ ( pre_ve3823154982699730361t_unit @ G4 ) )
& ( ( out_ar3101079854450576208t_unit @ G4 @ V2 )
= bot_bo1839476491465656141t_unit ) ) ) ) ).
% pre_digraph.leaf_def
thf(fact_849_scc__disj,axiom,
! [C2: pre_pr7278220950009878019t_unit,D: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( digraph_pre_sccs_a_b @ g ) )
=> ( ( member6939884229742472986t_unit @ D @ ( digraph_pre_sccs_a_b @ g ) )
=> ( ( C2 != D )
=> ( ( inf_inf_set_a @ ( pre_ve642382030648772252t_unit @ C2 ) @ ( pre_ve642382030648772252t_unit @ D ) )
= bot_bot_set_a ) ) ) ) ).
% scc_disj
thf(fact_850_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_851_graph__del__vert,axiom,
! [X: a] : ( graph_a_b @ ( pre_del_vert_a_b @ g @ X ) ) ).
% graph_del_vert
thf(fact_852_strongly__connected__imp__induce__subgraph__strongly__connected,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H @ g )
=> ( ( digrap8691851296217657702ed_a_b @ H )
=> ( digrap8691851296217657702ed_a_b @ ( digrap7873285959652527175ph_a_b @ g @ ( pre_ve642382030648772252t_unit @ H ) ) ) ) ) ).
% strongly_connected_imp_induce_subgraph_strongly_connected
thf(fact_853_subgraph__refl,axiom,
digraph_subgraph_a_b @ g @ g ).
% subgraph_refl
thf(fact_854_graph__axioms,axiom,
graph_a_b @ g ).
% graph_axioms
thf(fact_855_reachable__mono,axiom,
! [H: pre_pr7278220950009878019t_unit,U: a,V: a] :
( ( reachable_a_b @ H @ U @ V )
=> ( ( digraph_subgraph_a_b @ H @ g )
=> ( reachable_a_b @ g @ U @ V ) ) ) ).
% reachable_mono
thf(fact_856_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_857_subgraph__del__vert,axiom,
! [U: a] : ( digraph_subgraph_a_b @ ( pre_del_vert_a_b @ g @ U ) @ g ) ).
% subgraph_del_vert
thf(fact_858_digraph__subgraph,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H @ g )
=> ( digraph_a_b @ H ) ) ).
% digraph_subgraph
thf(fact_859_IntI,axiom,
! [C2: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C2 @ A2 )
=> ( ( member_set_a @ C2 @ B3 )
=> ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_860_IntI,axiom,
! [C2: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C2 @ A2 )
=> ( ( member6939884229742472986t_unit @ C2 @ B3 )
=> ( member6939884229742472986t_unit @ C2 @ ( inf_in1092213268631476299t_unit @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_861_IntI,axiom,
! [C2: b,A2: set_b,B3: set_b] :
( ( member_b @ C2 @ A2 )
=> ( ( member_b @ C2 @ B3 )
=> ( member_b @ C2 @ ( inf_inf_set_b @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_862_IntI,axiom,
! [C2: a,A2: set_a,B3: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ( member_a @ C2 @ B3 )
=> ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_863_Int__iff,axiom,
! [C2: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B3 ) )
= ( ( member_set_a @ C2 @ A2 )
& ( member_set_a @ C2 @ B3 ) ) ) ).
% Int_iff
thf(fact_864_Int__iff,axiom,
! [C2: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( inf_in1092213268631476299t_unit @ A2 @ B3 ) )
= ( ( member6939884229742472986t_unit @ C2 @ A2 )
& ( member6939884229742472986t_unit @ C2 @ B3 ) ) ) ).
% Int_iff
thf(fact_865_Int__iff,axiom,
! [C2: b,A2: set_b,B3: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A2 @ B3 ) )
= ( ( member_b @ C2 @ A2 )
& ( member_b @ C2 @ B3 ) ) ) ).
% Int_iff
thf(fact_866_Int__iff,axiom,
! [C2: a,A2: set_a,B3: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B3 ) )
= ( ( member_a @ C2 @ A2 )
& ( member_a @ C2 @ B3 ) ) ) ).
% Int_iff
thf(fact_867_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_868_sccs__verts__disjoint,axiom,
! [S2: set_a,T2: set_a] :
( ( member_set_a @ S2 @ ( digrap2871191568752656621ts_a_b @ g ) )
=> ( ( member_set_a @ T2 @ ( digrap2871191568752656621ts_a_b @ g ) )
=> ( ( S2 != T2 )
=> ( ( inf_inf_set_a @ S2 @ T2 )
= bot_bot_set_a ) ) ) ) ).
% sccs_verts_disjoint
thf(fact_869_induced__subgraph__altdef,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ H @ g )
= ( ( digraph_subgraph_a_b @ H @ g )
& ! [H2: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H2 @ g )
=> ( ( ( pre_ve642382030648772252t_unit @ H2 )
!= ( pre_ve642382030648772252t_unit @ H ) )
| ( ord_less_eq_set_b @ ( pre_ar1395965042833527383t_unit @ H2 ) @ ( pre_ar1395965042833527383t_unit @ H ) ) ) ) ) ) ).
% induced_subgraph_altdef
thf(fact_870_subgraph__induce__subgraphI,axiom,
! [V5: set_a] :
( ( ord_less_eq_set_a @ V5 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( digraph_subgraph_a_b @ ( digrap7873285959652527175ph_a_b @ g @ V5 ) @ g ) ) ).
% subgraph_induce_subgraphI
thf(fact_871_Int__subset__iff,axiom,
! [C: set_pr5411798346947241657t_unit,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ C @ ( inf_in1092213268631476299t_unit @ A2 @ B3 ) )
= ( ( ord_le8200006823705900825t_unit @ C @ A2 )
& ( ord_le8200006823705900825t_unit @ C @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_872_Int__subset__iff,axiom,
! [C: set_b,A2: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ C @ ( inf_inf_set_b @ A2 @ B3 ) )
= ( ( ord_less_eq_set_b @ C @ A2 )
& ( ord_less_eq_set_b @ C @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_873_Int__subset__iff,axiom,
! [C: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A2 @ B3 ) )
= ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_eq_set_a @ C @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_874_Int__insert__left__if0,axiom,
! [A: set_a,C: set_set_a,B3: set_set_a] :
( ~ ( member_set_a @ A @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B3 ) @ C )
= ( inf_inf_set_set_a @ B3 @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_875_Int__insert__left__if0,axiom,
! [A: pre_pr7278220950009878019t_unit,C: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ A @ C )
=> ( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A @ B3 ) @ C )
= ( inf_in1092213268631476299t_unit @ B3 @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_876_Int__insert__left__if0,axiom,
! [A: b,C: set_b,B3: set_b] :
( ~ ( member_b @ A @ C )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B3 ) @ C )
= ( inf_inf_set_b @ B3 @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_877_Int__insert__left__if0,axiom,
! [A: a,C: set_a,B3: set_a] :
( ~ ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B3 ) @ C )
= ( inf_inf_set_a @ B3 @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_878_Int__insert__left__if1,axiom,
! [A: set_a,C: set_set_a,B3: set_set_a] :
( ( member_set_a @ A @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B3 ) @ C )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B3 @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_879_Int__insert__left__if1,axiom,
! [A: pre_pr7278220950009878019t_unit,C: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ A @ C )
=> ( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A @ B3 ) @ C )
= ( insert6864688055023459379t_unit @ A @ ( inf_in1092213268631476299t_unit @ B3 @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_880_Int__insert__left__if1,axiom,
! [A: b,C: set_b,B3: set_b] :
( ( member_b @ A @ C )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B3 ) @ C )
= ( insert_b @ A @ ( inf_inf_set_b @ B3 @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_881_Int__insert__left__if1,axiom,
! [A: a,C: set_a,B3: set_a] :
( ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B3 ) @ C )
= ( insert_a @ A @ ( inf_inf_set_a @ B3 @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_882_insert__inter__insert,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A @ A2 ) @ ( insert6864688055023459379t_unit @ A @ B3 ) )
= ( insert6864688055023459379t_unit @ A @ ( inf_in1092213268631476299t_unit @ A2 @ B3 ) ) ) ).
% insert_inter_insert
thf(fact_883_insert__inter__insert,axiom,
! [A: a,A2: set_a,B3: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ ( insert_a @ A @ B3 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B3 ) ) ) ).
% insert_inter_insert
thf(fact_884_Int__insert__right__if0,axiom,
! [A: set_a,A2: set_set_a,B3: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B3 ) )
= ( inf_inf_set_set_a @ A2 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_885_Int__insert__right__if0,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ~ ( member6939884229742472986t_unit @ A @ A2 )
=> ( ( inf_in1092213268631476299t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ B3 ) )
= ( inf_in1092213268631476299t_unit @ A2 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_886_Int__insert__right__if0,axiom,
! [A: b,A2: set_b,B3: set_b] :
( ~ ( member_b @ A @ A2 )
=> ( ( inf_inf_set_b @ A2 @ ( insert_b @ A @ B3 ) )
= ( inf_inf_set_b @ A2 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_887_Int__insert__right__if0,axiom,
! [A: a,A2: set_a,B3: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( inf_inf_set_a @ A2 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_888_Int__insert__right__if1,axiom,
! [A: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B3 ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_889_Int__insert__right__if1,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ A @ A2 )
=> ( ( inf_in1092213268631476299t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ B3 ) )
= ( insert6864688055023459379t_unit @ A @ ( inf_in1092213268631476299t_unit @ A2 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_890_Int__insert__right__if1,axiom,
! [A: b,A2: set_b,B3: set_b] :
( ( member_b @ A @ A2 )
=> ( ( inf_inf_set_b @ A2 @ ( insert_b @ A @ B3 ) )
= ( insert_b @ A @ ( inf_inf_set_b @ A2 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_891_Int__insert__right__if1,axiom,
! [A: a,A2: set_a,B3: set_a] :
( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_892_induce__subgraph__ends,axiom,
! [G2: pre_pr7278220950009878019t_unit,S2: set_a] :
( ( arc_to_ends_a_b @ ( digrap7873285959652527175ph_a_b @ G2 @ S2 ) )
= ( arc_to_ends_a_b @ G2 ) ) ).
% induce_subgraph_ends
thf(fact_893_insert__disjoint_I1_J,axiom,
! [A: set_a,A2: set_set_a,B3: set_set_a] :
( ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ B3 )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B3 )
& ( ( inf_inf_set_set_a @ A2 @ B3 )
= bot_bot_set_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_894_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B3 )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B3 )
& ( ( inf_inf_set_a @ A2 @ B3 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_895_insert__disjoint_I1_J,axiom,
! [A: b,A2: set_b,B3: set_b] :
( ( ( inf_inf_set_b @ ( insert_b @ A @ A2 ) @ B3 )
= bot_bot_set_b )
= ( ~ ( member_b @ A @ B3 )
& ( ( inf_inf_set_b @ A2 @ B3 )
= bot_bot_set_b ) ) ) ).
% insert_disjoint(1)
thf(fact_896_insert__disjoint_I1_J,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A @ A2 ) @ B3 )
= bot_bo1839476491465656141t_unit )
= ( ~ ( member6939884229742472986t_unit @ A @ B3 )
& ( ( inf_in1092213268631476299t_unit @ A2 @ B3 )
= bot_bo1839476491465656141t_unit ) ) ) ).
% insert_disjoint(1)
thf(fact_897_insert__disjoint_I2_J,axiom,
! [A: set_a,A2: set_set_a,B3: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ ( insert_set_a @ A @ A2 ) @ B3 ) )
= ( ~ ( member_set_a @ A @ B3 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_898_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B3: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B3 ) )
= ( ~ ( member_a @ A @ B3 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_899_insert__disjoint_I2_J,axiom,
! [A: b,A2: set_b,B3: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ ( insert_b @ A @ A2 ) @ B3 ) )
= ( ~ ( member_b @ A @ B3 )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A2 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_900_insert__disjoint_I2_J,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( bot_bo1839476491465656141t_unit
= ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A @ A2 ) @ B3 ) )
= ( ~ ( member6939884229742472986t_unit @ A @ B3 )
& ( bot_bo1839476491465656141t_unit
= ( inf_in1092213268631476299t_unit @ A2 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_901_disjoint__insert_I1_J,axiom,
! [B3: set_set_a,A: set_a,A2: set_set_a] :
( ( ( inf_inf_set_set_a @ B3 @ ( insert_set_a @ A @ A2 ) )
= bot_bot_set_set_a )
= ( ~ ( member_set_a @ A @ B3 )
& ( ( inf_inf_set_set_a @ B3 @ A2 )
= bot_bot_set_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_902_disjoint__insert_I1_J,axiom,
! [B3: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B3 @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B3 )
& ( ( inf_inf_set_a @ B3 @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_903_disjoint__insert_I1_J,axiom,
! [B3: set_b,A: b,A2: set_b] :
( ( ( inf_inf_set_b @ B3 @ ( insert_b @ A @ A2 ) )
= bot_bot_set_b )
= ( ~ ( member_b @ A @ B3 )
& ( ( inf_inf_set_b @ B3 @ A2 )
= bot_bot_set_b ) ) ) ).
% disjoint_insert(1)
thf(fact_904_disjoint__insert_I1_J,axiom,
! [B3: set_pr5411798346947241657t_unit,A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit] :
( ( ( inf_in1092213268631476299t_unit @ B3 @ ( insert6864688055023459379t_unit @ A @ A2 ) )
= bot_bo1839476491465656141t_unit )
= ( ~ ( member6939884229742472986t_unit @ A @ B3 )
& ( ( inf_in1092213268631476299t_unit @ B3 @ A2 )
= bot_bo1839476491465656141t_unit ) ) ) ).
% disjoint_insert(1)
thf(fact_905_disjoint__insert_I2_J,axiom,
! [A2: set_set_a,B: set_a,B3: set_set_a] :
( ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ B @ B3 ) ) )
= ( ~ ( member_set_a @ B @ A2 )
& ( bot_bot_set_set_a
= ( inf_inf_set_set_a @ A2 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_906_disjoint__insert_I2_J,axiom,
! [A2: set_a,B: a,B3: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B @ B3 ) ) )
= ( ~ ( member_a @ B @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_907_disjoint__insert_I2_J,axiom,
! [A2: set_b,B: b,B3: set_b] :
( ( bot_bot_set_b
= ( inf_inf_set_b @ A2 @ ( insert_b @ B @ B3 ) ) )
= ( ~ ( member_b @ B @ A2 )
& ( bot_bot_set_b
= ( inf_inf_set_b @ A2 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_908_disjoint__insert_I2_J,axiom,
! [A2: set_pr5411798346947241657t_unit,B: pre_pr7278220950009878019t_unit,B3: set_pr5411798346947241657t_unit] :
( ( bot_bo1839476491465656141t_unit
= ( inf_in1092213268631476299t_unit @ A2 @ ( insert6864688055023459379t_unit @ B @ B3 ) ) )
= ( ~ ( member6939884229742472986t_unit @ B @ A2 )
& ( bot_bo1839476491465656141t_unit
= ( inf_in1092213268631476299t_unit @ A2 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_909_Diff__disjoint,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ A2 @ ( minus_3777555517894451474t_unit @ B3 @ A2 ) )
= bot_bo1839476491465656141t_unit ) ).
% Diff_disjoint
thf(fact_910_Diff__disjoint,axiom,
! [A2: set_a,B3: set_a] :
( ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B3 @ A2 ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_911_Diff__disjoint,axiom,
! [A2: set_b,B3: set_b] :
( ( inf_inf_set_b @ A2 @ ( minus_minus_set_b @ B3 @ A2 ) )
= bot_bot_set_b ) ).
% Diff_disjoint
thf(fact_912_inout__arcs__arc__simps_I6_J,axiom,
! [E: b,G2: pre_pr7278220950009878019t_unit,U: a] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ G2 ) )
=> ( ( inf_inf_set_b @ ( in_arcs_a_b @ G2 @ U ) @ bot_bot_set_b )
= bot_bot_set_b ) ) ).
% inout_arcs_arc_simps(6)
thf(fact_913_inout__arcs__arc__simps_I3_J,axiom,
! [E: b,G2: pre_pr7278220950009878019t_unit,U: a] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ G2 ) )
=> ( ( inf_inf_set_b @ ( out_arcs_a_b @ G2 @ U ) @ bot_bot_set_b )
= bot_bot_set_b ) ) ).
% inout_arcs_arc_simps(3)
thf(fact_914_graphI,axiom,
( ( symmetric_a_b @ g )
=> ( graph_a_b @ g ) ) ).
% graphI
thf(fact_915_Diff__Int__distrib2,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ C )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ ( inf_inf_set_a @ B3 @ C ) ) ) ).
% Diff_Int_distrib2
thf(fact_916_Diff__Int__distrib2,axiom,
! [A2: set_b,B3: set_b,C: set_b] :
( ( inf_inf_set_b @ ( minus_minus_set_b @ A2 @ B3 ) @ C )
= ( minus_minus_set_b @ ( inf_inf_set_b @ A2 @ C ) @ ( inf_inf_set_b @ B3 @ C ) ) ) ).
% Diff_Int_distrib2
thf(fact_917_Diff__Int__distrib,axiom,
! [C: set_a,A2: set_a,B3: set_a] :
( ( inf_inf_set_a @ C @ ( minus_minus_set_a @ A2 @ B3 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C @ A2 ) @ ( inf_inf_set_a @ C @ B3 ) ) ) ).
% Diff_Int_distrib
thf(fact_918_Diff__Int__distrib,axiom,
! [C: set_b,A2: set_b,B3: set_b] :
( ( inf_inf_set_b @ C @ ( minus_minus_set_b @ A2 @ B3 ) )
= ( minus_minus_set_b @ ( inf_inf_set_b @ C @ A2 ) @ ( inf_inf_set_b @ C @ B3 ) ) ) ).
% Diff_Int_distrib
thf(fact_919_Diff__Diff__Int,axiom,
! [A2: set_a,B3: set_a] :
( ( minus_minus_set_a @ A2 @ ( minus_minus_set_a @ A2 @ B3 ) )
= ( inf_inf_set_a @ A2 @ B3 ) ) ).
% Diff_Diff_Int
thf(fact_920_Diff__Diff__Int,axiom,
! [A2: set_b,B3: set_b] :
( ( minus_minus_set_b @ A2 @ ( minus_minus_set_b @ A2 @ B3 ) )
= ( inf_inf_set_b @ A2 @ B3 ) ) ).
% Diff_Diff_Int
thf(fact_921_Diff__Int2,axiom,
! [A2: set_a,C: set_a,B3: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ ( inf_inf_set_a @ B3 @ C ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C ) @ B3 ) ) ).
% Diff_Int2
thf(fact_922_Diff__Int2,axiom,
! [A2: set_b,C: set_b,B3: set_b] :
( ( minus_minus_set_b @ ( inf_inf_set_b @ A2 @ C ) @ ( inf_inf_set_b @ B3 @ C ) )
= ( minus_minus_set_b @ ( inf_inf_set_b @ A2 @ C ) @ B3 ) ) ).
% Diff_Int2
thf(fact_923_Int__Diff,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ C )
= ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B3 @ C ) ) ) ).
% Int_Diff
thf(fact_924_Int__Diff,axiom,
! [A2: set_b,B3: set_b,C: set_b] :
( ( minus_minus_set_b @ ( inf_inf_set_b @ A2 @ B3 ) @ C )
= ( inf_inf_set_b @ A2 @ ( minus_minus_set_b @ B3 @ C ) ) ) ).
% Int_Diff
thf(fact_925_disjoint__iff__not__equal,axiom,
! [A2: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A2 @ B3 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ! [Y5: a] :
( ( member_a @ Y5 @ B3 )
=> ( X3 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_926_disjoint__iff__not__equal,axiom,
! [A2: set_b,B3: set_b] :
( ( ( inf_inf_set_b @ A2 @ B3 )
= bot_bot_set_b )
= ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ! [Y5: b] :
( ( member_b @ Y5 @ B3 )
=> ( X3 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_927_disjoint__iff__not__equal,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ( inf_in1092213268631476299t_unit @ A2 @ B3 )
= bot_bo1839476491465656141t_unit )
= ( ! [X3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ A2 )
=> ! [Y5: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ Y5 @ B3 )
=> ( X3 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_928_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_929_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_930_Int__empty__right,axiom,
! [A2: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ A2 @ bot_bo1839476491465656141t_unit )
= bot_bo1839476491465656141t_unit ) ).
% Int_empty_right
thf(fact_931_Int__empty__left,axiom,
! [B3: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B3 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_932_Int__empty__left,axiom,
! [B3: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ B3 )
= bot_bot_set_b ) ).
% Int_empty_left
thf(fact_933_Int__empty__left,axiom,
! [B3: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ bot_bo1839476491465656141t_unit @ B3 )
= bot_bo1839476491465656141t_unit ) ).
% Int_empty_left
thf(fact_934_disjoint__iff,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B3 )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ~ ( member_set_a @ X3 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_935_disjoint__iff,axiom,
! [A2: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A2 @ B3 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_936_disjoint__iff,axiom,
! [A2: set_b,B3: set_b] :
( ( ( inf_inf_set_b @ A2 @ B3 )
= bot_bot_set_b )
= ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ~ ( member_b @ X3 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_937_disjoint__iff,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ( inf_in1092213268631476299t_unit @ A2 @ B3 )
= bot_bo1839476491465656141t_unit )
= ( ! [X3: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X3 @ A2 )
=> ~ ( member6939884229742472986t_unit @ X3 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_938_Int__emptyI,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ~ ( member_set_a @ X4 @ B3 ) )
=> ( ( inf_inf_set_set_a @ A2 @ B3 )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_939_Int__emptyI,axiom,
! [A2: set_a,B3: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ~ ( member_a @ X4 @ B3 ) )
=> ( ( inf_inf_set_a @ A2 @ B3 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_940_Int__emptyI,axiom,
! [A2: set_b,B3: set_b] :
( ! [X4: b] :
( ( member_b @ X4 @ A2 )
=> ~ ( member_b @ X4 @ B3 ) )
=> ( ( inf_inf_set_b @ A2 @ B3 )
= bot_bot_set_b ) ) ).
% Int_emptyI
thf(fact_941_Int__emptyI,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ! [X4: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X4 @ A2 )
=> ~ ( member6939884229742472986t_unit @ X4 @ B3 ) )
=> ( ( inf_in1092213268631476299t_unit @ A2 @ B3 )
= bot_bo1839476491465656141t_unit ) ) ).
% Int_emptyI
thf(fact_942_Int__Collect__mono,axiom,
! [A2: set_set_a,B3: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B3 @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_943_Int__Collect__mono,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit,P: pre_pr7278220950009878019t_unit > $o,Q: pre_pr7278220950009878019t_unit > $o] :
( ( ord_le8200006823705900825t_unit @ A2 @ B3 )
=> ( ! [X4: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le8200006823705900825t_unit @ ( inf_in1092213268631476299t_unit @ A2 @ ( collec8000012497822511960t_unit @ P ) ) @ ( inf_in1092213268631476299t_unit @ B3 @ ( collec8000012497822511960t_unit @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_944_Int__Collect__mono,axiom,
! [A2: set_b,B3: set_b,P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ! [X4: b] :
( ( member_b @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ ( collect_b @ P ) ) @ ( inf_inf_set_b @ B3 @ ( collect_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_945_Int__Collect__mono,axiom,
! [A2: set_a,B3: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B3 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_946_Int__greatest,axiom,
! [C: set_pr5411798346947241657t_unit,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ C @ A2 )
=> ( ( ord_le8200006823705900825t_unit @ C @ B3 )
=> ( ord_le8200006823705900825t_unit @ C @ ( inf_in1092213268631476299t_unit @ A2 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_947_Int__greatest,axiom,
! [C: set_b,A2: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ C @ A2 )
=> ( ( ord_less_eq_set_b @ C @ B3 )
=> ( ord_less_eq_set_b @ C @ ( inf_inf_set_b @ A2 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_948_Int__greatest,axiom,
! [C: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B3 )
=> ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_949_Int__absorb2,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ B3 )
=> ( ( inf_in1092213268631476299t_unit @ A2 @ B3 )
= A2 ) ) ).
% Int_absorb2
thf(fact_950_Int__absorb2,axiom,
! [A2: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ( inf_inf_set_b @ A2 @ B3 )
= A2 ) ) ).
% Int_absorb2
thf(fact_951_Int__absorb2,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( inf_inf_set_a @ A2 @ B3 )
= A2 ) ) ).
% Int_absorb2
thf(fact_952_Int__absorb1,axiom,
! [B3: set_pr5411798346947241657t_unit,A2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ B3 @ A2 )
=> ( ( inf_in1092213268631476299t_unit @ A2 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_953_Int__absorb1,axiom,
! [B3: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B3 @ A2 )
=> ( ( inf_inf_set_b @ A2 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_954_Int__absorb1,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_955_Int__lower2,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] : ( ord_le8200006823705900825t_unit @ ( inf_in1092213268631476299t_unit @ A2 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_956_Int__lower2,axiom,
! [A2: set_b,B3: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_957_Int__lower2,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_958_Int__lower1,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] : ( ord_le8200006823705900825t_unit @ ( inf_in1092213268631476299t_unit @ A2 @ B3 ) @ A2 ) ).
% Int_lower1
thf(fact_959_Int__lower1,axiom,
! [A2: set_b,B3: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B3 ) @ A2 ) ).
% Int_lower1
thf(fact_960_Int__lower1,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ A2 ) ).
% Int_lower1
thf(fact_961_Int__mono,axiom,
! [A2: set_pr5411798346947241657t_unit,C: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit,D2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A2 @ C )
=> ( ( ord_le8200006823705900825t_unit @ B3 @ D2 )
=> ( ord_le8200006823705900825t_unit @ ( inf_in1092213268631476299t_unit @ A2 @ B3 ) @ ( inf_in1092213268631476299t_unit @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_962_Int__mono,axiom,
! [A2: set_b,C: set_b,B3: set_b,D2: set_b] :
( ( ord_less_eq_set_b @ A2 @ C )
=> ( ( ord_less_eq_set_b @ B3 @ D2 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B3 ) @ ( inf_inf_set_b @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_963_Int__mono,axiom,
! [A2: set_a,C: set_a,B3: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B3 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_964_pre__digraph_OUnion_Ocong,axiom,
digrap6752193522309670266on_a_b = digrap6752193522309670266on_a_b ).
% pre_digraph.Union.cong
thf(fact_965_subgraph__trans,axiom,
! [G2: pre_pr7278220950009878019t_unit,H: pre_pr7278220950009878019t_unit,I5: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ G2 @ H )
=> ( ( digraph_subgraph_a_b @ H @ I5 )
=> ( digraph_subgraph_a_b @ G2 @ I5 ) ) ) ).
% subgraph_trans
thf(fact_966_subgraph_Osub__G,axiom,
! [T2: pre_pr7278220950009878019t_unit,G2: pre_pr7278220950009878019t_unit] :
( ( shorte3657265928840388360ph_a_b @ T2 @ G2 )
=> ( digraph_subgraph_a_b @ T2 @ G2 ) ) ).
% subgraph.sub_G
thf(fact_967_IntE,axiom,
! [C2: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B3 ) )
=> ~ ( ( member_set_a @ C2 @ A2 )
=> ~ ( member_set_a @ C2 @ B3 ) ) ) ).
% IntE
thf(fact_968_IntE,axiom,
! [C2: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( inf_in1092213268631476299t_unit @ A2 @ B3 ) )
=> ~ ( ( member6939884229742472986t_unit @ C2 @ A2 )
=> ~ ( member6939884229742472986t_unit @ C2 @ B3 ) ) ) ).
% IntE
thf(fact_969_IntE,axiom,
! [C2: b,A2: set_b,B3: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A2 @ B3 ) )
=> ~ ( ( member_b @ C2 @ A2 )
=> ~ ( member_b @ C2 @ B3 ) ) ) ).
% IntE
thf(fact_970_IntE,axiom,
! [C2: a,A2: set_a,B3: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B3 ) )
=> ~ ( ( member_a @ C2 @ A2 )
=> ~ ( member_a @ C2 @ B3 ) ) ) ).
% IntE
thf(fact_971_IntD1,axiom,
! [C2: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B3 ) )
=> ( member_set_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_972_IntD1,axiom,
! [C2: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( inf_in1092213268631476299t_unit @ A2 @ B3 ) )
=> ( member6939884229742472986t_unit @ C2 @ A2 ) ) ).
% IntD1
thf(fact_973_IntD1,axiom,
! [C2: b,A2: set_b,B3: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A2 @ B3 ) )
=> ( member_b @ C2 @ A2 ) ) ).
% IntD1
thf(fact_974_IntD1,axiom,
! [C2: a,A2: set_a,B3: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B3 ) )
=> ( member_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_975_IntD2,axiom,
! [C2: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C2 @ ( inf_inf_set_set_a @ A2 @ B3 ) )
=> ( member_set_a @ C2 @ B3 ) ) ).
% IntD2
thf(fact_976_IntD2,axiom,
! [C2: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( member6939884229742472986t_unit @ C2 @ ( inf_in1092213268631476299t_unit @ A2 @ B3 ) )
=> ( member6939884229742472986t_unit @ C2 @ B3 ) ) ).
% IntD2
thf(fact_977_IntD2,axiom,
! [C2: b,A2: set_b,B3: set_b] :
( ( member_b @ C2 @ ( inf_inf_set_b @ A2 @ B3 ) )
=> ( member_b @ C2 @ B3 ) ) ).
% IntD2
thf(fact_978_IntD2,axiom,
! [C2: a,A2: set_a,B3: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B3 ) )
=> ( member_a @ C2 @ B3 ) ) ).
% IntD2
thf(fact_979_Int__assoc,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ C )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B3 @ C ) ) ) ).
% Int_assoc
thf(fact_980_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_981_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B5: set_a] : ( inf_inf_set_a @ B5 @ A4 ) ) ) ).
% Int_commute
thf(fact_982_Int__left__absorb,axiom,
! [A2: set_a,B3: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B3 ) )
= ( inf_inf_set_a @ A2 @ B3 ) ) ).
% Int_left_absorb
thf(fact_983_Int__left__commute,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B3 @ C ) )
= ( inf_inf_set_a @ B3 @ ( inf_inf_set_a @ A2 @ C ) ) ) ).
% Int_left_commute
thf(fact_984_pre__digraph_Oreachable__mono,axiom,
! [H: pre_pr7278220950009878019t_unit,U: a,V: a,G2: pre_pr7278220950009878019t_unit] :
( ( reachable_a_b @ H @ U @ V )
=> ( ( digraph_subgraph_a_b @ H @ G2 )
=> ( reachable_a_b @ G2 @ U @ V ) ) ) ).
% pre_digraph.reachable_mono
thf(fact_985_Int__insert__right,axiom,
! [A: set_a,A2: set_set_a,B3: set_set_a] :
( ( ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B3 ) )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ A2 @ B3 ) ) ) )
& ( ~ ( member_set_a @ A @ A2 )
=> ( ( inf_inf_set_set_a @ A2 @ ( insert_set_a @ A @ B3 ) )
= ( inf_inf_set_set_a @ A2 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_986_Int__insert__right,axiom,
! [A: pre_pr7278220950009878019t_unit,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ( member6939884229742472986t_unit @ A @ A2 )
=> ( ( inf_in1092213268631476299t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ B3 ) )
= ( insert6864688055023459379t_unit @ A @ ( inf_in1092213268631476299t_unit @ A2 @ B3 ) ) ) )
& ( ~ ( member6939884229742472986t_unit @ A @ A2 )
=> ( ( inf_in1092213268631476299t_unit @ A2 @ ( insert6864688055023459379t_unit @ A @ B3 ) )
= ( inf_in1092213268631476299t_unit @ A2 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_987_Int__insert__right,axiom,
! [A: b,A2: set_b,B3: set_b] :
( ( ( member_b @ A @ A2 )
=> ( ( inf_inf_set_b @ A2 @ ( insert_b @ A @ B3 ) )
= ( insert_b @ A @ ( inf_inf_set_b @ A2 @ B3 ) ) ) )
& ( ~ ( member_b @ A @ A2 )
=> ( ( inf_inf_set_b @ A2 @ ( insert_b @ A @ B3 ) )
= ( inf_inf_set_b @ A2 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_988_Int__insert__right,axiom,
! [A: a,A2: set_a,B3: set_a] :
( ( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B3 ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_989_Int__insert__left,axiom,
! [A: set_a,C: set_set_a,B3: set_set_a] :
( ( ( member_set_a @ A @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B3 ) @ C )
= ( insert_set_a @ A @ ( inf_inf_set_set_a @ B3 @ C ) ) ) )
& ( ~ ( member_set_a @ A @ C )
=> ( ( inf_inf_set_set_a @ ( insert_set_a @ A @ B3 ) @ C )
= ( inf_inf_set_set_a @ B3 @ C ) ) ) ) ).
% Int_insert_left
thf(fact_990_Int__insert__left,axiom,
! [A: pre_pr7278220950009878019t_unit,C: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ( member6939884229742472986t_unit @ A @ C )
=> ( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A @ B3 ) @ C )
= ( insert6864688055023459379t_unit @ A @ ( inf_in1092213268631476299t_unit @ B3 @ C ) ) ) )
& ( ~ ( member6939884229742472986t_unit @ A @ C )
=> ( ( inf_in1092213268631476299t_unit @ ( insert6864688055023459379t_unit @ A @ B3 ) @ C )
= ( inf_in1092213268631476299t_unit @ B3 @ C ) ) ) ) ).
% Int_insert_left
thf(fact_991_Int__insert__left,axiom,
! [A: b,C: set_b,B3: set_b] :
( ( ( member_b @ A @ C )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B3 ) @ C )
= ( insert_b @ A @ ( inf_inf_set_b @ B3 @ C ) ) ) )
& ( ~ ( member_b @ A @ C )
=> ( ( inf_inf_set_b @ ( insert_b @ A @ B3 ) @ C )
= ( inf_inf_set_b @ B3 @ C ) ) ) ) ).
% Int_insert_left
thf(fact_992_Int__insert__left,axiom,
! [A: a,C: set_a,B3: set_a] :
( ( ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B3 ) @ C )
= ( insert_a @ A @ ( inf_inf_set_a @ B3 @ C ) ) ) )
& ( ~ ( member_a @ A @ C )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B3 ) @ C )
= ( inf_inf_set_a @ B3 @ C ) ) ) ) ).
% Int_insert_left
thf(fact_993_induced__imp__subgraph,axiom,
! [H: pre_pr7278220950009878019t_unit,G2: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ H @ G2 )
=> ( digraph_subgraph_a_b @ H @ G2 ) ) ).
% induced_imp_subgraph
thf(fact_994_digraph_Odigraph__subgraph,axiom,
! [G2: pre_pr7278220950009878019t_unit,H: pre_pr7278220950009878019t_unit] :
( ( digraph_a_b @ G2 )
=> ( ( digraph_subgraph_a_b @ H @ G2 )
=> ( digraph_a_b @ H ) ) ) ).
% digraph.digraph_subgraph
thf(fact_995_graph_Oaxioms_I1_J,axiom,
! [G2: pre_pr7278220950009878019t_unit] :
( ( graph_a_b @ G2 )
=> ( digraph_a_b @ G2 ) ) ).
% graph.axioms(1)
thf(fact_996_graph_Oaxioms_I2_J,axiom,
! [G2: pre_pr7278220950009878019t_unit] :
( ( graph_a_b @ G2 )
=> ( pseudo_graph_a_b @ G2 ) ) ).
% graph.axioms(2)
thf(fact_997_subgraph__imp__subverts,axiom,
! [H: pre_pr7278220950009878019t_unit,G2: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H @ G2 )
=> ( ord_less_eq_set_a @ ( pre_ve642382030648772252t_unit @ H ) @ ( pre_ve642382030648772252t_unit @ G2 ) ) ) ).
% subgraph_imp_subverts
thf(fact_998_image__Int__subset,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] : ( ord_le3724670747650509150_set_a @ ( image_7466199892558553556_set_a @ F @ ( inf_in1092213268631476299t_unit @ A2 @ B3 ) ) @ ( inf_inf_set_set_a @ ( image_7466199892558553556_set_a @ F @ A2 ) @ ( image_7466199892558553556_set_a @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_999_image__Int__subset,axiom,
! [F: a > set_a,A2: set_a,B3: set_a] : ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ ( inf_inf_set_a @ A2 @ B3 ) ) @ ( inf_inf_set_set_a @ ( image_a_set_a @ F @ A2 ) @ ( image_a_set_a @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_1000_image__Int__subset,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a,B3: set_set_a] : ( ord_le8200006823705900825t_unit @ ( image_6801035452528096924t_unit @ F @ ( inf_inf_set_set_a @ A2 @ B3 ) ) @ ( inf_in1092213268631476299t_unit @ ( image_6801035452528096924t_unit @ F @ A2 ) @ ( image_6801035452528096924t_unit @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_1001_image__Int__subset,axiom,
! [F: a > pre_pr7278220950009878019t_unit,A2: set_a,B3: set_a] : ( ord_le8200006823705900825t_unit @ ( image_5713294457175270716t_unit @ F @ ( inf_inf_set_a @ A2 @ B3 ) ) @ ( inf_in1092213268631476299t_unit @ ( image_5713294457175270716t_unit @ F @ A2 ) @ ( image_5713294457175270716t_unit @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_1002_image__Int__subset,axiom,
! [F: a > b,A2: set_a,B3: set_a] : ( ord_less_eq_set_b @ ( image_a_b @ F @ ( inf_inf_set_a @ A2 @ B3 ) ) @ ( inf_inf_set_b @ ( image_a_b @ F @ A2 ) @ ( image_a_b @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_1003_image__Int__subset,axiom,
! [F: a > a,A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F @ ( inf_inf_set_a @ A2 @ B3 ) ) @ ( inf_inf_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_1004_Int__Diff__disjoint,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ ( inf_in1092213268631476299t_unit @ A2 @ B3 ) @ ( minus_3777555517894451474t_unit @ A2 @ B3 ) )
= bot_bo1839476491465656141t_unit ) ).
% Int_Diff_disjoint
thf(fact_1005_Int__Diff__disjoint,axiom,
! [A2: set_a,B3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ ( minus_minus_set_a @ A2 @ B3 ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_1006_Int__Diff__disjoint,axiom,
! [A2: set_b,B3: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ A2 @ B3 ) @ ( minus_minus_set_b @ A2 @ B3 ) )
= bot_bot_set_b ) ).
% Int_Diff_disjoint
thf(fact_1007_Diff__triv,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] :
( ( ( inf_in1092213268631476299t_unit @ A2 @ B3 )
= bot_bo1839476491465656141t_unit )
=> ( ( minus_3777555517894451474t_unit @ A2 @ B3 )
= A2 ) ) ).
% Diff_triv
thf(fact_1008_Diff__triv,axiom,
! [A2: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A2 @ B3 )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A2 @ B3 )
= A2 ) ) ).
% Diff_triv
thf(fact_1009_Diff__triv,axiom,
! [A2: set_b,B3: set_b] :
( ( ( inf_inf_set_b @ A2 @ B3 )
= bot_bot_set_b )
=> ( ( minus_minus_set_b @ A2 @ B3 )
= A2 ) ) ).
% Diff_triv
thf(fact_1010_subgraph__induce__subgraphI2,axiom,
! [H: pre_pr7278220950009878019t_unit,G2: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H @ G2 )
=> ( digraph_subgraph_a_b @ H @ ( digrap7873285959652527175ph_a_b @ G2 @ ( pre_ve642382030648772252t_unit @ H ) ) ) ) ).
% subgraph_induce_subgraphI2
thf(fact_1011_spanningE,axiom,
! [H: pre_pr7278220950009878019t_unit,G2: pre_pr7278220950009878019t_unit] :
( ( digraph_spanning_a_b @ H @ G2 )
=> ( ( digraph_subgraph_a_b @ H @ G2 )
& ( ( pre_ve642382030648772252t_unit @ G2 )
= ( pre_ve642382030648772252t_unit @ H ) ) ) ) ).
% spanningE
thf(fact_1012_spanning__def,axiom,
( digraph_spanning_a_b
= ( ^ [H3: pre_pr7278220950009878019t_unit,G4: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H3 @ G4 )
& ( ( pre_ve642382030648772252t_unit @ G4 )
= ( pre_ve642382030648772252t_unit @ H3 ) ) ) ) ) ).
% spanning_def
thf(fact_1013_digraph_OgraphI,axiom,
! [G2: pre_pr7278220950009878019t_unit] :
( ( digraph_a_b @ G2 )
=> ( ( symmetric_a_b @ G2 )
=> ( graph_a_b @ G2 ) ) ) ).
% digraph.graphI
thf(fact_1014_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_1015_Digraph_Ograph__def,axiom,
( graph_a_b
= ( ^ [G4: pre_pr7278220950009878019t_unit] :
( ( digraph_a_b @ G4 )
& ( pseudo_graph_a_b @ G4 ) ) ) ) ).
% Digraph.graph_def
thf(fact_1016_graph_Ointro,axiom,
! [G2: pre_pr7278220950009878019t_unit] :
( ( digraph_a_b @ G2 )
=> ( ( pseudo_graph_a_b @ G2 )
=> ( graph_a_b @ G2 ) ) ) ).
% graph.intro
thf(fact_1017_induced__subgraphI_H,axiom,
! [H: pre_pr7278220950009878019t_unit,G2: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H @ G2 )
=> ( ! [H4: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H4 @ G2 )
=> ( ( ( pre_ve642382030648772252t_unit @ H4 )
!= ( pre_ve642382030648772252t_unit @ H ) )
| ( ord_less_eq_set_b @ ( pre_ar1395965042833527383t_unit @ H4 ) @ ( pre_ar1395965042833527383t_unit @ H ) ) ) )
=> ( digrap5251062021860773499ph_a_b @ H @ G2 ) ) ) ).
% induced_subgraphI'
thf(fact_1018_pre__digraph_Oinduced__subgraph__altdef,axiom,
( digrap5251062021860773499ph_a_b
= ( ^ [H3: pre_pr7278220950009878019t_unit,G4: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H3 @ G4 )
& ! [H2: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H2 @ G4 )
=> ( ( ( pre_ve642382030648772252t_unit @ H2 )
!= ( pre_ve642382030648772252t_unit @ H3 ) )
| ( ord_less_eq_set_b @ ( pre_ar1395965042833527383t_unit @ H2 ) @ ( pre_ar1395965042833527383t_unit @ H3 ) ) ) ) ) ) ) ).
% pre_digraph.induced_subgraph_altdef
thf(fact_1019_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_1020_sym__digraph_Oscc__decomp__unique,axiom,
! [G2: pre_pr7278220950009878019t_unit,S2: set_pr5411798346947241657t_unit] :
( ( sym_digraph_a_b @ G2 )
=> ( ( ord_le8200006823705900825t_unit @ S2 @ ( digraph_pre_sccs_a_b @ G2 ) )
=> ( ( ( pre_ve642382030648772252t_unit @ ( digrap6752193522309670266on_a_b @ G2 @ S2 ) )
= ( pre_ve642382030648772252t_unit @ G2 ) )
=> ( S2
= ( digraph_pre_sccs_a_b @ G2 ) ) ) ) ) ).
% sym_digraph.scc_decomp_unique
thf(fact_1021_subgraph__cycle,axiom,
! [H: pre_pr7278220950009878019t_unit,P2: list_b] :
( ( digraph_subgraph_a_b @ H @ g )
=> ( ( arc_pre_cycle_a_b @ H @ P2 )
=> ( arc_pre_cycle_a_b @ g @ P2 ) ) ) ).
% subgraph_cycle
thf(fact_1022_graph_Oconnected__iff__reachable,axiom,
! [G2: pre_pr7278220950009878019t_unit] :
( ( graph_a_b @ G2 )
=> ( ( digrap8783888973171253482ed_a_b @ G2 )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( pre_ve642382030648772252t_unit @ G2 ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( pre_ve642382030648772252t_unit @ G2 ) )
=> ( reachable_a_b @ G2 @ X3 @ Y5 ) ) )
& ( ( pre_ve642382030648772252t_unit @ G2 )
!= bot_bot_set_a ) ) ) ) ).
% graph.connected_iff_reachable
thf(fact_1023_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_1024_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_1025_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_1026_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_1027_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_1028_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_1029_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_1030_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_1031_inf__bot__right,axiom,
! [X: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ X @ bot_bo1839476491465656141t_unit )
= bot_bo1839476491465656141t_unit ) ).
% inf_bot_right
thf(fact_1032_le__inf__iff,axiom,
! [X: real,Y: real,Z4: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z4 ) )
= ( ( ord_less_eq_real @ X @ Y )
& ( ord_less_eq_real @ X @ Z4 ) ) ) ).
% le_inf_iff
thf(fact_1033_le__inf__iff,axiom,
! [X: set_pr5411798346947241657t_unit,Y: set_pr5411798346947241657t_unit,Z4: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ X @ ( inf_in1092213268631476299t_unit @ Y @ Z4 ) )
= ( ( ord_le8200006823705900825t_unit @ X @ Y )
& ( ord_le8200006823705900825t_unit @ X @ Z4 ) ) ) ).
% le_inf_iff
thf(fact_1034_le__inf__iff,axiom,
! [X: set_b,Y: set_b,Z4: set_b] :
( ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ Y @ Z4 ) )
= ( ( ord_less_eq_set_b @ X @ Y )
& ( ord_less_eq_set_b @ X @ Z4 ) ) ) ).
% le_inf_iff
thf(fact_1035_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z4 ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z4 ) ) ) ).
% le_inf_iff
thf(fact_1036_le__inf__iff,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z4 ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z4 ) ) ) ).
% le_inf_iff
thf(fact_1037_inf_Obounded__iff,axiom,
! [A: real,B: real,C2: real] :
( ( ord_less_eq_real @ A @ ( inf_inf_real @ B @ C2 ) )
= ( ( ord_less_eq_real @ A @ B )
& ( ord_less_eq_real @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_1038_inf_Obounded__iff,axiom,
! [A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit,C2: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A @ ( inf_in1092213268631476299t_unit @ B @ C2 ) )
= ( ( ord_le8200006823705900825t_unit @ A @ B )
& ( ord_le8200006823705900825t_unit @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_1039_inf_Obounded__iff,axiom,
! [A: set_b,B: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ A @ ( inf_inf_set_b @ B @ C2 ) )
= ( ( ord_less_eq_set_b @ A @ B )
& ( ord_less_eq_set_b @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_1040_inf_Obounded__iff,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C2 ) )
= ( ( ord_less_eq_set_a @ A @ B )
& ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_1041_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C2 ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_1042_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_1043_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_1044_inf__bot__left,axiom,
! [X: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ bot_bo1839476491465656141t_unit @ X )
= bot_bo1839476491465656141t_unit ) ).
% inf_bot_left
thf(fact_1045_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_1046_bot__set__def,axiom,
( bot_bot_set_b
= ( collect_b @ bot_bot_b_o ) ) ).
% bot_set_def
thf(fact_1047_bot__set__def,axiom,
( bot_bo1839476491465656141t_unit
= ( collec8000012497822511960t_unit @ bot_bo8537066411596906360unit_o ) ) ).
% bot_set_def
thf(fact_1048_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1049_wf__digraph_Obranching__points_Ocong,axiom,
graph_4596510882073158607ts_a_b = graph_4596510882073158607ts_a_b ).
% wf_digraph.branching_points.cong
thf(fact_1050_wf__digraph_Omerging__points_Ocong,axiom,
graph_2957805489637798020ts_a_b = graph_2957805489637798020ts_a_b ).
% wf_digraph.merging_points.cong
thf(fact_1051_wf__digraph_Olast__branching__points_Ocong,axiom,
graph_1747835947655717337ts_a_b = graph_1747835947655717337ts_a_b ).
% wf_digraph.last_branching_points.cong
thf(fact_1052_wf__digraph_Olast__merging__points_Ocong,axiom,
graph_2659413520663303054ts_a_b = graph_2659413520663303054ts_a_b ).
% wf_digraph.last_merging_points.cong
thf(fact_1053_wf__digraph_Ois__chain_Ocong,axiom,
graph_3890552050688490787in_a_b = graph_3890552050688490787in_a_b ).
% wf_digraph.is_chain.cong
thf(fact_1054_wf__digraph_Ois__chain_H_Ocong,axiom,
graph_8150681439568091980in_a_b = graph_8150681439568091980in_a_b ).
% wf_digraph.is_chain'.cong
thf(fact_1055_inf__sup__ord_I2_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1056_inf__sup__ord_I2_J,axiom,
! [X: set_pr5411798346947241657t_unit,Y: set_pr5411798346947241657t_unit] : ( ord_le8200006823705900825t_unit @ ( inf_in1092213268631476299t_unit @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1057_inf__sup__ord_I2_J,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1058_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1059_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1060_inf__sup__ord_I1_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_1061_inf__sup__ord_I1_J,axiom,
! [X: set_pr5411798346947241657t_unit,Y: set_pr5411798346947241657t_unit] : ( ord_le8200006823705900825t_unit @ ( inf_in1092213268631476299t_unit @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_1062_inf__sup__ord_I1_J,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_1063_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_1064_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_1065_inf__le1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_1066_inf__le1,axiom,
! [X: set_pr5411798346947241657t_unit,Y: set_pr5411798346947241657t_unit] : ( ord_le8200006823705900825t_unit @ ( inf_in1092213268631476299t_unit @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_1067_inf__le1,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_1068_inf__le1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_1069_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_1070_inf__le2,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_1071_inf__le2,axiom,
! [X: set_pr5411798346947241657t_unit,Y: set_pr5411798346947241657t_unit] : ( ord_le8200006823705900825t_unit @ ( inf_in1092213268631476299t_unit @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_1072_inf__le2,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_1073_inf__le2,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_1074_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_1075_le__infE,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ A @ B ) )
=> ~ ( ( ord_less_eq_real @ X @ A )
=> ~ ( ord_less_eq_real @ X @ B ) ) ) ).
% le_infE
thf(fact_1076_le__infE,axiom,
! [X: set_pr5411798346947241657t_unit,A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ X @ ( inf_in1092213268631476299t_unit @ A @ B ) )
=> ~ ( ( ord_le8200006823705900825t_unit @ X @ A )
=> ~ ( ord_le8200006823705900825t_unit @ X @ B ) ) ) ).
% le_infE
thf(fact_1077_le__infE,axiom,
! [X: set_b,A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ A @ B ) )
=> ~ ( ( ord_less_eq_set_b @ X @ A )
=> ~ ( ord_less_eq_set_b @ X @ B ) ) ) ).
% le_infE
thf(fact_1078_le__infE,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A )
=> ~ ( ord_less_eq_set_a @ X @ B ) ) ) ).
% le_infE
thf(fact_1079_le__infE,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_nat @ X @ A )
=> ~ ( ord_less_eq_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_1080_le__infI,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_eq_real @ X @ A )
=> ( ( ord_less_eq_real @ X @ B )
=> ( ord_less_eq_real @ X @ ( inf_inf_real @ A @ B ) ) ) ) ).
% le_infI
thf(fact_1081_le__infI,axiom,
! [X: set_pr5411798346947241657t_unit,A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ X @ A )
=> ( ( ord_le8200006823705900825t_unit @ X @ B )
=> ( ord_le8200006823705900825t_unit @ X @ ( inf_in1092213268631476299t_unit @ A @ B ) ) ) ) ).
% le_infI
thf(fact_1082_le__infI,axiom,
! [X: set_b,A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ X @ A )
=> ( ( ord_less_eq_set_b @ X @ B )
=> ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ A @ B ) ) ) ) ).
% le_infI
thf(fact_1083_le__infI,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X @ B )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% le_infI
thf(fact_1084_le__infI,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_1085_inf__mono,axiom,
! [A: real,C2: real,B: real,D: real] :
( ( ord_less_eq_real @ A @ C2 )
=> ( ( ord_less_eq_real @ B @ D )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ ( inf_inf_real @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_1086_inf__mono,axiom,
! [A: set_pr5411798346947241657t_unit,C2: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit,D: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A @ C2 )
=> ( ( ord_le8200006823705900825t_unit @ B @ D )
=> ( ord_le8200006823705900825t_unit @ ( inf_in1092213268631476299t_unit @ A @ B ) @ ( inf_in1092213268631476299t_unit @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_1087_inf__mono,axiom,
! [A: set_b,C2: set_b,B: set_b,D: set_b] :
( ( ord_less_eq_set_b @ A @ C2 )
=> ( ( ord_less_eq_set_b @ B @ D )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ ( inf_inf_set_b @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_1088_inf__mono,axiom,
! [A: set_a,C2: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_1089_inf__mono,axiom,
! [A: nat,C2: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_1090_le__infI1,axiom,
! [A: real,X: real,B: real] :
( ( ord_less_eq_real @ A @ X )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_1091_le__infI1,axiom,
! [A: set_pr5411798346947241657t_unit,X: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A @ X )
=> ( ord_le8200006823705900825t_unit @ ( inf_in1092213268631476299t_unit @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_1092_le__infI1,axiom,
! [A: set_b,X: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ X )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_1093_le__infI1,axiom,
! [A: set_a,X: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_1094_le__infI1,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_1095_le__infI2,axiom,
! [B: real,X: real,A: real] :
( ( ord_less_eq_real @ B @ X )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_1096_le__infI2,axiom,
! [B: set_pr5411798346947241657t_unit,X: set_pr5411798346947241657t_unit,A: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ B @ X )
=> ( ord_le8200006823705900825t_unit @ ( inf_in1092213268631476299t_unit @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_1097_le__infI2,axiom,
! [B: set_b,X: set_b,A: set_b] :
( ( ord_less_eq_set_b @ B @ X )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_1098_le__infI2,axiom,
! [B: set_a,X: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_1099_le__infI2,axiom,
! [B: nat,X: nat,A: nat] :
( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_1100_inf_OorderE,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( A
= ( inf_inf_real @ A @ B ) ) ) ).
% inf.orderE
thf(fact_1101_inf_OorderE,axiom,
! [A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A @ B )
=> ( A
= ( inf_in1092213268631476299t_unit @ A @ B ) ) ) ).
% inf.orderE
thf(fact_1102_inf_OorderE,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( A
= ( inf_inf_set_b @ A @ B ) ) ) ).
% inf.orderE
thf(fact_1103_inf_OorderE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( A
= ( inf_inf_set_a @ A @ B ) ) ) ).
% inf.orderE
thf(fact_1104_inf_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( inf_inf_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_1105_inf_OorderI,axiom,
! [A: real,B: real] :
( ( A
= ( inf_inf_real @ A @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% inf.orderI
thf(fact_1106_inf_OorderI,axiom,
! [A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( A
= ( inf_in1092213268631476299t_unit @ A @ B ) )
=> ( ord_le8200006823705900825t_unit @ A @ B ) ) ).
% inf.orderI
thf(fact_1107_inf_OorderI,axiom,
! [A: set_b,B: set_b] :
( ( A
= ( inf_inf_set_b @ A @ B ) )
=> ( ord_less_eq_set_b @ A @ B ) ) ).
% inf.orderI
thf(fact_1108_inf_OorderI,axiom,
! [A: set_a,B: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% inf.orderI
thf(fact_1109_inf_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( inf_inf_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_1110_inf__unique,axiom,
! [F: real > real > real,X: real,Y: real] :
( ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( F @ X4 @ Y3 ) @ X4 )
=> ( ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( F @ X4 @ Y3 ) @ Y3 )
=> ( ! [X4: real,Y3: real,Z3: real] :
( ( ord_less_eq_real @ X4 @ Y3 )
=> ( ( ord_less_eq_real @ X4 @ Z3 )
=> ( ord_less_eq_real @ X4 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_real @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1111_inf__unique,axiom,
! [F: set_pr5411798346947241657t_unit > set_pr5411798346947241657t_unit > set_pr5411798346947241657t_unit,X: set_pr5411798346947241657t_unit,Y: set_pr5411798346947241657t_unit] :
( ! [X4: set_pr5411798346947241657t_unit,Y3: set_pr5411798346947241657t_unit] : ( ord_le8200006823705900825t_unit @ ( F @ X4 @ Y3 ) @ X4 )
=> ( ! [X4: set_pr5411798346947241657t_unit,Y3: set_pr5411798346947241657t_unit] : ( ord_le8200006823705900825t_unit @ ( F @ X4 @ Y3 ) @ Y3 )
=> ( ! [X4: set_pr5411798346947241657t_unit,Y3: set_pr5411798346947241657t_unit,Z3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ X4 @ Y3 )
=> ( ( ord_le8200006823705900825t_unit @ X4 @ Z3 )
=> ( ord_le8200006823705900825t_unit @ X4 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_in1092213268631476299t_unit @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1112_inf__unique,axiom,
! [F: set_b > set_b > set_b,X: set_b,Y: set_b] :
( ! [X4: set_b,Y3: set_b] : ( ord_less_eq_set_b @ ( F @ X4 @ Y3 ) @ X4 )
=> ( ! [X4: set_b,Y3: set_b] : ( ord_less_eq_set_b @ ( F @ X4 @ Y3 ) @ Y3 )
=> ( ! [X4: set_b,Y3: set_b,Z3: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y3 )
=> ( ( ord_less_eq_set_b @ X4 @ Z3 )
=> ( ord_less_eq_set_b @ X4 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_b @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1113_inf__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y3 ) @ X4 )
=> ( ! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y3 ) @ Y3 )
=> ( ! [X4: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( ord_less_eq_set_a @ X4 @ Z3 )
=> ( ord_less_eq_set_a @ X4 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1114_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y3 ) @ X4 )
=> ( ! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y3 ) @ Y3 )
=> ( ! [X4: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_nat @ X4 @ Z3 )
=> ( ord_less_eq_nat @ X4 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1115_le__iff__inf,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y5: real] :
( ( inf_inf_real @ X3 @ Y5 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_1116_le__iff__inf,axiom,
( ord_le8200006823705900825t_unit
= ( ^ [X3: set_pr5411798346947241657t_unit,Y5: set_pr5411798346947241657t_unit] :
( ( inf_in1092213268631476299t_unit @ X3 @ Y5 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_1117_le__iff__inf,axiom,
( ord_less_eq_set_b
= ( ^ [X3: set_b,Y5: set_b] :
( ( inf_inf_set_b @ X3 @ Y5 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_1118_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y5: set_a] :
( ( inf_inf_set_a @ X3 @ Y5 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_1119_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y5: nat] :
( ( inf_inf_nat @ X3 @ Y5 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_1120_inf_Oabsorb1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( inf_inf_real @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_1121_inf_Oabsorb1,axiom,
! [A: set_pr5411798346947241657t_unit,B: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ A @ B )
=> ( ( inf_in1092213268631476299t_unit @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_1122_inf_Oabsorb1,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( inf_inf_set_b @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_1123_inf_Oabsorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_1124_inf_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_1125_inf_Oabsorb2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( inf_inf_real @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1126_inf_Oabsorb2,axiom,
! [B: set_pr5411798346947241657t_unit,A: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ B @ A )
=> ( ( inf_in1092213268631476299t_unit @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1127_inf_Oabsorb2,axiom,
! [B: set_b,A: set_b] :
( ( ord_less_eq_set_b @ B @ A )
=> ( ( inf_inf_set_b @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1128_inf_Oabsorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1129_inf_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1130_inf__absorb1,axiom,
! [X: set_pr5411798346947241657t_unit,Y: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ X @ Y )
=> ( ( inf_in1092213268631476299t_unit @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1131_inf__absorb1,axiom,
! [X: set_b,Y: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( inf_inf_set_b @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1132_inf__absorb1,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( inf_inf_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1133_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_1134_closed__w__imp__cycle,axiom,
! [P2: list_b] :
( ( arc_wf_closed_w_a_b @ g @ P2 )
=> ? [X_1: list_b] : ( arc_pre_cycle_a_b @ g @ X_1 ) ) ).
% closed_w_imp_cycle
thf(fact_1135_max__subgraphI,axiom,
! [P: pre_pr7278220950009878019t_unit > $o,X: pre_pr7278220950009878019t_unit] :
( ( P @ X )
=> ( ( digraph_subgraph_a_b @ X @ g )
=> ( ! [Y3: pre_pr7278220950009878019t_unit] :
( ( X != Y3 )
=> ( ( digraph_subgraph_a_b @ X @ Y3 )
=> ( ( digraph_subgraph_a_b @ Y3 @ g )
=> ~ ( P @ Y3 ) ) ) )
=> ( digrap4729920478598810909ph_a_b @ g @ P @ X ) ) ) ) ).
% max_subgraphI
thf(fact_1136_max__subgraph__cong,axiom,
! [H: pre_pr7278220950009878019t_unit,H5: pre_pr7278220950009878019t_unit,P: pre_pr7278220950009878019t_unit > $o,P3: pre_pr7278220950009878019t_unit > $o] :
( ( H = H5 )
=> ( ! [H6: pre_pr7278220950009878019t_unit] :
( ( digraph_subgraph_a_b @ H5 @ H6 )
=> ( ( digraph_subgraph_a_b @ H6 @ g )
=> ( ( P @ H6 )
= ( P3 @ H6 ) ) ) )
=> ( ( digrap4729920478598810909ph_a_b @ g @ P @ H )
= ( digrap4729920478598810909ph_a_b @ g @ P3 @ H5 ) ) ) ) ).
% max_subgraph_cong
thf(fact_1137_max__subgraph__mp,axiom,
! [Q: pre_pr7278220950009878019t_unit > $o,X: pre_pr7278220950009878019t_unit,P: pre_pr7278220950009878019t_unit > $o] :
( ( digrap4729920478598810909ph_a_b @ g @ Q @ X )
=> ( ! [X4: pre_pr7278220950009878019t_unit] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ( P @ X )
=> ( digrap4729920478598810909ph_a_b @ g @ P @ X ) ) ) ) ).
% max_subgraph_mp
thf(fact_1138_max__subgraph__prop,axiom,
! [P: pre_pr7278220950009878019t_unit > $o,X: pre_pr7278220950009878019t_unit] :
( ( digrap4729920478598810909ph_a_b @ g @ P @ X )
=> ( P @ X ) ) ).
% max_subgraph_prop
thf(fact_1139_subgraphI__max__subgraph,axiom,
! [P: pre_pr7278220950009878019t_unit > $o,X: pre_pr7278220950009878019t_unit] :
( ( digrap4729920478598810909ph_a_b @ g @ P @ X )
=> ( digraph_subgraph_a_b @ X @ g ) ) ).
% subgraphI_max_subgraph
thf(fact_1140_max__subgraph__subg__eq,axiom,
! [P: pre_pr7278220950009878019t_unit > $o,H1: pre_pr7278220950009878019t_unit,H22: pre_pr7278220950009878019t_unit] :
( ( digrap4729920478598810909ph_a_b @ g @ P @ H1 )
=> ( ( digrap4729920478598810909ph_a_b @ g @ P @ H22 )
=> ( ( digraph_subgraph_a_b @ H1 @ H22 )
=> ( H1 = H22 ) ) ) ) ).
% max_subgraph_subg_eq
thf(fact_1141_max__subgraph__def,axiom,
! [P: pre_pr7278220950009878019t_unit > $o,H: pre_pr7278220950009878019t_unit] :
( ( digrap4729920478598810909ph_a_b @ g @ P @ H )
= ( ( digraph_subgraph_a_b @ H @ g )
& ( P @ H )
& ! [H2: pre_pr7278220950009878019t_unit] :
( ( ( H2 != H )
& ( digraph_subgraph_a_b @ H @ H2 ) )
=> ~ ( ( digraph_subgraph_a_b @ H2 @ g )
& ( P @ H2 ) ) ) ) ) ).
% max_subgraph_def
thf(fact_1142_induced__subgraphI__arc__mono,axiom,
! [P: pre_pr7278220950009878019t_unit > $o,H: pre_pr7278220950009878019t_unit] :
( ( digrap4729920478598810909ph_a_b @ g @ P @ H )
=> ( ( digraph_arc_mono_a_b @ P )
=> ( digrap5251062021860773499ph_a_b @ H @ g ) ) ) ).
% induced_subgraphI_arc_mono
thf(fact_1143_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N2: nat] :
( X
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_1144_some__unvis__vert_I2_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 ) )
=> ! [X5: a] :
( ( member_a @ X5 @ ( graph_2016941059203891550ts_a_b @ g @ U @ U2 ) )
=> ( ord_le1083603963089353582_ereal @ ( shortest_wf_mu_a_b @ g @ W @ U @ X ) @ ( shortest_wf_mu_a_b @ g @ W @ U @ X5 ) ) ) ) ) ).
% some_unvis_vert(2)
thf(fact_1145_subgraph__in__degree,axiom,
! [H: pre_pr7278220950009878019t_unit,V: a] :
( ( digraph_subgraph_a_b @ H @ g )
=> ( ord_less_eq_nat @ ( in_degree_a_b @ H @ V ) @ ( in_degree_a_b @ g @ V ) ) ) ).
% subgraph_in_degree
thf(fact_1146_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_1147_sp__non__neg__if__w__non__neg,axiom,
! [W: b > real,U: a,V: a] :
( ! [X4: b] :
( ( member_b @ X4 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( W @ X4 ) ) )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( shortest_wf_mu_a_b @ g @ W @ U @ V ) ) ) ).
% sp_non_neg_if_w_non_neg
thf(fact_1148_sp__to__self__if__w__non__neg,axiom,
! [W: b > real,U: a] :
( ! [X4: b] :
( ( member_b @ X4 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( W @ X4 ) ) )
=> ( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( shortest_wf_mu_a_b @ g @ W @ U @ U )
= zero_z2744965634713055877_ereal ) ) ) ).
% sp_to_self_if_w_non_neg
thf(fact_1149_unvis__nearest__vert__contr,axiom,
! [W: b > real,U: a,N: nat,U2: set_a,X: a,Y: a] :
( ( graph_3148032005746981223ts_a_b @ g @ W @ U @ N @ U2 )
=> ( ( member_a @ X @ U2 )
=> ( ( X != U )
=> ( ( member_a @ Y @ ( graph_2016941059203891550ts_a_b @ g @ U @ U2 ) )
=> ~ ( ord_le1188267648640031866_ereal @ ( shortest_wf_mu_a_b @ g @ W @ U @ Y ) @ ( shortest_wf_mu_a_b @ g @ W @ U @ X ) ) ) ) ) ) ).
% unvis_nearest_vert_contr
thf(fact_1150_ex__unvis__vert,axiom,
! [U: a,U2: set_a,W: b > real] :
( ( ( graph_2016941059203891550ts_a_b @ g @ U @ U2 )
!= bot_bot_set_a )
=> ? [X4: a] :
( ( member_a @ X4 @ ( graph_2016941059203891550ts_a_b @ g @ U @ U2 ) )
& ! [Xa2: a] :
( ( member_a @ Xa2 @ ( graph_2016941059203891550ts_a_b @ g @ U @ U2 ) )
=> ( ord_le1083603963089353582_ereal @ ( shortest_wf_mu_a_b @ g @ W @ U @ X4 ) @ ( shortest_wf_mu_a_b @ g @ W @ U @ Xa2 ) ) ) ) ) ).
% ex_unvis_vert
thf(fact_1151_ex__sp__eq__dia,axiom,
! [F: b > real] :
( ( ( pre_ve642382030648772252t_unit @ g )
!= bot_bot_set_a )
=> ? [X4: a] :
( ( member_a @ X4 @ ( pre_ve642382030648772252t_unit @ g ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( pre_ve642382030648772252t_unit @ g ) )
& ( ( shortest_wf_mu_a_b @ g @ F @ X4 @ Xa )
= ( graph_926353876199057498er_a_b @ g @ F ) ) ) ) ) ).
% ex_sp_eq_dia
thf(fact_1152_dia__lowerB,axiom,
! [U: a,V: a,W: b > real] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ord_le1083603963089353582_ereal @ ( shortest_wf_mu_a_b @ g @ W @ U @ V ) @ ( graph_926353876199057498er_a_b @ g @ W ) ) ) ) ).
% dia_lowerB
thf(fact_1153_ex__sp__eq__fin__dia,axiom,
! [F: b > real] :
( ( ( pre_ve642382030648772252t_unit @ g )
!= bot_bot_set_a )
=> ? [X4: a] :
( ( member_a @ X4 @ ( pre_ve642382030648772252t_unit @ g ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( pre_ve642382030648772252t_unit @ g ) )
& ( ( shortest_wf_mu_a_b @ g @ F @ X4 @ Xa )
= ( graph_1932031826008834157er_a_b @ g @ F ) ) ) ) ) ).
% ex_sp_eq_fin_dia
thf(fact_1154_dia__eq__fin__dia__if__strongly__con,axiom,
( ( digrap8691851296217657702ed_a_b @ g )
=> ( ( graph_926353876199057498er_a_b @ g )
= ( graph_1932031826008834157er_a_b @ g ) ) ) ).
% dia_eq_fin_dia_if_strongly_con
thf(fact_1155_fin__dia__lowerB,axiom,
! [U: a,V: a,W: b > real] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( ord_le1188267648640031866_ereal @ ( shortest_wf_mu_a_b @ g @ W @ U @ V ) @ extend1530274965995635425_ereal )
=> ( ord_le1083603963089353582_ereal @ ( shortest_wf_mu_a_b @ g @ W @ U @ V ) @ ( graph_1932031826008834157er_a_b @ g @ W ) ) ) ) ) ).
% fin_dia_lowerB
thf(fact_1156_sp__triangle,axiom,
! [A: a,B: a,C2: a,W: b > real] :
( ( member_a @ A @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( member_a @ B @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( member_a @ C2 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ! [X4: b] :
( ( member_b @ X4 @ ( pre_ar1395965042833527383t_unit @ g ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( W @ X4 ) ) )
=> ( ord_le1083603963089353582_ereal @ ( shortest_wf_mu_a_b @ g @ W @ A @ C2 ) @ ( plus_p7876563987511257093_ereal @ ( shortest_wf_mu_a_b @ g @ W @ A @ B ) @ ( shortest_wf_mu_a_b @ g @ W @ B @ C2 ) ) ) ) ) ) ) ).
% sp_triangle
thf(fact_1157_strongly__con__imp__sp__finite,axiom,
! [U: a,V: a,W: b > real] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( digrap8691851296217657702ed_a_b @ g )
=> ( ord_le1188267648640031866_ereal @ ( shortest_wf_mu_a_b @ g @ W @ U @ V ) @ extend1530274965995635425_ereal ) ) ) ) ).
% strongly_con_imp_sp_finite
thf(fact_1158_shortest__path__inf,axiom,
! [U: a,V: a,F: b > real] :
( ~ ( reachable_a_b @ g @ U @ V )
=> ( ( shortest_wf_mu_a_b @ g @ F @ U @ V )
= extend1530274965995635425_ereal ) ) ).
% shortest_path_inf
thf(fact_1159_fin__diameter__finite,axiom,
! [F: b > real] : ( ord_le1188267648640031866_ereal @ ( graph_1932031826008834157er_a_b @ g @ F ) @ extend1530274965995635425_ereal ) ).
% fin_diameter_finite
thf(fact_1160__092_060mu_062__reach__conv,axiom,
! [F: b > real,U: a,V: a] :
( ( ord_le1188267648640031866_ereal @ ( shortest_wf_mu_a_b @ g @ F @ U @ V ) @ extend1530274965995635425_ereal )
= ( reachable_a_b @ g @ U @ V ) ) ).
% \<mu>_reach_conv
thf(fact_1161_dia__eq__fin__dia__if__finite,axiom,
! [F: b > real] :
( ( ord_le1188267648640031866_ereal @ ( graph_926353876199057498er_a_b @ g @ F ) @ extend1530274965995635425_ereal )
=> ( ( graph_926353876199057498er_a_b @ g @ F )
= ( graph_1932031826008834157er_a_b @ g @ F ) ) ) ).
% dia_eq_fin_dia_if_finite
thf(fact_1162_ereal__add__strict__mono,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal,D: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A )
=> ( ( A != extend1530274965995635425_ereal )
=> ( ( ord_le1188267648640031866_ereal @ C2 @ D )
=> ( ord_le1188267648640031866_ereal @ ( plus_p7876563987511257093_ereal @ A @ C2 ) @ ( plus_p7876563987511257093_ereal @ B @ D ) ) ) ) ) ) ).
% ereal_add_strict_mono
thf(fact_1163_ereal__infty__less_I1_J,axiom,
! [X: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X @ extend1530274965995635425_ereal )
= ( X != extend1530274965995635425_ereal ) ) ).
% ereal_infty_less(1)
thf(fact_1164_ereal__less__PInfty,axiom,
! [A: extended_ereal] :
( ( A != extend1530274965995635425_ereal )
=> ( ord_le1188267648640031866_ereal @ A @ extend1530274965995635425_ereal ) ) ).
% ereal_less_PInfty
thf(fact_1165_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1166_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1167_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_1168_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1169_ereal__minus_I4_J,axiom,
! [X: extended_ereal] :
( ( minus_2816186181549245109_ereal @ extend1530274965995635425_ereal @ X )
= extend1530274965995635425_ereal ) ).
% ereal_minus(4)
thf(fact_1170_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_1171_diff__diff__left,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_1172_ereal__minus_I7_J,axiom,
! [X: extended_ereal] :
( ( minus_2816186181549245109_ereal @ X @ zero_z2744965634713055877_ereal )
= X ) ).
% ereal_minus(7)
thf(fact_1173_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1174_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1175_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1176_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1177_diff__Suc__diff__eq2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1178_diff__Suc__diff__eq1,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1179_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_1180_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1181_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1182_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_1183_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_1184_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1185_add__le__mono,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_1186_add__le__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_1187_trans__le__add1,axiom,
! [I2: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_le_add1
thf(fact_1188_trans__le__add2,axiom,
! [I2: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_le_add2
thf(fact_1189_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M4 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1190_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1191_trans__less__add2,axiom,
! [I2: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_less_add2
thf(fact_1192_trans__less__add1,axiom,
! [I2: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_less_add1
thf(fact_1193_add__less__mono1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_1194_not__add__less2,axiom,
! [J2: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_1195_not__add__less1,axiom,
! [I2: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ I2 ) ).
% not_add_less1
thf(fact_1196_add__less__mono,axiom,
! [I2: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_1197_add__lessD1,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_1198_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1199_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1200_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1201_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1202_less__imp__add__positive,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I2 @ K2 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_1203_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1204_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M4: nat,N4: nat] :
? [K3: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M4 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1205_less__add__Suc2,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_1206_less__add__Suc1,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M ) ) ) ).
% less_add_Suc1
thf(fact_1207_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_1208_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1209_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1210_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1211_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1212_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1213_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1214_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1215_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1216_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1217_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1218_less__diff__conv,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_1219_le__diff__conv,axiom,
! [J2: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_1220_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1221_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1222_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1223_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I2 )
= K )
= ( J2
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1224_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1225_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1226_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1227_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1228_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1229_less__diff__conv2,axiom,
! [K: nat,J2: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1230_ereal__minus__mono,axiom,
! [A2: extended_ereal,B3: extended_ereal,D2: extended_ereal,C: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A2 @ B3 )
=> ( ( ord_le1083603963089353582_ereal @ D2 @ C )
=> ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ A2 @ C ) @ ( minus_2816186181549245109_ereal @ B3 @ D2 ) ) ) ) ).
% ereal_minus_mono
thf(fact_1231_ereal__diff__add__assoc2,axiom,
! [X: extended_ereal,Y: extended_ereal,Z4: extended_ereal] :
( ( minus_2816186181549245109_ereal @ ( plus_p7876563987511257093_ereal @ X @ Y ) @ Z4 )
= ( plus_p7876563987511257093_ereal @ ( minus_2816186181549245109_ereal @ X @ Z4 ) @ Y ) ) ).
% ereal_diff_add_assoc2
thf(fact_1232_diff__add__eq__ereal,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ ( minus_2816186181549245109_ereal @ A @ B ) @ C2 )
= ( minus_2816186181549245109_ereal @ ( plus_p7876563987511257093_ereal @ A @ C2 ) @ B ) ) ).
% diff_add_eq_ereal
thf(fact_1233_add__diff__eq__ereal,axiom,
! [X: extended_ereal,Y: extended_ereal,Z4: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ X @ ( minus_2816186181549245109_ereal @ Y @ Z4 ) )
= ( minus_2816186181549245109_ereal @ ( plus_p7876563987511257093_ereal @ X @ Y ) @ Z4 ) ) ).
% add_diff_eq_ereal
thf(fact_1234_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M4: nat,N4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_1235_less__ereal_Osimps_I2_J,axiom,
! [A: extended_ereal] :
~ ( ord_le1188267648640031866_ereal @ extend1530274965995635425_ereal @ A ) ).
% less_ereal.simps(2)
thf(fact_1236_ereal__diff__le__mono__left,axiom,
! [X: extended_ereal,Z4: extended_ereal,Y: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ X @ Z4 )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Y )
=> ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ X @ Y ) @ Z4 ) ) ) ).
% ereal_diff_le_mono_left
thf(fact_1237_ereal__diff__positive,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ A @ B )
=> ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( minus_2816186181549245109_ereal @ B @ A ) ) ) ).
% ereal_diff_positive
thf(fact_1238_ereal__diff__le__self,axiom,
! [Y: extended_ereal,X: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ Y )
=> ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ X @ Y ) @ X ) ) ).
% ereal_diff_le_self
thf(fact_1239_less__eq__ereal__def,axiom,
( ord_le1083603963089353582_ereal
= ( ^ [X3: extended_ereal,Y5: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% less_eq_ereal_def
thf(fact_1240_ereal__diff__gr0,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ ( minus_2816186181549245109_ereal @ B @ A ) ) ) ).
% ereal_diff_gr0
thf(fact_1241_ereal__0__less__1,axiom,
ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ one_on4623092294121504201_ereal ).
% ereal_0_less_1
thf(fact_1242_ereal__one__not__less__zero__ereal,axiom,
~ ( ord_le1188267648640031866_ereal @ one_on4623092294121504201_ereal @ zero_z2744965634713055877_ereal ) ).
% ereal_one_not_less_zero_ereal
thf(fact_1243_ereal__add__strict__mono2,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal,D: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ A @ B )
=> ( ( ord_le1188267648640031866_ereal @ C2 @ D )
=> ( ord_le1188267648640031866_ereal @ ( plus_p7876563987511257093_ereal @ A @ C2 ) @ ( plus_p7876563987511257093_ereal @ B @ D ) ) ) ) ).
% ereal_add_strict_mono2
thf(fact_1244_ereal__less_I5_J,axiom,
ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ extend1530274965995635425_ereal ).
% ereal_less(5)
thf(fact_1245_ereal__mono__minus__cancel,axiom,
! [C2: extended_ereal,A: extended_ereal,B: extended_ereal] :
( ( ord_le1083603963089353582_ereal @ ( minus_2816186181549245109_ereal @ C2 @ A ) @ ( minus_2816186181549245109_ereal @ C2 @ B ) )
=> ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ C2 )
=> ( ( ord_le1188267648640031866_ereal @ C2 @ extend1530274965995635425_ereal )
=> ( ord_le1083603963089353582_ereal @ B @ A ) ) ) ) ).
% ereal_mono_minus_cancel
thf(fact_1246_ereal__le__epsilon,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ! [E3: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ E3 )
=> ( ord_le1083603963089353582_ereal @ X @ ( plus_p7876563987511257093_ereal @ Y @ E3 ) ) )
=> ( ord_le1083603963089353582_ereal @ X @ Y ) ) ).
% ereal_le_epsilon
thf(fact_1247_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_1248_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 ) )
=> ( ! [W3: a] :
( ( member_a @ W3 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( U != W3 )
=> ( ( V != W3 )
=> ( ( in_degree_a_b @ g @ W3 )
= ( out_degree_a_b @ g @ W3 ) ) ) ) )
=> ( ( ( 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_1249_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_1250_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_1251_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_1252_closed__euler,axiom,
( ( ? [U4: a,P5: list_b] : ( pre_euler_trail_a_b @ g @ U4 @ P5 @ U4 ) )
= ( ( digrap8783888973171253482ed_a_b @ g )
& ! [X3: a] :
( ( member_a @ X3 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( in_degree_a_b @ g @ X3 )
= ( out_degree_a_b @ g @ X3 ) ) ) ) ) ).
% closed_euler
thf(fact_1253_closed__euler1,axiom,
( ( digrap8783888973171253482ed_a_b @ g )
=> ( ! [U6: a] :
( ( member_a @ U6 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( in_degree_a_b @ g @ U6 )
= ( out_degree_a_b @ g @ U6 ) ) )
=> ? [U6: a,P4: list_b] : ( pre_euler_trail_a_b @ g @ U6 @ P4 @ U6 ) ) ) ).
% closed_euler1
thf(fact_1254_open__euler,axiom,
( ( ? [U4: a,P5: list_b,V2: a] :
( ( pre_euler_trail_a_b @ g @ U4 @ P5 @ V2 )
& ( U4 != V2 ) ) )
= ( ( digrap8783888973171253482ed_a_b @ g )
& ? [U4: a] :
( ( member_a @ U4 @ ( pre_ve642382030648772252t_unit @ g ) )
& ? [V2: a] :
( ( member_a @ V2 @ ( pre_ve642382030648772252t_unit @ g ) )
& ! [X3: a] :
( ( member_a @ X3 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( U4 != X3 )
=> ( ( V2 != X3 )
=> ( ( in_degree_a_b @ g @ X3 )
= ( out_degree_a_b @ g @ X3 ) ) ) ) )
& ( ( plus_plus_nat @ ( in_degree_a_b @ g @ U4 ) @ one_one_nat )
= ( out_degree_a_b @ g @ U4 ) )
& ( ( plus_plus_nat @ ( out_degree_a_b @ g @ V2 ) @ one_one_nat )
= ( in_degree_a_b @ g @ V2 ) ) ) ) ) ) ).
% open_euler
thf(fact_1255_arc__set__balanced__all,axiom,
! [U: a,V: a] :
( ( pre_ar5931435604406180204ed_a_b @ g @ U @ ( pre_ar1395965042833527383t_unit @ g ) @ V )
= ( ( ( U = V )
=> ! [X3: a] :
( ( member_a @ X3 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( in_degree_a_b @ g @ X3 )
= ( out_degree_a_b @ g @ X3 ) ) ) )
& ( ( U != V )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( pre_ve642382030648772252t_unit @ g ) )
=> ( ( ( X3 != U )
& ( X3 != V ) )
=> ( ( in_degree_a_b @ g @ X3 )
= ( out_degree_a_b @ g @ X3 ) ) ) )
& ( ( 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_1256_diff__diff__commute__ereal,axiom,
! [X: extended_ereal,Y: extended_ereal,Z4: extended_ereal] :
( ( minus_2816186181549245109_ereal @ ( minus_2816186181549245109_ereal @ X @ Y ) @ Z4 )
= ( minus_2816186181549245109_ereal @ ( minus_2816186181549245109_ereal @ X @ Z4 ) @ Y ) ) ).
% diff_diff_commute_ereal
thf(fact_1257_empty__imp__dia__minf,axiom,
! [W: b > real] :
( ( ( pre_ve642382030648772252t_unit @ g )
= bot_bot_set_a )
=> ( ( graph_926353876199057498er_a_b @ g @ W )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ).
% empty_imp_dia_minf
thf(fact_1258_empty__imp__fin__dia__minf,axiom,
! [W: b > real] :
( ( ( pre_ve642382030648772252t_unit @ g )
= bot_bot_set_a )
=> ( ( graph_1932031826008834157er_a_b @ g @ W )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ).
% empty_imp_fin_dia_minf
thf(fact_1259_ereal__minus__less__minus,axiom,
! [A: extended_ereal,B: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ A ) @ ( uminus27091377158695749_ereal @ B ) )
= ( ord_le1188267648640031866_ereal @ B @ A ) ) ).
% ereal_minus_less_minus
thf(fact_1260_ereal__MInfty__lessI,axiom,
! [A: extended_ereal] :
( ( A
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ A ) ) ).
% ereal_MInfty_lessI
thf(fact_1261_ereal__infty__less_I2_J,axiom,
! [X: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ X )
= ( X
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ).
% ereal_infty_less(2)
thf(fact_1262_neg__0__less__iff__less__erea,axiom,
! [A: extended_ereal] :
( ( ord_le1188267648640031866_ereal @ zero_z2744965634713055877_ereal @ ( uminus27091377158695749_ereal @ A ) )
= ( ord_le1188267648640031866_ereal @ A @ zero_z2744965634713055877_ereal ) ) ).
% neg_0_less_iff_less_erea
thf(fact_1263_ereal__minus_I5_J,axiom,
( ( minus_2816186181549245109_ereal @ ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) @ extend1530274965995635425_ereal )
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ).
% ereal_minus(5)
thf(fact_1264_ereal__minus_I8_J,axiom,
! [X: extended_ereal] :
( ( minus_2816186181549245109_ereal @ zero_z2744965634713055877_ereal @ X )
= ( uminus27091377158695749_ereal @ X ) ) ).
% ereal_minus(8)
thf(fact_1265_ereal__minus_I6_J,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( minus_2816186181549245109_ereal @ X @ ( uminus27091377158695749_ereal @ Y ) )
= ( plus_p7876563987511257093_ereal @ X @ Y ) ) ).
% ereal_minus(6)
thf(fact_1266_minus__ereal__def,axiom,
( minus_2816186181549245109_ereal
= ( ^ [X3: extended_ereal,Y5: extended_ereal] : ( plus_p7876563987511257093_ereal @ X3 @ ( uminus27091377158695749_ereal @ Y5 ) ) ) ) ).
% minus_ereal_def
thf(fact_1267_ereal__minus__diff__eq,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ( X = extend1530274965995635425_ereal )
=> ( Y != extend1530274965995635425_ereal ) )
=> ( ( ( X
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
=> ( Y
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) )
=> ( ( uminus27091377158695749_ereal @ ( minus_2816186181549245109_ereal @ X @ Y ) )
= ( minus_2816186181549245109_ereal @ Y @ X ) ) ) ) ).
% ereal_minus_diff_eq
thf(fact_1268_ereal__minus__eq__minus__iff,axiom,
! [A: extended_ereal,B: extended_ereal,C2: extended_ereal] :
( ( ( minus_2816186181549245109_ereal @ A @ B )
= ( minus_2816186181549245109_ereal @ A @ C2 ) )
= ( ( B = C2 )
| ( A = extend1530274965995635425_ereal )
| ( ( A
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( B
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
& ( C2
!= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) ) ) ) ) ).
% ereal_minus_eq_minus_iff
thf(fact_1269_ereal__minus__eq__PInfty__iff,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( ( minus_2816186181549245109_ereal @ X @ Y )
= extend1530274965995635425_ereal )
= ( ( Y
= ( uminus27091377158695749_ereal @ extend1530274965995635425_ereal ) )
| ( X = extend1530274965995635425_ereal ) ) ) ).
% ereal_minus_eq_PInfty_iff
thf(fact_1270_ereal__add__uminus__conv__diff,axiom,
! [X: extended_ereal,Y: extended_ereal] :
( ( plus_p7876563987511257093_ereal @ ( uminus27091377158695749_ereal @ X ) @ Y )
= ( minus_2816186181549245109_ereal @ Y @ X ) ) ).
% ereal_add_uminus_conv_diff
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( suc @ ( minus_minus_nat @ ( finite_card_a @ ( pre_ve642382030648772252t_unit @ g ) ) @ one_one_nat ) )
= ( finite_card_a @ ( pre_ve642382030648772252t_unit @ g ) ) ) ).
%------------------------------------------------------------------------------